Symmetry, invariants, topology edited by Louis Michel editor: E. Brezin Contents Chapter I< L. Michel, Fundamental concepts for the study of crystal symmetry
265
11
Chapter < J.S. Kim, L. Michel, B.I. ZhilinskimH , The ring of invariant real functions on the Brillouin zone
337
Chapter II B.I. ZhilinskimH , Symmetry, invariants, and topology in molecular models
85
Chapter
377
Chapter III L. Michel, B.I. ZhilinskimH , Rydberg states of atoms and molecules. Basic group theoretical and topological analysis
173
Obituary Avant-Propos Symmetry, invariants, topology Chapter I L. Michel, B.I. ZhilinskimH , Symmetry, invariants, topology. Basic tools
Elsevier Science B.V.
5
7
Physics Reports 341 (2001) 5}6
Obituary With the untimely death of Louis Michel on December 30, 1999, the world Physics community has lost one of its prominent members. Louis was born in 1923, in Roanne, France and graduated from Ecole Polytechnique, where he later founded the Center for Theoretical Physics. Louis has been a Member of the French Academy of Science since 1979, and is a recipient of the Wigner Medal (1984). In 1962, Louis was appointed as Professor of Physics at l'Institut des Hautes Etudes Scienti"ques (IHES), where he worked until his death. Among his scienti"c achievements Louis is best known for the `Michel parameter,a in the muon decay into an electron and two neutrinos. In the "eld of lepton polarisation Louis is very well known for the discovery of `Isotopic paritya (later known as `G-paritya). Louis is also coauthor of the Bargmann}Michel}Telegdi Equation for the description of the relativistic spin precession. The `Michel parametera which was introduced by Louis in 1953 has been widely used in the interpretation of experiments until today. During his recent visit to Canada, in November 1999, Louis was invited to the Triumf Cyclotron facility at the University of British Columbia in Vancouver, Canada, where experimental work was in progress utilizing the `Michel parameter.a In the later years of his career, Louis has worked in applications of symmetry and topology in condensed matter physics, in mathematical crystallography and in various other "elds. Louis liked collaborating with people and to share his almost unlimited knowledge, and it was a great privilege to work with him. In collaboration with a long list of researchers, Louis had contributions of great importance to a broad variety of "elds, including Landau theory of second-order phase transitions, topological classi"cation of defects, invariant polynomials of crystallographic groups, van Hove singularities, geometry of lattices, band representations of space groups, Rydberg series for atoms and molecules and others. In all these "elds of research, Louis was playing a leading role. For example, the work in this Volume of Physics Reports is a product of Louis' initiative, his guidance and his unlimited enthusiasm for teaching future generations. Louis' research activities were not limited to IHES, he liked to travel and also to invite people to his home institution. Accompanied by his wife Therese, Louis has visited many institutes in more than 30 countries all over the world for di!erent lengths of time, and in the earlier days they spent a number of years in the most prestigious centers for Theoretical Physics, the Niels Bohr Institute in Copenhagen (1950}1953) and The Institute of Advanced Studies in Princeton (1953}1955) were among them. Louis was not only an eminent scientist, he was also very well known for his active involvement with people and events on an international scale. Being famous as he was, Louis had his heart and door open not only for well-established scientists but also for students in the beginning of their career, and he was very warm and considerate with practically everybody around him. With Louis' 0370-1573/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 0 0 ) 0 0 0 8 6 - 7
6
Obituary
death, we lost a very good friend, a coworker of many years and a great teacher. Louis is survived by his wife Therese, by six children and eleven grandchildren. Jai Sam Kim Josuah Zak Boris Zhilinskii
Physics Reports 341 (2001) 7}9
Avant-Propos
Symmetry, invariants, topology Nearly seventy years have elapsed since the publication of the two great classic books by Weyl (1931) and by Wigner (1959), dealing with group theory and its application to quantum mechanics. More than a hundred books have appeared on the same subject; they are also centered on linear representations of groups. In this Volume, the emphasis is on group actions and their decomposition into orbits and strata; the study of the corresponding orbit space and of the set of strata is basic in physics. Linear representations are only a particular case of group actions; their decomposition into strata, made under di!erent names in the di!erent domains of physics, is here uni"ed. Except for some part of Chapter III dealing with atoms, the "eld of applications is restricted to molecules and crystals whose symmetry groups are essentially discrete. This seems to make paradoxal the appearance of the word topology in the title, but the source of the topological problems comes from the topology of the space on which the symmetry group G acts: phase space, surface of constant energy, Brillouin zone2. The topological consequences are very varied and often new in physics. They give a `qualitativea study of phenomena. The qualitative approach is reviewed mainly for molecules in Chapters II and III and for crystals in IV (Section 6), V and VI. Chapter I is a necessary introduction to topics not found in the hundreds of books we mentioned. Probably many readers, after a "rst reading which will acquaint them with the presented background, will use it more as a dictionary to be consulted if needed for the study of the other Chapters or in their future works! In Chapter I, group actions are introduced and their properties, which will be used later, are explained; for their proofs we refer to the original literature but diverse examples of group actions are studied, most of them to be used later. The action of G on a space M is naturally transferred to the action of G on the functions de"ned on M. In all the applications presented in this issue, there exist `critical orbitsa which are the orbits of extrema for all invariant functions (Michel, 1971) and, for each function, the existence of other orbits of extrema on some strata can be predicted (Michel, 1971). For G-invariant Morse functions, the predictions of Morse theory in the presence of symmetry are explained; they require the existence of a minimum number of extrema of di!erent nature (minima, or maxima, or saddle points). Important examples are given in the di!erent chapters. In Chapter I, we also study the rings of invariant functions and more specially those of invariant polynomials with several examples. This study also allows a `qualitativea approach of many phenomena; a simple method is the study of level functions on orbit spaces (it has been introduced by Kim (1982, 1984) in high-energy physics and by Zhilinskii (1989a, 1989b) for molecules). Some work, for some space groups, had been made since forty years on types of invariant functions on the Brillouin zone. In Chapter V, the problem is completely solved for all space groups; the results are 0370-1573/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 0 0 ) 0 0 0 8 7 - 9
8
Avant-Propos
given as modules of invariant polynomials and presented in a dozen of short tables. It is remarkable that most of these modules have so few generators. In Chapter VI, the same philosophy is applied to the symmetry and topology of electron energy bands in crystals. The basic hypotheses have been made by Zak (1980, 1982a, 1982b). It has been possible to deduce their full consequences only in the last four years; this is a general presentation of the results. It must be emphasized that this monograph is not written for specialists; for instance, to help readers who have never worked in solid state physics, Chapter IV is an introduction to crystallography (using concepts de"ned in Chapter I). It has also to be read by solid state physicists because it emphasizes basic concepts (e.g. arithmetic class) and tools (necessary for the next two chapters) which are not even mentioned in usual text books. Moreover, some of the sections (e.g. 6 and 7) of this chapter contain completely new material. This monograph is written for readers who want to learn methods, either new or not enough common, for studying symmetry in physics and the `qualitativea or topological approach of the rigorous consequence of symmetries. We hope that they will enjoy the applications to molecular and crystal physics presented here. But the main aim of the authors of this monograph is to suggest to the reader to apply these tools to other domains of physics! For this, we sometimes include (generally in smaller print or in appendix) glimpses of theories not strictly necessary for the studied examples but which may be necessary for other applications. Seeing analogy between some well understood and some new puzzling phenomena is a natural way of discovery in science; and it is very fruitful when the two classes of phenomena can be described by the same type of mathematics. For a more advanced "eld of science, `qualitativea mathematical approach is a way of `thinkinga about phenomena for the next generation of scientists. We see a deep truth in the sentence written by Galilei (1623) at the dawn of modern science:
&&La "loso"a e` scritta in questo grandissimo libro, che continuamente ci sta aperto innanzi agli occhi (io dico l'universo), ma non si puo` intendere se prima non s'impara a intender la lingua, a conoscer i caratteri, ne` quali e` scritto. Egli e` scritto in lingua mathematica2''
References Galilei, G., 1623. Il Saggiatore. Lincean Adaemy, Rome. Kim, J.S., 1982. General methods for analysing Higgs potentials. Nucl. Phys. B 196, 285}300. Kim, J.S., 1984. Orbit spaces of low-dimensional representations of simple compact connected Lie groups and extrema of a group-invariant scalar potential. J. Math. Phys. 25, 1694}1717. Michel, L., 1971. Points critiques des fonctions invariantes sur une G-varieteH , C. R. Acad. Sci. Paris 272, 433}436. Weyl, H., 1931. The Theory of Groups and Quantum Mechanics, 1st Edition, 1928. Dover Publications, Dover. Wigner, E., 1959. Group Theory and its Application to the Quantum Mechanics of Atomic Spectra, 1st Edition, 1931. Academic Press, New York. A summary in English: `The great book of the universe stays open before our eyes; but to understand it, we have "rst to learn the language in which it is written, the mathematics.a
9
Avant-Propos Zak, J., 1980. Symmetry speci"cation of bands in solids. Phys. Rev. Lett. 45, 1025}1028. Zak, J., 1982a. Continuity chords of bands in solids. The diamond structure. Phys. Rev. B 25, 1344}1357. Zak, J., 1982b. Band representation of space groups. Phys. Rev. B 26, 3010}3023. Zhilinskii, B., 1989a. Qualitative analysis of vibrational polyads: N mode case. Chem. Phys. 137, 1}13. Zhilinskii, B., 1989b. Theory of Complex Molecular Spectra. Moscow University Press, Moscow.
Louis Michel Institut des Hautes E! tudes Scientixques, 91440 Bures-sur-Yvette, France Jai Sam Kim Department of Physics, Pohang University of Science and Technology, Pohang 790-784, South Korea Josuah Zak Department of Physics, Technion, Israel Institute of Technology, 32000 Haifa, Israel Boris Zhilinskii Universite& du Littoral, BP 5526, 59379 Dunkerque Ce& dex, France
Physics Reports 341 (2001) 11}84
Symmetry, invariants, topology. I
Symmetry, invariants, topology. Basic tools L. Michel , B.I. ZhilinskimH * Institut des Hautes E! tudes Scientixques, 91440 Bures-sur-Yvette, France Universite& du Littoral, BP 5526, 59379 Dunkerque Ce& dex, France
Contents 1. Introduction 2. Group actions and their strata 2.1. Constructing more group actions from given ones 3. Examples of groups and group actions 3.1. Action of the group G on itself or on the lattice of its subgroups 3.2. Action of group G on another group K 3.3. Action of G on its set of elements G 3.4. Action of the group G on a manifold 3.5. Representations. Non-e!ective actions. Kernels and images of the irreducible representations of 3-D-point groups 4. Compact group smooth actions; their critical orbits; their linearization 4.1. Examples of critical orbits for group actions 5. Rings of G-invariant functions 5.1. Molien function manipulations 5.2. Integrity basis, syzygies, and other related notions 5.3. Extension to continuous groups 5.4. Invariant polynomials and integrity bases for 3-D crystallographic point groups
13 14 16 17 22 25 25 26
31 33 36 36 40 44 47
5.5. Ring of C invariant polynomials. DescripG tion in terms of generators and syzygies 5.6. Representation of the orbit space in terms of invariant polynomials 6. Morse theory 6.1. Examples of Morse theory applications. Stationary points of the simplest Morsetype functions 6.2. Modi"cations of the system of stationary points. Bifurcations 7. Physical applications 7.1. Action of the Lorentz group on the Minkowski space 7.2. Physical examples of systems and phenomena with continuous subgroups of O(3);T as symmetry groups 7.3. Geometrical con"guration of N-particle systems. Shape coordinates and their invariant description 7.4. Landau theory of phase transitions Appendix A. Group theory: Glossary Appendix B. Morse}Bott theory B.1. Compilation of Bott and Kirwan about the construction of equivariant homology References
49
* Corresponding author. E-mail address:
[email protected] (B.I. ZhilinskimH ). Deceased 30 December 1999. 0370-1573/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 0 0 ) 0 0 0 8 8 - 0
51 53 60
63 65 66 66
66
70 72 72 78 80 82
12
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
Abstract Elementary concepts of group actions: orbits and their stabilizers, orbit types and their strata are introduced and illustrated by simple examples. We give the uni"ed description of these notions which are often used in the di!erent domains of physics under di!erent names. We also explain some basic facts about rings of invariant functions and their module structure. This leads to a geometrical study of the orbit space and of the level surfaces of invariant functions (e.g. energy levels of Hamiltonians). Combining these tools with Morse theory we study the extrema of invariant functions. Some physical applications (not studied in other chapters) are sketched. 2001 Elsevier Science B.V. All rights reserved. PACS: 03.65.Fd; 31.15.Md Keywords: Group actions; Invariant polynomials; Critical orbits; Morse theory
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
13
1. Introduction This chapter introduces the basic concepts and the basic tools to be used in this issue of Physics Reports. The linear representations of groups are a particular case of group actions. The non-linear actions appear naturally in physics: for instance the action of the Euclidean group on the space is not a linear action! That is generally the case for the action of the symmetry group G of a physical problem on the con"guration space or on the space of dynamical variables. In Section 2 we explain the fundamental concepts for the study of group actions. The action of a group G on a space M (let us say for instance a manifold) decomposes M into strata which are themselves union of orbits of the same type; i.e. they correspond to the di!erent type of `locala symmetry. In all examples we study, the number of strata is "nite, so it is so natural to list them. For instance the action of the two-dimensional Euclidean group Eu on the plane de"nes an action of Eu on the triangles ("triples of distinct points) in this plane; there are three strata: the equilateral, the isosceles and the other triangles. There is only one stratum for the action of Eu on the segments (" pair of distinct points) of the plane and this stratum contains a continuous set of orbits (labeled by the length of the segment), while for the action of the PoincareH group (" Lorentz inhomogeneous group) on the segments of the #at space-time there are three strata: space like, time like, and light like. Indeed the word `stratuma was coined by Thom (1954, 1962, 1969) 50 years ago, in a more general situation of di!erential topology and seems to have been "rst applied to group actions by one of us (Michel 1971), thirty years ago. More examples of strata will be discussed in Sections 3 and 7. In Section 4 we study the G-invariant functions (e.g. Hamiltonian or Lagrangian); they play a great role. At each point of M, their gradient is tangent to the stratum; so the study of strati"cation of the orbits determines the `critical orbitsa which are orbits of extrema for all G-invariant functions. One can also determine for any G-invariant function the minimum number of orbits of extrema on each stratum. In Section 5 we review what is presently known on the nature of the ring of G-invariant functions on the space of a "nite-dimensional linear representations and how to "nd their generators. We give tables of examples which will be used in the later chapters. In some chapters we shall show how to extend the techniques of Section 5 for describing the ring of functions invariant under some non linear group actions. All these examples also yield illustrations of the predictions of the theorems of Section 4. In Section 6 we explain how one can apply Morse theory to G-invariant functions; this leads to the complete information on the minimal number of their extrema and on the nature of these extrema. In Section 7 we shall sketch some physical applications of this general approach not studied in the other chapters. Appendix A summarizes several important group-theoretical de"nitions which can be found in classical textbooks on group theory. We give them here to make this issue more self-contained. Appendix B includes some introduction into more complicated mathematical subject related with generalization of Morse theory to the case of function with continuous symmetry.
That is done in the di!erent domains of physics under di!erent names!
14
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
2. Group actions and their strata De5nition (Group action). An action of a group G on a mathematical structure M is de"ned by M Aut M into the group of automorphisms of M. a homomorphism GP For instance if M is a vector space <, then Aut M"G¸(<), the linear group on <, and o is a linear representation. In the more particular case of a Hilbert space H, the Aut H is the unitary group ;(H) and o is a (linear) unitary representation of G. We generally assume that Ker o is trivial: i.e. any element g3G, except the identity (gO1), moves at least one point of M. In that case the action is called ewective. Otherwise we will specify the kernel Ker o and remark that this non-e!ective G action de"nes an e!ective action of the quotient G/Ker o on M. (See Section 3.5 for examples of non-e!ective actions.) Next comes naturally the MG Aut M , i"1, 2 are equivalent De5nition (Equivalence of group actions). The two actions GP G F M which is equivariant, i.e. it commutes with the two if there exists an isomorphism: M P actions: ∀g3G: ho (g)"o (g) h 0 o (g)"h o (g) h\ .
(1)
The usual equivalence of group linear representations is a particular case of the equivalence of group actions. When we consider a unique action of G on M we generally shorten the notation o(g) (m), the transform of m3M by g3G, into g ) m. De5nition (Group orbit). The set of transforms of m, that we denote by G ) m, is the orbit of m. An orbit may have a unique element! this is a `"xed pointa. More generally, to be on the same orbit is an equivalence relation for the elements of M. So M is a disjoint union of its orbits. The set of orbits is called the orbit space and we denote it by M " G. In our applications the orbit space can be a manifold; more generally it is an orbifold, i.e. a manifold with singular points or sub-manifolds. De5nition (Stabilizer). The set G "+g3G, g ) m"m, of elements of G which leave m "xed, is the K stabilizer of m; it is a subgroup of G. In physical applications a stabilizer is sometimes called a local symmetry group or a little group. For a "xed point o, G "G. In the general case, it is easy to prove that G "gG g\; so the set M E K K of stabilizers of the elements of an orbit is a conjugacy class [H] of subgroups of G (H is one of the % stabilizers).
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
15
M M de"nes implicitly an action of A very simple, but important remark, is that the action GP any strict subgroup H(G, by restricting o to H (the restriction is denoted by o" ). When the & M M is restricted to that of a subgroup H the stabilizer H of the H-action follows action GP K immediately from G-action M Aut M: H "H5G . H(GP K K
(2)
In general the G-orbits split into a disjoint union of H-orbits and there is a natural (surjective) map M"HPM"G .
(3)
When H is an invariant subgroup of G (we write here H¢G, as usual in the mathematical literature on groups) there is a natural action of the quotient group G/H on the orbit space M"H. This general theorem is very intuitive when H is a G-subgroup of index 2, i.e. G/H&Z : then there are two types of Z orbits on M"H, those of one point and those of two points. We shall use several times this theorem, most often in the simplest case. De5nition (Orbit type). Orbits with the same conjugacy class of stabilizers are of the same type. One such type with G as stabilizer is the set of "xed points. Similarly the orbits with a trivial stabilizer 1 (then all stabilizers of the orbit are 1), are called principal orbits. The action of the group G on M is free if all orbits have trivial stabilizers (i.e. all orbits are principal). We shall often use the notation G : H for the type of a G-orbit whose stabilizers are the G-subgroups conjugate to H. For instance a principal orbit can be noted G:1. For a "nite set F we denote by "F" its number of elements. When G is "nite, the number of elements of an orbit of type G:H is "G" . "G:H"" "H"
(4)
For the smooth actions ("in"nitely di!erentiable) of Lie groups, the orbit G:H is a manifold of dimension: dim(G:H)"dim(G)!dim(H) .
(5)
If dim(G:H)"0, i.e. the orbit has a "nite number of points, this number is the quotient of the number of connected components of G and H.
The traditional notation for subset is X-M and, for strict subset, XLM. For subgroups of groups we "nd it convenient to use the less traditional notation H4G (or for strict subgroups: H(G). This notation recalls also the fact that to be a subgroup of a group is a partial order relation (by inclusion) among the subgroups of G.
16
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
Group G acts on the space M transitively if there exists only one orbit, the space M itself. That is for instance the case of the action of an Euclidean group on its space. De5nition (Stratum). In a group action, a stratum is the union of orbits of the same type. Equivalently, two points belong to the same stratum iw ("if and only if ) their stabilizers are conjugate. When they exist, the "xed points form one stratum and the principal orbits form another one. The set of strata is called the stratum space and is denoted by M""G. To belong to the same stratum is an equivalence relation for the elements of M or for those of the orbit space M"G. The set of strata M""G is a (rather small in our applications) subset of the set of conjugacy classes of subgroups of G. Notice that there exists a natural partial order between these conjugacy classes of subgroups: one is smaller than the other if it contains a group which is a strict subgroup of a group of the other. That gives to the set of strata M""G a structure of partially ordered set: S 3M""G, S (S means that the local symmetry of S is smaller than that of S ; i.e. the G stabilizers of the points of S are, up to a conjugation, subgroups of those of S . Remark that relation (3) between the orbit spaces of G and of the subgroup H obtained by restriction cannot be extended to the stratum spaces: indeed there is no natural map between them and often the set of strata is larger for the larger group. 2.1. Constructing more group actions from given ones Given one or several G actions, one can consider other actions which are automatically de"ned M Aut M de"nes automatically the action of G on from the given ones. For instance the action GP MG M the subsets of M or, in particular, on families of these subsets. Similarly, from the actions GP G on di!erent spaces M one de"nes `naturala actions on the structures built from the M , e.g. their G G topological product. An important example is the action on F(M , M ), the space of function from M to M . Denoting by g ) f the transformed by g3G of the function f3F(M , M ) one has the de"nition ∀g3G, ∀m3M , (g ) f )o (g) (m)"o (g)f (m) 0 (g ) f ) (m)"o (g) ) f (o (g\) ) m) .
(6)
The function f is G-invariant when g ) f"f for all g3G. We will meet often the simpler case when G acts trivially on M ; then the G-invariant functions satisfy ∀g3G, ∀m3M , f (m) "f (g ) m).
Beware that the less symmetric stratum might have a larger dimension than the more symmetric ones. So we emphasize that the partial order on the set of strata deals with their local symmetry and not with their size. In the introduction we gave the example of the Euclidean group acting on the triangles and of the Lorentz group acting on the segments of space time.
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
17
3. Examples of groups and group actions We have "rst to recall the two standard notations used for subgroups of O(d) (d"2, 3) in molecular physics and chemistry and in solid state physics and crystallography. These are SchoK n#ies notation (Landau and Lifshitz, 1965; Hamermesh, 1964) and International Tables of Crystallography (ITC, 1996) notations. (In what follows we will use the abbreviation ITC for International Tables of Crystallography (ITC, 1996).) In contrast to the abstract mathematical de"nition of groups (up to an isomorphism) the natural physical classi"cation of point groups ("nite or continuous subgroups of O(3) group) is done up to conjugation in O(3). We call them geometric classes. In crystallography one also needs to consider arithmetic classes, i.e. the classes of subgroups conjugated in G¸(3, Z), the group of 3;3 matrices with integer elements. More generally one may have to consider the classi"cation of subgroups H of G up to conjugation in G. To see better the correspondence between di!erent classi"cations of subgroups let us start with subgroups of O(2), the group of orthogonal transformations of two-dimensional space. The matrices of O(2) of determinant 1 (respectively, !1) are the rotations r(h) by an angle h around the origin (respectively, the re#ections s( ) through the axis of azimuth ):
r(h)"
s( )"
cos h
!sinh
sin h
cos h
,
h(mod 2p) ,
cos(2 )
sin(2 )
sin(2 )
!cos(2 )
,
(7)
(mod p) .
They satisfy the following relations: r(h)r(h )"r(h#h ), s( )s( )"r(2( ! )) , r(h)s( )"s(h# )"s( )r(!h) .
(8)
In particular r(h)r(!h)"I, s( )"I, s( )r(h)s( )"r(!h) , r(h)s( )r(h)\"s( #2h) .
(9)
We denote by c the n-element group formed by the rotations r(2pk/n), 04k4n!1; it is L isomorphic to Z , cyclic group of order n, and c ¢O . We denote by c the conjugation class of L L Q subgroups c &Z generated by the re#ection s( ). The two re#ections s( ) and s( #p/n) Q( generate a 2n element group that we denote by c , with the simple notation c for the LT( LT conjugacy class [c ] . These groups are isomorphic to the `dihedrala groups, de"ned by the LT( generators and relations s "s "1"(s s )L; we denote them by d (Coxeter and Moser, 1972); L they are non-Abelian for n'2. The group c is the symmetry group of a regular polygon of LT n vertices and edges.
The notations c , c , c are those of SchoK n#ies. We use small letters for 2-D groups and capital letters for 3-D groups. L Q LT
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
18
Fig. 1. The partially ordered sets of conjucacy classes of the subgroups of C , C , C . When a subgroup is not T T T invariant we indicate before its name, between [ ], the number of subgroups in its conjugacy classes. The order of subgroups is indicated at the left.
The set c , n51 and c , n51 give the complete list of "nite subgroups of O(2). Remark that L LT c is a trivial `no symmetrya group, c is the c group. To exhaust all subgroups of O(2) we need to T Q add c which is the SO(2) and c which is the complete O(2) group. T In Fig. 1 we give the partially ordered set of the conjugacy classes of subgroups of c , c , c T T T which are the symmetry groups of the square, the regular hexagon, and the regular pentagon, respectively. When n is prime, c has only four conjugacy classes (including 1 and the whole LT group). When n is even there are pairs of conjugacy classes which belong to the same geometric classes. To distinguish them, we have added prime ` a at a member of each pair. However we do not need to invent a new notation since one already exists and is known by some of the readers. The symmetry point group of crystals are "nite subgroups of O(d ), (here d"2, 3). Their conjugation class in O(d) is called the `geometry classa. For d"2, 3 there are 10, 32 geometric classes. This classi"cation is generally su$cient for describing the symmetry of the macroscopic physical properties of crystals. It is not su$cient for the symmetry classi"cation of microscopic properties. The action of a point group P on ¸, the lattice of translations of the crystals is given by the injective homomorphism (see the "rst de"nition of Section 2) PPAut ¸"G¸(d, Z). Here we label the image of P by PX, it de"nes a conjugacy class in G¸(d, Z). These classes are called arithmetic classes. For d"2, 3 there are 13, 73 of them. There are two maximal conjugacy classes of "nite subgroups of G¸(2, Z). Their geometric classes are c and c and they coincide with the arithmetic classes which are denoted by p4mm, p6mm. T T The symmetry group of the square c &O(2, Z) contains four re#ections; they are represented T by the two pairs of matrices:
1
0
$
0 !1
and $
0 1 1 0
,
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
19
which are denoted, respectively, by $pm,$cm in crystallography since they are not conjugate in G¸(2, Z). The same symbols are also used for the two-element groups they generate. More generally, for the eight conjugacy classes of subgroups of the symmetry group of the square, we give here the dictionary between the SchoK n#ies notations used in Fig. 1 and the ITC, (1996) ones: 1p1, c p2, c pm, c cm, c p4 , Q Q c p2mm, c c2mm, c p4mm . (10) T T T As we will show in Chapter IV, the "rst letters p, c distinguish between two Bravais classes of lattices. The dictionary between the notations for the conjugacy classes of subgroups of c is T c "cmm, c p3, c "p3m1, c "p31m , (11) T T T c "p , c "p6mm . (12) T The conjugacy classes cm and cm of c are conjugated in G¸(2, Z) and not distinguished by the T ITC notation. The ITC notation will be partly de"ned below for d"3 and fully explained in Chapter IV. Remark that for d"2, 3 the orders of the elements of the "nite groups of G¸(d, Z) are 1, 2, 3, 4, 6. We have decided to use both notations: that of SchoK n#ies used by molecular physicists and chemists and that of ITC, so well adapted to crystallography and solid state physicist need. That will help each tribe of scientists to read the literature of another tribe. In this chapter, to help the reader, when we use a group we use both symbols. The description of three-dimensional symmetry groups can be naturally based on the above established list of 2-D symmetry groups. The two-dimensional subgroups, c , c , c which act trivially in the perpendicular direction of their L Q LT plane, are denoted in three-dimensions by C , C , C . One adds C , the two-element group generated L Q LT G by !I , the symmetry through the origin, which can be equivalently described as S group with the S element being the rotation}re#ection of order two, i.e. the product of C rotation and the re#ection in the perpendicular plane. Generalizing C to C with n even leads to S groups L L isomorphic to Z , where we use again the notation Z for the m-element cyclic group. L K Historically, the index v is for vertical; it is the direction of the rotation axis of C , C . If one adds L LT to C the symmetry through the horizontal plane, one obtains the group C . Remark the L LF isomorphisms: C &Z ;Z ; but when n is odd that is also C &Z . If one adds to C a rotation LF L LF L L by p around a horizontal axis (containing the origin), one obtains the group D (SO ; note that it L is isomorphic to C . Adding the symmetry through the horizontal plane to D or C one obtains LT L LT the same group which is denoted by D . When one adds to D the vertical symmetry planes LF L bisectors of neighbor axes of rotation by p, one obtains D . LB We have still to explain the irreducible subgroups of O : The symmetry groups of the regular tetrahedron is denoted by ¹ , of the cube (or regular B octahedron) by O , of the regular icosahedron (or dodecahedron) by > . Then we de"ne F F ¹"¹ 5SO , O"O 5SO , >"> 5SO ; note that ¹;C "¹ (also O;C "O , B F F G F G F >;C "> ). G F It would have been better to use the notation ¹ , O , > instead of ¹ , O , > but we follow the tradition. G G G F F F
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
20
Thus the complete list of "nite 3-D geometrical classes includes the following groups given here in SchoK n#ies notation. There are seven in"nite sequences C , C , S , C , D , D , D , and seven L LF L LT L LB LF `exceptionala groups ¹, ¹ , ¹ , O, O , >, > . There are also "ve one-dimensional Lie subB F F F groups C , C , C , D , D , and SO(3), and O(3) itself. C is another notation for SO(2), the F T F group of rotations around the `verticala axis and C is O(2), i.e. it is generated by C and T a re#ection through any vertical plane containing the vertical rotation axis. Similarly C is F generated by C and a re#ection through the horizontal plane; it would be noted C , indeed G it contains the re#ection through the origin !I (notice that C is Abelian). D is generated F by C and a rotation by p around an azimuthal axis passing through the origin. Finally, D "D ;C . F G The other standard notation, used in ITC (1996), is necessary for the study of crystals (Chapters IV}VI). It is based on the description of groups in terms of generators. In O(3) there are two conjugacy classes of elements of "nite order n. If its determinant is 1, it is a rotation by 2p/n which is denoted by n; if its determinant is !1 (n has to be even), then it is the product of a rotation by the matrix !I and it is denoted by n with the exception that 2 is replaced by m, for mirror (indeed it is a re#ection through a plane). The cyclic groups generated by these "nite-order elements are denoted by the same symbol: hence the translation ITC SchoK n#ies: nC , 1 C , mC , L G Q
(13)
n,0 mod 4: n S , L
(14)
64n,2 mod 4: n C , L F
n odd: n C &S . LG L
(15)
For the "nite subgroups of O(3) with two or three generators one writes these generators together; for example: nmC , n2D . Beware that, to distinguish di!erent groups with the same generLT L ators, a convention is made for the order of generators, e.g. 32D , 23¹, 3 mD , m3 "¹ . B F The ITC notation can be used as well for one-dimensional Lie subgroups of O(3): C "R, C "Rm, C "R/m, D "R2, D "Rmm (see ITC, 1996, p. 783). T F F The full list of the 32 geometric classes was established (independently) by Frankenheim (1826) and Hessel (1830) before the word group was created by Galois in 1830 (Galois' work was published only in 1846 (Galois, 1846)). Table 1 gives the list of these 32 geometric classes in the two notations and classes them among the 18 isomorphy classes they form; note that half of them are isomorphy classes of Abelian groups. Among all crystallographic 3-D-geometrical classes there are two maximal ones: O "m3 m and F D "6/mmm. F In particular the symmetry group O of the three-cube has 48 elements. It is generated by nine F re#ections forming two conjugacy classes. In the coordinate system we de"ned above (in dimension d), O is represented by O(3, Z) and the matrices representing its re#ections are the three (Pm ) and F G the six (Cm ) and (Cm ): G G
1 0
0
0 , (Pm )" 0 1 0 0 !1
!1 0 0
(Pm )"
1
0 0
0 1 0 , (Pm )" 0 !1 0 , 0 0 1 0 0 1
(16)
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
21
Table 1 The 32 crystallographic geometric classes and their 18 isomorphy classes. The isomorphy classes are listed in columns 1, 3 and are de"ned as direct products of cyclic groups Z , dihedral groups c , permutation group of four objects S , and L LT its subgroup of even permutations A . In column 2, 4 the geometric classes are listed in ITC and SchoK n#ies notations Isom.
Geometric
Isomorphic
Geometric
1 Z Z Z c T c T c ;Z T A S
1"1 2/m"C , mm2"C , 222"D F T 3"C 4"C , 4 "S 3m"C , 32"D T 4mm"C , 422"D , 4 m2"D T B 6mm"C , 622"D , 3 m"D , 6 m2"D T B F 23"¹ 4 3m"¹ , 432"O B
Z Z Z ;Z Z ;Z Z ;Z c ;Z T c ;Z T A ;Z S ;Z
1 "C , m"C , 2"C G Q mmm"D F 6"C , 3 "C , 6 "C G F 4/m"C F 6/m"C F 4/mmm"D F 6/mmm"D F m3 "¹ F m3 m"O F
0 1 0
1 0 0
0 0 1
(Cm )" 1 0 0 , (Cm )" 0 0 1 , (Cm )" 0 1 0 , 0 0 1 0 1 0 1 0 0 (Cm )"!(Pm ) (Cm ) . G G G We also introduce the orthogonal matrices:
(17)
(18)
(P4)"(Pm )(Cm )"!(P4 ) , (R3)"(Cm )(Cm )"(Cm )(Cm )"(Cm )(Cm )"!(R3 ) . (19) O(3, Z) is generated by the three matrices Pm , (R3 ), (Cm ); that explains the ITC notation Pm3 m for O(3, Z). Let us recall that the symmetry group of the three-cube corresponds to a unique geometry class (conjugacy class in O or G¸(3, R)) which is denoted by O "m3 m, but it is F isomorphic to three arithmetic classes in G¸(3, Z) which are denoted in ITC Pm3 m, Fm3 m, Im3 m; the "rst one will be studied here and the last two in Chapter IV. The group O "m3 m has 98 subgroups including c "1 and O itself falling into 33 conjugacy F F classes. For O(3, Z)"Pm3 m these classes coincide with the conjugacy classes in the larger group G¸(3, Z), so they are arithmetic classes. We shall denote them by their ITC labels. These 33 arithmetic classes form only 25 crystallographic geometric classes, so their SchoK n#ies notation is ambiguous. For each arithmetic class 4O(3, Z) we choose a representative subgroup and give in Table 2 a set of matrices which generate it. In Fig. 2 we give the partial ordered set of the 33 conjugacy classes of subgroups of O(3, Z)"Pm3 m in SchoK n#ies and ITC notation. These classes are arithmetic classes; each one is explicitly de"ned by matrices generating its subgroups. Of course the notation used for arithmetic classes is that of ITC. Explicitly, when more than one arithmetic class correspond to a geometric class 4O , we list them in Table 3 with gc for geometric class, ac for arithmetic class: F
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
22
Table 2 Set of generators for arithmetic classes. For each arithmetic class 4O(3, Z) we choose a representative subgroup and give a set of matrices which generate it. The matrices are !I and those de"ned in Eqs. (16)}(19) P1 : (!I) C2: !(Cm ) Pmm2: !(Pm ), (Pm ) Amm2: !(Cm ), (Pm ) P4/m: (P4), (Pm ) P4 m2: !(P4), (Pm ) R32: (R3),!(Cm ) G Pm3 : (Pm ), (R3) G
P2: !(Pm ) Cm: (Cm ) Pmmm: (Pm ) G Cmmm: $(Cm ), (Pm ) P422: (P4),!(Pm ) P4/mmm: (P4), (Pm ) G R3m: (R3), (Cm ) G P432: !(Pm ),!(Cm ) G G
Pm: (Pm ) C2/m: $(Cm ) C222: !(Cm ),!(Pm ) P4: (P4) P4mm: (P4), (Pm ) R3: (R3) R3 m: !(R3),$(Cm ) G P4 3m: !(Pm ), (Cm ) G G
P/m: $(Pm ) P222: !(Pm ),!(Pm ) Cmm2: (Cm ),!(Pm ) P4 : !(P4) P4 2m: !(P4),!(Pm ) R3 : !(R3) P23: !(Pm ), (R3) G Pm3 m: (Pm ), (Cm ) G G
Table 3 Correspondence between geometric and arithmetic classes for subgroups of O &Pm3 m F gc: ac:
C P2, C2
C Q Pm, Cm
C F P2/m, C2/m
gc: ac:
D P222, C222
D F Pmmm, Cmmm
D B P4 2m, P4 m2
C T Pmm2, Cmm2, Amm2
To distinguish between P4 m2 and P4 2m we note that the "rst symbol after 4 indicates the symmetry element in the coordinate planes: they are the re#ections through the planes for 4 m2 and the axes of rotation of order 2 in 4 2m. It is this last group which is a subgroup of the tetrahedron group 4 3m. Similar analysis can be done for another maximal "nite subgroup of O(3), namely D "P6/mmm. The diagram of the partially ordered conjugacy classes of subgroups of F P6/mmm"D is given in Fig. 3. The 32 conjugacy classes of subgroups correspond to 28 F conjugacy classes in G¸(3, Z); for them we use the names of ICT. Only 16 of them are not orthogonal arithmetic classes (O(3, Z). They all contain 3"C which is the derived group. There are three conjugate subgroups Cmmm; this arithmetic class is the largest of all arithmetic classes contained also in Pm3 m"O(3, Z). Remark that there are four pairs of isomorphic hexagonal arithmetic classes: D "P321, P312, C "P3m1, P31m , T D "P3 m1, P3 1m, D "P6 m2, P6 2m B F among the subgroups of P6/mmm.
(20) (21)
3.1. Action of the group G on itself or on the lattice of its subgroups The homomorphism GPAut G has for kernel C(G), the center of G and for image the group In Aut G of inner automorphisms, i.e. G acts on itself by conjugation: g ) x"gxg\. The stabilizer
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
23
Fig. 2. Lattice of conjugated subgroups of O "Pm3 m. Two parts of the "gure are identical except for the notation. F SchoK n#ies notation is used in the upper part of the "gure and the ITC notation in the lower part. The 33 conjugacy classes of subgroups of Pm3 m"O(3, Z)&O correspond to 33 arithmetic classes 4O(3, Z). Invariant subgroups are F underlined.
G is the set of group elements commuting with x; it is a subgroup (the whole group for Abelien V group) called the centralizer of x and denoted by C (x). The orbit G ) x is the conjugacy class of x. % For "nite groups "C (x)""G ) x"""G". We will often use the corresponding action of G on the set of % its subgroups. The centralizer of the subgroup H is the set +g3G, gHg\"H,. It is a subgroup called the normalizer of H in G and denoted by N (H); it is the largest subgroup of G which % contains H as an invariant subgroup. The orbit is the conjugacy class [H] of H in G. The orbit %
24
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
Fig. 3. Lattice of conjugated subgroups of P6/mmm.
space is the set of conjugacy classes of subgroups; it is a partially ordered set. Such partially ordered sets are shown in Fig. 1 for the subgroups of c , c , c . T T T Classes of conjugated subgroups of the O group are given in Fig. 2 for the convenience of reader F using both SchoK n#ies (Hamermesh, 1964; Landau and Lifshitz, 1965; Lyubarskii, 1957; ITC, 1996) and international crystallographic notation (ITC, 1996). There are several invariant subgroups ¹ "P4 3m, O"P432, ¹ "Pm3 , ¹"P23, D "Pmmm, D "P222, C "P1 , C "P1, i.e. B F F G corresponding classes of conjugated subgroups include only one subgroup. We remark the existence of three non-conjugated C subgroups (Pmm2, Cmm2, Amm2) and six pairs of nonT conjugated subgroups D , (P4 2m, P4 m2); D , (Cmmm, Pmmm); D , (C222, P222); C , (P2/m, B F F C2/m); C , (P2, C2); C , (Pm, Cm). Fifteen classes of conjugated subgroups include each three Q subgroups (P2, Pm, P4, P4 , Pmm2, Cmm2, C222, P2/m, P4mm, P4/m, P4 2m, P4 m2, P422, Cmmm, P4/mmm). All "ve R classes include each four conjugated subgroups and four classes (C222, C2/m, Cm, C2) consist of six subgroups. All 33 classes of conjugated subgroups (we count the O "Pm3 m group itself and the trivial F C "P1 subgroup as well) remain inequivalent (non-conjugated) even if we consider them as subgroups of the larger group G¸(3, Z). This means that all these subgroups correspond to di!erent arithmetic classes of 3D-crystals. (Arithmetic classes will be more fully discussed in Chapter IV.) At the same time three non-conjugated C (Pmm2, Cmm2, Amm2) subgroups as well as all menT tioned above pairs of non-conjugated subgroups become conjugated as subgroups of O(3). Non-conjugated in O(3) classes of subgroups correspond to geometrical classes of crystals. There
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
25
are 25 di!erent geometrical classes which are the subgroups of O "Pm3 m. Remind that there are F 32 di!erent geometric classes. At the same time only two isomorphy classes (6/m&C and F 6/mmm&D ) are absent among subgroups of O "Pm3 m. F F Another maximal subgroup P6/mmm&D and its classes of conjugated subgroups are given in F Fig. 3. The complete list of arithmetic classes will be constructed in Chapter IV. 3.2. Action of the group G on another group K M Aut K gives an action of G on K respecting its group structure. A group homomorphism GP We will meet many examples of this construction. Given such an action we can build a new group on the Cartesian product (K, G) of the sets of elements of K and G. This group is called the semi-direct product of K by G and is usually denoted by K ) G; its group law is (k , g )3(K, G), (k , g ) ) (k , g )"(k (g ) k ), g g ) . (22) G ? The d-dimensional Euclidean group Eu and the PoincareH group P are the semi-direct products B Eu "RB ) O(d), P "R ) O(3, 1) (23) B where O(3, 1)&O(1, 3) is the Lorentz group, i.e. the group which leaves invariant the quadratic form (x !x !x !x ). It is useful to know a linearization of the Euclidean group (and of the PoincareH group), but for this we have to use (d#1);(d#1) matrices
(t, A)3RB ) O(d) C
A t 0
1
,
(24)
acting on the (d#1) variables (x ,2, x ,1). B Using Fourier transform one can also work in the space of momenta (or energy momenta in special relativity). We remind the reader that this space is a vector space (containing a null vector p"0). So the action of Eu on the momentum space is linear; indeed it is not faithful: the B translations act trivially. In crystallography each arithmetic class (" conjugacy classes of "nite subgroups of G¸(3, Z)) de"nes an action of a "nite subgroup (the point group) on a lattice (isomorphic to the Abelian group Z), so 73 of the crystallographic space groups in dimension 3 are semi-direct products. The direct product K;G is the particular case of semi-direct product when the action of G on K is trivial. 3.3. Action of G on its set of elements G Beware that G is only a set (the corresponding automorphism group is the permutation group of its elements). The natural action of g3G is by multiplication on the left g ) x"gx (or on the right
These inhomogeneous groups are written as subgroups of the a$ne group RB ) G¸(d, R)"A+ (G¸(d#1, R). B Beware that for n'2 the adjoint representation G¸(n, R) U gCg ,(g\)?"(g?)\ is not equivalent to the natural one. So one can extend Eu to two A+ non-conjugate in G¸(3, R).
26
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
Fig. 4. Action of the O(n) group on n-dimensional space de"ned by its natural (vector) representation. n"1, 2 are shown. For each n there are two types of orbits: one point orbit with the O(n) stabilizer (the origin) and a continuous family of orbits with the O(n!1) stabilizer.
Fig. 5. Action of the O(3) group on three-dimensional space de"ned by its natural (vector) representation. Two orbits with the O(2) stabilizer are shown.
Fig. 6. Space of orbits for the natural action of the O(n) group on n-dimensional space. Two di!erent strata are indicated by their stabilizers O(n) and [O(n!1)] . -L
g ) x"xg\). The orbit is G itself: it is a principal orbit. For the corresponding action of G on its subgroups, g ) H"gH, the left coset of H by g; so the stabilizer is H itself. The orbit is the set of H left cosets and is often denoted by G : H; this orbit is often considered as the prototype of the orbits of type [H] . % 3.4. Action of the group G on a manifold 1. The natural (also called vector) representation of O(n) on the orthogonal space < has two L strata: the origin, a "xed point and a one parameter family of (n!1)-dimensional spheres S with stabilizers [O(n!1)] . Examples of orbits are shown in Figs. 4 and 5 for n"1, 2, and L\ -L 3. Corresponding spaces of orbits are given in Fig. 6. For every n the space of orbits is onedimensional. If we restrict the action of O(n) to its strict subgroup SO(n) the system of orbits does not change but their stabilizers do. (Stabilizers become SO(n) and SO(n!1).)
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
27
Fig. 7. Action of C "R and C "Rm on S induced by natural action of O(3). There are two one-point orbits T (north and south poles of the sphere) and a one-parameter family of circle orbits (each one being a parallel).
Fig. 8. Orbifold for C "R and C "Rm group action on S induced by natural action of O(3). The stabilizer for T one-point orbits is C (C ) and for generic S orbits is C (C ) for the symmetry groups C (C ), respectively. T Q T
2. The vector representation of O(3) de"nes an action of its one-dimensional subgroups on S , with S being one orbit of the natural O(3) action. For the action of C "R and C "Rm T subgroups there are two strata: one consists of two "xed points (" the two poles), the other includes a one-parameter family of circles (the `parallelsa) with stabilizers 1 and C for C and Q C subgroups, respectively. A coordinate of the one-dimensional orbit space is called the latitude T j with !9034j4903. See Fig. 7 for the representation of orbits and Fig. 8 for the space of orbits (orbifold). For the action of C "R/m, D "R2, D "Rmm subgroups, there are three strata. Two F F strata consist of one orbit: one contains the two poles (stabilizers C , C , C , respectively), the T other, the equator (stabilizers C , C , C ). The third stratum is the generic one, containing Q T a one-parameter family of pairs of parallels with the same absolute value j of latitude, i.e. 0(j(903. Fig. 9 gives orbits for these groups. Notice that for these "ve groups, the orbit space is a closed line segment. (See Figs. 8 and 10.) 3a. Vector representation of O "Pm3 m, the symmetry group of the cube; see e.g. Jaric et al. F (1984), and Michel and Mozrzymas (1978). This group contains the re#ections through nine symmetry planes intersecting at the cube symmetry center. These planes fall into two O -orbits (for F the O -induced action on the set of two-dimensional subspaces " planes of the representation F space < ): one orbit contains the three planes P , parallel to the faces, the other orbit contains the G six planes P , each one containing two opposite edges. Each re#ection generates a two-element H group conjugate in O(3) to C . We denote, respectively, by [C ] F "Pm and [C ] F "Cm the two Q Q Q corresponding conjugacy classes of, respectively, 3, 6 two-element subgroups. The group O conF tains three families of rotation axes which are, respectively, made up of three axes of rotations by p/2 (they contain the center of opposite faces), four axes of rotations by 2p/3 (they contain opposite
28
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
Fig. 9. Action of C "R/m, (D "R2, and D "Rmm) on S induced by natural action of O(3). There are one F F two-point-orbit (north and south poles of the sphere) with the stabilizer C , (C , C ), one S circle orbit with the T stabilizer C , (C , C ), and a one-parameter family of orbits (each one being a pair of circles) with the stabilizer Q T C , (C , C ). Q
Fig. 10. Orbifold for C "R/m, D "R2, and D "Rmm group action on S induced by natural action of O(3). F F The stabilizer for two-point-orbit is, respectively, C , C , and C . The stabilizer for one circle orbit is C , C , and C . T Q T The stabilizer for two-circle-orbit is C , C , and C . Q
vertices), six axes of rotations by p (they contain the middle of opposite edges). Hence the existence of seven strata: E the generic three-dimensional stratum contains all points which do not belong to a symmetry plane: stabilizer C "1; each orbit includes 48 points; E the two-dimensional strata whose stabilizers are [C ] F "Pm and [C ] F "Cm, respectively: Q Q they contain all points belonging to a unique symmetry plane: P and P , respectively; each orbit G H consists of 24 points; E the stratum of stabilizers [C ] F "R3m, it contains all points which are at the intersection of T only three symmetry planes of type P and it has an alternative description as the union of four H three-fold rotation axes minus the origin; each orbit includes eight points; E the stratum of stabilizers [C ] F "Amm2, it contains all points which are at the intersection of T only two symmetry planes, one of type P the other of type P and it is the union of six two-fold G H rotation axes minus the origin; each orbit includes 12 points; E the stratum of stabilizers [C ] F "P4mm, it contains all points which are at the intersection of T only four symmetry planes, two of type P and the other two of type P , that stratum is the union G H of three four-fold rotation axes minus the origin; each orbit consists of six points; E the maximal stratum contains only one point, "xed by O "Pm3 m: the origin. F The nine symmetry planes partition the space < into 48 convex cones; any one closed cone can be identi"ed with an orbit space. The corresponding space of orbits is given in Fig. 11.
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
29
Fig. 11. Orbifold for O "Pm3 m group action on 3-D-space induced by natural action of O(3). Seven di!erent strata are F given by their stabilizers. Fig. 12. Orbifold for O group action on S deduced from its action on 3-D-space. Six di!erent strata are given by their F stabilizers.
3b. The non-linear action of O "Pm3 m on S is deduced from the previous action. There are no F "xed point. The three maximal strata have for stabilizers: [C ] F "P4mm, [C ] F "R3m, T T [C ] F "Amm2 and each contains a unique orbit of 6, 8, 12 points, respectively. The orbit space is T a two-dimensional orbifold schematically represented in Fig. 12 as a "lled triangle. Each of its three vertices represents the orbit of a maximal stratum; the open edge between the vertices [C ] F "P4mm and [C ] F "Amm2 represents the one-parameter family of 24-point orbits T T with stabilizers forming [C ] F "Pm. The other two open edges represent the stratum whose Q 24-point orbits have [C ] F "Cm as stabilizers. The inside of the triangle represents the "O ""48 F Q point orbits of the generic stratum. 3c. The action of two-element inversion group C "P1 on two-dimensional sphere S induced G by its natural action on three-dimensional space. This action is free. All orbits formed by two opposite points on the sphere are principal and there is only one stratum. The space of orbits is a manifold, namely the real projective space RP . 4. The action of SO(2) group on the direct product of two-dimensional spheres S ;S . To de"ne such an action we introduce explicitly spheres in three-dimensional space as x"R and G y"R and consider the diagonal action of the SO(2) group as a simultaneous rotation around G x and y axes of two spheres. It is clear that this action creates two kind of orbits. There are four exceptional one-point orbits with the stabilizer SO(2). These four orbits correspond to the choice of north or south pole on each of two spheres. All other orbits are circles which have a trivial stabilizer and can be characterized by specifying the relative orientation of two vectors de"ned on two spheres. We can specify each orbit by two projections of corresponding vectors on axes x and y , by scalar product of these vectors, and by additional parameter which distinguishes the orientation of the triple formed by two vectors and the axis of rotation. This construction arises quite naturally in the classical analysis of the Rybderg
30
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
atoms and molecules (see Chapter III) and in the problem of coupling of two angular momenta (Sadovskii and Zhilinskii, 1999). 5. Action of the SO(2) group on the N-dimensional complex space C . Let us de"ne the action of , SO(2) group on N complex variables through (z , z ,2, z )P(z e (, z e (,2, z e () . (25) , , This action leaves invariant the norm "z ". For each non-zero norm the space of orbits gives the G G complex projective space CP (Cox et al., 1992; Mumford, 1976). ,\ The construction of the complex projective space as a space of orbits of the SO(2) group acting on N complex variables will be studied in more details in Chapter II devoted to molecular models because this construction is naturally related to the reduction of the dynamic symmetry in the study of the internal structure of the so-called vibrational polyads (Zhilinskii, 1989a). One sees immediately that the restriction of the action of the SO(2) group de"ned on C to , 2N!1 real-dimensional sphere possesses N exceptional one point orbits whose homogeneous coordinates are (z , 0,2, 0), (0, z , 0,2, 0), 2, (0,2, 0, z ) . (26) , 6. Action of a "nite group on complex projective space CP . The action of a "nite group on the complex projective space can be naturally induced from its action on initial complex variables (Zhilinskii, 1989a) after constructing the complex projective space as a space of orbits of SO(2) action on C given by Eq. (25). As the simplest example here we de"ne the action of the , two-elements group C "+E, C , on three complex variables (z , z , z ) as E(z , z , z )P(z , z , z ) , (27) C (z , z , z )P(!z , z , z ) . (28) This action induces the action on the points of CP space given by their homogeneous coordinates (z , z , z ). There is one isolated orbit with the stabilizer C , namely the point (1, 0, 0) of C (having the homogeneous coordinates (z , 0, 0)), and a continuous set of orbits with the stabilizer C (having the homogeneous coordinates (0, z , z )). All these C invariant points form CP &S invariant two-dimensional sub-manifold of the four-dimensional CP space. This example of the group action is in some sense trivial but it is interesting because the C stratum is the union of a two-dimensional manifold and an isolated point (Fig. 13).
Fig. 13. Schematic representation of the strati"cation of CP space under the action of C group. Isolated point and S surface formed by points with C local symmetry. All other orbits are generic two-point orbits with trivial stabilizer.
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
31
Another interesting and important example of the action of two-element group on CP space is the action through complex conjugation. Set of CP space points invariant with respect to complex conjugation form two-dimensional invariant sub-manifold which is a real projective space RP from the topological point of view. All other points form generic two-point orbits. The space of orbits is the S space from the topological point of view (Kuiper, 1974; Massey, 1973; Arnol'd, 1988). 3.5. Representations. Non-ewective actions. Kernels and images of the irreducible representations of 3-D-point groups Very often in physics under the presence of a symmetry group G we analyze properties of functions constructed from variables which span themselves some representation C (we can limit ourselves with the study of irreducible representations) of the symmetry group G (the model space < of the representation C). In such a case the action of the symmetry group G of the initial physical problem on < depends on the representation and can be described as the natural vector representation of another group ImC (G) acting in the space <. The group Im (C) is the image of the group % G in the representation C. From the mathematical point of view it is the quotient group G/Ker C of the initial group G over the kernel of the representation C (Ker C is an invariant subgroup of G represented in C by identity). In many cases it is su$cient to study only di!erent images. The model is independent of the initial group and representation if their images are the same (Michel and Mozrzymas, 1978; Michel, 1980; Zhilinskii, 1989b; Izyumov and Syromyatnikov, 1984). The extreme case is the trivial representation of any group G; its image is C "1 and the kernel is the group itself. Image of group G in any of its one-dimensional real (but not trivial) representation is always a group z of order 2 and the kernel enables us to distinguish di!erent representations. In particular, Table 2 de"nes the vector representation which is also a faithful representation for O "Pm3 m group. This representation is used further in Table 4 below (Section 5.4) to give the F system of invariants for all subgroups of O "Pm3 m. F More generally one needs to know both the image and the kernel of the representation but this information is not su$cient to completely describe di!erent irreducible representations. From Table 1 we can immediately de"ne the unitary irreducible representations (unirreps for short) of the 32 crystallographic geometric classes of point groups. First we recall that geometric classes form only 18 isomorphic classes; nine of them are Abelian. The others are the direct product of one of the four groups D , D , ¹, O with Abelian groups. We recall that the images of unirreps of "nite groups are cyclic (i.e. 1, Z ) and that the unirreps of a direct product of two groups is the L tensor product of their unirreps. Many books list these unirreps with their characters. It is unnecessary to reproduce them here. But it is very useful to characterize these unirreps by their kernel and their image. These 163 unirreps of crystallographic point groups have only 13 distinct images: 1, z , z , z , z , c , c , c , T T T ¹, ¹ , ¹ , O, O . Remark that among these images there are "ve one-dimensional, three twoF B F dimensional, and "ve three-dimensional. For 32 crystallographic groups, the couple Ker, Im characterizes all their unirreps except those of images z , z , z , O which correspond to pairs of unirreps obtained from each others through an F involutive outer automorphism of the image which does not correspond to equivalent representations with the same kernel. For the three cyclic images z , z , z it corresponds to their complex
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
32
conjugation. For O "Pm3 m it is the automorphism which exchanges the following F conjugacy classes of its subgroups: P4 3mP432, P4 2mP422, P4 m2P4mm, P4 P4, R3mR32, Cmm2C222, CmC2. We distinguish these two representations of O "Pm3 m F with the same image by using notation O for the vector representation and O for the other F F one. The unirreps of the 32 geometric point groups are characterized below by their images and kernels. (We omit the trivial group C "1 which has the only trivial representation.) 15 non-trivial Abelian geometrical classes forming eight isomorphy classes give
1 "C G 2"C m"C Q 1
Ker Ker Ker Im Ker Im Ker Ker Im
mmm"D F 1 4"C 4 "S 1
Ker
3"C
Im
1
Ker Im
1
Ker
1
Ker
1
Ker
z
Im
mm2 V z
mm2 W z
2
1
2
1
z
z , z
Im
222 z
1
2
m
m V 2 V z
m 2 W 2 W z
2 X 2 X z
(2/m) V z
4/m"C F 1
Ker 6"C Ker 3 "C G Ker 6 "C F Im 1
z , z
6/m"C F 1
mm2 X z
Ker
1
2/m"C F mm2"C T 222"D 1
4
4
m
1
z
z
z
z , z
z , z
2
1
3
1
1
3
m
1
z
z , z
z , z
3
6
6
2/m
m
2
1
z
z
z
z , z
z , z
z , z
z , z
(2/m) X z
(2/m)
3
(2/m) W z
For 16 non-Abelian geometrical classes forming nine isomorphy classes we have the following description of unirreps by their images and Kernels: Ker Ker Ker Im
4mm"C T 422"D 4 m2"D B 1
4
mm2
mm2
1
4
222
222
1
4
222
mm2
1
z
z
z
c T
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
Ker Im Ker Ker Im
Ker Im
4/mmm"D 4/m F 1 z 3m"C T 32"D 1
mmm 422 4mm z z z Ker 6mm"D T Ker 622"D Ker 3 m"D B Ker 6 m2"D F Im 1
4 m2 z 6
4 m2 z 3m
1 c T 3m
6
32
3
3 m
3 m
622
6mm
z
z
z
z
mmm z
3
1
3
1
z
c T
6/mmm"D 6 F 1 z
Ker
23"¹
222
1
Im
1
z , z
¹
Ker m3 "¹ F Im 1
Ker 4 3m"¹ B Ker 432"O
23 222
1
1
23 222
1
1
Im
z
O ¹
Ker
m3 m"O
Im
1
F
33
1
c T
m c T
2
1
32
2
1
32
3m
1
1
6
32
3m
2
1
z
z
z
6 m2
6 2m
2/m
1
z
z
c T
c T
23
mmm
222
z
z , z
z , z
¹ ¹
1
c
T
c
T
2m c c T T
1 F
B
m3
432
4 3m
mmm
222
1
1
1
1
z
z
z
c T
c T
O
¹ B
O F
O F
4. Compact group smooth actions; their critical orbits; their linearization One can say more on the strata in the cases of our applications; indeed we shall deal most often with smooth actions of compact Lie groups or "nite groups (which are the particular cases of Lie groups of dimension zero) on "nite-dimensional manifolds M. We quote here some theorems that we shall need in our applications: Theorem 4a (Montgomery and Yang, 1957). In the smooth action of a compact (or xnite) group G on a xnite-dimensional manifold, the set of strata is xnite. There exists a unique stratum with minimal symmetry; it is open dense in M. The maximal strata are closed; more generally the union of a stratum S and of all the strata 'S is a closed set.
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
34
The open dense stratum is often called the generic stratum. From this theorem we can prove (see also, (Michel, 1971) for a direct proof ) the Corollary 4a. In the smooth action of a compact (or xnite) group G on a xnite-dimensional manifold M, at each point m3M one can dexne a neighborhood V such that x3V NG 4G up to K K V K a conjugation. Given a smooth action of G on the manifold M, at m3M, the stabilizer G acts linearly on the K tangent hyper-plane ¹ (M) of M at m. A G-invariant vector "eld *(m) on M has to be a "xed vector K of G , which leads to K Lemma. In the smooth action of a compact (or xnite) group G on a xnite-dimensional manifold M, a G-invariant vector xeld on M must at each point m3M be tangent to the closure of the stratum of m; i.e. v (m)3¹ (S(m)): K This lemma imposes no condition for the gradient at points of the generic stratum. In many examples we have seen, the closure of strata are symmetry axes, symmetry planes, etc.; it is obvious from symmetry that at points of a symmetry axis, of a symmetry plane, etc., an invariant vector "eld must be along the symmetry axis, inside the symmetry plane, etc. Let f be a function on M; if G is "nite, "G"\ f (g\ ) m) is the average of f on G; it is EZ% easy to verify that it is a G-invariant function. This average can be generalized to a compact Lie group; indeed on such a group there exists, up to a factor an invariant measure k (g) which can be normalized by k (g)"1. Then the average of f on G is f (g\ ) m)k (g). By taking the average on % % G of an arbitrary Riemann metric we obtain a G-invariant Riemann metric on M. Since a Ginvariant function f is constant on each orbit, at each point m of the orbit, the gradient of f has to be in the normal plane N (G ) m) to the orbit. This condition is trivial for "nite groups. K We can always take a local system of geodesic coordinates. In this system we can identify ¹ (M) K with M and N (G ) m)"¹ (G ) m), is called the slice in the mathematical literature. Indeed every K K orbit in a neighborhood of G ) m cuts the slice transversally. Moreover the linear representation of G on ¹ (M) is orthogonal. Combining the condition f (m) (" the gradient of f at m) with that K K imposed by Lemma 4a we obtain
f (m)3F(m)"¹ (S(m))5N (G ) m) . (29) K K Equivalently, the orthogonal representation of G at m can be decomposed into the direct K sum of three representations and that on F(m) is a trivial representation. It is of dimension zero (F(m)"m) if the tangent plane to the stratum, ¹ (S(m)) coincides with the tangent plane to the K orbit. That means that the orbit is isolated in its stratum (" there are no other orbits of the same stratum in a neighborhood of m). In that case f (m)"0 and this is independent of the function: Theorem 4b (Michel, 1971). In the smooth action of a compact (or xnite) group G on a xnitedimensional manifold M, the gradient of every G-invariant functions vanishes on the orbits which are isolated in their strata. These orbits are called critical. A corollary is labeled by the number of the theorem to which it relates.
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
35
That is the case when a stratum contains a "nite number of orbits. Then it is closed, since the orbits are closed. The converse of that theorem was also proven by Michel (1971) as well as the next theorem. As we saw (Theorem 4a), it is equivalent for a stratum to be of maximum symmetry or to be closed; if moreover M is compact, then the closed strata are compact. The "rst proof of the theorem was given in Michel (1970) (Theorem 2, p.133). Theorem 4c (Michel, 1970). In the smooth action of a compact (or xnite) group G on a xnitedimensional compact manifold M, on a closed stratum containing an inxnity of orbits, the gradient of each G-invariant function vanishes at least on two orbits of the stratum. These orbits might be di!erent for di!erent functions. The proof can be sketched as follows: every continuous function on a compact has at least one maximum and one minimum; that occurs for the restriction to the stratum of every G-invariant function on M. Since the gradient of the whole function is tangent to the stratum, on it the zeroes of the gradients of the restriction and of the whole function coincide. We can reformulate part of the last two theorem into Corollary 4c. In the smooth action of a compact (or xnite) group G on a xnite-dimensional compact manifold M, every G-invariant smooth function has orbits of extrema on every stratum with maximal symmetry. When such a stratum has a xnite number of orbits all its orbits are critical; on the other maximal symmetry strata, every G-invariant function has at least two orbits of extrema on each connected component. In Section 6 we will recall all information we can obtain on the nature of each extremum. When a physicist, interested for instance by spontaneous symmetry breaking, makes a model whose Lagrangian or any other function to be varied has an extremum on a maximal symmetry stratum, he veri"es this corollary, but cannot claim that it is a speci"c success of his model! An excellent monograph on the compact group smooth actions is that of Palais (1960) (see also, p M"G of M on the orbit space is open Bredon, 1972). In it he proves that the continuous map MP and closed, so the Montgomery Theorem 2a (Montgomery and Yang, 1957) applies also to the images of the strata in the orbit space. Palais (1961) has also shown that all the properties given here for compact groups are also true in the more general case of the action of a non compact Lie group (or countable discrete group) when all stabilizers are compact (or "nite). We shall make two applications of this result in Chapter IV. One is for the action of a crystallographic space group on our space, the other for its action on the corresponding momentum space, called the Brillouin zone; instead to be described by a real vector space, it is a ;B group. There are remarkable theorems on the compact group actions on compact manifolds; Palais (1970) proved that such a di!erentiable (in short C) action is equivalent to an in"nitely di!erentiable (in short C or `smootha) action. In that case Mostow (Mostow, 1957a, b) proved that the number of strata (""(M""G)") is "nite and that there exists an orthogonal representation of G on
That is: the images of open sets are open sets, the image of closed sets are closed sets.
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
36
a "nite-dimensional orthogonal space < containing M in such a way that the action of G on M can be obtained as a restriction of the orthogonal representation of G on <. In physical applications studied below the group actions are either linear representations on a real (orthogonal) vector space < or can be considered as their restriction to an invariant manifold contained in <. For example, in Chapter V we transform the non-linear action of an arithmetic class on the Brillouin zone to the orthogonal representation on higher-dimensional space. So in all our applications the study of the group action on M can be obtained by the restriction of the action of an orthogonal linear representation to an invariant manifold ML<, the carrier space of the representation. That explains our interest in the next section. 4.1. Examples of critical orbits for group actions Let us now return to examples of group actions on manifold cited in 3.4 from the point of view of critical orbits. 1. There is one critical orbit for the action of O(n) group on orthogonal space < . It is the origin L which is a "xed point (a one-point orbit with the stabilizer O(n)). 2. Action of C "R and C "Rm subgroups of the O(3) group on a two-dimensional sphere T induced by the natural action of O(3) group on 3-D space leads to two one-point critical orbits with the same stabilizer (two poles of the sphere). At the same time action of C "R/m and F D R/mm subgroups induces appearance of one two-point critical orbit with the stabilizer C or F C (two poles) and one critical manifold (the equator, having S topology). T 3. Action of the point group O "Pm3 m on the 3-D space gives only one critical orbit, namely the F origin with the stabilizer O . At the same time the restriction of this action on the two-dimensional F sphere leads to several critical orbits: one six-point orbit with stabilizer C "P4mm, one eight-point T orbit with stabilizer C "R3m, and one 12-point orbit with stabilizer C "Amm2. T T 4. Action of the point group C "1 on the 3-D space gives only one critical orbit, namely the G origin with the stabilizer C "1 . At the same time the restriction of this action on the twoG dimensional sphere possesses no critical orbits. 5. Natural diagonal action of the SO(2) group on the direct product of two two-dimensional spheres S ;S possesses four critical orbits which are stationary points for any smooth SO(2) invariant function de"ned over S ;S . 6. Natural diagonal action of the SO(2) group on the space of N complex variables restricted to the sphere "z ""const (see Eq. (25)) possesses N isolated critical points. G G 5. Rings of G-invariant functions Given an n-dimensional linear orthogonal representation on < of a compact (or "nite) group G, L all C (i.e. in"nitely di!erentiable) G-invariant functions are C functions of invariant polynomials
Unfortunately the proof does not give an upper bound on dim(<).
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
37
(this is the statement of the Schwarz theorem (Schwarz, 1975)). So we can restrict our study to the ring P% of invariant polynomials on < (P% is a ring since the sum and the product of invariant L L L polynomials are invariant polynomials). For more details on the structure of P% see the reviews L (Stanley, 1979; Jaric et al., 1984). We must "rst recall some results on the action of G on P , the ring of polynomials on < (they L L depend on n variables). P is also an in"nite-dimensional vector space of functions. We denote by L PK the subspace of homogeneous polynomials of degree m. That is a "nite-dimensional subspace L of P : L m#n!1 . (30) PK, dim(PK)" P " L L L m K Let *3< and p3P . The action of G on P is de"ned by L L L g ) p (*)"p (g\ ) *) . (31)
This equation shows in particular that if the representation of G on < contains the matrix !I , L L every invariant polynomial must contain only monomials of even degree. It is easy to obtain an invariant polynomial starting from an arbitrary polynomial. One simply sums all polynomials of a G-orbit: q" g ) p(*). This construction is well known in physical and chemical applications as EZ% a projection operator (Hamermesh, 1964) whereas in mathematics it is often referred as Reynolds operator (Sturmfels, 1993). Another systematic procedure for the invariant construction (especially for high-order invariants) is based on the coupling of irreducible tensors using vector coupling coe$cients (mainly known to physicists and chemists as Clebsch}Gordan or Wigner coe$cients (Wigner, 1959; Hamermesh, 1964; Biedenharn and Louck, 1981a, b)). Let us consider before going to a generalization a simple example of the construction of the system of invariants for an Abelian group c acting in the natural way on the two-dimensional L vector representation x, y. Changing the variable x, y into the complex linear combinations x$iy we diagonalize the two-dimensional representation. Now the construction of invariants is straightforward. We can construct one invariant of degree 2, namely (x#iy) (x!iy)"x#y and two linearly independent invariants of degree n, which can be chosen as Re(x#iy)L and Im(x#iy)L. It is important to note that the three invariants are not algebraically independent. There is an algebraic relation between them (x#y)L"(Re(x#iy)L)#(Im(x#iy)L) .
(32)
The existence of this relation (in theory of invariants, this type of relations is named syzygy) means that to construct all linearly independent invariants we can take all polynomials in two algebraically independent invariants (say x#y and Re(x#iy)L, we will name them basic invariants) and form arbitrary polynomials P(x#y, Re(x#iy)L) and similar arbitrary polynomial multiplied by the third invariant (we will name it auxiliary invariant) P(x#y, Re(x#iy)L)Im(x#iy)L. This means that all c invariants form the module of dimension two over the ring formed by two basic L invariants. A module is similar to a vector space except that its scalars do not form a "eld, but only a ring (not every scalar is divisible by another one O0). The property that every vector space has bases extends only to free modules. For a very simple counter-example see Eq. (27) in Ref. (Jaric et al., 1984).
38
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
As soon as we have the system of c invariants we can immediately construct the system of L c invariants using the fact that c is an invariant subgroup of c (see Section 3). The action of the LT L LT quotient group z "c : c on c invariants shows that the basic invariants of c remain invariants LT L L L of the larger group c . At the same time the auxiliary invariant of c becomes the pseudoinvariant LT L of c because it changes the sign under the z action. Consequently to construct all invariants of LT c it is su$cient to take an arbitrary polynomial in two basic invariants P(x#y, Re(x#iy)L). LT We see that the module of invariants of the higher group c is a submodule of the module of LT invariants of a smaller group c . We give here this simple example because many more complicated L examples treated further in this and following chapters are based essentially on the same construction. Now we want to proceed to a systematic knowledge of P% in all orders. To perform such an analysis we remark that the action de"ned in Eq. (31) transforms the PK into themselves. The L character sK(g) of the linear representation of G on PK was "rst given by Molien (1897) more than L one hundred years ago through the generating function sK(g)jK"det(I !jg)\ , (33) L K where j is a dummy variable. We label by an index a the di!erent equivalent classes of irreducible representations of the compact or "nite group G (with a"0 for the trivial representation) and we denote by s (g) the corresponding character. Let cK be the multiplicity of the irreducible ? ? representation a which appears in the G representation on PK; when G is "nite, its value can be L computed from the Molien generating functions (Burnside, 1911; Weyl, 1939): M (j), cKjK""G"\ s (g)det(I !jg)\ . (34) ? ? ? L K EZ% To extend this formula to Lie groups, one must replace the group average "G"\ EZ% by the integral on the group dk(g) where k(g) is an invariant measure on G whose integral % is normalized to 1. The Molien functions are rational fractions (i.e. ratios of polynomials) in j. It is quite important that these rational functions give symbolic information about the structure of the ring of invariant polynomials rather than the numbers of invariants in every degree. To explain the symbolic meaning of Molien generating functions we start with an example of one speci"c class of groups. Chevalley (1955) proved that for "nite groups generated by re#ections the following theorem takes place. Theorem Chevalley 1. The ring P% of invariant polynomials of a xnite reyection group G acting on the orthogonal space < , is an n-variable polynomial ring. L In the same paper Chevalley (1955) proved the second theorem: Theorem Chevalley 2. For a xnite reyection group G, the ring P of the polynomials on < is a free L module on the ring P% of invariant polynomials; its dimension is "G" and it carries the regular representation of G.
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
39
More explicitly, P% is the ring of all n variable polynomials whose variables are n algebraically independent polynomials h , 14k4n. The degrees d of the h are obtained from the Molien I I I function which has the form L with N"1, D" (1!j)BI . (35) I Parameters d are related with the structure of the group G itself, namely, I d ""G", (d !1)"b , (36) I I I I where "G" is the order of group G and b is a number of re#ections in G. More generally (as was found empirically by Shephard, 1956 and proved by Solomon, 1963) N(j) M(j)" D(j)
L L (1#(d !1)t)" b tH , (37) I H I H where b is the number of elements of G whose space of invariant vectors has dimension n!j. H For "nite groups which are not generated by re#ections, the Molien function M has still a denominator of the form (35). The numerator N(j) is a polynomial with positive coe$cients N(j)" l jB, l '0 . (38) B B B In the case of a Molien function for invariants we always have N(0)"l "1. In contrast, the Molien function for covariants does not have a constant term in the numerator. For each power d which appears in the numerator polynomial (38), there exist l G-invariant B linearly independent homogeneous polynomials u of degree d; of course these polynomials have to ? be algebraic functions of the n `denominator invariantsa h (see below). The ring P% is a free I module and the N(1) polynomials u 's form a basis of it. That is, there is a unique way to write any ? G-invariant polynomial as a linear combination of the u 's: ? , \ p" p (h (x ))u (x ) with u "1 , (39) ? I H ? H ? where the coe$cients p 's are n variable polynomials. Moreover every n-variable polynomial (with ? the h 's as variable) can appear in such a decomposition. A similar decomposition is valid for G covariants but now there is no u "1 term. For any "nite group G there exists a general relation between numbers and degrees of denominator and numerator invariants and the structure of group G, namely with its order "G" and the number of re#ections r contained in G (see p. 487 of Stanley, 1979). Let d , d ,2, d be L the degrees of denominator invariants and e , e ,2, e the degrees of numerator invariants R
This formula was proved by Stanley (1979) in the more general case of complex groups with re#exions replaced by pseudore#exions, i.e. symmetry operations with one eigenvalue sO1, "s""1 di!erent from 1.
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
40
(The number t of numerator invariants includes unity as a numerator invariant of degree 0). Then the following relation generalizes relation (36) t"G""d d 2d , L rt#2(e #2#e )"t(d #2#d !n) . R L
(40) (41)
5.1. Molien function manipulations We want in this section to give some useful formula for Molien function calculations which will be used later in this issue. Some of them are well known, others are less frequently used, especially in physical applications. Much more still can be found in mathematical literature. We start with rewriting the Molien generating function (34) using more detailed notation which speci"es explicitly the group G and the initial C and "nal C representations. G D (42) M%(C ; C ; j)""G"\ sCD (g)det(I !jC (g))\ . L G D G EZ% In fact the Molien function depends only on the image of the group G in the representation C , G but for physical applications often it is useful to construct the Molien functions for all di!erent irreducible representations of the same group. Moreover very often the space of variables span the reducible representation. In such a case, if the C is reducible and decomposes as C "C C we have G G G G M%(C ; C C ; j)" nCCDD CD M%(C ; C ; j)M%(C ; C ; j) , (43) D G G D G D G C C D D where n CCDD CD are the multiplicities of C in the decomposition of the product C C into D D D irreducible representations (Hamermesh, 1964) C C " n CCDD CD C . D D D C
(44)
D
In the particular case of invariants (we put C "C for invariants) formula (43) simpli"es into D (45) M%(C ; C C ; j)" M%(C; C ; j)M%(CH; C ; j) . G G G G C If the initial representation is a direct sum of two representations we can distinguish C and C by G G di!erent auxiliary variables thus constructing the Molien function with two parameters. For example, instead of (43) we can write (46) M%(C ; C C ; j, k)" n CCDD CD M%(C , C ; j)M%(C , C ; k) . G D G D G D G C C D D This formula will be quite useful for the construction of the integrity basis for reducible representations from known integrity bases for irreducible components. In some cases one needs to calculate the total number of terms of several di!erent symmetry types. In such a case one is interested in generating functions for "nal representations
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
41
C being the direct sum of representations C "C C , the corresponding generating function has D D the form M%(C C ; C ; j)"M%(C ; C ; j)#M%(C ; C ; j) . (47) G G G In particular if the "nal representation is the regular representation C " "C "C , we simply I I I count the total sum of polynomials which can be constructed from initial variables (see Theorem Chevalley 2), this means 1 . M%(C "C ; C ; j)" D G (1!j) CG
(48)
Generating functions for the number of invariants and all possible covariants for all "nite groups can be found in Ref. (Patera et al., 1978; Desmier and Sharp, 1979) Below, we give several simple examples. The generating function for the initial trivial representation C (Im C "C "1) of any group G is trivial 1 . M%(C ; C ; j)" 1!j
(49)
In the case of real one-dimensional but not totally symmetric representation C (Im C "z ) the generating function for invariants has the form 1 , M%(C ; C ; j)" 1!j whereas the generating function for the covariants of type C is j . M%(C ; C ; j)" 1!j
(50)
(51)
Generating functions for the invariants and the two types of covariants constructed from twodimensional representations C with Im G"c have the form T 1 , (52) MAT (C ; C ; j)" (1!j) (1!j) j MAT (C ; C ; j)" , (1!j) (1!j)
(53)
j#j . MAT (C ; C ; j)" (1!j) (1!j)
(54)
We can immediately verify relation (48) for this simple example. (55) MAT (C ; C ; j)#MAT (C ; C ; j)#2MAT (C ; C ; j)"(1!j)\ . Let us now consider the situation where the initial representation C is reducible. We take as an G example the six-dimensional representation of the Abelian group C which will be used later in Chapter V for the description of the ring of invariant functions on the Brillouin zone for the
42
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
tetragonal 3-D arithmetic classes. This particular representation C has the following decomposi tion into irreducible representations of C group. C "c p; c"[(1) (1) (!1)] , p"[(1) (!i) (i)] . (56) We use for the four complex irreducible representations of the group C the following notation (1), (i), (!1), (!i). From the physical reason (see Chapter V, Section 5.3) it is useful to treat separately two three-dimensional reducible representations p and c. To do that we use Molien function with two parameters and "rst remark the identity M! (C "1; C "c p; j, k) D G "M! (C "(1); C "c; j)M! (C "1; C "p; k) D G D G (57) #M! (C "(!1); C "c; j)M! (C "(!1); C "p; k) D G D G which follows from (46). Remark that only C "1, (!1) give non-zero contributions. Intermediate D Molien functions can be easily calculated: 1 M! (C "(1); C "c; j)" , D G (1!j) (1!j)
(58)
j , M! (C "(!1); C "c; j)" D G (1!j) (1!j)
(59)
1#2k#k M! (C "(1); C "p; k)" , D G (1!k) (1!k)
(60)
k#2k#k . M! (C "(!1); C "p; k)" D G (1!k) (1!k)
(61)
Thus, for the Molien function (57) we get the expression 1#jk#2jk#2k#k#jk . M! (C "1; C "c p; j, k)" D G (1!j)(1!j) (1!k)(1!k)
(62)
Somewhat di!erent interpretation of the information encoded in Molien function turns out to be quite useful if in physical applications we are interested in number of tensors of di!erent symmetry types of various degrees. We remind that the power series expansion of the Molien function gives these numbers M%(j)" C j, . (63) , , Direct calculation of the formal series expansion gives these numbers numerically but in fact it is possible to give the function C(N) which equals C for integer values of N. Moreover this function , turns out to be relatively simple. For su$ciently high N it can be written in the form of a quasi-polynomial as discussed, for example, by Stanley (1986, Chapter 4.4). A quasi-polynomial of degree d is a function f : NPC f (n)"c (n)nB#c (n) dL\#2#c (n) B B\
(64)
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
43
with each term c (n) being a periodic function with integer period. To illustrate the construction of G such quasi-polynomial let us consider the generating function of the form 1 g(j)" , (1!jL )2(1!jLQ )
(65)
which has in the denominator the product of s terms (1!jLG ), i"1,2, s with positive integers n ,2, n . The formal expansion of g(j) gives the series Q g(j)" C j, (66) , , and coe$cients C can be considered as values of the function C(N) de"ned for arbitrary N but , taken for integer N. C(N) is a quasi-polynomial (64) of degree s!1 with oscillating coe$cients. The period of oscillations equals the least common multiple, D, of all n . In fact the period can be di!erent for G di!erent terms C I C (N)"C (N#D ) , (67) I I I but all D divide D. The regular part of the quasi-polynomial can be unambiguously de"ned after I averaging over the period D: 1 ?>D C (N) . d(N)" I I D I?> The coe$cients of the regular part of the quasi-polynomial
(68)
Q\ C(N)" dNI (69) I I can be related explicitly with the n numbers of the initial generating function. Several initial terms G look like 1 (n , , n )" , Q\ 2 Q (s!1)!n G n G d (n ,2, n )" , Q\ Q 2(s!2)!n G 3( n )! n G G , d (n ,2, n )" Q\ Q 24(s!3)!n G ( n )! n n G G G , d (n ,2, n )" Q\ Q 48(s!4)!n G 15( n )#5( n)!30( n ) n#2 n G G G G G , d (n ,2, n )" Q\ Q 8 ) 6!(s!5)!n G 3( n )!10( n ) n#5 n ( n)#2 n n G G G G G G G . d (n ,2, n )" Q\ Q 16 ) 6!(s!6)!n G d
(70) (71) (72) (73) (74) (75)
44
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
Let us remark that these coe$cients are proportional to Todd polynomials, ¹d de"ned through G the generating function as given, for example by Gri$ths and Harris (1978, Section 3.4)
det A "(!1)Lt\L ¹d (P(a),2, PG(a))tG , (76) G det (I!e\R ) G where PG are symmetric functions over eigenvalues of the matrix A. This means that in a formal way expressions (70)}(75) can be obtained from the generating function (65) by replacing jPexp(!w) and taking initial coe$cients of the Laurent series (see Chapter II for some examples). This procedure can be generalized to calculate the oscillatory part. 5.2. Integrity basis, syzygies, and other related notions We will use the name integrity basis for the basis formed by `denominatora and `numeratora invariants in order to follow notation used initially by Weyl (1939) and accepted in considerable part of physical literature (Gilmore and Draayer, 1985; Bickerstu! and Wyborne, 1976; Jaric and Birman, 1977; Judd et al., 1974; Schmelzer and Muller, 1985; Izyumov and Syromyatnikov, 1984). The same polynomial basis is referred in mathematical literature as homogeneous system of parameters (Stanley, 1979, 1996) or as Hironaka decomposition (Sturmfels, 1993). From now on we shall write this structure of module as P%"P[h , h ,2, h ]E(1, u ,2, u ) . (77) L I We have to emphasize that this notation represents a ring and a module structure. Indeed any polynomial in the u 's is an invariant polynomial and should be written in a unique way as a linear ? combination of the u's with coe$cients in P[h ,h ,2,h ]. So Eq. (77) describes a structure of L module algebra, i.e. an algebra whose vector space formed by its elements is replaced by a module of ring of scalars P[h ,h ,2,h ] and of basis +u ,. In the computation of the algebra module it L ? might be useful to check its structure by computing its structure `constantsa; they are polynomials in the h 's: G u u " pA u , pA "pA with u "1 . (78) ? @ ?@ A ?@ @? A Throughout this issue we shall use the notation of Eq. (77); the h , (i"1,2, n) within the brackets G are n algebraically independent (`denominatora) invariants and u , (a"0,2, k) with u "1 are ? algebraically dependent (`numeratora or auxiliary) invariants which form the basis of the module of invariant polynomials. Given a ring P%, it can be represented by di!erent modules. For example the obvious equivalence P%"P[h , h ,2, h ]E(1, u ,2, u ) L I "P[h , h ,2, h]E(1, u ,2, u )(1, h ) . (79) L I L enables one to change the system of denominator invariants (to go from h to h) with simultaneous L L doubling of the number of numerator invariants. The Molien functions in both cases are the same, but to pass from the "rst form to the second form, the numerator and denominator of the "rst one are both multiplied by (1#jB), where d is the degree of h . That possibility is quite useful when one L wants to impose constraints on h rather than on h , for example. L L
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
45
A similar, but more sophisticated equivalence appears in our discussion of Rydberg problems (see Chapter III). One should also note that there is a di$culty: it may happen that the generating function M(j) (35) could not be used under its most reduced form with positive coe$cients in the numerator. Such a possibility is given by Sloane (1977, Eq. (47)), the reduced form cannot represent a module. The symbolic interpretation of the Molien function introduced above is related with the structure of the integrity basis. There is another possibility of the symbolic interpretation (Hilbert, 1890, 1893). One can use all generators of the ring of invariants as denominator invariants. The number of generators is "nite but it is larger than the number of algebraically independent polynomials. In such a case the numerator of the generating function has both positive and negative integer coe$cients. The symbolic meaning of the numerator in this case is the relations (syzygies) between generators of the ring. For example the representation of the generating function for invariant polynomials in the form 1! jDG # jEG !2$ jIG , M(j)" (1!jBG )
(80)
can be interpreted in the following way (see example in Section 5.4 below). There are s generators having the degree d , d ,2, d . The generators are algebraically dependent and there are t poly Q nomial relations between these generators which are, respectively, of degree f , f , 2, f with respect R to initial variables (these relations are called syzygies of the "rst kind). Further, the set of syzygies of the "rst kind is not generically independent. There possibly exist some relations between syzygies of the "rst kind (syzygies of the second kind) characterized by degrees g , g ,2, g , etc. This S approach introduced in invariant theory by Hilbert (Hilbert, 1890, 1893) enabled him to prove many important theorems, in particular to show that the number of generators and syzygies is "nite. Description of the ring of invariant polynomials in terms of generators and syzygies is in some aspects complementary to the analysis based on the integrity basis construction. We will mainly use the explicit description of the system of invariants and the form of generating functions giving information about the integrity basis. The non-trivial example of the set of syzygies arising in very simple group action (spatial inversion in three-dimensional space) will be discussed in the example below. One of the goals is to use the integrity basis to describe the space of orbits (the idea is to use invariant polynomials as coordinates on the space of orbits, see Section 5.6). By de"nition an invariant polynomial is constant along an orbit. Given two orbits, there is at least one polynomial of P% (the ring of G-invariant polynomials whose variables are the coordinates of the vector space of the linear representation of G) which takes di!erent values on them (see e.g. Michel, 1979). Then that has also to be true for the "nite set of polynomials +h , u ,, forming an integrity basis. So one G ? can label the points of the orbit space <"G by the set of values of these polynomials. The number n of di!erent h is equal to the dimension of the orbit space and these n h's form a global system of G coordinates of <"G when there are no numerator invariants. The latter, when they exist, resolve the ambiguities left by the value of the h's. As values of n#k real polynomials of n real variables, the coordinates of the orbit space must satisfy polynomial equalities and inequalities; this is summarized by `the orbit space <"G is semi-algebrica. Moreover as image of polynomials on the (connected) orthogonal vector space <, the orbit space <"G is connected.
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
46
In the case that G is an invariant subgroup of a re#ection group G and G /G is Abelian, P P Stanley (1977) has given a systematic way for building the module of invariant polynomials for such a subgroup of re#ection unitary groups. In the case of orthogonal groups which interest us here, the method is obvious. By the second theorem of Chevalley (see the paragraph preceding Eq. (35) each non-trivial one-dimensional representation of G appears once in P. We call them P the pseudo invariants of G . We keep among them only those which are invariant for G. They P are products of the linear forms whose zeros de"ne the symmetry planes of G . For instance for the P unimodular subgroup of G (i.e. the subgroup of elements of determinant 1), there is only one P non-trivial numerator invariant: the product of the equations of all symmetry planes; it was known that it is, up to a factor, the Jacobian of the G invariants. P Instead of smooth invariant functions it might also be useful to consider the "eld F% of invariant rational fractions over < . In Chapter XVII of his famous book Burnside (1911) proved that the L number of generators of F% is either n or n#1. Notice that the numerator and the denominator of an invariant rational function are not necessarily invariant. However we should also point out, as a general theorem, that an invariant rational fraction can always be written as a quotient of invariant polynomials. For re#ection groups this number is n since F%"F[h ,2, h ]. For some groups in 3D-space L which are not generated by re#ections (for example the group O) we can prove that it is 4. At the same time even for some groups which are not generated by re#ections the number of generators is only 3. The group C gives such an example G x x (81) F!G "F x , , . x x It can be easily veri"ed that all generators for F!G may always be rewritten as a quotient of invariant polynomials: x /x "x x /x and x /x "x x /x . The group of linear transformations of coordinates of a space < , i.e. G¸(n, R), transforms L polynomials into polynomials. Notice that the Laplacian (which appears in the SchroK dinger equation) commutes with all matrices of the orthogonal representation of a group G. So
*PK%LPK\% ,
(82)
remember that P is the one-dimensional space of constants; it carries the trivial representation of G. General coordinate transformations of a smooth manifold M preserve only properties of smooth functions (they do not generally transform polynomials into polynomials). But the Schwarz theorem (Schwarz, 1975) shows the interest to consider G-invariant polynomials of the coordinates on M. When the action of G on M has been embedded in an orthogonal representation of G on < , L it is not di$cult to compute, from the module P% of invariant polynomials on < , the module of L L their restriction on M. Let M, of dimension n!d, be de"ned by d algebraic equations q "0. M That is equivalent to G 4G(G where the `derived groupa group G is the group generated by the commutators of G . P P P P Very natural to physicists, that method was immediately applied to all "nite subgroups of O(3) by Michel (1977). Unfortunately the last table of this paper is marred by many misprints; they are corrected in Jaric et al. (1984). In the general case some inequalities might also be necessary, but that will not be needed in this paper.
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
47
Since M is transformed into itself by G, the q must be (pseudo)invariant polynomials. Note that M the equation can be replaced by qK"0 where m is the smallest integer 52 such that qK is M M invariant. The simplest situation corresponds to the case when these polynomials q belong to the M ring of scalars of the module, they de"ne an ideal of it; let Q be the corresponding quotient. Then P%" is a module on Q, either with the same basis as the module P% or smaller if some u become + proportional. The denominator of M of P%" has only n!d factors which is the maximal number + of polynomials on < which are algebraically independent on M. L 5.3. Extension to continuous groups The theory of invariants of "nite groups can be extended to continuous groups with serious caution. We give in this section several applications in the case of the SO(2) symmetry group and show that the module of covariants is not free even in rather simple cases. Let us consider the four-dimensional space with the action of SO(2) group on it which is equivalent to twice the natural vector representation. More formally the initial representation is C "(m"#1)#(m"!1)#(m"#1)#(m"!1) . Molien function for invariants and covariants of SO(2) reads
(83)
[sK( )]H d
1 p M[(m)Q(C ); j]" (1!j exp(i ))(1!j exp(!i )) 2p
(84)
sK( )"exp(im ) .
(85)
with
An explicit calculation gives for invariants and covariants the following generating functions: 1#j , M[(0)Q(C ); j]" (1!j)
2j M[(1)Q(C ); j]" , (1!j)
(86)
3j!j , M[(2)Q(C ); j]" (1!j)
4j!2j M[(3)Q(C ); j]" , (1!j)
(87)
with the general expression for "m"52 ("m"#1)j K !("m"!1)j K > . M[(m)Q(C ); j]" (1!j)
(88)
We see that for "m"52 the numerator includes negative terms. One can easily verify that the sum over all covariants gives the generating function for polynomials in four independent variables. 1 M[(0)Q(C ); j]#2 M[(m)Q(C ); j]" . (89) (1!j) K Remark that the substitution j"1 into the numerators of all generating functions for invariants and covariants gives always 2. This fact shows that all unirreps are one-dimensional.
48
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
Numbers of invariants and covariants can be calculated for any form of Molien functions (irrespective of positive or negative terms in the numerator). They are, respectively, M[(0)Q(C ); j]"1#4j#9j#16j#2 , (90) M[(1)Q(C ); j]"2j#6j#12j#20j#2 , (91) M[(2)Q(C ); j]"3j#8j#15j#2 . (92) Let us now introduce four dynamical variables x , y and x , y in such a way that +x , y , span the two-dimensional vector representation (m"!1)#(m"#1) of SO(2) and +x , y , span the same representation. Integrity basis for invariants can be easily written explicitly. It includes three quadratic denominator polynomials h , h , h and one numerator (auxiliary) polynomial u (with u expressible in ? @ A terms of denominator invariants). h "x #y , h "x #y , h "x x #y y , ? @ A u "x y !x y , u "h h !h . ? @ A All invariant polynomials can be unambiguously represented in the form
(93) (94)
P(h , h , h )#u R(h , h , h ) , (95) ? @ A ? @ A where P(h , h , h ) and R(h , h , h ) are arbitrary polynomials. This means that invariant poly? @ A ? @ A nomials form a free module P(h , h , h )E(1, u ) . ? @ A Covariants of type m"1 also form a free module with two generators
(96)
u "x #iy , u "x #iy , (97) ? @ P(h , h , h )E(u , u ) . (98) ? @ A ? @ The situation becomes more complicated in the case of covariants with m"2. We have now three linearly independent quadratic covariants u "(x #iy ), u "(x #iy ), u "(x #iy ) (x #iy ) . (99) ? @ A At the same time these three quadratic covariants are algebraically dependent. There exists one quartic relation between quadratic covariants and quadratic basic invariants 2u h "u h #u h . (100) A A ? ? @ @ This means that the module of m"2 covariants is not free. To keep only uncorrelated terms we should write the general polynomial expansion for m"2 covariants, for example, in the following form: u P (h , h , h )#u P (h , h , h )#u P (h , h ) . (101) ? ? ? @ A @ @ ? @ A A A ? @ Remark that P and P are arbitrary polynomials in three variables while P is a polynomial in ? @ A two variables only. This particular form is speci"c to the problem considered. But the general recipe seems to be the same. In the case of presence of negative terms in numerator for covariants the general polynomial expansion of covariants will not include all polynomials in denominator invariants.
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
49
5.4. Invariant polynomials and integrity bases for 3-D crystallographic point groups Invariant polynomials for "nite groups and the integrity bases for di!erent irreducible and certain reducible representations were calculated by many authors (see Patera et al., 1978; Gaskel et al., 1979; Michel, 1977; Jaric et al., 1984; Sturmfels, 1993; Worfolk, 1994, and references therein). The explicit form of the invariant polynomials depends on the choice of coordinate system and it remains ambiguous even in an unambiguously chosen coordinate system because an arbitrary linear transformation within the space of basic polynomials of the same degree is possible. The choice of a coordinate frame is of particular importance when the geometric symmetry is analyzed together with periodic symmetry imposed by crystal lattice. This choice is essentially done when the arithmetic class is speci"ed. In Table 4 we give the module of invariant polynomials for the 33 arithmetic classes 4O(3, Z)"Pm3 m. To help the reader we give the translation of the ITC notation of these arithmetic classes into the ambiguous SchoK n#ies notation (which is valid only for geometrical classes). In all cases we use the orthonormal system of coordinates de"ned by three C axes of the O group. When the center of a cube is at the origin, the coordinate axes are orthogonal to the faces F of the cube and pass through the symmetry centers. The orientation of the coordinate system with respect to symmetry elements for each arithmetic class can be reconstructed by using Table 2 for a list of generators and expressions (16)}(19) for the matrix representation of these generators. We will make for D "P6/mmm the same study we have done for O(3, Z)"Pm3 m. The 32 F conjugacy classes of subgroups correspond to 28 conjugacy classes in G¸(3, Z); for them we use the names of ICT. Only 16 of them are not orthogonal arithmetic classes (O(3, Z). All these contain 3"C which is the derived group. There are three conjugate subgroups Cmmm; this arithmetic class is the largest of all arithmetic classes contained also in Pm3 m"O(3, Z). The system of invariant polynomials for 16 hexagonal arithmetic classes is given in Table 5 using an orthonormal coordinate system +x, y, z, with axes z aligned along the C or C symmetry axes. We can rewrite invariant polynomials in the non-orthogonal coordinate system associated with the non-orthogonal lattice. Such transformation will be discussed and used in Chapter IV. Let us give two examples for the two other cubic arithmetic classes. For F and I arithmetic classes (see Chapters IV, V for details) it could be interesting to use non-orthogonal coordinate system to give the invariant polynomials. We can easily transform the invariant polynomials forming integrity basis from an orthogonal system to a non-orthogonal one. For example, in the non-orthogonal system with +x, y, z, axes coinciding with three C axes of O F (F arithmetic class) the basic invariants of O can be chosen as F h$ "x#y#z#xy#yz#zx ,
(102)
h$ "xyz(x#y#z) ,
(103)
h$ "(x#y)(y#z)(z#x) .
(104)
In fact, the analytical form given in Eqs. (102) and (105) for h$ and h' di!ers from the straightforward transformation of the h invariant given in Table 4. We give the simpli"ed form by substracting the h invariant with an appropriate coe$cient.
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
50
Table 4 Invariant polynomials for the 33 conjugacy classes of subgroups of O "O(3, Z). Column 1 gives the SchoK n#ies notations F (they are ambiguous); column 2 gives the eventual arithmetic class with the same matrices. Column 3 give the number of groups in each conjugacy class of subgroups. In the three cases C "P1 , S "P4 , C "R3 there are three non-trivial G G numerator invariants Geom.
Arithm.
C
h
h
h
u
C C G
P1 P1
1 1
x x
y y
z z
C C Q C F C C Q C F
P2 Pm P2/m C2 Cm C2/m
3 3 3 6 6 6
x x x x!y x#y x#y
y y y xy xy xy
z z z z z z
C T D D F C T C T D D F
Pmm2 P222 Pmmm Cmm2 Amm2 C222 Cmmm
3 1 1 3 6 3 3
x x x x#y x!y x#y x#y
y y y xy xy xy xy
z z z z z z z
C C T S C F D B D B D D F
P4 P4mm P4 P4/m P4 2m P4 m2 P422 P4/mmm
3 3 3 3 3 3 3 3
x#y x#y x#y x#y x#y x#y x#y x#y
xy xy xy xy xy xy xy xy
z z z z z z z z
xy(x!y)
C C T D C G
R3 R3m R32 R3
4 4 4 4
x#y#z x#y#z (x#y#z) (x#y#z)
x#y#z x#y#z x#y#z x#y#z
xyz xyz
xyz
"(x!y)(y!z)(z!x)
D B
R3 m
4
(x#y#z)
x#y#z
xyz
¹ ¹ B ¹ F O O F
P23 P4 3m Pm3 P432 Pm3 m
1 1 1 1 1
x#y#z x#y#z x#y#z x#y#z x#y#z
x#y#z x#y#z x#y#z x#y#z x#y#z
xyz xyz xyz xyz xyz
xy"yz"zx xy xy (x#y)z (x#y)z
xyz
(x!y)z
xyz"z(x!y)"xy(x!y) xy(x!y) xyz z(x!y) xyz(x!y)
(x#y#z)xyz (x#y#z)xyz" (x#y#z) " xyz
(x#y#z)xyz (x!y)(y!z)(z!x) (x!y)(y!z)(z!x) xyz(x!y)(y!z)(z!x)
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
51
Table 5 Module of invariant polynomials for the 16 hexagonal arithmetic classes. Column 1 gives the SchoK n#ies notation of the geometric class; column 2 gives the arithmetic classes. The three h generate the module ring, and 1 and the u's form the G bases of the modules. Six groups are generated by re#ections (one-dimensional module), the other modules are of dimension 2 except that of P3 "C which is of dimension 4 G Geom.
Arithm.
h
h
h
u
C C T C T D D C G
P3 P3m1 P31m P321 P312 P3
x#y x#y x#y x#y x#y x#y
x(x!3y) x(x!3y) y(3x!y) y(3x!y) x(x!3y) Re(x#iy)
z z z z z z
y(3x!y)
D B D B C F D F D F C C F C T D D F
P3 m1 P3 1m P6 P6 m2 P6 2m P6 P6/m P6mm P622 P6/mmm
x#y x#y x#y x#y x#y x#y x#y x#y x#y x#y
Re(x#iy) Re(x#iy) x(x!3y) x(x!3y) y(3x!y) Re(x#iy) Re(x#iy) Re(x#iy) Re(x#iy) Re(x#iy)
z z z z z z z z z z
zx(x!3y) zy(3x!y) xz(x!3y) yz(3x!y) xy(3x!y)(x!3y) zx(x!3y) zy(3x!y) y(3x!y)
xy(3x!y)(x!3y) xy(3x!y)(x!3y) xyz(3x!y)(x!3y)
In another non-orthogonal system with +x, y, z, axes coinciding with three C axes of O (I F arithmetic class) the basic invariants are h' "3(x#y#z)!2(xy#yz#zx) , h' "xy#yz#zx!xyz(x#y#z) , h' "(x#y!z)(y#z!x)(z#x!y).
(105) (106) (107)
5.5. Ring of C invariant polynomials. Description in terms of generators and syzygies G Let us now analyze the description of the ring of invariant function in terms of generators and syzygies (as it was generally outlined earlier, see Eq. (80)) using as example the C natural action on G 3-D-space and its induced action on the two-dimensional sphere. C "1 is the two-element G group generated by !I , the symmetry through the origin. The smallest re#ection group which contains it is the eight element group D "mmm whose three symmetry planes are the planes of F
In molecular physics and chemistry the group C is often denoted as S . We prefer to avoid this notation because it G can be easily confused with the S notation for two-dimensional sphere frequently used in this paper.
52
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
coordinates. The group D has eight one-dimensional inequivalent representations. Only four of F them have character 1 for !I . They yield the four numerator invariant of C . With the G convention that the indices i, j, k form a permutation of 1, 2, 3, the Molien function, the denominator and numerator invariant polynomials of C are G 1#3j , h "x, u "1, u "x x . (108) M G (j)" I I G H I ! (1!j) The structure of the module of the C invariant functions on the 3-D-space corresponding to the G Molien function in Eq. (108) may be written as (109) P!G "P[x , x , x ]E(1, x x , x x , x x ) . From Eq. (109) we deduce immediately the module structure for the ring of invariant polynomials after restriction of the 3-D action on the two-dimensional subspace given by equation x #x #x "r. (110) P!G " "P[x , x ]E(1, x x , x x , x x ) . 1 The corresponding Molien function for the C invariant polynomials on the sphere S is G (1#3j)(1!j)\. The natural action of C on 3-D-space gives us the possibility to illustrate the non-trivial G character of syzygies even for very small groups (see Example 6.6 in Stanley, 1979). The minimal set of generators of the ring of C invariants includes six generators of degree 2 each (including all G denominator and numerator invariants corresponding to the most reduced form of the Molien function (108)). To get the symbolic information about the set of syzygies we can multiply both numerator and denominator of the Molien function (108) by (1!j) to get the expression with six denominator invariants of degree two. Such a transformation gives 1!6j#8j!3j M G" . ! (1!j)
(111)
Naturally, the numbers of invariant polynomials in each degree do not alter upon such a transformation but we have new symbolic information. There are six syzygies of the "rst kind, eight syzygies of the second kind and three syzygies of the third kind. One should not think that it is always easy to interpret the numerator of a Molien function in terms of numbers of syzygies of di!erent kinds (sometimes this is even impossible without independent explicit calculation of syzygies as shown for example by Stanley (1996, Chapter I-11)). At the same time the symbolic interpretation is quite useful to see the complexity of the invariant ring and to verify the system of relations. We end this example with the complete list of all syzygies between generators of the ring of invariants for the C action of 3-D space. G Let us denote the generators by g "x , g "x , g "x , g "x x , g "x x , g "x x . Six syzygies of the "rst kind can be written as s"g g !g , s"g g !g , s"g g !g , ? @ A s"g g !g g , s"g g !g g , s "g g !g g . B C D
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
53
All these relations are of degree 4 in x variables. These six relations are not independent. Eight G relations between them (syzygies of second kind) can be given as s"g s#g s#g s, s"g s#g s#g s , ? ? B C @ ? B C s"g s#g s#g s , s"g s#g s#g s , A @ B D B @ B D s"g s#g s#g s , s "g s#g s#g s , C A C D D A C D s"g s!g s#g s!g s , s"g s!g s#g s!g s . (112) E ? @ C D F ? A B D These eight syzygies of second type are of degree 6 in x variables. They are dependent as well. G Three syzygies of the third kind have the form s"g s!g s#g s!g s#g s!g s , ? ? @ A C E F s"g s!g s!g s!g s #g s , @ ? A B D F s"g s!g s#g s!g s #g s . @ @ B C D E These three syzygies of third type are of degree 8 in x variables. We remark at the end of this G example that the construction we have realized above is well known in commutative algebra construction of a "nite free resolution (Cox et al., 1998; Stanley, 1996). 5.6. Representation of the orbit space in terms of invariant polynomials We have de"ned the orbit space for the G-action on M in Section 2 and denoted it by M"G. Invariant functions on M have constant values on an orbit. For the linear orthogonal representations of a compact group from Schawrz theorem (Schwarz, 1975) (see the beginning of Section 5) we need only to consider the invariant polynomials. Another theorem says that invariant polynomials separate the orbits, i.e. given two di!erent orbits, there is an invariant polynomial taking di!erent values on them. This implies that any system of generators of the ring of invariant polynomials must separate the orbits. This fact enables one to use invariant polynomials to represent orbit spaces geometrically (Kim, 1984; Sartori, 1991). Let us consider the system of generators containing the d basic invariants h of the denominator G and n generators of the numerator n 4N(t"1). Every orbit can be de"ned by the values of G these generators and represented by a point in a RB>LY space. The orbit space is a semi-algebraic set of points, i.e. it is de"ned by algebraic equations (the algebraic relations between the and h ) and G H inequalities, since we work on the real. The largest dimension of connected components is d (the number of algebraically independent polynomials). In all examples, we give here, the orbit space is connected. Then it is an orbifold. In Chapter V, for the non-linear action of space groups on the Brillouin zone (BZ), we show that for a natural global coordinate system on BZ we have a module structure for all arithmetic classes. So all we said extends to this non-linear action. The description of the ring of invariant polynomials in terms of integrity bases is advantageous because typically the invariant functions are given in terms of the same invariant polynomials and thus we can easily study functions geometrically on the orbit space through their level sets. The role of basic and auxiliary polynomials is quite di!erent. We illustrate the geometrical construction in several simple examples.
54
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
5.6.1. O and O natural action on 3-D-space F The space of orbits for the O or O natural group action on the 3-D-space arises in many di!erent F physical problems. We can just cite the triply degenerate molecular vibrations of octahedral molecules described by irreducible representation with Ker 1 and Im O (F irreducible represF S entation as listed in most tables of characters used by physicists and chemists (Landau and Lifshitz, 1965)) or rotation of spherical top molecules described by three rotational angular momentum (F E representation of O with Ker 1 and Im O). F It is well known (e.g., Michel, 1977) that the O "m3 m group is generated by re#ections and the F values of the three d (degrees of denominator invariants) are 2, 4, 6. The corresponding Molien I function has the form 1 . M F (j)" (1!j) (1!j) (1!j)
(113)
The explicit form of the invariants depends on the choice of basis in < . We choose an orthonormal L basis whose coordinates axes are identical to the four-fold rotation axes (orthogonal to the cube faces). Then we can choose (114) k"1, 2, 3, h " xI . I G G The polynomial h is invariant for every orthogonal group representation. The choice of the other h's is not unique. For instance we could choose instead of h any linear combination h "h #ah , a real; e.g. for a"!1, h " xx. Often instead of x#y#z one uses G$H G H xyz as one of basic invariant polynomials of degree six. (See expression in Eq. (117) below for the relation between these two choices.) It is easy now to restrict the above-de"ned action of O group on 3-D-space to the sphere S of F equation h ,x #x #x "r and to characterize the polynomial ring of invariants on this sub-manifold. Let us denote by h , h the polynomials in x , x obtained from h , k"2, 3, de"ned I in Eq. (114), by replacing their monomial xI by (r!x !x )I. Then P-F " "P[h , h ], the 1 polynomial ring of all the polynomials in these two variables. The transformation of the Molien function corresponding to this restriction consists in simple elimination from the denominator of (113) the factor (1!j). The group O"432 is the rotational (unimodular) subgroup of the group O "m3 m. It possesses F the only non-trivial numerator polynomial u 1 D(h , h , h ) "u"x x x (x !x ) (x !x ) (x !x ) . (115) 24 D(x , x , x ) The polynomial u is a pseudoinvariant of O ; it is the (non-trivial) numerator invariant of O, the F unimodular subgroup of O . This means that the u is a polynomial in basic invariants, namely if F we take as basic invariants h "x#y#z, u"h (h !27h !9h h h !h h #5h h #h h !h ) . (116) We call pseudoinvariant of a group G, every quantity transforming as a non-trivial one-dimensional representation of G.
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
55
Table 6 Strata equations for the O group action on S . Basic invariant polynomials h "x#y#z and h "xyz are used F as coordinates for the orbifold representation Stabilizer
Equations and/or inequalities
C "P4mm T C "R3m T C "Amm2 T C "Pm Q C "Cm Q
h "1, h "0 h "1/3, h "1/27 h "1/2, h "0 1/2(h (1, h "0 h "1!2s#3s/2, h "s(1!s)/4, 0(s(1, sO2/3 Internal points
C "1
Fig. 14. Orbifold for the natural action of the O group on the two-dimensional sphere S . Invariant polynomials are F speci"ed in Table 6. Singular points C , C , C correspond, respectively, to six-point, eight-point, and 12-point orbits T T T on S . Notice that cusp points C , C persist under the change of the form in invariant polynomials whereas the T T geometrical form itself varies.
This relation should be simpli"ed in the case of the O action on the S subspace by imposing F h "1. The resulting relation for u becomes inhomogeneous in two basic invariants. Remark, that in Table 6 and Fig. 14 which describe the action of the O group on the S sphere we use only F two basic invariants speci"ed in the caption to Table 6. Schematic representation of the orbifold of the O action on 3-D space and on two-dimensional F S subspace was given in a previous section (see Fig. 12). Here we represent the same space of orbits using invariant polynomials as coordinates. The polynomial ring of O invariants is generated by F two basic polynomials (one of fourth and another of sixth degree) which can be chosen as h "x#y#z and h "xyz. These two invariant polynomials can be used to label orbits. Fig. 14 shows the geometrical form of the orbifold in the space of invariant polynomials. Di!erent strata are de"ned by equalities and inequalities summarized in Table 6. The boundary of the orbifold in the space of invariant polynomials shown in Fig. 14 is de"ned by the relation in Eq. (116) after imposing u"0, h "1 and changing the notation for invariant polynomials h Ph and h Ph because on the S subspace we have only two basic invariants. One could use instead
56
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
of the xyz invariant polynomial another polynomial of degree 6, namely x#y#z. In fact, on the S surface the linear relation exist between these two polynomials: (117) x#y#z"3xyz#(x#y#z)! . Consequently, an alternative choice of basic invariants corresponds to linear transformation in the space of basic invariants. The geometrical form of the orbifold in the invariant polynomial variables re#ects certain important aspects of the behavior of invariant functions. Two vertexes of the orbifold (C and C ) T T are formed by two boundary lines with the same slope at singular cusp point. This means that the corresponding critical orbit (C and C ) should be always maximum or minimum for any generic T T function possessing only non-degenerate stationary points. (This is the Morse-type function, see next section for de"nitions). At the same time the critical orbit C which lies at the intersection of T two boundary lines with di!erent slopes can be a saddle point as well. Using the orbifold representation in the invariant polynomials we can use simple geometrical analysis to judge the existence of stationary points of a function de"ned over the manifold. In order to do that it is su$cient to plot the contour lines of a function (level set of a function) directly on the orbifold and to analyze the topology of di!erent level sets. Stationary points may exist only for level sets with exceptional topological structure. Let us consider, for example, an O invariant function written in the form of a linear combination F of two basic polynomials ah #bh . (118) Apart from a trivial scalar factor this is the most general O invariant function up to the sixth F degree in initial variables. At the same time this is just a linear function in terms of invariant polynomials and its level set on the orbifold is the set of straight lines. Figs. 15 and 16 show examples of contour plots (levels of constant value of functions) for two qualitatively di!erent types of invariant functions. Function plotted in Fig. 15 has three orbits of stationary points. These orbits are critical and consequently this function has a minimal possible number (namely 26) of stationary points. Three level sets (a, c, e) of this function have exceptional topology. The topological structure of level sets (b) and (d) does not vary if the value of the function varies slightly, i.e. these sets are regular. The function plotted in Fig. 16 has four exceptional level sets (a, c, e, g). The level sets (a, e, g) include critical orbits. The level set (e) is exceptional because it touches the boundary and the touching point is a stationary non-critical point. Its position is not "xed but this point persists on the boundary even after a small perturbation of the function. We should remark that among di!erent functions of the form (118) there are functions with the minimal possible number of stationary points (only critical orbits), functions with one additional orbit of non-critical stationary points, and exceptional type functions corresponding to the qualitative phenomenon known as bifurcation of stationary points. More exact terminology for these di!erent types of functions based on the Morse theory will be introduced in the next section. The space of orbits for the O group action follows immediately from the analysis made above for the O group action. The system of denominator invariants are the same for two groups. To F completely characterize orbits for O group we should add the value of auxiliary invariant polynomial. All principal orbits of O (internal points of the orbifold) are split into two orbits of the F
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
57
Fig. 15. Contour plot of the level set of the invariant function in Eq. (118) on the orbifold. The function corresponds to a/b&!1/10. It has only three critical levels (a, c, e) and two di!erent types of regular levels (b, d). Fig. 16. Contour plot of the level set of the invariant function in Eq. (118) with a/b& on the orbifold. The function possesses stationary points outside of critical orbits. There are four critical levels (a, c, e, g) and three di!erent types of regular levels (b, d, f ). The non-critical stationary point located at the touching point of level c and the boundary.
O group which have the opposite values of the numerator invariant u. All O orbits lying on the F boundary of the orbifold correspond to the zero value of auxiliary O invariant u. This means that the auxiliary O invariant u de"nes the form of the O orbifold and the complete representation of F the O orbifold consists of two parts (two O orbifolds with identi"cation of corresponding points on F the orbifold boundary). 5.6.2. Orbifold for the C natural action on 3-D-space G Integrity basis includes in this case three basic polynomial invariants and three auxiliary invariants (see Eq. (108)). Basic polynomial invariants can be used to label unambiguously the orbits within any region where all auxiliary polynomials keep their sign. The boundary of those regions is de"ned as a set of points where at least one of the auxiliary polynomials equals zero. Consequently, the space of orbits can be represented by a set of smaller orbifolds that have to be glued together by identi"cation of some boundaries. Each part corresponds to one region where all auxiliary polynomials have the constant sign (with the boundary of the region corresponding to zero value of some of the auxiliary polynomials). Boundary points of di!erent parts should be identi"ed to represent properly the topology of the whole space of orbits. Fig. 17 shows the representation of the orbifold as a four-part decomposition with identi"cation of side-planes labeled by identical letters. It is a connected orbifold. 5.6.3. Finite group action on 2-D-torus Let us start with a trivial example of the C group action on a torus. We treat this case in order to demonstrate the representation of a manifold with non-trivial topology in terms of polynomial functions de"ned on it. 2-D-torus can be represented in (k , k ) variables restricted, for example as 04k , k (2p and taken modulo 2p. This is a standard representation of a torus as a square with identi"cation of
58
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
Fig. 17. Orbifold for the natural action of the C group on three-dimensional Euclidean space. To understand the G topology of the orbifold four 3-D-parts should be glued together through identi"cation of coordinate planes marked by the same letters.
Fig. 18. Tore without symmetry in k , k variables. Opposite sides should be glued together respecting the letters on the boundary to get the torus. Internal letters and auxiliary decomposition will be useful to make the comparison with the representation of torus in terms of invariant polynomials constructed from cos(k ) and sin(k ) functions. See G G Fig. 19.
opposite sides (see Fig. 18) To introduce an alternative representation of the same torus let us introduce the ring of smooth functions de"ned on it and the associated integrity basis. It is clear that any smooth function on the torus can be expresses as formal series in 2p periodic functions c "cos(k ), c "cos(k ), s "sin(k ), s "sin(k ) . (119) Naturally these four functions are algebraically dependent and the generating function for the number of linearly independent homogeneous polynomials in c , c , s , s has the form 1#2t#t . (120) (1!t)
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
59
Fig. 19. Tore without symmetry in c , c variables. The space of trivial orbits is represented in terms of integrity basis polynomials. Four parts correspond to di!erent signs of auxiliary invariants s , s which are marked inside each part. To construct the whole orbifold (the torus) four parts should be glued together through identi"cation of their boundaries. Partial identi"cation of boundaries EFG, ELG, CME, and CDE leads to the representation of the torus as a square with identi"ed opposite sides as in Fig. 18.
It re#ects the structure of integrity basis of invariant polynomials depending on four variables c , c , s , s and de"ned on the two-torus. The two basic (denominator) invariants can be taken as c "cos(k ), c "cos(k ), while three auxiliary (numerator) invariants can be chosen as s "sin(k ), s "sin(k ), and s s . The module of polynomials which can be considered as invariant polynomials with respect to the trivial C symmetry group acting on the torus can be written as P. "P[c , c ]E(1, s , s , s s ) .
(121)
Now we can use the denominator basic polynomials as continuous variables to represent the trivial orbits of the C group action on the torus. To represent completely the orbifold we need to add auxiliary invariant polynomials. This leads us to representation of the orbifold as four body decomposition with the each body being the same square in c , c variables but characterized by di!erent sets of signs of auxiliary polynomials. In fact, it is su$cient to use only signs of two numerator invariants because the third invariant is simply the product of the two "rst. The geometrical representation of orbifold in terms of c which shows explicit correspondence with the G representation of the same orbifold in k , k variables is given in Fig. 19. Remark that if we consider the "nite group p2mm action on the torus its ring of invariant polynomials includes only basic invariants c and c . This means that the orbifold for the p2mm action on the torus coincides with one sub-orbifold drawn in Fig. 19. This situation is quite general. It was illustrated in previous example of O and O group actions on S and still will be discussed in F Chapters II and V.
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
60
To describe the ring of functions on the torus we could use instead of the integrity basis construction the alternative approach based on the system of generators and syzygies and leading to the so-called "nite free resolution widely used in mathematical literature (Stanley, 1979; Cox et al., 1992, 1998). To construct the minimal free resolution we take "rst four generators c , c , s , s and form polynomial ring with four variables. Next step is to introduce relations (syzygies of the "rst kind) s"1!c, (i"1, 2), s s "z . (122) G G This gives us two restrictions but the third relation gives new variables and we can further consider the polynomial ring with three variables c , c , z. Again these three variables are not independent. There exist syzygies of the second kind z"(1!c )(1!c ) , (123) which is in fact of the fourth degree in initial generators. To see the presence of syzygies in the symbolic representation we can rewrite the generating function (120) as 1#2t#t (1#2t#t)(1!t) (1!t) 1!2t#t " " " . (1!t) (1!t) (1!t) (1!t)
(124)
This form shows that there are two quadratic syzygies of the "rst kind and one quartic syzygies of the second kind. This example illustrates once more the relation between the description of the ring of invariant functions in terms of integrity basis and in terms of generators and syzygies. While generators}syzygies construction is more appropriate for abstract mathematical study and theorem proving, the integrity basis approach is more suited for detailed analysis of concrete examples.
6. Morse theory In this work we also need to apply some results of Morse theory (Morse, 1925; Seifert and Threlfall, 1938; Milnor, 1963; Poe`naru, 1976; Palais and Terng, 1980; Bott, 1982; Dubrovin et al., 1990; Fomenko, 1983). Consider a smooth ("in"nitely di!erentiable) real-valued function f on a real compact manifold M with a coordinate system +x ,, 14i4d"dim M. If at a point I m3M of coordinates x the function satis"es the equations: vanishing gradient, i.e. *f/*x "0, and I G non-vanishing determinant of the Hessian, i.e. det(*f (x)/*x *x )O0, we say that it has a nonG H degenerate extremum. Then by a change of coordinates +x ,C+y ,, in a neighborhood of m the G G function can be transformed into f" e y with e "$1. The number of minus signs is G G G G independent of the coordinate transformation and it is called the Morse index k of this nondegenerate extremum: for instance k"0 for a minimum, k"d for a maximum and the intermediate values correspond to the di!erent types of saddle points. A function on M with all its extrema
In general a coordinate system cannot be de"ned on the whole manifold.
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
61
non-degenerate is called a Morse function. Let c be the number of its extrema of Morse index k; I these numbers are "nite and may be used to construct the Morse polynomial M (t) for a Morse D function which counts all stationary points of the function f with their indices B M (t)" c t I with d"dim(M) . (125) D I I More generally the Morse polynomial M (t) for a Morse function on M may be de"ned as D M (t)" tIN, M (1)""C( f )" , (126) D D NZ!D where the sum is taken over the set C( f ) of extrema and k(p) is the Morse index of the point p. The polynomial M (t) is a quantitative measure of the critical behavior of f. D The essence of the Morse theory is the relation between the numbers of extrema and the Betti numbers of the manifold M. The Betti number b is de"ned as the rank of the kth homology group I of M. Intuitively b is the maximal number of k-dimensional sub-manifolds of M which cannot be I transformed one into another or into a sub-manifold of a smaller dimension; for instance for the sphere S of dimension d, b "b "1 and all the others b vanish. B B I To formulate Morse inequalities for a given d-dimensional manifold M we introduce the `PoincareH polynomiala P (t) in which the coe$cient of tI is the Betti number b of M: + I B P (t)" b tI with d"dim(M) . (127) + I I The Euler}Poincare& characteristic s of the manifold M is de"ned by + s "P (t"!1) . (128) + + The PoincareH polynomial of a topological product M"; M is the product of the PoincareH G G polynomials: (129) M"; M NP (t)" P G (t) N s " s G . + + G G + + G G For instance the PoincareH polynomial of the manifold R"S ;S has the following explicit form: PR (t)"(1#t), b "b "1, b "b "0, b "2, sR "4 , (130) whereas for d-dimensional torus SB we see immediately that the Betti numbers are binomial coe$cients.
B d tI . PR (t)"(1#t)B" k I
(131)
If there were an in"nite number of extrema on the compact manifold M there would be an accumulation point which would be a degenerate extremum. An example of S is the boundary of the unit ball in the (d#1)-dimensional Euclidean space. B
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
62
Using the PoincareH polynomial of the manifold, P (t), and the Morse polynomial M (t) for + D a Morse function f we can formulate the relations between the Betti numbers and the numbers of stationary points of the function f. The precise statement is as follows. Morse inequalities. For every Morse function f there exists a polynomial Q (t)"q #q t#2 D with non-negative coezcients such that M (t)!P (t)"(1#t)Q (t) . D + D This statement may be rewritten in terms of d inequalities
(132)
l
04l(d: Nc 5b I I and one equality
(!1)l\I(c !b )50 I I I
(133) (134)
B B B s , (!1)I (c !b )"0 0 (!1)I c " (!1)I b " (135) I I I I + I I I where s is the Euler}PoincareH characteristic of the manifold M. The inequalities (134) are not + equivalent to Morse inequalities (133), but they give lower bounds to the number of extrema of a Morse function. There exists another limit on the number of stationary points of any smooth function on the manifold which is valid for an arbitrary non-Morse function with degenerate stationary points. The number of stationary points cannot be larger than the Lusternik}Schnirelmann category of the manifold (initially introduced in Lusternik and Schnirelmann (1930), two chapters of which were translated into French (Lusternik and Schnirelmann, 1934). Lusternik}Schnirelmann category is the homotopy invariant which is not easy to calculate in general case. For n-dimensional sphere S this invariant is equal to 2 and for an n-dimensional torus it is equal n#1 (Seifert and Threlfall, L 1938; Fomenko and Fuks, 1989). In the absence of any symmetry requirements many su$ciently good manifolds allow the existence of a so-called perfect Morse function which reduces all Morse inequalities into equalities. In other words for the perfect Morse function Q ( f )"0, i.e. M (t)"P (t)&c "b . In particular, R D + I I the number of stationary points "C( f )" of a perfect Morse function f equals the sum of Betti numbers ("P (1)). When a "nite symmetry group G acts on the manifold M, there are further requirements + on the c 's of G-invariant Morse function: all points of a G-orbit should be simultaneously either I extrema with the same index or not be extrema. Some simple physical applications of the Morse theory under the presence of symmetry which are close in spirit to the present analysis were given by Michel and Mozrzymas (1978), Michel (1979) and Zhilinskii (1989a) (see also Zhilinskii, 1989b). As we saw in Section 4, very often the group action itself insures (Corollary 4c) that several orbits (the critical orbits) should be stationary for any G-invariant function f. If apart from that there exist closed strata (those with maximal symmetry) and if the orbits they contain are not all critical, then Theorem 4c (see Section 4) applies and requires for any function at least two orbits of extrema in these strata. In any case, among all invariant Morse functions there exist some of them with the minimal possible number of extrema. We will call such functions the simplest Morse-type functions.
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
63
We can still introduce the natural measure of the complexity of a given Morse-type function just by counting the total number of stationary points. In case of a symmetry realized by the e!ective action of a compact Lie group, orbits are themselves continuous compact manifolds (some of them might be of dimension zero " "nite set of points). The situation becomes more complicated when some of these orbits are orbits of extrema of an invariant function f: indeed these extrema of f are degenerated along the directions tangent to the orbit. So the Morse theory in its original form is not applicable. There exists an extension of Morse theory for G-invariant function with non-zero-dimensional orbits of extrema. Atiyah and Bott (1982), Bott (1982) and Fomenko (1983) give the basic mathematical facts with some ideas of physical applications. Analysis of one particular physical problem using this technique is made, for example, by Kirwan (1988). We give some hint of this theory in Appendix B. We can de"ne now the simplest Morse (Morse}Bott)-type functions as a class of functions possessing the minimal number of non-degenerate stationary points (or manifolds). This minimal number cannot be less than the number of critical orbits but it is allowed to be larger. To give the classi"cation of Morse (Morse}Bott)-type functions due to complexity of the system of stationary points we introduce the measure of complexity to be the number of stationary points and the number of stationary manifolds excluding those associated with critical orbits. Further classi"cation of the Morse-type functions within one class of the same complexity (i.e. with the same number of stationary points (manifolds)), takes into account possible di!erent distributions of stationary points (manifolds) over strata or within one stratum. Now the general aim of the qualitative analysis may be formulated: (i) To describe qualitatively di!erent generic Hamiltonians invariant under given symmetry group. (ii) To describe possible qualitative generic changes of Hamiltonians occurring under the variation of a given number of parameters. (iii) To relate qualitative changes of classical functions with corresponding changes of quantum Hamiltonians.
6.1. Examples of Morse theory applications. Stationary points of the simplest Morse-type functions Morse theory enables one to "nd restrictions on the numbers of stationary points of a generic function de"ned over compact manifolds. Table 7 shows examples of minimal sets of stationary points for several manifolds which are relevant for molecular and solid-state physical applications. Perfect Morse-type functions exist on all cited manifolds. 6.1.1. Function on the sphere in the presence of xnite symmetry Normally under the presence of symmetry the minimal possible number of stationary points of an invariant function is larger than that in the absence of symmetry. The simplest example is given by functions de"ned over S in the presence of a point symmetry group with natural action induced by its linear action on three-dimensional ambient space (see Table 8). Remark that for some cases all stationary points of the simplest Morse-type function belong to critical orbits, sometimes they
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
64
Table 7 Stationary points of the simplest Morse functions de"ned on some manifolds Manifold
Betti numbers
Stationary points
R
S CP
b "b "1, b "0
c "c "1
2
2
¹ "S ;S
b "b "1, b "2
c "c "1, c "2
0
4
¹ "S ;S ;S
b "b "1, b "b "3
c "c "1, c "c "3
0
8
CP
b "b "b "1 b "b "0
c "c "c "1
3
3
CP ,
b "1 I b "0 I>
c "1 I
N#1
N#1
S ;S
b "b "1, b "2, b "b "0
c "c "1, c "2
4
4
c G G
c is the number of stationary points of Morse index k. I R" (!1)Ib is the Euler}PoincareH characteristics of the manifold (cf. Eq. (135)). I Minimal number of stationary points. Table 8 Simplest Morse functions de"ned on S in the presence of symmetry c I
Group C ,C L LT
C ,C ,D ,D G F F
D ,D ,D L LB LF
¹, ¹ B
O, O , ¹ F F
I, I F
c c c
1 0 1
2 2 2
2 or n n n or 2
4 6 4
6 or 8 12 8 or 6
12 or 20 30 20 or 12
Q(t)
0
1#t
(n!1)t#1 or (n!1)#t
3#3t
7#5t or 5#7t
19#11t or 11#19t
c is the number of stationary points of Morse index k. I
lie on the close strata as well, and for C "S "1 they are on the generic stratum. Table 8 gives G examples of the simplest Morse-type functions de"ned over the S sphere in the presence of symmetry. The polynomial Q(t) (see Eq. (132)) is given to show that except for C , C the simplest L LT functions are not the perfect ones. 6.1.2. Functions on the d-torus in the presence of symmetry Most physical properties of crystals are described by functions on the Brillouin zone invariant by the crystal symmetry and time reversal. Chapters IV}VI give detailed analysis of this symmetry.
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
65
Here we just note that the characterization of the system of stationary points of a function de"ned over the torus in the presence of the symmetry follows the same ideas based on the Morse theory as a simple example cited above for invariant functions over the two-dimensional sphere. The complete description of the system of stationary points for functions invariant under crystal arithmetic classes and time reversal was "rst given in Michel (1996) and reproduced in Chapter IV with the misprints corrected. 6.2. Modixcations of the system of stationary points. Bifurcations Very often in applications we work with families of functions depending on one or several control parameters rather than with a single speci"c function. In such a case instead of qualitative characterization of one chosen function we need to describe possible qualitative modi"cations occurring under the variation of the control parameters. One of the simplest cases to study is the possible modi"cations of the system of stationary points of a function under the variation of a control parameter. Morse theory naturally imposes severe restrictions on the type of possible modi"cation of the system of stationary points. The simplest consequence is the impossibility to add just one stationary point (without removing the requirement for the function to be of Morse type). The classi"cation of possible modi"cations depends on the number of parameters and on the presence of symmetry. Bifurcations of the stationary points are responsible for the qualitative modi"cations of a function and to the appearance of new features for physical models. We will see in Chapter II many examples of the manifestation of the bifurcations of stationary points in molecular problems, in both classical and quantum pictures. In order to supply here one simple example let us return to the analysis of O invariant functions F de"ned over S sphere (see Section 5.6.1). If we consider the one-parameter family of functions (similar to Eq. (118) but with slightly di!erent parametrization) H(a)"sin ah #cos ah ,
(136)
it is easy to verify that H(a) is a Morse-type function for tan aO0,!,!. Moreover for !R(tan a(! we have the simplest Morse-type function. For !(tan a(R the function has the "rst level of complexity, i.e. the number of stationary points equals 50. One additional non-critical stationary orbit formed by 24 points exists on a one-dimensional stratum C "Cm. It is possible even to precise the position of stationary orbit. For !(tan a(! the Q stationary point lies between C "Amm2 and C "R3m, whereas for !(tan a(R the T T stationary point lies between C "R3m and C "P4mm. Exceptional values tan a" T T !!,$R correspond to functions which are not of the Morse type. tan a"! corresponds to the function with a degenerate stationary point at the C "Amm2 critical orbit. In other words T if we consider the family of functions H(a) and vary a from (!!d) to (!#d), the number of stationary orbits changes and the bifurcation of stationary points takes place at d"0 (i.e. at a"!). d"! corresponds to another typical bifurcation for a one-parameter dependent family of functions. These two bifurcations can be easily characterized geometrically. At a bifurcation point the contour plot of the function on the orbifold becomes tangent to the boundary at C or at C critical orbit. The exceptional value tan a"$R corresponds to the function which T T
66
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
possesses a continuum set of stationary points. Such behavior is not typical and slight perturbation by functions of higher degree removes this non-generic situation.
7. Physical applications We conclude this chapter with general remarks about several physical applications which are not studied in details in the following chapters of this issue. 7.1. Action of the Lorentz group on the Minkowski space The Minkowski space M is the space of energy E"p and three-momentum p in the theory of special relativity. It is mathematically described by a four-dimensional real vector space carrying the invariant scalar product (in a unit system with c"1) s (p)"p ! p invariant under the G G linear action of the Lorentz group [&O(1, 3)]. The origin p"0 is the unique "xed point; so it is its own stratum. The three other strata are the set of four-vectors p with, respectively, s'0, called time-like vectors; their stabilizers are conjugated to O , s(0, called space-like vectors; their stabilizers are conjugated to O(1, 2), s"0, pO0, called light-like vectors (they form the light cone minus its vertex); their stabilizers are conjugated to a subgroup of O(1, 3) isomorphic to Eu . 7.2. Physical examples of systems and phenomena with continuous subgroups of O(3);T as symmetry groups In 1894 P. Curie wrote the remarkable paper (Curie, 1894) which is concerned to the symmetry of physical phenomena and gives the realization of the one-dimensional Lie subgroups of the O(3) group as symmetry groups of di!erent physical phenomena. The study of O(3) group is quite important for physical applications because it re#ects the isotropy of our 3-D-physical space. We extend here the O(3) group by adding the time reversal symmetry operation. The resulting group O(3);T has larger number of continuous symmetry subgroups and consequently we can give the more detailed description of physical systems and phenomena in the spirit of Curie analysis taking into account the presence or breaking of time reversal invariance. The connected group SO(3) is a `completea group, i.e. it has no center and no outer automorphisms. The following theorem for complete groups can be found in mathematical literature on group theory (Hall 1959, Section 6.4): if the complete group H is an invariant subgroup of group G, then G&H;Q with Q"G/H. So the combined action of the geometric O(3) and time reversal T symmetry on classical or quantum states with integral spin is that of the group The permutation groups S , for nO6 give another example of `completea groups. L Probably the "rst application in physics of this theorem has been made with the group generated by SO(3) of isospin and Q"Z (C), where C is the charge conjugation; the two groups do not commute. Michel (1953) proved that their action on isospin states is that of a group isomorphic to O(3) and called isotopic parity (now physicists say `isoparitya) the corresponding new quantum number.
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
67
G"O(3);Z (T)"SO(3);Z (i);Z (T) where i is one of the traditional notations used in some domains of physics for the symmetry through the origin. We remind that we use the notation H"Kg to say that the group H is generated by the subgroup K and its element g. In this section we will consider physical states whose symmetry is that of a Lie subgroup of G. To study their symmetry, we have to consider all conjugacy classes of Lie subgroups of G. The group G&SO(3);Q, with Q&z has four connected components. The three-dimensional Lie subgroups of G are the counter image of the "ve subgroups of Q for the group homomorphism GPQ. So the three Lie subgroups of G with two connected components are: O(3), SO(3)T, and SO(3)T "SO(3)T , where T "Ti and T "Ts where the s is any re#ection through G Q G Q a plane. In order to determine all one-dimensional Lie subgroups of O(3);T (up to a conjugation) we remark that the connected component of the identity for all one-dimensional Lie subgroups of O(3);T is C and it is an invariant subgroup of any of them. There is a partial ordering of their conjugacy classes. The maximal one is D ;T. The quotient group (D ;T)/C "Z has 16 subgroups, seven F F of two elements, seven of four elements, itself Z (eight elements), and the identity (trivial subgroup). The counter image of these 16 subgroups yields the 16 one-dimensional Lie subgroups of O(3);T. Those which have two connected components are the union of two cosets of C . The one di!erent from C is of the form gC . We can take for g the elements (the C axis will be considered as vertical and denoted by v) v, h, vh"2 , T, T "vT, T "hT, T "2 T, (137) F T F F where v, h are, respectively, re#ections in a plane containing the C axis and in a plane perpendicular to it. (We could have taken, instead of h, the space inversion i"h2 . We follow the T tradition of molecular physicists who prefer the notation C to C .) The corresponding notation F G for these seven groups with two connected components is C , C , D , C T, C T , C T , C T . (138) T F T F All other one-dimensional subgroups with four and eight connected components (" 4 or " 8 cosets) are given in Table 9. Remark that Curie (1894) have considered "ve one-dimensional subgroups C , C , C , D , D . T F F Several groups from the complete list naturally arise as symmetry groups of simple quantum systems. C T is the symmetry group for the atom in static electric "eld or for the Rydberg states of T heteronuclear diatomic molecule. C T is the symmetry group of an atom in constant magnetic "eld (Zeeman e!ect). F T C T is the maximal common subgroup of C T and C T . Thus it corresponds to T T F T the symmetry of an atom under the simultaneous presence of two parallel magnetic and electric "elds. D T group is the symmetry group for Rydberg states of homonuclear diatomic molecule, or F of the quadratic Zeeman e!ect. Another interesting quantum example can be given by circularly polarized photon. It is characterized by the group C T . Let us now add the physical realization for three-dimensional Lie subgroups of O(3)T.
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
68
Table 9 Description of one-dimensional subgroups of O(3);T with four connected components in terms of continuous and discrete symmetry elements. Sign # means that the corresponding element belongs to the symmetry group C
C T
C F
C
# # # # # # #
# #
#
#
#
#
# #
T
T T
# # #
#
T F
#
#
# # #
#
#
#
T
# #
#
#
#
# # #
#
Group
Perturbations
D F C T T C T F D T C T T F C T F T D T T
Dilation (compression) Axis asymmetry Double twist Uniform twist Axial current Angular current Two opposite angular current
D T F
Unperturbed cylinder
Alternative notation for C T is C T , for C T is C T , for D T is D T , and for D T is T T F T F T F F D T with T "T ,T , or T . F ? ? T F
Electric charge has complete O(3)T symmetry. In physically reasonable approximation He atom in its ground state gives another example of a system with O(3)T symmetry. Spherically symmetric outgoing wave is the system with O(3) symmetry. We can consider space dilation or compression as an example of perturbation breaking the time reversal symmetry. Such a perturbation physically can be realized if an atom is supposed to be in the spherical box with varying radius. He atom within the fullerene cage under varying pressure gives approximate physical realization of the e!ect in the molecular domain. To realize the SO(3)T symmetry group as the symmetry group of a physical system we can take a `relativelya stable particle which is a pseudoscalar. p meson is an example. p! can be considered as stable for times much smaller than &10\ s. A macroscopic example of this symmetry group is given by isotropic optically active liquid. The magnetic monopole gives an example of a system with the invariance symmetry group SO(3)T . Q Finally the disintegration of a n meson is the process which is invariant with respect to the SO(3) symmetry group only. As soon as Curie did not consider explicitly the time reversal he made no di!erence between static electric "eld and electric current, for example. We can propose more formal physical realizations of di!erent one-dimensional subgroups of O(3)T. Let us consider a homogeneous "xed cylinder as an initial classical object. Its symmetry group is the D T group which is the maximal one-dimensional subgroup of O(3)T. Further F we introduce several elementary excitations (physical perturbations) which correspond to di!erent broken symmetries. In Table 9 we introduce these perturbations, the seven one-dimensional symmetry groups with four connected components are listed in this table. We explain them shortly below.
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
69
(i) Homogeneous dilation or compression of the system considered as a time dependent phenomenon. This perturbation breaks the time reversal invariance because time reversal interchanges dilation and compression. The associated symmetry group is D . F (ii) Axis asymmetry can be introduced by a static form of asymmetry (transformation of cylinder into pear-like object) or by a constant electric "eld parallel to the symmetry axis. The corresponding symmetry group is C T. T (iii) Double twist is the static perturbation of the cylinder such that the middle section of the cylinder is twisted in one direction whereas two ends are twisted in the same opposite direction. The corresponding symmetry group is C T. F (iv) Uniform twist is just a simple static twist of two opposite ends of the cylinder in opposite directions. The symmetry group is D T. Much simpler physical realization of the same symmetry group can be reached with a cylinder "lled by isotropic optically active liquid. (v) Axial current can be realized for example as an electric or thermic current along the axis of the cylinder. This perturbation breaks in particular the time reversal invariance. The symmetry group is C T . T F (vi) Angular current can be realized as uniform rotation of the cylinder, or as constant magnetic "eld parallel to the axis (solenoid magnetic "eld). The symmetry group is C T . F T (vii) Two opposite angular currents can be constructed with two rotations, two electric circular currents or a magnetic "eld parallel to the axis and varying linearly (or more generally as an odd function) along the axis and passing through the zero in the equatorial plane of the cylinder. The symmetry group is D T . T There are four plus signs in each column of Table 9. One # stands to unperturbed cylinder and three others correspond to three di!erent four-component groups possessing the same twocomponent group as a subgroup. Thus each two component symmetry group can be obtained by three di!erent pairs of perturbations taken from Table 9. Physical interpretation of this construction is as follows. Let us associate with each elementary perturbation the physical situation characterized by the same symmetry. Three di!erent situations are associated with each one component symmetry group. In principle each situation can be considered as a cause or as an e!ect. The application of Curie principle means that if we apply two (among three possible) situations as a cause, the third could be the e!ect. Following realizations can be proposed for illustration. (a) `Angular currenta, `axial currenta, and `uniform twista have C T symmetry group as common subgroup. Simultaneous application of `angular currenta and `axial currenta which we consider as a cause could produce the appearance of a `uniform twista as an e!ect. Realization of angular current as a magnetic "eld and axial current as an electric current enables us to formulate the following physical statement: mechanical twist of a cylinder could be observed if electric current passes along the axis of the cylinder situated along the axis of magnetic "eld. We can choose two other perturbations as a cause and the third perturbation will be the e!ect. Two complementary physical statements will be: (i) Mechanical twist of a cylinder with axial electric current can produce appearance of the axial magnetic "eld. (ii) Mechanical twist of an axial magnet can result in the appearance of electric current along the axis of the cylinder.
70
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
Naturally each formal perturbation can be realized by di!erent physical situations. For example, instead of electric current and magnetic "eld we can consider the propagation of a circular polarized light parallel to the cylinder axis. (b) Another triple of elementary perturbations associated with the symmetry group C T are F F `two opposite angular currentsa, `axial currenta, and `double twista. Here and below we give only one (among three possible) choice of cause and e!ect. Two other e!ects can be easily deduced. Simultaneous application of `two opposite angular currentsa and axial current gives a system with C T symmetry could result in the appearance of a double twist as an e!ect. F F (c) Uniform twist and axis asymmetry together reduce the symmetry to C T and produce the double twist. (d) Axis asymmetry together with angular current give the symmetry group C T and result T in appearance of two opposite angular currents. (e) Axis asymmetry together with axial current give the C symmetry group which indicates the T presence of the dilation. (f) Double twist plus angular current lead to C symmetry group and again produce the F dilation as an e!ect. (g) Uniform twist plus two opposite angular currents yield the D symmetry group and are accompanied by the dilation as well. (h) To obtain the C symmetry group we can apply three elementary excitations, for example, static axis asymmetry, axial current, and angular current. 7.3. Geometrical conxgurations of N-particle systems. Shape coordinates and their invariant description To characterize an instantaneous geometrical con"guration of the N-particle system one should introduce the internal coordinates (shape coordinates in terminology of Littlejohn and Reinsch (1997)) which give the unambiguous description of di!erent geometrical con"gurations. Naturally, two geometrical con"gurations of N points in 3-D-space are equivalent if they have the same center of gravity and they can be transformed one into another through orthogonal transformation. In 3-D-space the N!1 Jacobi vectors (3N!3 scalar components can be used as variables) are su$cient to characterize the geometrical position of N particle with respect to their center of mass. But at the same time only 3N!6 variables are algebraically independent as shape coordinates. The problem of the construction of a suitable set of shape coordinates can be reformulated as a description of variables needed to classify the space of orbits of group SO(3) acting on a system of N!1 3-vectors. Each vector belongs to the irreducible representation of SO(3) for J"1, where J is the total angular momentum quantum number. In what follows it will be denoted by (1). Thus, we should construct a system of SO(3) invariants for an initial reducible representation C"(1) (1) 2 (1). Invariant polynomials forming integrity basis for this initial representation can be used as shape coordinates for the N-body problem. Description of the algebra of invariant polynomials for this particular problem was given by Weyl (1939) in terms of generators and relations (syzygies) between them. With increasing N the algebra of invariants becomes very complicated but for low N rather simple and physically clear integrity bases can be proposed. We list below generating functions for invariants for low N before formulating the general result.
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
71
Let us consider the trivial case of one vector (N"2) which corresponds to the (1)-representation of the SO(3) group. The generating function for invariants has the form 1 g(0Q1)" . 1!t
(139)
We have one basic invariant which is of the second order I "x. It can be used to label the orbits. This means that the shape of a two-particle system can be completely characterized by a one polynomial variable, the square of the distance between particles, which is clearly a trivial result. In the case of a three-particle system we should analyze the system of SO(3) invariants for two vectors (initial representation is (1) (1)). The generating function for invariants has the form 1 . g(0Q1 1)" (1!t)
(140)
So to represent invariant functions we have three basic invariants (polynomials of second degree). We can take them (for two vectors x , x ) as I "x , I "x , I "(x ) x ). It is clear that I 50 and I 50, whereas I should satisfy the restriction I 4I I . For the four-particle system the generating function for invariants is 1#t , g(0Q1 1 1)" (1!t)
(141)
which for the "rst time has a non-trivial numerator. This means that there exist six linearly independent invariants of degree 2 and one auxiliary invariant of degree three. This numerator invariant is algebraically dependent on six basic invariants. Itself, it cannot be represented as a polynomial in basic invariants, but its square can. Natural choice of six basic invariants I"(I ,2, I ) (for three vectors, say y , y , y ) is I"(y , y , y ,( y ) y ), ( y ) y ), ( y ) y )) . (142) As an auxiliary invariant of the third degree it is natural to take the mixed product
"( y ) ( y ;y )). It is clear that is a polynomial in basic invariants. When O0 the coordinates I , I , I , I , I , I do not give the unique description of the con"guration. The same set I ,2, I describes two di!erent con"gurations (right and left) which can be distinguished by the sign of the auxiliary invariant . This means that in the six-dimensional space of basic invariants all internal points of the orbifold have two points as an image in the real con"guration space, whereas points on the surface "0 are unique. For the "ve-particle system the generating function for invariants starts looking complicated 1#t#4t#t#t . g(0Q1 1 1 1)" (1!t) We have for the "ve-particle system seven non-trivial numerator invariants.
(143)
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
72
For general N the generating function for invariants can be written in terms of an integral over SO(3) group
1 p p p sin(h) dh d ds , (144) (N; t)" det[I!tM(h, , s)] 8p where the matrix M(h, , s) is block diagonal with N!1 identical blocks each being the 3;3 Euler rotation matrix a(h, , s). This integral can be calculated explicitly and the generating function has the following form (Collins and Parsons, 1993) g
1-
g
1-
F(t) (N; t)" (1!t),\
(145)
with (n#k)! 1 ,\ L tI(1!t),\\L\I(1#t),\>L\I . F(t)" (k!)(n!k)! N!2 L I
(146)
7.4. Landau theory of phase transitions About 60 years ago Landau (1937a, b) has formulated the basic principles of the phenomenological theory of second-order phase transitions which were based on the crucial idea about spontaneous symmetry breaking under phase transition (Landau and Lifshitz, 1958). This approach was generalized later and led to the uni"ed description of completely di!erent physical systems within the unique mathematical formalism. The essential point of the Landau-type theory is the symmetry breaking occurring under the continuous variation of the physical state of the system. The thermodynamic potential or any other physical quantity which characterizes the phase transition is represented near the transition points as a series expansion in invariant functions for the high-symmetry phase. That is why the mathematical theory of continuous phase transitions is closely related to the description of invariant and covariant polynomials for di!erent chains of groups and that is why the "rst physical applications of this theory was related with phase transitions in crystal solids. We do not want to enter into this vast "eld of applications and just restrict ourselves to some basic references which cover this "eld (Landau and Lifshitz, 1958; Lyubarskii, 1957; Stanley, 1971; Ma, 1976; Izyumov and Syromyatnikov, 1984). Appendix A. Group theory: glossary We give in this appendix some group-theoretical de"nitions which can be found in various textbooks at di!erent level of abstraction. As group theory for physicists books we recommend Wigner (1959), Weyl (1931), Lyubarskii (1957), and Hamermesh (1964), for more mathematical aspects Hall (1959), Lang (1965), Serre (1977), and Brown (1982). Group A group G is a set with a composition law: 䉫3M(G;G, G), which is associative: ∀g, h, k3G, (g䉫h)䉫k"g䉫(h䉫k) ,
(A.1)
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
73
which has a neutral element e: ∀g3G, e䉫g"g"g䉫e
(A.2)
and every element has an inverse one: ∀g3G, g , g 䉫g"e"g䉫g .
(A.3)
Remark that the neutral element is unique. There are two usual notations for the group law and the neutral element: #, 0. Examples: the additive group of integers 9, of real or complex numbers, 1 or ", the additive group M of m;n matrices with real (respectively, complex) elements. This notation is KL generally restricted to Abelian groups, i.e. the groups with a commutative law: a#b"b#a. The inverse element of a is denoted by !a and is called the opposite. ;, (1 or I ). Examples: the multiplicative groups 1",""; the n-dimensional linear groups G¸(n, R), G¸(n, C), i.e. the multiplicative groups of n;n matrices on 1 or " with non-vanishing determinant. The inverse element of g is denoted by g\. Often we will omit the sign ;. In all examples we have just given, the groups have an in"nite number of elements. An example of a xnite group is S the group of permutations of n objects. We shall denote by "G" the number of L elements of G. Subgroup When a subset HLG of elements of G form a group (with the composition law of G restricted to H), we say that H is a subgroup of G and we shall denote it by H4G (this is not a general convention) or by H(G when we want to emphasize that H is a strict G-subgroup, i.e. H is subgroup of G and HOG. Note that A4B, B4GNA4G. Examples: The subset ;(n) of matrices of G¸(n, C) which satisfy mH"m\ is a subgroup of G¸(n, C); it is called the n-dimensional unitary group. In particular ;(1)("". Note also that G¸(n, R)(G¸(n, C). Since the determinant of the product of two matrices is the product of their determinants, in a group of matrices the matrices of determinant one form the `unimodular subgroupa. Another general example of a subgroup is the one generated by one element. Let g3G and consider its successive powers: g, g, g,2; The order of g is the smallest integer such that gL"I. If no such n exists, we say that g is of in"nite order. When g is of "nite order n the subgroup generated by g is formed of the distinct powers of g; it is called a cyclic group of order n and it is usually denoted by Z . For example ep L3"" generates the cyclic group L Z "+ep I L, 04k4n!1,(;(1)("" . (A.4) L Remark that the intersection of subgroups of G is a G-subgroup. Generally, the union of subgroups is not a subgroup.
For n"1, 1""G¸(1, R), """G¸(1, C). We denote by m? the transpose of the matrix m, i.e. (m?) "m and by mH the Hermitean conjugate of m, i.e. GH HG (mH) "m , the complex conjugate of m . GH HG HG We shall sometimes use the notation C , but Z is very traditional; it comes from the German `Zyklischa. Similarly L L S is traditional and comes from `substitutiona. L
74
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
Lattice The set of subgroups of a group is an example of a lattice. A lattice is a partially ordered set such that for any two given elements, x, y, there exists a unique minimal element among all elements z such that z'x and z'y, and similarly for any two given elements, x, y, there exists a unique maximal element among all elements z such that z (x and z (y. In particular, a lattice has unique minimal and maximal elements. G is the maximal element of the lattice of its subgroups, the trivial subgroup +1, is the minimal one. Cosets Let H(G. The relation among the elements of G:x3yH is an equivalence relation; it is re#exive: x3xH, symmetric: x3yH0y3xH, transitive: x3yH, y3zHNx3zH. The equivalence classes are called cosets. We denote by G:H the quotient set, i.e. the set of cosets. Invariant subgroup We could have pointed out that we were using left cosets; similarly we can introduce right cosets Hx. In general xHOHx. When left and right cosets are identical, H is called an invariant subgroup. We will also write this property H¢G: H¢G" H4G, ∀g3G, gH"Hg .
(A.5)
Evidently, every subgroup of an Abelian group is invariant. Every non-trivial group has two invariant subgroups: +1, and G itself. The set of invariant G-subgroups forms a lattice (sublattice of the subgroup lattice). Beware that K¢H, H¢G does not imply K¢G. Quotient group When K¢G0 there is a natural group structure on G:K; indeed gK䉫hK"(gK)(hK)" gKhK"(gh)K. We call this group the quotient group of G by K and we denote it by G/K. Since the determinant of a matrix is invariant by conjugacy by an invertible matrix, the unimodular subgroups are invariant. The corresponding quotient groups are G¸(n, C)/S¸(n, C)""", G¸(n, R)/S¸(n, R)"1" , ;(n)/S;(n)";(1), O(n)/SO(n)"Z . Remark that an index 2 subgroup is always an invariant subgroup. Conjugacy classes Two elements x, y3G are conjugate if there exists a g3G such that y"gxg\. Conjugacy is an equivalence relation among the elements of a group. A group is therefore a disjoint union of its conjugacy classes. Remark that gh is conjugate to hg. The elements of a conjugacy class have the same order.
An often used synonym is normal subgroup. Sometimes this is called the `factora group; this is illogical, and even confusing: generally the quotient group cannot be identi"ed with a G-subgroup.
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
75
Center An element which commutes with every element of a group G forms a conjugacy class by itself (this is always the case of the identity); these elements form a subgroup called the center of G and often denoted by C(G). The center is an invariant subgroup of G. If the group A is Abelian: C(A)"A. Examples: C(G¸(n, C))"""I, C(G¸(n, R))"1"I, C(;(n))";(1)I, C(O(n))"Z I. The matrices of these centers are multiples of the identity matrix I. Conjugated subgroups Two G-subgroups H, H are said to be conjugate if there exists g3G such that H "gHg\. This is an equivalence relation among subgroups of a group. We denote by [H] the conjugacy class of % H. When a subgroup is alone in its conjugacy class, this is an invariant subgroup. We have seen that the subgroups of a group form a lattice. Beware that this is not true for the set of subgroup conjugacy classes. Centralizer The centralizer of XLG is the set of elements of G which commute with every element of X; this set is a G-subgroup, we denote it by C (X); e.g. the center of G: C(G)"C (G). % % Normalizer The normalizer of XLG is a G-subgroup N (X)"+g3G, gXg\"X, . (A.6) % Note that C (X)¢N (X). % % From the de"nition of the normalizer, when H4G, N (H) is the largest G-subgroup such that % H¢N (H). For instance: N (H)"G0H¢G. % % Homomorphism M H A group homomorphism or, shorter, a group morphism between the groups G, H is a map GP compatible with both group laws M H, o(xy)"o(x)o(y), o(1)"13H . GP
(A.7)
This implies o(x\)"o(x)\ .
(A.8)
A morphism of a group G into the groups G¸(n, C), G¸(n, R), ;(n), O(n) respectively is called a n-dimensional (complex, real) linear, unitary, orthogonal representation of G. Kernel and image M H is denoted by Im o. It is a subgroup of H, Im o4H which The image of the morphism GP includes images of all elements of G. It has even a stronger property: it is a characteristic subgroup. Indeed it is generated by all G-subgroups which contain H as an invariant subgroup.
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
76
M H is the set K3G which is mapped on 13H. Ker o is an The kernel Ker o of the morphism GP invariant subgroup of G. We have the important relation Im o"G/ Ker o. Sequence of homomorphisms Let us consider the following sequence of homomorphisms of Abelian groups: M M ML\ ML G P G P 2G PG PG 2. L\ L L>
(A.9)
Such a construction, named complex, is quite useful to relate topological and group-theoretical properties. If for all n we have Im o "Ker o , the sequence in Eq. (A.9) is an exact sequence of L\ L homomorphisms. M G/H we can write Examples: If H¢G and GP G GP M G/HP1 , 1PHP
(A.10)
G G is the injection map, i.e. ∀x3H, i(x)"x3G. An exact sequence of this type is named where HP G G means that Ker i"1, i.e. i is injective. 1PH is the injection of short exact. The diagram 1PHP M G/HP1. For any homomorphism the unit into H. The fact that o is surjective is expressed by GP o there is always a short exact sequence G GP M Im oP1 . 1PKer oP
(A.11)
Homology groups Let us now consider a sequence of homomorphisms (A.9) with less restrictive condition on consecutive homomorphisms Im (o o )"0 . L> L That is
(A.12)
Im o LKer o . (A.13) L L> Let Z "Ker o and B "Im o . Elements of G are called n-(co)chains, those of Z n-(co)cycles, L L L L\ L L and those of B n-(co)boundaries. The groups L H "Z /B . (A.14) L L L are called the (co)-homology groups of the complex (A.9). We do not discuss here the di!erence between homology and cohomology, chains and cochains, etc. and just suggest the reader to consult mathematical literature. If we had H "0 for all n, the sequence (A.9) would have been exact. Thus the (co)homology L group measures the `lack of exactnessa of the sequence. Euler-Poincare& characteristics Let (H ) be rank of the ith homology group. The Euler}PoincareH characteristics of the complex G (A.9) is s" (!1)G (H ) . G
(A.15)
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
77
Isomorphism A bijective morphism o is called an isomorphism. In other words if Ker o"1 , and Im o"H % than o is an isomorphism and G&H. When we want to classify groups, this will be done up to isomorphism, except if we precise explicitly a more re"ned classi"cation. Often when we write about `abstract groupsa we mean an isomorphism class of groups. Example. For every prime number p there is (up to isomorphism) only one group of order p: this is Z . N Automorphism An isomorphism from G to G is called an automorphism of G. The composition of two automorphisms is an automorphism. Moreover ∀g3G, I(g)"g is the identity automorphism and every automorphism has an inverse; so the automorphisms of G form a group, Aut G. The conjugation by a "xed element g3G induces a G-automorphism: (gxg\) (gy\g\)"g(xy\)g\ which is an `innera automorphism. The set of inner automorphisms forms a subgroup of Aut G that we denote by In Aut G. Note that the elements c3C(G) of the center of G induce the trivial automorphism I , so we have the exact sequence: % F In Aut GP1 . 1PC(G)PGP
(A.16)
An automorphism which is not inner, is called outer automorphism. We remark that In Aut G is an invariant subgroup of Aut G. Direct products Given two groups G , G , one can form a new group, G ;G , the direct product of G and G : the set of elements of G ;G is the product of the set of elements of G and of G , i.e. the set of ordered pairs: +(g , g ), g 3G , g 3G ,, the group law is: (g , g ) (h , h )"(g h , g h ) . (A.17) When G OG , G ;G OG ;G , but they are isomorphic i.e. G ;G &G ;G . Semi-direct products F K one de"nes the semi-direct product as the group whose elements are Given a morphism QP the pairs (k, q), k3K, q3Q, and the group law is (using q ) k as a short for (h(q)) (k)): (k , q ) (k , q )"(k q ) k , q q ) . (A.18) Here we denote this semi-direct product by K ) Q. The semi-direct product of 1L and G¸(n, R) is called the a$ne group: A+ (R)"1L ) G¸(n, R) . L Similarly we can de"ne the complex a$ne group: A+ (C)""L ) G¸(n, C). L
In that case we may often use"for isomorphic groups.
(A.19)
78
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
The Euclidean group E is the semi-direct product (we write simply A+ for A+ (R)): L L L E "1L ) O(n)4A+ . (A.20) L L Group extensions Given two groups K, Q, a very natural problem is to "nd all groups E such that K¢E and Q"E/K. E is called an extension of Q by K. The extension can be represented by a diagram 1PKPEPE/KP1 ,
(A.21)
which is not an exact sequence. The main problem is to classify di!erent extensions up to equivalence. When E is determined, the action of the group Q"E/K in K is de"ned. The inner automorphisms of E induce automorphisms of K in a natural way: ∀x3E, a3K, aPxax\"f (x)a, D Aut K. EP So the problem of "nding all group extensions of Q by K can be decomposed in two steps: (i) (ii)
Find all homomorphisms g of Q into Out K, they form the set Hom(Q, Out K). Given Q, K, g3Hom(Q, Out K) "nd all non-equivalent extensions.
The solution of this mathematical problem is known and given in terms of cohomology groups. The semi-direct product (and it particular case, the direct product) are particular examples of an extensions. But in general case of an extension E of Q by K there is no subgroup of E isomorphic to the quotient Q. Such an example is given by S;(2) as an extension of SO(3) by Z : 1PZ PS;(2)PSO(3)P1 . (A.22) S;(2) is the group of two-by-two unitary matrices of determinant 1. Its center Z has two elements, the matrices 1 and !1. These matrices are the only square-root of the unit. The three-dimensional rotation group SO(3) is isomorphic to S;(2)/Z . This group has an in"nity of square roots of 1: the rotations by p around arbitrary axis. So SO(3) is not a subgroup of S;(2). Appendix B. Morse}Bott theory First of all we extend the concept of non-degeneracy of a function f de"ned over the manifold M. If N is a connected sub-manifold of M, it will be called a non-degenerate stationary manifold for f if and only if at every point of N, df"0 and the Hessian is non-degenerate on the normal bundle. It is natural to extend the notion of the Morse index to non-degenerate stationary manifolds by counting the number of negative eigenvalues of the Hessian on the normal bundle. A function f will be called non-degenerate Morse}Bott-type function if its set C( f ) of extrema is a union of non-degenerate stationary manifolds and non-degenerate isolated extrema. These isolated extrema will also be called stationary points of the function, as it is often done in mechanics.
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
79
Let now describe shortly what should be done with Morse theory to take into account a priori symmetry of a function f under the action of a (continuous) compact Lie group G on a manifold M. If the action is free (i.e. all stabilizers G "1 so there is only one stratum, and all its orbits are K principal) the solution is very simple. In this case the orbit space M"G is itself a manifold and we can apply the usual Morse theory to the corresponding function f/G de"ned over M"G. Unfortunately this is not the case for the most part of physically interesting examples. To extend Morse inequalities to the case of functions possessing a non-degenerate stationary manifold there are two possibilities. There exist more sophisticated Morse inequalities written in terms of usual homology groups for the whole manifold M and for stationary (non-degenerate) sub-manifolds. We have also to consider PoincareH polynomial P ? (t) for each connected com! ponent C of the set of stationary points. These last polynomials are needed to write down ? the polynomial counting stationary points and stationary manifolds of a Morse}Bott function, M (t) D (B.1) M (t)" tI!? P ? (t) , ! D Here k(C ) is the Morse index of the non-degenerate stationary submanifold C . If C is an isolated ? ? ? stationary point the corresponding PoincareH polynomial reduces to 1. If for example, C is ? a one-dimensional sphere S , such a stationary submanifold is counted by a (1#t) poly nomial. The Morse inequalities now can be formulated in the following statement. All coe$cients of the polynomial Q (t) de"ned by Eq. (B.2) D (B.2) tI!? P ? (t)!P (t)"(1#t)Q (t) , ! + D are non-negative. Another way of extending Morse theory to G-invariant functions is based on the possibility to replace everywhere the usual homology by their equivariant analogs. The following theorem is valid. The equivariant Morse inequalities. For any non-degenerate G invariant f on the compact manifold M, the Morse inequalities hold in the equivariant sense, i.e. Eq. (B.2) can be rewritten as M%(t)!P%(M)"(1#t)Q%(t) (B.3) D R D with all coe$cients of the polynomial Q%(t) being non-negative. In spite of the fact that this D extension seems to be rather simple, the non-triviality lies in the fact that equivariant homology should be calculated. We will not go here into equivariant theory (see Section B.1 for further examples) but just note that naturally the de"nition of the perfect function may be given in the equivariant sense. Namely, the function is perfect in the equivariant sense if Q%(t) is zero. As it was D remarked by Bott (1982), the function may be perfect in the equivariant sense but not in the normal sense. That is why we prefer to use the notion of the simplest Morse}Bott function in the case of continuous symmetry as well and to use the complexity measure by counting the number of connected components of the set of stationary points of the function. It should be noted that even in the presence of continuous symmetry group the symmetry group action may result in a set of critical orbits which are the isolated points and the simplest Morse}Bott function may possess in such a case only isolated stationary points.
80
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
B.1. Compilation of Bott and Kirwan about the construction of equivariant homology We follow in this section the equivariant method developed mainly by Bott (1982), Atyah and Bott (1982) and Kirwan (1984, 1988). The starting point for the construction of the equivariant theory is a simple remark that if the group action would be free all problems disappear. Now the idea is to extend the group action to larger space where its action is free and make necessary calculations on this larger space. First observation. If ; is any space on which G acts freely, then the diagonal action of G on =;; is free. So, to transform our problem into another one with the free group action we should "nd a space ; in which G acts freely and whose homotopy is trivial (i.e. ; is contractible). It is known that such spaces exist, be essentially unique, and play an absolutely essential role in all modern topology. Bott gives examples. For G";(1)"S , ; is a unit sphere in a complex in"nite-dimensional Hilbert space. For G";(2) the corresponding ; space is the space of two-frames in a complex in"nite-dimensional Hilbert space. As soon as ; is known for a given G, the homotopy quotient = of any action is de"ned by % = "(;;=)/G , (B.4) % assuming that G acts diagonally on the product. The construction of = enables one to introduce the equivariant version, F%, of any functor F by % imposing F%(=)"F(= ) . (B.5) % This means that the equivariant functor on the initial space = is de"ned through the ordinary functor on the quotient space = . % The space ;/G plays the role of the homotopy quotient of the trivial action of G on a point. This space is usually denoted as BG and is referred to as the classifying space of G. It is a topological space which somehow re#ects both the algebraic and the topological properties of G. In the equivariant Morse series a non-degenerate critical point contributes the expression P% (t)"P (t) . (B.6) % Here P (t) is the ordinary PoincareH polynomial for BG, the classifying space of G. % If N is a non-degenerate critical manifold consisting of a single orbit N"G/H then this critical submanifold contributes the expression P% (t)"P (t) . (B.7) % & & In particular, if the group G acts freely on the critical orbit N, this orbit counts in terms of its index and the ordinary Poincare` polynomial of N/G which in this case is a point. The classifying spaces and their ordinary homology play an essential role in the equivariant theory. Concrete example. For G";(1), the classifying space ;/G"BG is a CP space. Its Poincare` polynomial has the form 1 . P (t)"1#t#t#t#2" !. (1!t)
(B.8)
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
81
Trivial example. Let us consider the action of the circle S on the two sphere S given by rotation about the z-axis in R, S : x#y#z"1 . (B.9) Let us write "rst Morse counting function and Morse inequalities using ordinary homology for three simple functions z, z, !z. M (t)"1#t , (B.10) DX (B.11) M (t)"(1#t)#2t , DX M (t)"t(1#t)#2 , (B.12) D\X As soon as the Poincare` polynomial for the S is (1#t), it is clear that all three functions satisfy Morse inequalities M (t)!(1#t)"0 , (B.13) DX (B.14) M (t)!(1#t)"(1#t)t , DX M (t)!(1#t)"(1#t) . (B.15) D\X The function z is perfect in ordinary sense whereas the two others are obviously not perfect. Equivariant calculations for z and z are given by Bott (1982). The equivariant polynomial for the S is 1#t P%(S )" , G"S . (B.16) R 1!t The Morse counting polynomial for the function z t 1 # , G"S , M% (t)" DX 1!t 1!t
(B.17)
includes two contributions from two isolated stationary points. The Morse counting polynomial for the function z 2t , G"S , M% (t)"1# DX 1!t
(B.18)
includes one contribution from the stationary one-dimensional manifold S corresponding to a circular minimum at z"0 and two contributions from two isolated stationary points which are both maxima. It is easy to see that both functions z, z are perfect in the equivariant sense. The equivariant Morse counting polynomial for the (!z) function includes contributions from two isolated stationary points (two minima) which are 2 . 1!t
(B.19)
The contribution from the circular maximum at z"0 should be t and 2 , G"S , M% (t)"t# D\X 1!t
(B.20)
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
82
but in such a case the di!erence between the Morse counting equivariant polynomial and the PoincareH equivariant polynomial is 2 1#t (t)!P%(t)"t# ! "t#1 . M% D\X 1!t 1!t
(B.21)
This result shows that the (!z) function satis"es equivariant Morse inequalities, but it is not perfect.
References Arnol'd, V., 1988. A branched covering of CPP S, hyperbolicity and projectivity topology. Sibirsk Mat. Zh. 29, 36}47. Atiyah, M., Bott, R., 1982. The Yang}Mills equations over Riemann surfaces. Philos. Trans. Roy. Soc. (London) A 308, 523}615. Bickerstu!, P., Wyborne, B., 1976. Integrity basis, invariant operators and state labelling problem for "nite subgroups of SO(3). J. Phys. A 9, 1051}1068. Biedenharn, L., Louck, J., 1981a. Angular Momentum in Quantum Physics. Addison-Wesley, New York. Biedenharn, L., Louck, J., 1981b. The Racah-Wigner Algebra in Quantum Theory. Addison-Wesley, New York. Bott, R., 1982. Lecture on Morse theory, old and new. Bull. Am. Math. Soc. 7, 331}358. Bredon, G., 1972. Introduction to Compact Transformation Groups. Academic press, New York. Brown, K.S., 1982. Cohomology of Groups. Springer, New York. Burnside, W., 1911. Theory of Groups of Finite Order. Cambridge Univ. Press, New York. Chevalley, C., 1955. Invariants of "nite groups generated by re#ections. Amer. J. Math. 77, 778}782. Collins, M., Parsons, D., 1993. Implications of rotation-inversion-permutation invariance for analytic molecular potential energy surfaces. J. Chem. Phys. 99, 6756}6772. Cox, D., Little, J., O'Shea, D., 1992. Ideals, Varieties, and Algorithms. Springer, Berlin. Cox, D., Little, J., O'Shea, D., 1998. Using Algebraic Geometry. Springer, Berlin. Coxeter, H., Moser, W., 1972. Generators and Relations for Discrete Groups. Springer, Berlin. Curie, P., 1894. Sur la symeH trie dans les pheH nome`nes physiques, symeH trie d'un champ eH lectrique et d'un champ magneH tique. J. Phys. (Paris) 3, 393}416. Desmier, P.E., Sharp, R., 1979. Polynomial tensors for double point groups. J. Math. Phys. 20, 74}82. Dubrovin, B., Novikov, S., Fomenko, A., 1990. Modern Geometry: Methods and Applications. Part III. Introduction to Homology Theory. Springer, New York. Fomenko, A., 1983. Di!erential Geometry and Topology. Additional Chapters. Moscow State University Press, Moscow. Fomenko, A.T., Fuks, D.B., 1989. Homotopic Topology. Nauka, Moscow. Frankenheim, M.L., 1826. Crystallonomische AufsaK tze. ISI enzyklopadische Zeitung von Oken 5, 497}515. Galois, E., 1846. Oeuvres de Galois. J. Liouville, Ser. 1 11. Gaskel, R., Peccia, H., Sharp, R., 1979. Generating functions for polynomial irreducible tensors. J. Math. Phys. 19, 727}733. Gilmore, R., Draayer, J., 1985. Dynamical group chain and integrity basis. J. Math. Phys. 26, 3053}3067. Gri$ths, P., Harris, J., 1978. Principles of Algebraic Geometry. Wiley-Interscience, New York. Hall, M., 1959. The Theory of Groups. Macmillan, New York. Hamermesh, M., 1964. Group Theory and its Application to Physical Problems. Addison-Wesley, Reading, MA. Hessel, J., 1830. Krystallometrie oder Krystallonomie und Krystallographie. Gehler's Phys. Worterbuch 5, 1023}1360. Hilbert, D., 1890. Ueber der Theorie der algebraischen Formen. Math. Ann. 36, 473}534. Hilbert, D., 1893. Ueber der vollen Invariantensystem. Math. Ann. 42, 313}373. ITC, 1996. In: Hahn, T. International Tables for Crystallography. Vol. A. Space Group Symmetry, 4th Revised Edition. Kluwer, Dordrecht. Izyumov, Y., Syromyatnikov, V., 1984. Phase Transitions and Symmetry of Crystals. Nauka, Moscow.
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
83
Jaric, M., Birman, J., 1977. Calculation of the Molien generating functions for invariants of space groups. J. Math. Phys. 18, 1459}1465. Jaric, M., Michel, L., Sharp, R., 1984. Zeros of covariant vector "elds for the point groups: invariant formulation. J. Phys. (Paris) 45, 1}27. Judd, S., Miller, W., Patera, J., Winternitz, P., 1974. Complete sets of commuting operators and O(3) scalars in the envelopping algebra of S;(3). J. Math. Phys. 15, 1787}1799. Kim, J.S., 1984. Orbit spaces of low dimensional representations of simple compact connected Lie groups and extrema of a group-invariant scalar potential. J. Math. Phys. 25, 1694}1717. Kirwan, F., 1984. Cohomology of Quotients in Simplectic and Algebraic Geometry, Princeton University Press, Princeton, New Jersey. Kirwan, F., 1988. The topology of reduced phase space of the motion of vortices on a sphere. Physica D 30, 99}123. Kuiper, N.H., 1974. The quotient space of CP(2) by complex conjugation is the 4-sphere. Math. Ann. 208, 175}177. Landau, L.D., 1937a. On the theory of phase transitions. I. Zh. Eksp. Teor. Fiz. 7, 19. Landau, L.D., 1937b. On the theory of phase transitions. II. Zh. Eksp. Teor. Fiz. 7, 627. Landau, L., Lifshitz, E., 1958. Statistical Physics. Pergamon Press, Oxford. Landau, L., Lifshitz, E., 1965. Quantum Mechanics, Nonrelativistic Theory. Pergamon Press, Oxford. Lang, S., 1965. Algebra. Addison-Wesley, Reading, MA. Littlejohn, R., Reinsch, M., 1997. Gauge "elds in the separation of rotations and internal motions in the n-body problem. Rev. Mod. Phys. 69, 213}276. Lusternik, L.A., Schnirelmann, L.G., 1930. Topological Methods in Variational Problems. Moscow State University Press, Moscow. Lusternik, L., Schnirelmann, L., 1934. Methodes topologiques dans problemes variationnels. ActualiteH s scient. et indust., Paris. Lyubarskii, G.Y., 1957. Theory of Groups and their Applications in Physics. Fizmatgiz, Moscow. Ma, S., 1976. Modern Theory of Critical Phenomena. W.A. Benjamin, London. Massey, W.S., 1973. The quotient space of the complex projective plane under conjugation is a 4-sphere. Geom. Dedicata 2, 371}374. Michel, L., 1953. Selection rules imposed by charge conjugation. Nuovo Cimento. 10, 319}339. Michel, L., 1970. Application of Group Theory to Quantum Physics, Lecture Notes in Physics, Vol. 6, Springer, Berlin, pp. 36}144. Michel, L., 1971. Points critiques des fonctions invariantes sur une G-varieteH . C. R. Acad. Sci. Paris 272, 433}436. Michel, L., 1977. Invariants polynomiaux des groups de symeH trie moleH culaire et crystallographique. In: Sharp, R.T., Kolman, B. (Eds.), Proceedings of 5th International Colloquism Group Theoretical Methods in Physics. Academic Press, New York, pp. 75}91. Michel, L., 1979. Minima of Higgs-Landau potentials. In: Bacry, H. (Ed.), Regards sur la Physique contemporaine. CNRS, Paris, pp. 157}203. Michel, L., 1980. Symmetry defects and broken symmetry. Rev. Mod. Phys. 52, 617}650. Michel, L., 1996. ExtreH ma des fonctions sur la zone de Brillouin, invariantes par le groupe de symeH trie du crystal et le renversement du temps. C. R. Acad. Sci. Paris B 322, 223}230. Michel, L., Mozrzymas, J., 1978. Application of Morse Theory to Symmetry Breaking in Landau theory of Second Order Phase Transitions, Lecture Notes in Physics, Vol. 79. Springer, Berlin, pp. 247}258. Milnor, J., 1963. Morse Theory. Princeton University Press, New Jersey. Molien, T., 1897. Uber die Invarianten der linearen Substitutionsgruppen. Sitzungber. Konig. Preuss. Akad. Wiss. 52, 1152}1156. Montgomery, D., Yang, C., 1957. The existence of a slice. Ann. Math. 65, 108}116. Morse, M., 1925. Relations between the critical points of a real function of n independent variables. Trans. Am. Math. Soc. 27, 345}396. Mostow, G., 1957a. Equivariant embedding in Euclidean space. Ann. Math. 65, 432}446. Mostow, G., 1957b. On a conjecture of Montgomery. Ann. Math. 65, 513}516. Mumford, D., 1976. Algebraic Geometry. I. Complex Projective Varieties. Springer, New York.
84
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 11}84
Palais, R., 1960. The classi"cation of G-spaces. Mem. Am. Math. Soc. 36, 1}72. Palais, R., 1961. On the existence of a slice for actions of non compact Lie groups. Ann. Math. 73, 295}323. Palais, R., 1970. C-action of compact Lie groups on compact manifolds are equivalent to C-actions. Am. J. Math. 92, 748}760. Palais, R., Terng, C., 1980. Critical Point Theory and Submanifold Geometry. Lecture Notes in Mathematics, Vol. 1353, Springer, Berlin. Patera, J., Sharp, R., Winternitz, P., 1978. Polynomial irreducible tensors for point groups. J. Math. Phys. 19, 2362}2376. Poe`naru, V., 1976. SingulariteH s C en PreH sence de SymeH trie. Springer, Berlin. Sadovskii, D., Zhilinskii, B., 1999. Monodromy, diabolic points and angular momentum coupling. Phys. Lett. A 256, 235}244. Sartori, G., 1991. Geometric invariant theory: a model independent approach to spontaneous symmetry and/or supersymmetry breaking. Riv. Nouvo Cimento. 14, 1}120. Schmelzer, A., Muller, J., 1985. The general analytic expression for S symmetry invariant potential function of tetratomic homonuclear molecules. Int. J. Quant. Chem. 28, 287}295. Schwarz, G., 1975. Smooth functions invariant under the action of a compact Lie group. Topology 14, 63}68. Seifert, H., Threlfall, W., 1938. Variationsrechnung im grossen. Leipzig. Serre, J.P., 1977. Linear representations of Finite Groups. Springer (Trans. from French, Hermann, Paris, 1967), New York. Shephard, G., 1956. Some problems of "nite re#ection groups. Enseignement Math. 2, 42}48. Sloane, N., 1977. Error correcting codes and invariant theory: new applications of a nineteenth-century technique. Am. Math. Mon. 84, 82}107. Solomon, L., 1963. Invariants of "nite re#ection groups. Nagoya J. Math. 22, 57}64. Stanley, H.E., 1971. Introduction to Phase Transition and Critical Phenomena. Clarendon, Oxford. Stanley, R.P., 1977. Relative invariants of "nite groups generated by pseudo-re#ections. J. Algebra 49, 134}148. Stanley, R., 1979. Invariants of "nite groups and their applications to combinatorics. Bull. Am. Math. Soc. 1, 475}511. Stanley, R., 1986. Enumerative Combinatorics, Vol. 1. Wadsworth Brooks, Montrey, CA. Stanley, R., 1996. Combinatorics and Commutative Algebra. Birkhauser, Boston. Sturmfels, B., 1993. Algorithms in Invariant Theory. Springer, New York. Thom, R., 1954. Quelques proprieH teH s globales des varieteH s di!erentiables. Commun. Math. Helv. 28, 17}86. Thom, R., 1962. La stabiliteH topologique des application polynomiales. Enseignement Math. 7, 24}33. Thom, R., 1969. Ensembles et morphysmes strati"eH s. Bull. Am. Math. Soc. 75, 240}284. Weyl, H., 1931. The Theory of Groups and Quantum Mechanics, (1st Edition 1928). Dover Publications, Dover. Weyl, H., 1939. The Classical Groups. Their Invariants and Representations. Princeton Univ., New Jersey. Wigner, E., 1959. Group Theory and its Application to the Quantum Mechanics of Atomic Spectra, (1st Edition 1931). Academic, New York. Worfolk, P., 1994. Zeros of equivariant vector "elds: Algorithms for an invariant approach. J. Symb. Comput. 17, 487}511. Zhilinskii, B., 1989a. Qualitative analysis of vibrational polyads: N mode case. Chem. Phys. 137, 1}13. Zhilinskii, B., 1989b. Theory of Complex Molecular Spectra. Moscow University Press, Moscow.
Physics Reports 341 (2001) 85}171
Symmetry, invariants, topology. II
Symmetry, invariants, and topology in molecular models B.I. ZhilinskimH Universite& du Littoral, BP 5526, 59379 Dunkerque Ce& dex, France Contents 1. Introduction 2. Qualitative analysis of molecular models. General principles 3. Rotational problem 3.1. E!ective quantum and classical rotational Hamiltonians and their symmetry 3.2. Strati"cation of the rotational phase sphere. Critical orbits and the simplest Morse-type e!ective Hamiltonians 3.3. Cluster structure of rotational energy levels 3.4. Quantum bifurcations of the rotational structure 3.5. Organization of quantum rotational bifurcations. Crossover 3.6. Symmetry breaking due to isotopic substitution and rotational cluster structure 3.7. Imperfect quantum bifurcations 4. Rotational structure for a N-quantum state system 4.1. E!ective quantum rotational Hamiltonian for an N-state problem and its classical matrix symbol 4.2. Isolated vibrational components and their rotational structure 4.3. Dynamical meaning of diabolic points and rearrangement of rotational multiplets 5. Vibrational problem 5.1. Vibrational polyads, resonances, and polyad quantum numbers 5.2. Generating functions for numbers of states in polyads 5.3. Density of states. Regular and oscillatory parts
87 87 89 89
91 98 99 102
103 108 111
111 115 117 123 126 129 131
5.4. Two polyad quantum numbers. Example of C H 5.5. Internal structure of polyads formed by two-quasi-degenerate modes 5.6. Vibrational quantum bifurcations and normal local mode transition 5.7. Internal structure of polyads formed by Nquasi-degenerate modes. Complex projective space as classical reduced phase space 5.8. Integrity bases for CP spaces , 5.9. Finite symmetry group action on CP 5.10. Continuous symmetry group action on CP 5.11. Nontrivial n : m resonances 5.12. Vibrational polyads for quasi-degenerate electronic states 6. Rovibrational problem 6.1. Model problem: coupling of two angular momenta. Quantum and classical monodromy 6.2. Rotational structure of bending overtones in linear molecule 6.3. Rotational structure of triply degenerate vibrations. Complete classical analysis 7. Microscopic models of qualitative phenomena 7.1. Microscopic theory of four-fold cluster formation in non-linear AB molecules 7.2. Rotational structure and intramolecular potential 8. Conclusions and perspectives Appendix A. Tables of the rotational energy surface types of individual vibrational components of CF References
E-mail address:
[email protected] (B.I. ZhilinskimH ). 0370-1573/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 0 0 ) 0 0 0 8 9 - 2
132 136 139
140 141 145 147 147 147 147
148 154 154 159 159 160 161
161 167
86
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
Abstract Description of intra-molecular dynamical behavior is usually made in terms of e!ective Hamiltonians for di!erent degrees of freedom. In such a way, rotational, vibrational, rovibrational, etc., dynamical systems arise in a natural way in the classical limit as corresponding to e!ective quantum Hamiltonians. The main idea of this paper is to answer the following general question: What kind of features of the quantum energy spectra can be predicted on the basis of qualitative (symmetry#topology) analysis of corresponding classical systems. 2001 Elsevier Science B.V. All rights reserved. PACS: 3.65.Fd; 31.15.Md; 11.30.Qc Keywords: Rotation-vibration of molecules; Rydberg states; Molecular symmetry
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
87
1. Introduction Description of intra-molecular dynamical behavior is usually made in terms of e!ective Hamiltonians for di!erent degrees of freedom. In such a way, rotational, vibrational, rovibrational, etc., dynamical systems arise in a natural way in the classical limit as corresponding to e!ective quantum Hamiltonians. The main idea of this chapter is to answer the following general question: What kind of features of the quantum energy spectra can be predicted on the basis of qualitative (symmetry#topology) analysis of corresponding classical systems. In the next section the general program of such qualitative analysis is brie#y formulated within the formalism introduced in the "rst chapter and then systematically applied to di!erent typical intra-molecular problems.
2. Qualitative analysis of molecular models. General principles The qualitative description of dynamical systems (Gilmore, 1981; Poston and Stewart, 1978; Smale, 1970a; Smale, 1970b; Thom, 1972) is based primarily on concepts of topology and symmetry. The key notion of the qualitative analysis is `structural stabilitya which means that the qualitative description obtained is stable with respect to small variations of the model. It assumes that we study some generic situation but this generic behavior may be restricted to some special class (some a priori symmetry requirements may be imposed by physical requirements). The idea of qualitative (topological) analysis of mechanical systems was initiated by PoincareH (1879) and Lyapunov (1892) in the 19th century. A program of topological analysis of simple mechanical systems with symmetry was formulated by Smale (1970a,b). Its partial concrete realization for the three-body problem culminates the classical book by Abraham and Marsden (1978) (see also (Arnol'd, 1981, 1988; Marsden and Ratiu, 1994)). During the several last decades the formal abstract theory was enormously developed. In particular, many mathematical results concerning in#uence of topology and symmetry on general dynamical systems and on Hamiltonian systems were formulated. Apart from some purely abstract models the most important applications of these developed mathematical techniques concern "eld theory, particle physics, nuclear physics and even other branches of science which are apparently rather far from physics (biology, 2). At the same time applications to quantum molecular physics problems are still relatively rare except, perhaps, for tentative studies of quantum chaos. The initial important step in the qualitative study of quantum problems is the transformation to and the analysis of corresponding classical objects. The transformation of a quantum problem into its classical limit is especially physically meaningful for large quantum numbers which play the role of the inverse of the Planck constant \ when going to the classical limit (Perelomov, 1986; Zhang et al., 1990). Very often in molecular problems this high quantum number limit is really accessible. We can cite the analysis of rotational spectra up to J&100 (Harter et al., 1978; Harter, 1988) and observation of Rydberg states of the H molecule with n&100 (Bordas and Helm, 1992). The reason why qualitative study of highly excited molecular systems through an analysis of classical limit is not so popular may probably be explained as follows. Molecular systems are described by a relatively simply formulated Hamiltonians with known potentials and the description of intra-molecular dynamics is considered often as a numerical problem whose solution depends mainly on the development of the computer facilities. From the other side, the
88
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
experimental study mainly due to the development of high and super-high-resolution laser spectroscopy enables one to collect an enormous amount of information about spectral lines in di!erent regions of spectra with the precision of about 10\}10\ in relative units. Thus, the spectroscopists are mainly concerned with the use of some quantitative phenomenological models which enable one to interpret and to reproduce observed spectra with high accuracy compatible with the experimental precision. The global qualitative features are considered sometimes as surprising facts which may be related to speci"c features of a single concrete model and are not recognized as universal. To support this point of view let us just cite an example: the description of the formation of the rotational cluster structure for isolated molecules. The presence of six- and eight-fold rotational clusters for spherical top molecules with cubic point group symmetry was discussed in 1977 by Fox et al. (1977) [see also the book by Biedenharn and Louck (1981)] on the basis of extensive numerical calculations for one concrete e!ective rotational operator (which is de"ned by symmetry arguments nevertheless). In spite of the fact that such clustering was remarked and even explained by simple classical arguments earlier (Dorney and Watson, 1972; Kirschner and Watson, 1973; Michelot and Moret-Bailly, 1975), it was only after receiving a huge amount of numerical results that the simple classical (or quantum) description of the clustering was fully explored (Harter and Patterson, 1977; Patterson and Harter, 1977; Harter and Patterson, 1984; Harter, 1988, 1993, 1996; Zhilinskii, 1978, 1979). Now, it is widely accepted that the essence of the clustering phenomena is the symmetry breaking e!ect related to the symmetry group action on the phase space of the problem considered. The type of clustering depends on the topology of the phase space and on the symmetry group action on this space (Michel, 1979; Zhilinskii, 1989b; Sadovskii and Zhilinskii, 1993a). Whereas the importance of the group action analysis for di!erent "elds of physics and its relation to di!erent kinds of phase transitions and spontaneous symmetry breaking e!ects was fully demonstrated in a series of papers by Michel (1972, 1979, 1980), the applications to realistic molecular examples was started signi"cantly later. We will follow in this paper the general scheme of the qualitative analysis to study di!erent molecular problems. The steps that we will follow can be summarized as follows: (A) Construction of a classical limit Hamiltonian system for a given model quantum Hamiltonian. At this stage we construct a classical phase space and a classical analog for the Hamiltonian. Two situations are quite di!erent from the point of view of subsequent analysis: the obtained classical phase space is compact or not. For a compact phase space we can make a global analysis and deduce possible numbers of stationary points of Hamiltonian functions from the topological characteristics of this space. For a non-compact phase space we are restricted to a local analysis around stationary points. (B) Analysis of the action of the invariance group of a particular problem on the phase space. (C) Qualitative classi"cation of corresponding classical Hamiltonian functions for a particular problem. This step assumes introduction of some qualitative equivalence between dynamical systems and their Hamiltonians. Several levels of qualitative, or topological, equivalence were introduced and studied for classical systems (Abraham and Marsden, 1978). The "nest classi"cation is based on the phase portrait equivalence. We use much cruder equivalence relations. First of all, for two classical systems to be qualitatively equivalent their phase spaces
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
89
should be at least topologically equivalent. A more detailed classi"cation of phase spaces takes the symmetry of the Hamiltonian into account. This is especially important for molecular systems which have many di!erent invariance groups. Starting from the invariance group of an initial molecular problem we can "nd the classi"cation of the points on the classical phase space by their symmetry, i.e. de"ne the group action on the phase space. At this point we introduce the equivalence relations which preserve the group action on the phase space. A further step involves the comparison of the Hamiltonian functions de"ned over the phase space by their systems of stationary points. Taking into account the action of a symmetry group, two Hamiltonian functions are considered to be equivalent if their numbers of stationary points of any index are the same. Moreover, we compare the numbers of stationary points of the same index and of the same local symmetry. (D) Analysis of bifurcations, or possible qualitative changes, of stationary points which occur due to the variation of some physical characteristics, such as strict or approximate integrals of motion. The latter are considered as parameters of the model Hamiltonian under study. (E) Re-interpretation of the qualitative features of the classical Hamiltonian function for the initial quantum operator. This scheme has been realized on several examples of molecular models. We give below the principal results with several particular applications to rotational, vibrational and rovibrational models. More detailed analysis of the Rydberg state problems is done in a separate Chapter III. 3. Rotational problem We start with qualitative analysis of molecular rotation for a system in a non-degenerate isolated electronic and vibrational state and in the absence of any external "elds. In such a case the classical-quantum correspondence naturally leads to the representation of e!ective rotational Hamiltonians as a classical function de"ned over reduced phase space which is a two-dimensional sphere. To understand the global qualitative features of quantum energy spectrum and of the corresponding classical dynamics the rotational energy surfaces should be analyzed qualitatively taking into account symmetry and topology of the reduced space. We start with formulating conditions on the number and type of extrema of classical rotational energy function which can be derived from the symmetry analysis and Morse theory. Related quantum phenomena of rotational energy-level clustering are further explained and demonstrated on concrete molecular examples. Next step is to describe the qualitative modi"cations of the rotational energy surface (RES) under the variation of control parameters. This leads naturally to the classi"cation of classical and quantum bifurcations in rotational dynamics in the presence of symmetry. Theoretical description of individual elementary bifurcations and of their organization is discussed in parallel with their manifestation in experimental studies. 3.1. Ewective quantum and classical rotational Hamiltonians and their symmetry For many molecular system in the ground state any electronic and vibrational excitations are much more energy consuming as compared with rotational excitations. Thus, to study the
90
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
molecular rotation the simplest physical assumption is to suppose that all electronic and all vibrational degrees of freedom are frozen. This means that a set of (or one) quantum numbers is given which have the sense of approximate integrals of motion, specifying the character of vibrational and electronic (vibronic to be short) motions in terms of these `gooda quantum numbers. At the same time for a free molecule in the absence of any external "elds due to the isotropy of the space the total angular momentum J and its projection on the laboratory "xed frame, J , are strict integrals of motion. Consequently, to describe the rotational motion for X a molecular system in a given electronic and vibrational state it is su$cient to analyze the e!ective problem with only one degree of freedom. The dimension of classical phase space in this case equals two and the two classical conjugate variables have the following physical interpretation in terms of classical action-angle variables: the projection of the total angular momentum on the body "xed frame and conjugate angle variable. The classical phase space is topologically a two-dimensional sphere, S . There is a one-to-one correspondence between the points on a sphere and the orientation of the angular momentum in the body-"xed frame. Such a representation gives a clear visualization of a classical rotational Hamiltonian as a function de"ned over a sphere (Harter, 1993). In quantum mechanics the rotation of molecules is traditionally described in terms of an e!ective rotational Hamiltonian which is constructed as a series in rotational operators J , J and J , the V W X components of the total angular momentum J, and in suitably chosen molecule-"xed frame can be written in the form H "AJ#BJ#CJ# c J? J@JA #2 , (1) V W X ?@A V W X where A, B, C and c are constants. The amplitude of the total angular momentum ?@A J "J#J#J"const."J(J#1) (2) V W X is an integral of motion and the dynamical parameter J in Eq. (2) can be absorbed in the coe$cients of H in Eq. (1) so that, for instance, c "c #c J #c J #2 . (3) ?@A ?@A ?@A ?@A To relate quantum and classical pictures we remark once again that J and energy E are integrals of the Euler's equations of motion for dynamical variables J , J and J . The phase space of the V W X classical rotational problem with constant "J " is S , the two sphere, and it can be parameterized with coordinates (h, u) such that the points on S de"ne the orientation of J, i.e., the position of the axis and the direction of rotation. To get the classical interpretation of the quantum Hamiltonian we introduce the classical analogs of the operators J , J and J , V W X J sin h cos u V (4) JP J " sin h sin u (J(J#1) W J cos h X and consider the rotational energy E as function of (h, u) and parameter J. Thus for an e!ective rotational quantum Hamiltonian the corresponding classical symbol is a function E (h, u) de"ned over S , two-dimensional phase space. In what follows, we will ( name E (h, u) the rotational energy surface (RES) in accordance with well accepted tradition ( (Harter, 1996).
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
91
The classical interpretation of the e!ective quantum rotational Hamiltonians has proven to be quite helpful in understanding the quasi-degeneracies of quantum rotational levels (Dorney and Watson, 1972; Harter et al., 1978; Harter and Patterson, 1984; Harter, 1988; Sadovskii and Zhilinskii, 1988). In fact, the analysis of the RES E (h, u) provides comprehensive information on ( the pattern of quantum energy levels and localization of quantum wave functions. The main qualitative characteristics of RES are the location and type of its stationary points. In the simplest case (Sadovskii and Zhilinskii, 1993a; Zhilinskii, 1996) these characteristics are mere consequences of the S topology of the phase space and the symmetry group of the problem (of the molecule). The qualitative study of molecular rotational dynamics is largely based on Morse theory of generic functions de"ned on S in the presence of certain a priori "nite symmetry group. Furthermore, it is natural to consider the whole parametric family of RESs and to study qualitative changes or bifurcations (Pavlichenkov and Zhilinskii, 1988; Sadovskii and Zhilinskii, 1993a; Zhilinskii, 1996) that occur in the system of stationary points of RES when the parameter J changes. Parameters of e!ective Hamiltonians can be regarded as phenomenological constants that can be obtained from the analysis of experimental data. At the same time, these parameters can, in principle, be derived theoretically by reducing the initial `fulla rotation}vibration Hamiltonian, known in molecular spectroscopy as Wilson}Howard}Watson Hamiltonian (Wilson et al., 1955; Watson, 1968; Louck, 1976). In fact, one of the goals of molecular spectroscopy is believed to be the inverse problem of recovery of molecular characteristics from phenomenological constants. Many theoretical formulae relating e!ective constants to the force "eld parameters, to the moment of inertia corrections, and to Coriolis constants can be found in the literature (Amat et al., 1971; Aliev and Watson, 1985). In our treatment here, we restrict ourselves to the qualitative analysis of phenomenological e!ective rotational operators. The microscopic approach which relates directly qualitative features of rotational spectra to internuclear adiabatic potential (microscopic approach) will be discussed later in Section 7. In our analysis of e!ective rotational Hamiltonian we should well distinguish the point symmetry G of the equilibrium con"guration of nuclei and the symmetry of the e!ective rotational Hamiltonian (or classical RES). E!ective rotational Hamiltonian for an isolated non-degenerate vibronic state is invariant as well with respect to time reversal. Thus, we should extend the point symmetry group G till G;T (where T is the two element group possessing the time-reversal T operation) and to "nd the image of this group in the representation spanned by three dynamical variables J , J , J used to construct the e!ective rotational Hamiltonian. The relation between V W X point symmetry group and the symmetry group of the e!ective rotational Hamiltonian is given in Table 1 for all 3D point symmetry groups. The "rst important conclusion is that only groups possessing the S ,C subgroup can be G realized as symmetry groups of e!ective rotational Hamiltonians for non-degenerate vibronic states. For each of these groups we "nd the system of strata, critical orbits, and the qualitative description of the simplest Morse function. 3.2. Stratixcation of the rotational phase sphere. Critical orbits and the simplest Morse-type ewective Hamiltonians Symmetry group of the e!ective rotational Hamiltonian induces strati"cation of the classical phase space. To visualize the system of strata we use two alternative approaches. First, one shows
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
92
Table 1 Correspondence between point symmetry group of the equilibrium con"guration and the symmetry group of the e!ective rotational Hamiltonian for non-degenerate vibronic states taking into account the time-reversal symmetry. Molien functions for invariants constructed from projections of rotational angular moments (J , J , J ) are given in the last V W X column Point group
Image
C ,S ,C N> N> N> G
S N>
C , C , S , p-even, C , p-odd N N F N NF
C N F
D , C , D , D , p-even, D , p-odd N N T N F N B NF
D N F
D ,C ,D N> N>T N>B
D N>B
¹, ¹
F
¹ F
O, ¹ , O B F
O F
>, > F
> F
C ,C F
C F
D ,C ,D T F
D F
SO(3), O(3)
O(3)
Molien function for invariants 1#2tN>#tN> (1!t)(1!tN>) 1#tN (1!t)(1!tN) 1 (1!t)(1!tN) 1#tN> (1!t)(1!tN>) 1#t (1!t)(1!t)(1!t) 1 (1!t)(1!t)(1!t) 1 (1!t)(1!t)(1!t) 1 (1!t) 1 (1!t) 1 (1!t)
the system of orbits as an elementary cell on the phase sphere (see Fig. 12 in Chapter I of such a representation of the O group action on the sphere). Such a schematic representation enables one F to relate in a simple and direct way the system of orbits and strata with points of the complete phase space. More sophisticated way for the representation of the space of orbits was introduced in Section 5.6 of Chapter I where the same space of orbits for the O action on the S sphere was F explicitly presented in Fig. 14. The geometrical representation of the space of orbits is based on the system of invariant polynomials forming an integrity basis. The choice of an integrity basis is ambiguous. One of the possible choices is given in Table 2 where the choice of axes system for the group image should be properly related with the system of axes of the initial point symmetry group for the equilibrium con"guration. For all symmetry groups one can choose J as one of the basic invariants. Thus, we list in Table 2 only two basic denominator invariants h , h for "nite groups and one h for one-dimensional continuous groups and all nontrivial auxiliary (numerator) invariants u . G To give the complete list of strata we are obliged to treat separately S and S (p51) groups. N> Tables 3 and 4 characterize strata, their stabilizers, list the number of points in orbits, and indicate critical orbits.
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
93
Table 2 Integrity bases for the representation of RES in the presence of symmetry. Re and Im stand for real and imaginary parts. h "(qJ!J)(qJ!J)(qJ!J), h"(J #J #J )(J !J !J )(J !J !J )(J !J !J ) (q\J!qJ) 7 V W W X X V 7 V W X V W X W X V X W V V W ;(q\J!qJ)(q\J!qJ ) W X X V Group image
h
h
u G
S N> C N F D N F D N>B ¹ F O F > F C ,D F F
Re(J )L > Re(J )L > Re(J )L > Re(J )L > J#J#J V W X J#J#J V W X h 7 J X
J X J X J X J X JJJ V W X JJJ V W X h 7 *
J Re(J )L, J Im(J )L, Im(J )L X > X > > Im(J )L > * J Im(J )L X > (J!J)(J!J)(J!J ) V W W X X V * * *
Table 3 Orbits and strata for the action of symmetry groups on the classical rotational phase sphere. R and R stand for oneand two-dimensional strata, respectively Group image
Stabilizer
Number of points per orbit
Number of orbits per stratum
Comments
S S (p51) N>
C C N> C C N C Q C C N T C T C T C Q C Q C Q C C N>T C C Q C
2
Generic
2 4p#2
R&RP 1 R
2 2p 4p
1 R&S R
Critical Close Generic
2 2p 2p 4p 4p 4p 8p
1 1 1 R R R R
Critical Critical Critical Open Open Open Generic
2 4p#2 4p#2 8p#4
1 1 R R
Critical Critical Open Generic
C (p51) N F
D (p51) N F
D (p51) N>B
Critical Generic
We brie#y discuss below the strati"cation of the rotational phase sphere for di!erent symmetry groups: 1. The simplest possible symmetry group of the e!ective rotational Hamiltonian is the order two group S (alternative notations are C or 2 ) which possesses only one non-trivial symmetry G
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
94
Table 4 Orbits and strata for the action of symmetry groups on the classical rotational phase sphere. Continuation Group image
Stabilizer
¹ F
C C T C Q C C T C T C T C Q C Q C C T C T C T C Q C C C Q C C T C T C Q
O F
> F
C F
D F
Number of points per orbit
Number of orbits per stratum
Comments
8 6 12 24
1 1 R R
Critical Critical Open Generic
6 8 12 24 24 48
1 1 1 R R R
Critical Critical Critical Open Open Generic
12 20 30 60 120
1 1 1 R R
Critical Critical Critical Open Generic
2 R&S R&S 6S 2 R&S R&S 6S
1 1 R
Critical Critical Generic
1 1 R
Critical Critical Generic
transformation of J variables, inversion, i. The action of the inversion i(J )P!J on the space ? ? ? spanned by J is due to time-reversal symmetry which changes the sign of components of angular ? momentum vector. The action of S group on the phase sphere is free. There is only one generic stratum formed by two-point orbits with trivial stabilizer. The space of orbits is the RP manifold (real projective space). There is no critical orbits. Minimal number of stationary points for a generic (Morse-type) function de"ned over the sphere in the presence of S symmetry equals six: one orbit (two points) of minima, one orbit (two points) of maxima, and one orbit (two points) of saddles. As soon as all symmetry groups for e!ective rotational Hamiltonians have S as a subgroup an immediate consequence is: any generic RES has at least two equivalent minima, two equivalent maxima, and two equivalent saddle points. The representation of the space of orbits in terms of invariant polynomials for C "S group was discussed in Chapter I (see Section 5.6.2 of Chapter I). G 2. E!ective Hamiltonian with C (n"1, 2, 3,2) symmetry group. There is one critical twoL F point orbit and one closed stratum (with S topology) formed by 2n-point orbits. The simplest Morse-type function necessarily possesses two stationary points on zero-dimensional stratum and two 2n-point orbits formed by stationary points on C closed stratum. There are at least (2#4n) Q stationary points and there is symmetry restriction on the location of these points. For the
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
95
Fig. 1. Orbifold of the C symmetry group action on S given in the space of two invariant polynomials h "J!J F V W and h "J. The sign of the auxiliary invariant u"J J distinguishes two parts which should be glued together through X V W identi"cation of boundary points (dotted boundary) with the same (h , h ) coordinates. Fig. 2. Orbifold of the D symmetry group action on S given in the space of two invariant polynomials h "J!J F V W and h "J. X
C group the minimal number of stationary points is six as in the case of a S group but two F stationary points are "xed on zero-dimensional stratum and four others should be on the one-dimensional C stratum. Q There is one auxiliary numerator invariant for C group (see Table 2). This means that the NF representation of the space of orbits in terms of invariant polynomials can be done with a two-body decomposition (see Fig. 1). Topologically, the space of orbits of the C group action on F S is a 2D-disk with one singular orbit inside (the C stratum) and the boundary formed by C stratum. Q 3. Hamiltonian with D symmetry group. This symmetry group is of particular importance F because it is the highest point symmetry group compatible with the e!ective rotational Hamiltonian for the asymmetric top molecule. D action on the S sphere possesses three critical F two-point orbits. Consequently, any D invariant function has three pairs of stationary points F "xed at these critical orbits (two maxima, two minima and two saddles). D group is generated by F re#ections and the corresponding orbifold has very simple form in terms of invariant polynomials. We can use the same basic polynomial invariants as for C group action but now there is no F auxiliary invariants and the orbifold is just a "lled triangle with three zero-dimensional strata and three one-dimensional strata forming the boundary and the generic orbits inside (see Fig. 2). The typical form of the rotational energy surface for a D invariant Hamiltonian is shown in F Fig. 3. Care should be taken that the choice of the zero point of the energy plotted in spherical coordinates along the radius is arbitrary and normally can be chosen in such a way that maxima, minima, and saddle points become clearly visualized in the "gure. 4. E!ective Hamiltonian with S (p"1, 2, 3,2) symmetry group. All these symmetry groups N> correspond to symmetric top molecules. There is one critical orbit for S group action on S for N> arbitrary p51. So, the simplest Morse-type Hamiltonian possesses one two-point critical orbit and two stationary generic orbits (including 4p#2 points) whose positions are not "xed by symmetry. 5. E!ective Hamiltonian with D (p"1, 2, 3,2) symmetry group. All these symmetry N>B groups correspond to symmetric top molecules. There are two critical orbits (two-point and 2(2p#1)-point ones). The simplest Morse-type Hamiltonian has additional stationary orbit
96
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
Fig. 3. Rotational energy surface for asymmetrical top molecule with D symmetry group of the e!ective rotational F Hamiltonian.
formed by 2(2p#1) points which belongs to C stratum. The space of orbits can be represented as Q a two-body decomposition (there is one auxiliary numerator invariant), (p"2, 3, 4,2) symmetry group. All these symmetry groups 6. E!ective Hamiltonian with D N F correspond to symmetric top molecules. This group possesses three critical orbits and consequently the simplest Morse-type function has all its stationary points situated on critical orbits only. The two-point orbits for D group with p52 should be stable (maximum or minimum). Two other N F 2p-point orbits can be either saddle or stable ones. 7. E!ective Hamiltonian with ¹ symmetry group. This symmetry group of the e!ective F rotational Hamiltonian corresponds to a spherical top molecules with equilibrium con"guration of ¹ or ¹ symmetry. There are not many examples of molecules with such a symmetry. (Be N O F is one of relatively rare examples.) The simplest Morse-type function invariant with respect to ¹ group has two critical orbits (formed by 6- and 8-point) and one non-critical orbit formed by 12 F points and situated on the C stratum. The number and type of stationary points for ¹ symmetriQ F cal molecules is identical to the simplest Morse-type function for much more wider case of O invariant rotational Hamiltonians except that the 12-point orbit is not "xed by symmetry and F belongs to an one-dimensional stratum C . Q 8. E!ective Hamiltonian with O symmetry group. This is the case of very well-known tetraF hedral and octahedral molecules. The rotational energy surfaces was mainly introduced and studied using these spherical tops as examples. The space of orbits of the O group action on S was F discussed in Chapter I as introductory pedagogical example. The O action on S has three critical F orbits (formed by 6-, 8- and 12-points). There exist two simplest Morse-type functions shown in Figs. 4 and 5 which have the same minimal number of stationary points (26 stationary points including 6#8 stable and 12 saddle points). An example of a more complicated Morse-type function with an additional non-critical orbit of stationary points situated on C stratum and Q formed by 24 points is shown in Fig. 6. 9. E!ective Hamiltonian with > symmetry group. The icosahedral symmetry was studied in the F past mainly by mathematicians (Klein, 1884) demostrated its relevance to di!erent branches of mathematics. The recent interest in chemical applications of icosahedral symmetry was largely stimulated by synthesis and detailed investigations of C molecule and its derivatives and general appearance of icosahedral structures for small clusters (Harter and Weeks, 1988; Weeks and Harter, 1988; Harter, 1993, 1996). All previous appearance of the > symmetry in boron chemistry F
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
97
Fig. 4. Rotational energy surface for a spherical top molecule with O symmetry group of the e!ective rotational F Hamiltonian. Simplest Morse-type function is shown with maxima at the C -axis. Fig. 5. Rotational energy surface for a spherical top molecule with O symmetry group of the e!ective rotational F Hamiltonian. Simplest Morse-type function is shown with maxima at the C -axis.
Fig. 6. Rotational energy surface for a spherical top molecule with O symmetry group of the e!ective rotational F Hamiltonian. Non-simplest Morse-type function is shown with maxima at the C - and C -axis and minima at the C -axis.
(B H units) or for a cage carbohydrides like C H was considered as some exotic fact. At the same time the orbifold representation of the > action on the S sphere is rather similar to the F O action. The orbifold is a topologically "lled disk with boundary formed by one-dimensional F C stratum and three zero-dimensional strata (each consists of one orbit). Q 10. E!ective Hamiltonian with C and D symmetry group. These two groups have the same F F integrity basis and the same space of orbits. The original Morse theory is not applicable in this case and should be extended to Morse}Bott theory for continuous symmetry groups. For simplest examples see Chapter I.
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
98
Table 5 Cluster structure for simplest Morse-type e!ective rotational Hamiltonians
min/max max/min
S , (p50) N>
C ,D (p51) N F N F
D , (p51) N>B
¹,O F F
> F
2 4p#2
2 2p
2 4p#2
6 8
12 20
3.3. Cluster structure of rotational energy levels It is natural to suppose that the rotational structure of an isolated vibrational state at relatively low rotational excitation can be described by some simplest Morse-type Hamiltonian, i.e. by the e!ective Hamiltonian with the minimal number of stationary points compatible with the topology of the rotational phase space (rotational phase sphere S ) and the imposed symmetry. If at the same time we suppose that the corresponding rotational quantum number J is, nevertheless, su$ciently high to apply the classical analysis, we can immediately describe the rotational cluster structure. The cluster structure is re#ected in classical mechanics as presence of several equivalent global or local maxima and minima on the rotational energy surface. The precession of the classical angular momentum near a stable stationary point (minimum or maximum) corresponds in quantum picture to a sequence of energy levels with the wave function localized near the classical stable point. The exact degeneracy of quantum states calculated for each individual minimum or maximum will be removed due to quantum tunneling between equivalent wells. The number of quantum states in each cluster is equal to the number of equivalent stable points, i.e. to the number of points in the orbit of the symmetry group action. The sequence of rotational clusters of each type near the stable point can be described in the harmonic approximation with the harmonic frequency approximately equal to the energy di!erence between consecutive clusters. Cluster structure for simplest Morse-type Hamiltonians for all possible symmetry types of rigid molecules (i.e. for molecules with one global minimum on the potential energy surface for the nuclear motion) is given in Table 5. Table 7 describes qualitatively di!erent types of Morse functions over an S manifold in the presence of the O symmetry. F 3.3.1. Symmetry of levels forming rotational cluster Symmetry of levels forming each rotational cluster can be easily constructed through the induced representation starting from the rotational function transforming according to an irreducible representation of the local symmetry group (stabilizer) of any one of stable axes of rotation. For C local symmetry group the rotational function "J, M2, with M being the projection of the , angular momentum on the symmetry axis, transforms according to an one-dimensional representation (M) mod N. To "nd the symmetry of levels in the rotational cluster one needs to construct the induced representation of the symmetry group G starting from the irreducible representation of (M) mod N of the local symmetry group C . Frobenius reciprocity theorem tells us that the , induced representations can be reconstructed from the table of reduction of irreducible representations of the group G into irreducible representations of the local symmetry group C if we read this ,
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
99
Table 6 Reduction of irreducible representations of the O group with respect to C , C , and C local subgroups (to read by lines). Representations of the O group induced by irreducible representations of C , C , and C local subgroups (to read by columns) 0 A A E F F
1 0 1 1 0
1
2
0 0 0 1 1
0 1 1 0 1
3
0
1
2
0
1
0 0 0 1 1
1 1 0 1 1
0 0 1 1 1
0 0 1 1 1
1 0 1 1 2
0 1 1 2 1
table by columns. The detailed analysis of this procedure can be found in di!erent books and reviews (Patterson and Harter, 1977; Harter et al., 1978, Biedenharn and Louck, 1981; Harter, 1988, 1993, 1996). We just give here in Table 6 the reduction of irreducible representations of group O with respect to its stabilizers C , C , C . Columns of this table give immediately the induced representations for six-fold (C ), eight-fold (C ), and 12-fold (C ) clusters. From Table 6 we get, for example, that the six-fold rotational clusters for the J multiplet of the octahedral molecule are formed by A #E#F levels for extreme M"40,(0 ) cluster, F #F levels for M"39,(3 ) cluster, A #E#F levels for M"38,(2 ) cluster, etc. For another sequence of eight-fold clusters the symmetry decomposition reads E#F #F levels for the extreme M"40,(1 ) cluster, A #A #F #F levels for M"39,(0 ) cluster, etc. The relative energy of individual levels within one rotational cluster can be obtained on the basis of a simple parametric model taking into account the possible tunneling between di!erent equivalent stable rotation axes (Patterson and Harter, 1977; Biedenharn and Louck, 1981). 3.4. Quantum bifurcations of the rotational structure As we have seen in the previous subsection the topology and symmetry arguments enable us to predict the qualitative features of the rotational structure of isolated vibrational states assuming that the e!ective rotational Hamiltonian is of the simplest Morse type, i.e. possesses the minimal possible number of stationary points. Naturally, under the rotational excitation the ro-vibrational interactions cause the modi"cation of the rotational energy surface (or the internal structure of rotational energy levels within the rotational multiplet). Natural way to describe the possible qualitative modi"cations of the rotational structure is to study the modi"cations of the system of stationary points of the RES as a function of rotational excitation. The value of the rotational angular momentum J which is a strict integral of motion (strict quantum number) can be considered in this case as the only parameter responsible for the qualitative modi"cations. The general mathematical answer about the possible qualitative modi"cations of a system of stationary points of a functions depending on some control parameters can be found in the bifurcation (or catastrophe) theory (Thom, 1972; Gilmore, 1981; Arnol'd, 1981; Golubitsky and Schae!er, 1985; Golubitsky and Stewart, 1987; Arnol'd, 1988). Generally the answer depends on the number of control parameters present for a parameterized set of functions and on the imposed
100
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
symmetry. We restrict ourselves to one-parametric families of functions which are the most interesting for our molecular applications: we consider an integral of motion (namely the rotational angular momentum) as a natural parameter to measure the excitation of the system. Very simple classi"cation of possible typical bifurcations of stationary points of a one-parameter family of functions under symmetry can be done for rotational energy surfaces (Zhilinskii and Pavlichenkov, 1987; Pavlichenkov and Zhilinskii, 1988; Pavlichenkov, 1993). The situation is particularly simple here because the rotational phase space is a two-dimensional sphere S and we have a rather modest list of possible local symmetry groups associated with the bifurcation of stationary points. The complete list of local symmetry groups includes only 2D-point groups completely described in Chapter I. We comment below on the typical bifurcations of stationary points which are possible for rotational energy surfaces and on the corresponding notation. All bifurcations of stationary points are described by the local symmetry. The global symmetry of the problem can be larger and due to this global symmetry the bifurcations occur simultaneously for all points forming one orbit of the global symmetry group: C! a non-symmetrical nonlocal bifurcation resulting in appearance (#), or disappearance (!) of a stable}unstable pair of stationary points with the trivial local symmetry C . This is the only possible bifurcation for a one-parameter family of functions in the absence of any symmetry requirements. C*! a local bifurcation with the broken C local symmetry resulting in appearance (#) or disappearance (!) of one unstable point with the C local symmetry and two stable points with the broken C local symmetry instead of one stable point with the C local symmetry. As a result of such bifurcation the number of stationary points increases or decreases by two. C,! a nonlocal bifurcation with the C broken local symmetry. It results in appearance (#) or disappearance (!) of two new unstable points with broken C symmetry, and simultaneous transformation of the initially stable (for #) or unstable (for !) stationary point into an unstable or stable one. The number of stationary points for this bifurcation increases or decreases by two. C,, (n"3, 4) a non-local bifurcation corresponding to n unstable stationary points passing L through a stable stationary point of the C local symmetry, and causing a maximum minimum L change for the stable point. The number of stationary points remains unchanged. C*!, (n54) a local bifurcation which results in appearance (#) or disappearance (!) of L n stable and n unstable stationary points with the broken C symmetry, and a simultaneous L minimum maximum change for a stable point with the C local symmetry. The number of L stationary points increases or decreases by 2n. The given classi"cation of generic bifurcations in the presence of symmetry enables us to describe generic qualitative changes of Morse-type functions under the variation of one parameter (see Table 7). There is only "ve really di!erent types of bifurcations of stationary points. The manifestation of classical bifurcations of stationary points on the rotational energy surfaces in quantum problems is related with the modi"cation of the cluster structure under rotational excitations. The most spectacular is the appearance of a new type of clusters. Several such qualitative changes are given in Table 8. Four-fold clusters in the rotational structure of non-linear triatomic molecules ABA is the consequence of a C* bifurcation at stable critical two-point orbit which results in the appearance of a new stable stationary orbit with lower symmetry formed by four points. The appearance of a 12-fold clusters in the rotational structure of spherical top
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
101
Table 7 Qualitatively di!erent types of Morse functions over an S manifold in the presence of the O symmetry F Type
N
6C T
8C T
12C T
24C Q
24C (a) 24C (b) Q Q
48C
0L 0G 1
26 26 50
min max min
max min max
sad sad max
* * *
* * *
* * sad
* * *
1
50
min
min
max
*
sad
*
*
1
50
max
min
max
sad
*
*
*
1G
50
max
min
min
*
*
sad
*
1G
50
max
max
min
*
sad
*
*
1G
50
min
max
min
sad
*
*
*
2 2 2
74 74 74
max max max
max max min
sad sad sad
sad min min
min sad *
* * sad
* * *
2
74
min 2
max 2
sad 2
sad#min * 2 2
* 2
* 2
`Genealogya
0L (C,>) 0G (C,>, C,>, C ) 0L (C,>, C ) 0G (C,>, C,) 0L (C,>, C , C,) 0G (C,>) 0G (C,>) 0L (C,>, C,>, C ) 0G (C,>, C ) 0L (C,>, C,) 0G (C,>, C , C,) 0L (C,>) 0L (C*>) 0L (C*>) 0L (C*>, C ) 0L (C>)
Total number of stationary points of the Morse function. If N"26#24k, C points are saddles for k"0, 2, 4,2, and stable points, minima or maxima, for k"1, 3, 5,2 . T By C (a) or (b) we denote the components of the C stratum which lie between the C and C , or the C - and C -axis. Q Q `Genealogya shows how a function of a given type can be reached from the simplest one by a series of bifurcations.
Table 8 Molecular examples of qualitative changes in the rotational structure of individual vibrational components under the variation of the angular momentum J Molecule
Component
J value G
Type
Bifurcation(s)
J value D
Type
Refs
SiH SnH CF H Se
l (#) l (!) l (#) "02
J(12 J(10 J(48 J(20
0 0G 0 0
C,> C,>, C , C,, C,\ C*> C*>
J'12 J'12 J"50253 J'20
1 0L 2 1
Only the level of complexity is indicated: see Table 7. Further details on the qualitative changes in the rotational structure may be found in the references. Sadovskii and Zhilinskii (1988) and Sadovskii et al. (1990). Krivtsun et al. (1990a,b). Zhilinskii et al. (1993) and Brodersen and Zhilinskii (1995). Zhilinskii and Pavlichenkov (1988), Pavlichenkov (1993) and Kozin and Pavlichenkov (1996).
102
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
molecules is a consequence of a bifurcation at 12-point unstable stationary orbit which is present for any simplest Morse-type rotational Hamiltonian for tetrahedral and octahedral molecules. It should be noted that the point of classical bifurcation is not exactly the same as the value of the control parameter corresponding to the appearance in the system of quantum energy levels of a new type of clustering. To see the clusters in the quantum spectra not only the new stable stationary point on the classical energy surface should be formed but the well around this surface should become su$ciently pronounced to ensure the formation of localized states near the stable stationary axis (localized precessional states). 3.5. Organization of quantum rotational bifurcations. Crossover Many molecular examples of the qualitative evolution of the internal structure of rotational multiplets under rotational excitations show the simultaneous presence of several bifurcations along with a variation of the control parameter (angular momentum J). This new interesting class of qualitative phenomena can be described as the organization of elementary bifurcations caused by symmetry. Such phenomena exist for the rotational problem with symmetry if the symmetry group action on the rotational phase sphere S produces the system of one-dimensional strata. Table 3 indicates that such symmetry groups of e!ective rotational Hamiltonians are: C ,D ,D ,¹ ,O ,> . N F N F N> B F F F Let us suppose that the sequence of bifurcations starts with one initial bifurcation and that after the "rst bifurcation the new stationary points move monotonously along the one-dimensional stratum. Under this supposition all organizations of bifurcations for di!erent symmetry groups of e!ective rotational Hamiltonians can be given. For an e!ective rotational Hamiltonian invariant with respect to C group the only initial LF bifurcation leading to new stationary points on the 1D stratum C is the non-symmetrical Q bifurcation C> resulting in the formation of two new orbits of stable and unstable points (each orbits is formed by n points). The monotone displacement of these stationary points should result in another non-symmetrical bifurcation C\ resulting in the disappearance of 2n stationary points. The organization of two bifurcations into the sequence C>PC\ gives as a result the rotational energy surface with the same number of stationary points as the initial one and the e!ect of this organization can be described as the "nite rotation of the energy surface. One should note that all other possible initial bifurcations for the C symmetry group lead to a formation of new stationary F points on the generic 2D stratum and consequently cannot cause the organization. The organization of bifurcations starting with C> bifurcation on the 1D stratum can be easily described for all other groups with 1D strata. In what follows only bifurcations of critical orbits leading to new stationary points on the 1D strata will be considered. For all groups D (n-even) the bifurcations C* or C, can be initial ones. In the case of n54 the LF bifurcation C> can also be the initial bifurcation for the organization. Initial C bifurcations can L result in C,> C,\ or C*> C*\ sequences. These both sequences lead to "nite rotations of the energy surface. For the D group the organization C,> C, C,\ is possible. This sequence is F equivalent to a simultaneous crossover and a "nite rotation. At last for the group D with n54 LF the organization C,\QC*>PC*\ with the initial bifurcation C*> results in crossover plus "nite L L rotation.
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
103
For all groups D , n-odd, only the local bifurcation at C critical orbit can produce stationary LB L points on the one-dimensional stratum. Such local bifurcation exists only for n'3. Consequently, there is no initial bifurcations for the D group. For D , n55 the initial local bifurcation C* can B LB L produce a sequence of bifurcations with two "nal C\ bifurcations C,\QC*>PC,\. The result L of this sequence is equivalent to crossover and "nite rotation of the energy surface. Organization of rotational bifurcations due to symmetry was mainly studied for spherical top molecules with O invariant Hamiltonians (Pierre et al., 1989; Davarashvili et al., 1990; Krivtsun F et al., 1990a; Zhilinskii et al., 1993). In this case all sequences of bifurcations result in crossover: C,>PC,PC,PC,\ , (5) C,>PC,PC,PC,\ , (6) C*>PC,P[C*\, C,\] . (7) Purely geometrical analysis of the energy level sections on the orbifold (see Chapter I) shows that during the crossover process it is more likely to see the stationary points between C and C strata than between C and C or between C and C . Geometrically, this is re#ected in the fact that the boundary of the orbifold is more curved between C and C strata and more serious modi"cations of the control parameters are necessary to move the stationary point along the boundary between C and C than between any other pairs of zero-dimensional strata. 3.6. Symmetry breaking due to isotopic substitution and rotational cluster structure We have seen that the simplest typical cluster structure strongly depends on the symmetry of the e!ective rotational Hamiltonian. For example, rotational cluster structure is completely di!erent for spherical top molecules and for asymmetrical tops. At the same time it is easy to imagine that slight isotopic substitution of a spherical top molecule (like Sb for which two isotopes of Sb are available with nearly equal abundance: Sb has 57% versus Sb which has 43%) will lead to symmetrical or asymmetrical tops. A natural question arises. How to correlate a typical spherical top rotational energy surface with 14 stable and 12 unstable stationary points with typical rotational energy surface for an asymmetrical top with four stable and two unstable stationary points. It is clear that formal qualitative analysis of the rotational energy surface as a function of the asphericity parameter should exhibit several bifurcations of stationary points. Instead of the asphericity parameter related to the mass modi"cation due to isotopic substitution we can imagine another physical model with more physical angular momentum playing the role of the same parameter. We take into account two di!erent physical e!ects responsible for the deviation from the ideal rigid spherical top for which all rotational levels within one J-multiplet are completely degenerate. The "rst e!ect is the centrifugal distortion of a spherical top and the second possible source of the deviation is the isotopic substitution e!ect. Depending on the domination of either of these e!ects two limiting cases are possible. When the centrifugal distortion e!ect is small compared to the mass asymmetry the energy-level structure of typical asymmetric (or symmetric) rigid rotor should dominate. When the centrifugal distortion becomes predominant (or equivalently the mass asymmetry becomes negligible) the structure of the rotational multiplet becomes typical for a non-rigid spherical top. The correlation between the two limits inevitably goes though
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
104
a sequence of bifurcations and this type of correlation will be brie#y discussed in this section. More detailed analysis can be found in Pavlichenkov and Zhilinskii (1985) and Zhilinskii et al. (1999). Let consider a tetrahedral A molecule as an example. The isotopic substitution of four masses by m "m "m!d , (8) m "m#d(1#d) , (9) m "m#d(1!d) (10) leads us to a A A A molecule with C point symmetry. The prime or double prime over an A-atom Q indicates slight isotopical modi"cation, i.e. d;m. There are two particular cases in Eqs. (8)}(10): the case d"0 is reduced to the A A molecule which has C point group and the case d"2 is T related to the A A molecule which has C point group. For any 04d42 the molecule is nearly T a spherical top and as a result the orientation of the principal inertia moment axes depend strongly on d. We can choose the reference frame in such a way that its axes coincide with the principal inertia axes of asymmetric top with d'0, d"0. Let x-, y-, and z-axis be, respectively, the axes of intermediate, maximal and minimal inertia momenta. This implies that the x-axis coincides with one of S -axis of the regular tetrahedron and y- and z-axis are orthogonal to the symmetry re#ection planes of the tetrahedron. If dO0 the inertia tensor is non-diagonal in the chosen frame. Its elements are given by the following expressions supposing that all nuclei are located at the distance r "r from each other being in the corners of the regular tetrahedron C ddr C , (11) I "rm! VV C 4m dr 1 I "rm! dr! C , WW C 2m 2 C
(12)
1 dr ddr C , I "rm# dr! C ! XX C C 2 2m 4m
(13)
I "I "0 , VX WX
(14)
(2 d I "! ddr 1! . VW C 4 m
(15)
Any changes in the equilibrium geometry due to isotopical substitution are assumed to be negligible. After inversion of the inertia tensor matrix, the rigid rotor part of the rotational Hamiltonian takes the following form: HK "B JK #B JK #B JK #B JK JK , (16) VV V WW W XX X VW V W where JK are the operators of projection of the total angular momentum in the molecular G coordinate frame. The rotational constants take simple expressions we restrict ourselves with linear in d/m terms. In such a case we can use the following phenomenological rotational Hamiltonian to characterize
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
105
the asymmetry of the mass distribution for the molecule A A A with the equilibrium con"guration being the regular tetrahedron
d d (2dd JK # 1! JK ! JK JK . HK "B JK # 1# VV V 2m V W 2m W 2m X
(17)
Note, that Hamiltonian (17) becomes the standard symmetric top Hamiltonian when d"2 through an appropriate rotation in the xy plane. Similarly, we can obtain the rigid rotor Hamiltonian for the isotopically substituted AB molecule. However it has exactly the form (17) in the linear approximation in d/m. The mass of the central nucleus A does not contribute to the approximate rotational constants since the displacement of the nucleus A from the center of the mass is proportional to d/m and consequently the central nucleus contributes to the rotational energy as square of the small parameter. The rotational constants for the AB molecule are not required for further discussion but the reader can easily obtain them using equality r "(8/3 r which comes from the geometrical consideration. The centrifugal correction is independent of d in the simplest approximation. We employ the centrifugal distortion term with tetrahedral symmetry which has maxima at C axes and minima T at S -axis of the regular tetrahedron. In the molecular frame introduced above it takes the form HK "4JK #3JK #3JK #4JK JK #10JK JK #4JK JK . (18) V W X V W W X X V Both e!ects of the mass asymmetry and the centrifugal distortion can now be combined into one e!ective rotational Hamiltonian
d d . HK "B JK # (JK !JK )# (2dJK JK !tHK X V W VV 2m W 2m
(19)
Qualitative classical analysis for the study of Hamiltonian (19) can be performed by constructing the classical symbol through the substitution of the operators JK by their classical analogies J . It is G G equally useful to introduce dimensionless normalized projections of the total angular momentum j "J /J, j "J /J and j "J /J. Since the scaling of rotational Hamiltonians does not change V V W W X X the energy structure of rotational multiplets, our scaled classical e!ective Hamiltonian can be written as H "1#cos a[( j!j)#(2dj j !2]!sin a W X V W ;( j # j# j#j j# jj#jj!) , VW WX XV V W X where the centrifugal distortion terms are estimated by the parameter a
(20)
2m tJ . (21) d B VV The sum in Hamiltonian (20) consists of three terms. The "rst one is the constant which gives a base energy and is not important in the analysis. The second one is the rigid rotor term. It gives asymmetry of the rigid rotor due to the inertia tensor. The third term is the spherical centrifugal distortion. The second and third terms have weights which are proportional to cos a and sin a correspondingly. It is obvious to see that there are two limiting cases. When a&0 the second term a"arctan
106
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
Table 9 Correlation between group orbits of O , D , and C symmetry groups F F F O F
D F
C F
C (6 points) T
C (2 points) T C (4 points) Q C (4 points) Q C (4 points) Q C (2 points) T C (2 points) T C (8 points) C #C (4#4 points) Q Q 2[C ] (8#8 points) 2[C ] (4#4 points) Q 2[C ] (8#8 points) 6[C ]
C (2 points) Q C (4 points) C (4 points) 2[C ] (2#2 points) Q C (2 points) C (2 points) Q 2[C ] (4#4 points) 2[C ] 4[C ] 2[C ] 4[C ] 12[C ]
C (8 points) T C (12 points) T
C (24 points) Q C (24 points) Q C (48 points)
is dominant and we have the rigid rotor limit. When a&p/2 the third term is leading and we obtain the spherical top limit. Varying a from 0 to p/2 the model in Eq. (20) can be continuously changed from one limit to other one. Several rotational bifurcations should appear when a varies between two di!erent physical limit. The concrete form of the bifurcation diagram depends on the model but we are again interested in some simplest sequence of bifurcations related to this correlation diagram. In the rigid asymmetric top limit (a&0) the RES has a well-known shape of three-axially deformed spheres (see Fig. 3). In the other limit (a&p/2) the centrifugal distortion e!ects become predominant and the mass asymmetry is negligible. The corresponding RES is typical for a spherical top (see Fig. 5 but pay attention to the choice of axes precised above). The rotational energy surfaces in the two limiting cases have di!erent systems of stationary points. There are six stationary points for the asymmetric rigid rotor and 26 stationary points for the spherical top. When d"0, the correlation is, in fact, between D and O invariant rotational F F energy surfaces, whereas for arbitrary masses (dO0,2) the correlation is between C and F O invariant Hamiltonians. The complete correlation between group orbits for the action of F O , D , and C groups on the rotational phase sphere is given in Table 9. F F F The minimal set of stationary points of D invariant function includes six points (three F two-point critical orbits). These six points have "xed positions in the phase space along the whole correlation diagram. On the other hand, the O invariant function has 26 stationary points (one F six-point orbit, one eight-point orbit, and one 12-point orbit). When the O symmetry breaks down F to D with the axes orientation as described above, the O orbits are split correspondingly into F F D orbits. For example, one C invariant eight-point orbit is split into two di!erent four-point F T orbits of the D group and so on. F The representation of the space of orbits of the D group action on the two-dimensional sphere F is given in Fig. 7 together with the evolution of the system of stationary points between the
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
107
asymmetric (D ) and spherical top (O ) limits. Critical orbits in the rigid rotor limit are given as F F empty circles. There are three of them since they are two-point orbits. Along the correlation diagram three critical orbits must be transformed into seven orbits of stationary points: three two-point critical orbits, three four-point orbits, and one eight-point orbit. This can be done through di!erent sequences of bifurcations which depend on the particular form of the Hamiltonian. But the minimal number of elementary bifurcations needed for that is four. The correlation diagram shown in Fig. 8 gives a particular example of such a bifurcation sequence. It corresponds to Hamiltonian (20) with d set equal to zero. The evolution of critical orbits is shown as solid lines in the "gure. Since critical orbits always correspond to extrema they give us a part of stationary points. The energies of other stationary points are given by dotted lines (for orbits which belong to one-dimensional strata) and dashed lines (for orbits in two-dimensional strata). Orbit stabilizers with respect to the D group are indicated together with a degeneracy in brackets when F necessary. On the right-hand side of Fig. 8, the stabilizers of the O critical orbits are shown in the F a"0 limit. In order to discuss the evolution of the system of stationary points we can split the whole variation range of a into four subintervals. The "rst bifurcation occurs at a"arctan(). Thus for a(arctan() the system of stationary points includes only critical orbits, i.e. the points along the symmetry axes: the RES has two maxima at the y-axis, two minima at the z-axis and two saddle points at the x-axis. The point a"arctan() corresponds to the bifurcation of the unstable stationary point at the x-axis. The x-axis becomes a stable axis of rotation and two new unstable stationary axes arise. Their positions undergo a shift in the xz plane as a increases further. The lines with arrows in Fig. 7 show as the new stationary points move upon the increase of a. Ten stationary
Fig. 7. Orbifold of the D group action in S for the Hamiltonian (20) with d"0 is presented in the space of two F invariant polynomials h and h (see text). Dashed lines show the strati"cation due to the action of the O group. Empty ? @ F circles indicate the stationary points at a"0 while "lled circles show the stationary points at the spherical top limit (a"p/2). The lines display the dynamic of the stationary points as a varies from 0 to p/2. Square denotes a bifurcation. The indicated symmetry of axes correspond to O group. F Fig. 8. Energy correlation diagram between asymmetric top (D symmetry) and spherical top (O symmetry). Solid lines F F represent critical orbits, dotted lines show in-plane orbits and dashed lines indicate orbits of general position (also marked by g).
108
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
Fig. 9. Energy correlation diagram between an asymmetric top (C symmetry) and a spherical top (O symmetry) for the F F case of d"1 in the model (20). Solid line represents the energy of the C critical orbit, dotted lines show the energy of F stationary points which belong to orbits in one-dimensional stratum and dashed lines indicate the energy of stationary points which belong to orbits in a two-dimensional stratum. Degeneracy of orbits is indicated in brackets where necessary.
points exist in the region arctan()(a(arctan(). When a reaches arctan() two simultaneous bifurcations occur. Both y- and z-axis become unstable and new stable axes appear. Similarly to the "rst bifurcation, the new axes move in the xy and yz planes, respectively (see Figs. 8 and 7). There are 18 stationary points on the RES in the region arctan()(a(p/4. Finally, the new stable axis in the xz plane bifurcates at a"p/4. For a'p/4 this orbit becomes again stable whereas unstable stationary points arise which correspond to the orbit in two-dimensional stratum. The last bifurcation is denoted as a square in Fig. 7. In total we have two local bifurcations with C broken symmetry and two non-local bifurcations with C and C broken symmetry. The dotted lines in Q Fig. 7 show strati"cation due to the O group. If the D orbifold were folded along them we would F F get the orbifold of the O group. That is why the limiting positions of the stationary points in F Fig. 7 are strictly de"ned. 3.7. Imperfect quantum bifurcations Generally speaking, the symmetry breaking results in decreasing of the number of allowed types of bifurcations. At the same time if the deviation from the symmetry is small in some sense, another type of bifurcation sequences is possible. It is related to the so-called imperfect bifurcations which are well known in the classical theory of bifurcations. The imperfect bifurcations correspond to the appearance of stationary points somewhere in a small region of the phase space near another stationary point which does not change its stability. The imperfect bifurcation is usually related to more complicated bifurcation which takes place in the presence of higher symmetry. We can illustrate the appearance of imperfect bifurcations on the same example of the correlation between asymmetric and spherical tops studied in the previous subsection but looking now for the C O correlation. F F The energy correlation diagram for the model in Eq. (20) with d"1 corresponding to the C O correlation is presented in Fig. 9. The orbifold representation is given in Fig. 10 and it F F
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
109
Fig. 10. Orbifold of the C group action on S is presented in the space of two invariant polynomials h "j!j and F ? V W h "j together with the sign of the auxiliary invariant u"j j . Dashed lines show the strati"cation due to the action of @ X VW the O group. Empty circles indicate the orbits of stationary points at a"0 for the Hamiltonian in Eq. (20) with d"1 F while "lled circles show the orbits of stationary points at the spherical top limit (a"p/2). Solid lines display the dynamic of the stationary points as a varies from 0 to p/2. Squares denote bifurcations. The indicated symmetry axes correspond to the O group. Points with u"0 in two graphs (left and right sides of big triangles) should be glued together. F
shows the positions of stationary points in the rigid rotor limit as empty circles. It is seen that x- and y-axis are not critical orbits any longer. The principal di!erence between D and F C symmetry groups is the number of critical orbits: three two-point orbits for D and only one F F two-point orbit for C . F The critical orbit (z-axis) displays the same type of behavior as the similar critical orbit in the case of the D symmetry group: the bifurcation with the C broken symmetry. Two other critical F orbits of the D symmetry group become non-critical C symmetry orbits in the C group and the F Q F corresponding stationary points are shifted in xy plane. Due to this they cannot change their stability without breaking the C symmetry and instead of two bifurcations at y- and x-axis with Q the C broken symmetry as in the case of the D symmetry we have now one bifurcation with the F C broken symmetry and one C -type fold in the xy plane. The C bifurcation occurs on the set of Q
110
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
Fig. 11. Part of the energy correlation diagram between asymmetric top (C symmetry) and spherical top (O symmetry) F F for the case of d"0.03 in the model of Eq. (20). Dotted lines represent the energy of stationary points which belong to orbits in one-dimensional stratum. Fig. 12. Easy recognizable pitchfork of the bifurcation in the xy plane for the model (20) with parameter d"0.03. The ordinate is the normalized projection of the angular momentum on the x-axis j "J /J. V V
C symmetrical orbits. The last bifurcation shown in Fig. 10 is generic C -type fold which results in Q the simultaneous appearance of four equivalent maxima and four saddle points. Its prototype in the D system is the bifurcation at a"p/4. The bifurcations are indicated as squares in Fig. 10. F Again, the limiting positions of stationary points at a"p/2 are pre-de"ned because of the tetrahedral symmetry. The general comparison of the energy correlation diagrams in Figs. 8 and 9 reveals that low-energy parts look similar. But relatively small isotopic substitution a!ects rather strongly the high-energy part of the diagrams. The comparison of two correlation diagrams between asymmetric and spherical top molecules upon isotopic substitution reveals a qualitative di!erence between Hamiltonians with D and F C symmetries. The di!erence is clearly seen in the classical limit when any violation of the F D symmetry results in the modi"cation of the type of observed bifurcations. We saw that in some F cases when the D symmetry breaks, non-symmetrical bifurcation occurs in the neighborhood of F original D symmetrical bifurcation (see Fig. 11). This phenomenon is known in the bifurcation F theory as imperfect bifurcation (Golubitsky and Schae!er, 1985; Golubitsky and Stewart, 1987). Its characteristic signature is the so-called perturbed `pitchforka in the plane in which the stationary point positions are given as a function of the parameter. Fig. 12 shows the j coordinate of V stationary points for a twice isotopically substituted molecule with d"0.03 as a changes from 0 to p/2. This is not a real case since d is too small but it makes perturbed pitchfork more obvious. When the perturbation is zero and two new maxima are identical (the case of D symmetry) the F pitchfork is symmetrical (cf. Fig. 12 of the present paper and Figs. 1.1 and 1.3 of Golubitsky and Schae!er, 1985). For any small d, the old maximum evolves with a without changing its stability and the newly appeared one is di!erent and non-symmetric. This is immediately re#ected in the
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
111
energy di!erence of these stationary points which is evident from the energy correlation diagram given in Fig. 11. 4. Rotational structure for a N-quantum state system We have analyzed in the previous section the rotational structure of one isolated quantum state. Natural generalization leads us to consider the rotational structure associated with several quantum states. The physical interpretation of the N quantum states under consideration can be quite di!erent. Rotational structure of several vibrational states for example should be treated simultaneously if the typical energy di!erence between vibrational states is smaller or comparable with the characteristic rotational excitation. In the limit of high rotational excitations the interesting possibility of constructing the classical limit for rotational variables only arises. The general idea is to keep the N-state description on the quantum level but to go to the classical limit in rotational variables. We will name such approach semi-quantum in order to make the distinction from the semi-classical approximation having naturally a di!erent meaning. The simplest semi-quantum models correspond to the description of the rotational structure of two vibrational states. Generalization to a N-state problem gives no principal di$culties in most cases because locally only the coupling between two states is important. In the case of relatively large number of states N, the problem can be analyzed on the completely classical footing using the classical limit in both types of variables. Comparison between the complete quantum description, the semi-quantum description and the complete classical description reveals many interesting qualitative e!ects and enables better understanding of the quantum classical correspondence. 4.1. Ewective quantum rotational Hamiltonian for an N-state problem and its classical matrix symbol A lot of high-resolution experimental data have been recently obtained for the rotational structure of di!erent groups of vibrational states of various polyatomic molecules. The interpretation of these experimental data and description of the rovibrational energy levels in most cases is done using e!ective phenomenological Hamiltonians which can be schematically represented in terms of coupled vibrational and rotational operators (22) H" t [V C;RC]C . H GH G G H In Eq. (22) V C are the vibrational operators having non-zero matrix elements only within the G considered block of vibrational states, RC are rotational operators, and t are the phenomenologiH GH cal coe$cients which are the parameters "tted to experimental data. In the presence of symmetry it is advantageous to use irreducible tensor vibrational and rotational operators and in this case the total Hamiltonian should be invariant with respect to the symmetry group of the problem. Di!erent approaches make di!erent choices of the basis of tensor operators, di!erent notation of the phenomenological parameters and di!erent schemes for the inclusion of the terms in the phenomenological expansion in Eq. (22). Nevertheless, in any case of the e!ective rotational operator for k vibrational states the e!ective quantum Hamiltonian can be written as a k;k matrix
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
112
with its elements being rotational operators. This leads us immediately to the construction of the classical limit over the rotational variables. It is su$cient to replace the rotational quantum operators by their classical analogs. The eigenvalues of this matrix should naturally be interpreted as rotational energy surfaces for di!erent quantum states treated together. The interpretation of k di!erent eigenvalues of the k;k matrix is quite simple if there is no degeneracy between di!erent eigenvalues but such degeneracy generically exists and it is related to a new not yet analyzed qualitative e!ect. To demonstrate the presence of degeneracy points let us consider the classical 2;2 matrix with each matrix element being the function of the J value (the absolute value of the angular momentum) and of two angles (h, ) showing the orientation of the angular momentum vector with respect to the molecule "xed frame (h and play the role of the rotational phase space variables):
R( (h, ) R( (h, ) . H " R( (h, ) R( (h, ) The eigenvalues are given by E (J; h, )"[R #R $((R !R )#"R "] . To have the degeneracy it is necessary to satisfy three equations
(23)
(24)
R !R "0, "R ""0 . (25) Remark that the second equation in (25) consists, in fact, in two independent equations } one for real and one for imaginary part. Each R depends on three parameters J, h, . This means that GH generically there are degeneracy points for some exceptional values of J and for some values of the angles h, and . Otherwise speaking the codimension of the subspace of degeneracy points is equal to three and such a statement is applicable to any hermitian k;k matrix. Some consequences of this general statement is well known in molecular physics and spectroscopy, for example the non-crossing rule of potential curves for diatomic molecules, or more general statement about intersections of multidimensional potential surfaces for di!erent electronic states (Landau and Lifshitz, 1965; Herzberg and Longuet-Higgins, 1963; Avron et al., 1988). If there is no symmetry requirements on the matrix elements R the two eigenvalues near the GH degeneracy point form a conical intersection point (sometimes called `diabolic pointa) shown in Fig. 13, left. At the same time additional symmetry which appears through the symmetry of two vibrational quantum states and imposes certain symmetry properties on the non-diagonal rotational operators can result in a higher-order touching point between two rotational energy surfaces. The second-order touching is shown in Fig. 13, right. In order to see the e!ect of symmetry on the formation of touching points let us consider the rovibrational problem for two vibrational states in the presence of the C symmetry group. In this case vibrational states are classi"ed according to irreducible representations of C and for the two vibrational states under study we get two irreps n , m for which either (i) (ii) (iii)
n "m mod 4, or n "m $1 mod 4, or n "m $2 mod 4.
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
113
Fig. 13. Typical behavior of two rotational energy surfaces near the touching point. Linear conical intersection (left) takes place in the absence of any symmetry restrictions. Special symmetry conditions on the symmetry of vibrational components corresponding to two di!erent rotational energy surfaces leads to second order touching (right).
It is easy to verify that these three cases are completely di!erent from the point of view of the formation of a touching point between two rotational energy surfaces for the C zero-dimensional stratum on the rotational phase sphere. If two representations are identical (case i), there are non-zero independent of h contributions near the C -axis for both diagonal and non-diagonal matrix elements: f (J) f (J) H (n , n )" . (26) f (J) f (J) Consequently, the formation of the degeneracy point on the symmetry axis is generically forbidden because f (J) depend only on one parameter J. GH In cases (ii) and (iii) the non-diagonal matrix elements are zero on the axis due to the symmetry and the formation of the degeneracy point is allowed because only one condition f !f "0 should be satis"ed to ensure the presence of the degeneracy point. To "nd the geometrical form of two RES near their intersection point it is necessary to take into account the most important contributions from non-diagonal and diagonal matrix elements. In case (ii) the non-diagonal matrix elements near the symmetry axis are proportional to sin h whereas the diagonal ones are proportional to sin h at least and may be dropped out for small h. The classical matrix has the form
0 f (J )sin h e ( (27) H (n , n $1; J"J )" f (J )sin h e\ ( 0 leading to the conical intersection point at those J values which satisfy the equation f (J )!f (J )"0. (Putting 0 on the diagonal is equivalent to the energy shift } to choose the zero for energy at the energy of the conical intersection point.) In case (iii) the non-diagonal matrix elements near the symmetry axis are proportional to sin h whereas the diagonal ones are proportional to sin h as well. So we must keep both leading contributions on the diagonal and o! diagonal. The classical matrix has the form
f (J )sin h f (J )sin h e ( . H (n , n $2; J"J )" f (J )sin h e\ ( f (J )sin h
(28)
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
114
Two corresponding eigenvalues (RES) near their degeneracy point are characterized by the following geometrical form: (29) E "[ f (J )#f (J )] sin h$([ f !f ]#f f sin h . We have in this case the second-order touching between two RES. The dependence could be analyzed more accurately but it is not very important because for the C zero-dimensional stratum the stationary point should be minimum or maximum rather than a saddle point. The general result that we have obtained so far can be reformulated as follows. Generically, there are no degeneracy points between rotational energy surfaces corresponding to di!erent vibrational states but such degeneracy points generically appear for a one-parameter family of rotational energy surfaces. The most natural parameter in this context is the absolute value of the rotational angular momentum, which is an integral of motion for an isolated molecule. Now, we will demonstrate how the existence of touching point between rotational energy surfaces manifests itself in the quantum energy-level patterns. The analysis of the evolution of the rotational multiplets under the variation of the parameter J over the interval containing the degeneracy point will be postponed to the next subsection. Here, we brie#y analyze the typical energy level pattern for an exceptional rovibrational Hamiltonian with the degeneracy point of two classical rotational surfaces. To study the energy level pattern close to the conical intersection point we cannot use simple quantization rules based on the harmonic approximation near the stable stationary point. We must take into account the existence of two sheets of the RES. Thus to "nd the characteristic system of energy levels near the conical intersection point we construct a simple quantum Hamiltonian which leads in the classical limit to RES with the conical intersection point and has an exact analytical solution. Such a Hamiltonian may be written in matrix form with matrix elements depending on rotational operators
H"
0
¸ !i¸ V W which leads to classical matrix
¸ #i¸ V W , 0
(30)
0 sin h e P . H ""L" sin h e\ P 0
(31)
Equivalently Eq. (30) can be given in the form of a spin rotational operator by introducing auxiliary spin operators corresponding to S" (in fact, this is possible because any 2;2 matrix may be represented in a form of a linear combination of S , S , S and identity auxiliary operators): > \ X H"2(S ¸ #S ¸ )"4(SL!S ¸ ) . (32) > \ \ > X X This is a particular case of the model operator studied by Pavlov-Verevkin et al. (1988) in order to demonstrate the dynamical meaning of the formation of the conical intersection points. Let us remark that the classical Hamiltonian symbol (31) corresponds to two rotational energy surfaces E "$"L""sin h".
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
115
An exact quantum solution of Eq. (32) or Eq. (30) may be easily found by noting that the projection of the formally constructed total angular momentum J "¸ #S is an integral of X X X motion. So wave function "¸ "¸, S "2 corresponding to J "¸# is the eigenfunction with X X X the energy E"0. In a similar way "¸ "!¸, S "!2 corresponding to J "!¸! is the X X X eigenfunction with the same energy E"0. There are two eigenfunctions corresponding to all other values of J (!¸#4J 4¸!). Each pair may be constructed from two wave functions with X X "¸ "M, S "2 and "¸ "M#1, S "!2. The energy for this pair is given by X X X X E"$2((J#M#1)(J!M), M"(¸!1), (¸!2),2, (!¸#1),!¸ .
(33)
We are interested in the solutions with "M" close to J only because only these energy levels are near the conical intersection point. We can introduce an index a to label the degenerate energy levels. Two levels with E"0 have a"0. All other energy levels which form pairs situated symmetrically with respect to zero are denoted correspondingly by a"1, 2,2 . Assuming that a is small compared to ¸!"M" we have a very simple expression for the energy levels E"$(8J(a .
(34)
It means that in the case of a conical intersection point instead of a system of equidistant energy levels characteristic for harmonic oscillator (the quantization near the stable stationary point) we have the sequence of energy levels separated from the central one (origin of the pseudo-symmetry) according to the square root rule: 1: (2 : (3 : 22
(35)
The most important conclusion is the presence of quantum energy level(s) located at the energy corresponding to the degeneracy point of classical energy surfaces and certain symmetry (pseudosymmetry or supersymmetry can be more properly used here) of energy levels around it. The modi"cation of a control parameter J will eliminate the presence of a degeneracy point in the classical limit and will destroy the symmetry of the quantum energy levels. Are there any general rules for such modi"cations? We turn now to this question. 4.2. Isolated vibrational components and their rotational structure We initially assumed in the preceding subsection that the rotational multiplet associated with the non-degenerate vibrational state has 2J#1 energy levels for given quantum number J of the angular momentum. We consider everywhere isolated molecules in the absence of external "elds and thus we neglect the additional (2J#1) degeneracy due to angular momentum projections on the axis of laboratory "xed frame. At the same time it is well known that for degenerate vibrational states the splitting of the total rovibrational multiplet into individual rotational multiplets for given J value can result in clearly seen rotational individual multiplets which consist of di!erent numbers of energy levels. The mostly known example is the Coriolis splitting of a triply degenerate vibrational state into three components with e!ective rotational quantum numbers R"J#1, J, J!1 and with the corresponding numbers of energy levels in each rotational sub-multiplet equal 2R#1"2J#3, 2J#1, 2J!1. In order to analyze possible decompositions of rovibrational
116
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
energy levels into rotational multiplets associated with individual vibrational components we start with more formal de"nition of such a decomposition. We "rst remark that all rotational levels of the group of vibrational states which span the vibrational representation C of the symmetry group G at a given J form a reducible representa tion of the group G which can be written symbolically as (36) C "C ;D(E , where D(E stands for the decomposition of the irreducible representation (J ) of the O(3) group E with respect to the symmetry group G of the problem. We say that all these rovibrational levels decompose into rotational branches corresponding to individual vibrational components if for all su$ciently high J values the reducible representation C is represented as a sum of "C " contributions, C (37) C " C ;D(>DG E , G G where D and C are independent on J, all C are one-dimensional irreducible representations G G G (the same representation can appear several times in the decomposition) and D(>DG E denotes an irreducible representation of the O(3) group with the e!ective weight J#D (more properly G speaking the decomposition of this representation into the irreducible representations of the symmetry group). Naturally D "0. Each number D de"nes an e!ective rotational quantum G G G number of the corresponding vibrational component (branch) i and therefore the number of rotational states in this component. The one-dimensional representations C of the group G, which enter in Eq. (37) are e!ective G symmetries of the corresponding vibrational components i. For example, the O group has four F one-dimensional representations and the corresponding four possible types of vibrational components are (38) A ;D(>DE ,D(>D , E E A ;D(>DE ,D(>D , (39) S S A ;D(>DE ,D(>D , (40) E E A ;D(>DE ,D(>D . (41) S S We introduce above a full and a shortened notation with two indices to distinguish between four types of irreducible representations. For example, in the case of the F triply degenerate vibrational state of an octahedral molecule S the well-known "rst-order Coriolis splitting into three components with e!ective quantum numbers R"J#1, J, and J!1 corresponds to a particular form of the decomposition in Eq. (37) F ;D(E " A ;D(>BE S S B ! "D(>#D(#D(\ . S S S
(42) (43)
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
117
Table 10 Characterization of the decomposition into vibrational components for F vibrational states of ¹ molecules B Vib.state.
Decomposition
F F F F F F F F F F F F F F F F F F
(#1) #(0) #(!1) S S S (!5) #(#6) #(!1) E E S (#1) #(!6) #(#5) S E E (#1) #(!4) #(#3) S S S (!5) #(#2) #(#3) E E S (#1) #(#2) #(!3) S E E (!3) #(#4) #(!1) S S S (#3) #(!2) #(!1) E E S (!3) #(!2) #(#5) S E E (!3) #(!4) #(#7) S S S (#3) #(#2) #(!5) E E S (!3) #(#2) #(#1) S E E (#5) #(0) #(!5) S S S (!1) #(#6) #(!5) E E S (#5) #(!6) #(#1) S E E (#5) #(#4) #(!9) S S S (!1) #(!2) #(#3) E E S (#5) #(!2) #(!3) S E E
The case of ¹ symmetry is simpler because there are only two di!erent one-dimensional B representations and as was initially introduced by Zhilinskii and Brodersen (1994) any vibrational component can be labeled by D(>D? with a"g, u due to equivalence A ;D0,D0. E S For tetrahedral ¹ molecules the list of possible decompositions of a rotational structure of B degenerate vibrational states into isolated vibrational components was given by Zhilinskii and Brodersen (1994). The general solution for the E vibrational state is particularly simple. There is an in"nite number of solutions E;D("D(>B#D(\B E E S
(44)
with d"$(6k#2),$(6k#4), k"0, 1, 2,2 . Some initial solutions corresponding to the decomposition of the F triply degenerate vibraG tional states of ¹ molecules are given in Tables 10 and 11. B 4.3. Dynamical meaning of diabolic points and rearrangement of rotational multiplets The rotational Hamiltonian for two vibrational states is formally equivalent to an e!ective Hamiltonian for two coupled angular momenta S and N with S". The semi-quantum approach in this case corresponds to a quantum description of the spin and the classical description of the rotation. To analyze the e!ect of the formation of a degeneracy point for two rotational energy
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
118
Table 11 Characterization of the decomposition into vibrational components for F vibrational states of ¹ molecules B Vib.state.
Decomposition
F F F F F F F F F F F F F F F F F F
(!5) #(0) #(#5) E E E (#1) #(!6) #(#5) S S E (!5) #(#6) #(!1) E S S (!5) #(!4) #(#9) E E E (#1) #(#2) #(!3) S S E (!5) #(#2) #(#3) E S S (#3) #(#4) #(!7) E E E (!3) #(!2) #(#5) S S E (#3) #(!2) #(!1) E S S (#3) #(!4) #(#1) E E E (!3) #(#2) #(#1) S S E (#3) #(#2) #(!5) E S S (!1) #(0) #(#1) E E E (#5) #(!6) #(#1) S S E (!1) #(#6) #(!5) E S S (!1) #(#4) #(!3) E E E (#5) #(!2) #(!3) S S E (!1) #(!2) #(#3) E S S
surfaces a very simple quantum Hamiltonian can be used: c 1!c S # (N ) S), 04c41 . H" X "N ""S" "S"
(45)
Here c is a coupling parameter. We consider this problem as a one-parameter family for arbitrary xxed amplitudes "N " and we "x for a moment "S"". When c varies between 0 and 1 the continuous transformation between two physically simple limits is performed. For arbitrary quantum numbers S and N the space of (2N#1)(2S#1) wavefunctions factors into a sum of subspaces of functions with a given quantum number J . (For S" the maximal X dimension of each term in the sum is 2.) The eigenvalues of the Hamiltonian in Eq. (45) can be easily given in the analytic form. At the same time it should be noted that the quantum number J does X not characterize the multiplet structure of the quantum spectrum of Eq. (45) because multiplets consist of states with di!erent J . This structure can be easily understood near the two limits c"0 X and c"1 using appropriate good (approximate) quantum numbers. When c is close to 0 the eigenvalues of S and N are good quantum numbers. S characterizes X X X the multiplet structure. For S" there are two multiplets or quasi-degenerate groups of levels with 2N#1 levels in each group. Within each such multiplet the levels have the same value of S and are X distinguished by the value of N . The "rst order splitting of levels within multiplets depends X linearly on N . We say that N describes the internal structure of multiplets. X X When c is close to 1 a di!erent pair of good quantum numbers (J, J ) exists. Here J"N#S is X the total angular momentum and J is the projection of J on the z-axis. In this limit J describes X
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
119
Fig. 14. Quantum energy spectrum (solid lines) for two-level (S") problems with Hamiltonian in Eq. (45) and N"4. Extrema of corresponding classical rotational energy surfaces are shown by dashed lines.
the multiplet structure. For S", there are two quasi-degenerate multiplets labeled by J"N#, N!. Within each multiplet, levels with the same J are distinguished by J so that the X "rst order splitting is a linear function of J (Landau and Lifshitz, 1965). X The transformation of the eigenfunctions of the Hamiltonian in Eq. (45) from the limit c"0 to the limit c"1 is a well-known transformation from the uncoupled to a coupled basis for two angular momenta, "NN SS 2P"NSJJ 2. When S", the number of multiplets in the two limits is X X X two but the number of levels within each multiplet is di!erent. Consequently, a number of levels is redistributed among the multiplets at intermediate values of the control parameter c. The redistribution phenomenon is illustrated in Fig. 14 on the example of S". Rotational structure of various groups of quasi-degenerate or degenerate by symmetry vibrational levels shows in a similar way the presence of redistribution phenomenon in the quantum energy-level patterns and the formation of the degeneracy points for the corresponding classical rotational energy surfaces constructed as eigenvalues of a classical matrix Hamiltonian in the semi-quantum model. A very spectacular example of this phenomenon was found in the rotational structure of the triply degenerate l vibrational band of the octahedral molecule Mo(CO) (Asselin et al., 2000; Dhont et al., 2000). Fig. 15 shows the redistribution phenomenon associated with the formation of the degeneracy point between upper and lower vibrational components of the triply degenerate vibrational state. The better visualization of the geometry of rotational energy surfaces can be reached through the 2D sections of all three surfaces which go across the three zero-dimensional strata (see Fig. 16). As long as the degeneracy point at C stratum belongs to the six-point orbit, six conical intersection points are formed at the same value of control parameter J in the classical picture. Naturally, the six-fold cluster is transferred from the middle component to the upper component as the J values go through J&8.
120
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
Fig. 15. Above: Quantum and classical energies for the triply degenerate l vibrational band of the octahedral molecule Mo(CO) in the interval 04J425 shown by grey horizontal bars and solid curves respectively. Below: Hessian values h( for the middle branch E (J) at points C , C , and C shown as arcsh(10;h((h, )) in order to increase the range of the plot.
The transfer of the six-fold cluster from the middle to the upper component naturally modi"es the type of these two vibrational components. It follows that di!erent good quantum numbers should be used for the values of J below and above J+8. The modi"cation of the good quantum
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
121
Fig. 16. Cuts of the rotational energy surfaces in the plane "p/4 for di!erent values of J; the energy E at the central Q point is taken E "2005.482 cm\ in the case J"11 where E "2005.481 cm\ for the internal surface. The presence of Q Q the degeneracy point at C stratum is seen at J"8.
numbers proceeds as follows for the three components: > 2(J#2)#1 , D(\PD(> : 2(J!1)#1P S S \ 2(J!3)#1 , D(PD(\ : 2J#1P S S D(>PD(> . S S In fact, only the two upper components are really changed. The upper component gains 6 levels while the central component looses 6 levels. This exchange of 6 levels can be symbolically represented as a `transitiona between the symmetry labels of vibrational components written in the spirit of chemical reactions as D(#D(\(3 P D(\#D(> . S S S S
(46)
Comparison of classical energies of critical orbits on the two RESs near the conical intersection (degeneracy or `diabolica point) and the corresponding part of the quantum energy-level spectrum clearly shows that the levels which transfer between the two components follow the classical energy
122
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
of the critical points for one of RESs involved in the intersection. On the other hand, the classical analysis of the corresponding RESs does not indicate any preference as to which critical point (associated with one or another vibrational component) the quantum level should `followa. Moreover, it turns out that two di!erent quantum e!ective Hamiltonians can be written in such a way that both quantum operators have the same classical symbol as a classical limit in the semi-quantum picture but the redistribution phenomenon occurs in the reverse direction (from the upper component to the middle one). To realize such a construction it is su$cient to change the sign at all imaginary matrix elements in the classical limit matrix. This alternation of sign can be interpreted as changing the sign of the commutation relation between the angular momentum operators [J , J ]"#e J P[J , J ]"!e J , ? @ ?@A A ? @ ?@A A with e the totally antisymmetric tensor and a, b, c standing for x, y or z. The `#a-relation holds in the laboratory "xed coordinate frame, whereas the `!a-relation applies in the molecule-"xed frame. The same classical Hamiltonian matrix symbol can be associated with two quantum operators which di!er in the signs of all odd-X terms. Fig. 17 demonstrates this e!ect using essentially the same e!ective operator that is used to describe the rotational structure for the l band of Mo(CO) . Very often the real rovibrational energy-level systems show several consecutive rearrangements resulting in considerable modi"cation of the system of vibrational components. For example, the bending modes of the AB (¹ ) molecules, i.e. two lowest in energy l (E), and l (F ), show B practically the same splitting into vibrational components at relatively low J values (see Table 12). The same structure of vibrational components is frequently observed in the whole region of accessible J values. (See Appendix A for the description of the qualitative structure of di!erent vibrational components for the CF molecule.) In contrast, the splitting of the 2l group of vibrational states of CF clearly shows several rearrangements between vibrational components under the variation of the J values between J"5 and 20. The results are given in Table 13. There are many di!erent rearrangements which are allowed but there are also some selection rules which indicate for a given symmetry group of the problem possible types of elementary rearrangements between vibrational components which can occur under the variation of the only control parameter. Such rearrangements can be described in terms of `reactionsa between the symmetry indices introduced above to characterize vibrational components. For the ¹ symmetry B molecules these symmetry indices of vibrational components are, in fact, the irreducible representations of the O(3) group. Elementary reactions are associated with the transfer of one 12-fold cluster, one eight-fold cluster, and one or two six-fold clusters. All possible `reactionsa for tetrahedral molecules are summarized in Table 14. At the end of our analysis of the rotational structure for a system of vibrational states we remark again that the qualitative rotational structure of each vibrational component can be analyzed in the same way as for isolated vibrational states (this can be done, in fact, for any J values which do not correspond to degeneracy points). Systems of stationary points on each rotational energy surface can be associated with the sequence of clusters or with the transition region between di!erent types of clusters exactly in the same way as for individual states. Appendix A summarizes in the form of tables examples of the rotational structure changes as a function of J for di!erent vibrational components of the CF molecule.
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
123
Fig. 17. Two possible rearrangements of rotational energy levels between vibrational components corresponding to the same classical limit.
Remark, however that the symmetry type of energy levels forming rotational clusters should be found from e!ective rotational quantum numbers associated with the vibrational components rather than the J value. More details about such description can be found by using the local symmetry indices (Zhilinskii and Brodersen, 1994). 5. Vibrational problem Usually, while considering the vibrational motion one assumes the existence of the additional integral of motion (extra quantum number) characterizing the electronic motion. The rotational
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
124
Table 12 Splitting of l , l dyad of tetrahedral molecules into vibrational components CF
(J!2) E (J#2) S (J#1) S (J) S (J!1) S
SiF
SnH
(J!1) S (J) S (J#1) S (J#2) S (J!2) E
(J!1) S (J) S (J#1) S (J#2) S (J!2) E
Table 13 Rearrangement of vibrational components for 2l state of CF under J variation. Transfer of clusters between neighboring components is indicated explicitly as n-fold. Vibrational components are ordered according to the energy increase from below to the top of the table Vib
J"5 (J#1)
F
(J) S (J!1)
A
S
S
(J) E
2
2
2
2
2
J"20
2
2
2
2
(J!2)
E
2
2
(J!4)
(J!1)
E
2 6-fold 2
(J!4)
E
2 8-fold 2
2 6-fold 2
(J) E
2
(J) E
(J#3)
S
(J) E
2
(J) E
2
(J#1)
E
2
2
(J#2)
E
(J!2)
S
2
2
2 6-fold 2
(J#2)
E
2
2
2
E
(J#1)
S
E
motion is suppressed either by putting J"0 or just by neglecting rotational degrees of freedom with the assumption that the rotational structure is too "ne to be discussed together with much more pronounced vibrational e!ects. Moreover, even among the vibrational modes it is sometimes very useful to restrict oneself by choosing only part of the possible modes. The natural principle of the separation of the vibrational modes is based on taking into account the resonance relations between them (Abraham and Marsden, 1978; Birkho!, 1966; Cushman and Rod, 1982; Cushman and Bates, 1997). The most important is surely the 1 : 1 resonance due to symmetry or quasidegeneracy. There are many molecules for which near resonances are to be expected on rather general footing. For example near degeneracy of symmetric and anti-symmetric stretching vibrations in bent triatomic molecule of the AB type takes place if atom B is signi"cantly heavier than the A one and the valence angle is close to p/2. Similar quasi-degeneracy takes place for three equivalent stretching vibrations in molecules with C or D point symmetry group. T F
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
125
Table 14 Di!erent possibilities of the transfer of clusters between two vibrational components. (aOb, a, b"u, g). Simple arrows (Q, P, ) indicate that the transfer of clusters does not lead to the change of parity of the vibrational components. Double arrows (=, N, 8) indicate that the transfer of clusters is associated with the change of parity of the vibrational components First vibrational component
Second vibrational component
Transfer of 1 cluster
(R) ? (R) ? (R) ? (R) ? (R) ? (R) ? (R) ?
(R#1#4t) ? (R#2#4t) ? (R#4t) @ (R#1#4t) @ (R#1#3t) ?Y (R#1#2t) ? (R#2t) @
=
Transfer of 2 clusters
= Q 8 8
First vibrational component
Second vibrational component
(R#3) @ (R$3) ? (R$3) ? (R#3) @ (R#4) ? (R$6) @ (R$6) @
(R!2#4t) @ (R#2G3#4t) ? (RG3#4t) @ (R!2#4t) ? (R!3#3t) ?Y (R#1G6#2t) @ (RG6#2t) @
Di!erent n : m resonances between vibrational modes are also quite typical for molecular systems. 1 : 2 resonances between non-degenerate or between doubly degenerate and non-degenerate vibrational modes initially studied by Fermi (and referred often in molecular physics as Fermi resonance) was extensively studied during the last years in many di!erent molecular systems. Tetrahedral AB molecules show an example of more complicated resonances. They have near degeneracies of four stretching modes and "ve bending modes with 1 : 2 resonance between them. Thus, taking into account the exact degeneracy of vibrations due to symmetry, this resonance may be described as l (A ) : l (F ) : l (E) : l (F )"2 : 2 : 1 : 1 or indicating explicitly the degeneracy of the vibrational modes as "2 : 2 : 2 : 2 : 1 : 1 : 1 : 1 : 1 resonance. Thus, the number of vibrational degrees of freedom which are to be studied in molecular models simultaneously may vary considerably from molecule to molecule and surely depends on the accuracy needed. The dimension of the phase space in any case is twice the number of degrees of freedom. The presence of resonance condition between harmonic frequencies ensures the formation of a group of quasi-degenerate vibrational levels, the so-called vibrational polyads. In the simplest case of the vibrational problem with K quasi-degenerate modes a natural possibility is to introduce an approximate integral of motion associated with the total number of vibrational quanta. This enables us to reduce the number of degrees of freedom by one and the dimension of the classical phase space by two. Assuming the existence of this extra integral of motion we can study the internal structure of vibrational polyads formed by K quasi-degenerate modes in (2K!2)dimensional phase space which is a subspace of the complete vibrational phase space (PavlovVerevkin and Zhilinskii, 1987, 1988a,b; Sadovskii et al., 1993; Sadovskii and Zhilinskii, 1993a,b; Zhilinskii, 1989a). Another important question is the relative energies of di!erent polyads and the numbers of energy levels within polyads (one prefers to know the numbers of states of each symmetry type within the polyad) (Sadovskii and Zhilinskii, 1995; Soldan and Zhilinskii, 1996). This more crude information is quite important for the estimation of the density of states needed for the calculation of thermodynamic properties and kinetic constants.
126
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
In a more general situation it is possible to introduce global polyad quantum numbers and more "ne sub-polyad quantum numbers producing a hierarchical description of the vibrational level system. In this section our qualitative analysis will be largely based on the vibrational polyad description. We will start with the study of a relatively simple question: How to describe the numbers of states of di!erent symmetry belonging to vibrational polyads and after that we return to the qualitative analysis of the internal structure of vibrational polyads. 5.1. Vibrational polyads, resonances, and polyad quantum numbers The structure of the vibrational energy-level system of many polyatomic molecules often exhibits isolated groups of vibrational levels, called vibrational polyads (Ja!eH , 1988; Fried and Ezra, 1987; Xiao and Kellman, 1989; Zhilinskii, 1989a; Kellman, 1990; Kellman and Chen, 1991; Kellman, 1995; Jonas et al., 1993). These polyads can be seen clearly when the ratio of the vibrational frequencies is close to a simple rational number. For example, consider a molecule with three vibrational modes. Near the equilibrium geometry the Hamiltonian of this system can be represented as a Hamiltonian of a three-dimensional anharmonic oscillator with frequencies l , l and l , 1 . (47) H" l (p#q)#< G G G 2 G In the simplest case of a nearly isotropic oscillator l +l +l , and, provided that the anhar monicity < is small, vibrational polyads obviously manifest themself. If we label these polyads by the polyad quantum number N"0, 1, 2,2, then the number of states (vibrational energy levels) in each polyad N(N) equals (N#1)(N#2)/2. The internal structure of polyads depends strongly on the nature of the anharmonic terms <. For instance, at low N this structure can be well described in terms of normal modes, while at high N the local mode description can be more physically meaningful (Child and Halonen, 1984; Mills and Robiette, 1985; Patterson, 1985; Kellman, 1985; Levine and Kinsey, 1986; Kellman and Lynch, 1986; Stefanski and Pollak, 1987). On the other hand, the polyads themselves exist for any su$ciently small anharmonic terms < regardless of the actual nature of these terms. Our initial purpose is to give a simple formula that can estimate the density of states using only the very basic initial information on the molecule, namely the frequencies and the symmetry types of the vibrational modes, and the resonance condition. Describing internal structure of the polyads might be di$cult, whereas the system of polyads as a whole can be analyzed in much simpler terms. Indeed, if we neglect the splitting of levels within each polyad then the energy of the polyad E(N) can be de"ned as a function of the polyad quantum number N. In the harmonic approximation E(N) is a linear function. We can introduce non-linear corrections, such that E(N)"l(1#a N#a N#2), to account for anharmonicity of the real potential. The simple non-linear corrections a to the energy E(N) should not be confused with numerous anharmonic G terms needed to reproduce the internal structure of each polyad. By ignoring any internal structure of polyads we essentially consider each polyad as a single level } thus, our anharmonic corrections are, in a sense, similar to those of a single-mode anharmonic oscillator with quantum number N. The precise de"nition of these corrections follows, for instance, from the normal form reduction of the complete Hamiltonian with respect to the total action I, the classical analog of the polyad
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
127
number N. This normal form may have many resonance terms needed to describe the dynamics at I"const. We, however, neglect such terms at this stage and leave only a I#a I#2 . Therefore, as long as the polyads exist (N is a good quantum number) and we are not interested in their internal structure, we can obtain a rough estimate of the average density of states by calculating the number of states in each polyad. Such an estimate will not display any #uctuations of the density of states due to the internal structure of polyads. Thus, as is well known (Landau and Lifshitz, 1965), the average density of states of a three-mode system with three frequencies of the same order of magnitude is a quadratic function of energy, and indeed, the number of states is a polynomial in N of degree 2. New features of the density of states arise when vibrational frequencies satisfy (approximately) less trivial resonance conditions. For example, let us consider a three-dimensional oscillator with a Fermi resonance l +2l +l (ratio l : l : l +2 : 1 : 2). This oscillator can serve as a zero order approximation to the vibrations of triatomic non-linear molecules, such as H O: the two stretching modes of H O, symmetric l and anti-symmetric l , are quasi-degenerate, and the frequency of the bending mode l is roughly one-half the frequency of the stretching modes. In this case, the energy gap between the neighboring polyads N and N#1 equals approximately hl . As shown below, the number of states in the Nth polyad,
5 11 5 (!1), 1 N (N)" N# N# ! N# 8 8 16 2 8
(48)
has a regular part, again a polynomial in N of degree 2, and an oscillatory part J(!1),. In more precise terms the number of states is a quasi-polynomial in N (see Chapter 1) and its regular part can be de"ned unambiguously if the oscillatory part is chosen in such a way that after being averaged over the period it gives zero contribution to the regular part. Hence, even though we ignore the internal structure of polyads, we still expect the quantum density of states to have oscillations in addition to the general parabolic behavior. Both the oscillations and the coe$cients in the regular part become more complicated when we use larger integer numbers n : n : n in order to reproduce more precisely the actual ratio l : l : l . Certainly, expressions become more complex as well with growing number of vibrational modes (degrees of freedom). Some simple combinatorics enables us to derive explicit general formulae for the density of states of a multidimensional quantum oscillator with arbitrary resonance condition n : n : 2 : n ) (Sadovskii and Zhilinskii, 1995). We also take into account the symmetry requirements that should be imposed if the molecule possesses some non-trivial symmetry. For the harmonic oscillator (the zero-order approximation) this problem can be completely solved by purely group-theoretical techniques based on the Molien generating function. The key point for real molecular application is that such formulae can be subsequently corrected to give a reliable quantitative estimation of the density of states of the actual anharmonic oscillator. This makes the approach useful for the study of such systems as polyatomic molecules (Quack, 1990b), and even for the analysis of `quantum chaosa (Gutzwiller, 1990). In the past the density of states of multidimensional anharmonic oscillators has been studied numerically for various model vibrational Hamiltonians with di!erent symmetry groups (Whitten and Rabinovich, 1963; Stein and Rabinovich, 1973; Quack, 1977; Lederman et al., 1983; Lederman and Marcus, 1984; Sinha and Kinsey, 1984; Quack, 1985, 1990a). The density of states of a given symmetry type for both a multidimensional oscillator system and
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
128
Table 15 Vibrational resonances in molecules Molecule
G
Vibrational modes l(C) N
K
Resonance l :l :2:l )
AB H O
C T (C ) (C ) T (C ) (C ) (C ) F D F D F (C ) T (¹ ) B (¹ ) B (¹ ) B
l (A ), l (A ), l (B ) l ,l l ,l ,l l ,l l ,l l ,l ,l l ,2, l
3 2 3
n :n :n 1:1 2:2:1
3 3 4 7
2:1:1 5:3:3 10 : 6 : 6 : 23 5:3:5:1:1:1:1
3 3
n :n :n 5:4:4
9 4 5
2:2:2:2:1:1:1:1:1 1:1:1:1 1:1:1:1:1
CO CS C H A H> CH SiH CD
l (A ), l (E ) V W l ,l l ,l ,l ,l l ,l l ,l
Polyad number N
N #N 2(N #N )#N 2N #N 5N #3N 10N #6N #23N
5N #4N 2(N #N )#N #N N #N N #N
Symmetry group and its image in the concrete vibrational representation (in brackets). For each mode we give spectroscopic notation l , symmetry type C, and components p for degenerate modes; p"(a, b) I for E-modes and (x, y, z) for F-modes. Total number of vibrational degrees of freedom that are considered. N is the number of quanta in mode l . G G For linear A B molecules the traditional symmetry labels of vibrational modes are l (R>), l (R>), l (R>), l (P ) , E E S E V W l (P ) . S V W
a quantum billiard with symmetry has been recently calculated using the semi-classical theory (Robbins, 1989; Creagh and Littlejohn, 1991; Weidenmuller, 1993). We are interested in molecular applications and therefore, we only analyze model vibrational Hamiltonians which can be initially approximated by a harmonic oscillator (small vibrations near the equilibrium). Resonances between the vibrational modes may be approximate or exact (due to symmetry). Vibrational structure of molecules provides a great number of examples of both kinds. Table 15 summarizes molecular examples with typical and quite interesting resonance conditions. In each case we consider K vibrational modes with frequencies l , i"1,2, K, and suppose a resonance G condition l : l : 2 : l +n : n : 2 : n . All n should be taken as positive integers; they can be ) ) G large in order to reproduce the ratio of the actual frequencies with desired accuracy. We draw attention to this de"nition because alternative de"nitions, with a similar notation, but with a completely di!erent meaning, are possible. To label the vibrational polyads we introduce the polyad quantum number N. The physical meaning of the polyad quantum number N can be understood in several ways. In a purely quantum approach, in the limit of uncoupled oscillators (vibrational modes) the de"nition of N is given in terms of the numbers of quanta in di!erent modes N , N ,2, N in accordance with the ) resonance condition. For instance, the polyads formed by two vibrations l : l +2 : 1, can be characterized by the number N"2N #N , with N and N , the number of quanta in modes
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
129
l and l . If the two modes couple, N still can remain a good quantum number while N and N can loose their meaning. A corresponding classical interpretation is in terms of total action I and individual actions I , I ,2, I (Ja!eH , 1988; Fried and Ezra, 1987; Xiao and Kellman, 1989). ) In terms of hyper-spherical coordinates, the quantum number N corresponds to the hyper-radial motion. The concept of vibrational polyads proves itself to be extremely useful in the interpretation of vibrational spectra and description of vibrational dynamics for highly vibrationally excited molecules. In what follows, we will consider two aspects of the qualitative description of vibrational polyads: (i) numbers of states in polyads and associated density of states; (ii) qualitative internal structure of vibrational polyads. 5.2. Generating functions for numbers of states in polyads To introduce the generating function method for the calculation of the number of states in vibrational polyads let us start with a trivial example of a K-dimensional isotropic harmonic oscillator with frequency l"l "l "2"l , and consider all states with energy l(N#K). ) This degenerate set of states for an isotropic harmonic oscillator becomes a quasi-degenerate polyad characterized by the quantum number N under a small perturbation breaking the S;(N) dynamic symmetry of the harmonic oscillator. Let N(N, K) be the total number of states in such a polyad. This number equals the number of partitions of N quanta into K parts (Landau and Lifshitz, 1965). From the group theoretical point of view N(N, K) is the dimension of the representation of the dynamical symmetry group of the K-dimensional isotropic harmonic oscillator (Kramer and Moshinsky, 1968), S;(K), characterized by the single-row Young diagram 䊐2䊐 with N boxes. It can be given either explicitly K(K#1)(K#2)2(K#N!1) N(N, K)" N! (N#1)(N#2)2(N#K!1) " (K!1)!
(49) (50)
or in the form of a generating function depending on an auxiliary variable j 1 . g (j)" ) (1!j))
(51)
To obtain N(N, K) from the generating function g (j) we expand the latter in the power series ) g (j)"C #C j#C j#2#C j,#2 . (52) ) , The coe$cient before j, gives the number of states in the polyad with polyad quantum number N, C "N(N, K) . (53) , The two alternative representations of N(N, K) in Eqs. (49) and (50) are equivalent; the form in Eq. (50) shows immediately that N(N, K) is a polynomial in N of degree (K!1). The generalization to the system of harmonic oscillators with the resonance condition l : l : 2 : l "d : d : 2 : d is straightforward. The generating functions for the number of ) )
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
130
states in such a polyad has the form g
B B
2
1 (j)" . B) (1!jB )(1!jB )2(1!jB) )
(54)
Let now suppose that the molecule possesses some symmetry group G. Vibrational modes of this molecule are classi"ed according to the irreducible representations +C , C ,2, C , of G. Some of 1 the vibrational modes can be degenerate and then S(K. In other words, 1 [C ]"K , (55) H H where [C] is the dimension of representation C. Together all the modes we consider span a (generally) reducible representation C "C C 2. In the zero-order harmonic ap proximation the vibrational states of the molecule are described by the basis functions "(N , a ), (N , a ),22, where N is the number of quanta in mode C , and a is a set of auxiliary H H H quantum numbers to distinguish the excited states of mode C with the same number of quanta N . H H We want to "nd the number of all excited vibrational states of symmetry C characterized by a given distribution of quanta +N , N ,2, or just by a given polyad quantum number if some resonance relation between modes is speci"ed. This is a standard group-theoretical problem which may be solved using Molien generating functions as explained brie#y in Chapter I (see also (Molien, 1897; Burnside, 1911; Weyl, 1939; Springer, 1977). For each possible C we "rst obtain the generating function gC (j , j ,2) whose auxiliary variables j , j ,2 correspond to the modes C , C ,2 we consider. The coe$cient C 2 of the ,, term j, j, 2 in the Taylor expansion of such a function gives the number of states of symmetry C with the distribution of quanta +N , N ,2,. Then we take into account the appropriate resonance condition n : n : 2 and introduce one single auxiliary variable j and the correspond ing polyad quantum number N. In all cases the main result is the set of generating functions gC (j), such that their sum equals g(j), the generating function for the total number of states introduced in the previous section, g(j)" [C] gC(j) . (56) C If we expand each of gC(j) in a power series similar to Eq. (52), the coe$cients of j, give NC(N), the number of vibrational states of given symmetry C in the polyad N. Of course, (57) N(N)" [C]NC(N) . C The density of states can be obtained from NC(N) by dividing the latter by l"E(N)!E(N#1), the energy gap between the neighboring polyads. Along with the number of states NC(N) and the corresponding density of states it is often useful to consider the partial density NC(N) (N)" . NC N(N)
(58)
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
131
Fig. 18. Partial numbers of vibrational states of di!erent symmetry types versus the polyad quantum number for tetrahedral molecules AB with the resonance relation l : l# : l$ : l$ "2 : 1 : 2 : 1.
The large-N asymptotic behavior of the partial density (58) is de"ned completely by the symmetry group: at large N the ratio of partial densities of states equals the ratio of the squares of the dimensions of the corresponding representations, NCI (N) [C ] " I . (59) lim NCG (N) [C ] G , This relation was formulated as a general conjecture by Quack (1977) and Lederman et al. (1983). Fig. 18 shows the convergence to the theoretical limit for tetrahedral molecules AB . A constructive proof of Eq. (59) can be given if for each "nite group G and C , we take generating functions gC(j) for all possible irreducible representations C of G and transform them into explicit expressions for NC(N). To realize such a transformation some more information about high N behavior of the numbers of states is needed. Particularly important is the separation of the expression for the number of states into regular and oscillatory parts. 5.3. Density of states. Regular and oscillatory parts The general form of a generating function giving the number of states of given symmetry in polyads can be written as a rational function with t#1 terms in the numerator and s factors (1!jBG ) in the denominator: jL #jL #2#jLR . g(j)" (1!jB )2(1!jBQ )
(60)
Some of n and d can be identical. In the case of a generating function for invariants one of G H n should be zero. The expansion of g(j) in Eq. (60) in a formal power series G g(j)" C(N)j, ,
(61)
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
132
leads for su$ciently big N (at least for N'max(n )) to the representation of C(N) as a quasiG polynomial (see Chapter I) of degree s!1. The regular part of this quasi-polynomial C(N)"a
NQ\#a NQ\#2#a #oscillatory part (62) Q\ Q\ can be directly found from Eq. (60) by replacing the formal parameter j by e\U and expanding the result in the Laurent series in w at w"0: e\UL #e\UL #2#e\ULR "b w\Q#b w\Q>#2#b w\#2 . Q\ Q\ (1!e\UB )2(1!e\UBQ )
(63)
The s initial coe$cients of this series are proportional to the coe$cients in Eq. (60): b /( j)!"a . H H In particular, for the "rst terms the expressions are rather simple:
(64)
Num(j"1) , (65) " d G Num(j"1) d !2 n G G , b " (66) Q\ 2 d G where Num(j"1) is the value of the numerator of the generating function in Eq. (60) for j"1. The oscillatory part of the quasi-polynomial has the period equal to the least common multiplier of (d ,2, d ). It can be always written in the form Q b
Q\
Q\ BG (67) N? hS d(u, N mod+lcm(d ),) . ? G S ? In fact for each particular value of a in Eq. (67) the real period can be shorter than the lcm(d ) but it G is always a divisor of it. 5.4. Two polyad quantum numbers. Example of C H
Acetylene molecule, C H , gives a quite interesting widely experimentally studied example of molecules for which the interpretation of data is largely based on the polyad concept (Herman et al., 1999). Moreover, this example shows the natural way of the generalization of the generating function approach to more complicated examples with several polyad quantum numbers. We remind that C H is a linear molecule with seven vibrational degrees of freedom and the D point group symmetry of the equilibrium con"guration in the ground electronic state. F Vibrational variables span the seven-dimensional representation A #A #A #E #E E E S E S (see the character Table 16 below for the notation of representations). The resonance condition between vibrational modes for C H can be approximated as (68) l E : l E : l S : l#E : l#S "5 : 3 : 5 : 1 : 1 . This particular ratio of vibrational frequencies enables one to introduce the polyad quantum number N "5n #3n #5n #n #n . The physical meaning of this polyad quantum number P is purely energetic. This quantum number does not take into account the presence of the most
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
133
Table 16 Character table for the D group F D F
E
2C( )
Rp
i
2[iC( )]
RC
R>"A E E R\"A E E EE I R>"A S S R\"A S S ES I
1 1 2 1 1 2
1 1 2 cos(k ) 1 1 2 cos(k )
1 !1 0 1 !1 0
1 1 2 !1 !1 !2
1 1 2 cos(k ) !1 !1 !2 cos(k )
1 !1 0 !1 1 0
T
k"1, 2,2 is a positive integer. Alternative notation: E? "P , E? "D , etc. ? ?
important dynamical resonance terms. As soon as dynamics is concerned another important polyad quantum number can be introduced, N "n #n #n , which has the physical meaning Q of the number of stretching quanta. This number is an invariant for the model Hamiltonian used to describe the vibrational energy levels for C H and it certainly can be considered as a good approximate quantum number for rather elaborated models. The numbers of states in N polyads are given through the generating function which has P a simple form 1 . g (j)" (1!j)(1!j)(1!j)
(69)
This generating function re#ects just the ratio of vibrational frequencies. The power series expansion of g (j) g (j)" c(N )j,P (70) P ,P gives the number c(N ) of vibrational states within one polyad with the N quantum number. In P P particular, several "rst terms give g (j)"1#4j#10j#21j#39j#68j#113j#2 . (71) The number c(N ) can be generally expressed as a quasi-polynomial using the general rule P formulated in the preceding section. It is possible to improve the density of states description by specifying the numbers of levels of di!erent symmetry within polyads. To realize such a construction we start with generating functions for a number of tensors constructed from various representations of the symmetry group (see Table 17). The most important and the only non-trivial part of this table is the series of generating functions g(C; P #P ; j). One can verify that the sum over all "nal representations E S C of these generating functions is (the dimension of the representations is denoted by [C]) 1 [C]g(C; P #P ; j)" E S (1!j) C
(72)
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
134 Table 17 Generating functions for the D
F
group
R> E
R> S
R\ E
R\ S
EE I
EE I>
ES I
ES I>
g(C; P ; j) E
1 1!j
0
0
0
jI 1!j
jI> 1!j
0
0
g(C; P ; j) S
1 1!j 1 Den 1 1!j 1 1!j
0
0
0
0
0
j Den
j Den
j Den
jI 1!j Num Den
Num Den
Num Den
jI> 1!j Num Den
0
0
0
0
0
0
0
j 1!j
0
0
0
0
0
0
g(C; P #P ; j) E S g(C; R>; j) E g(C; R>; j) S
Den"(1!j)(1!j); Num (k)"(k#1)jI!(k!1)jI>; Num(k)"(k#1)jI>#jI>!kjI>; Num(k)"kjI#2jI>!kjI>.
and it gives the generating function for the number of states of the four-dimensional isotropic harmonic oscillator. Generating functions for numbers of states of de"nite symmetry within N polyads follows P immediately from Table 17 taking into account the resonance relation between frequencies. The simplest is the generating function for D invariants (i.e. for states of k"0, g,# type): F 1#j g (k"0, g,#; j)" . (73) (1!j)(1!j)(1!j)(1!j)(1!j) Power series expansion gives, for example g (k"0, g,#; j)"1#2j#j#4j#3j#7j#7j#12j#2 . (74) Naturally, generating functions for all other symmetry types can be written in a similar way: j#j g (k"0, g,!; j)" , (1!j)(1!j)(1!j)(1!j)(1!j)
(75)
j#j , g (k"0, u,#; j)" (1!j)(1!j)(1!j)(1!j)(1!j)
(76)
j#j , g (k"0, u,!; j)" (1!j)(1!j)(1!j)(1!j)(1!j)
(77)
(k#1)jI!(k!1)jI>#j(kjI#2jI>!kjI>) , k"1, 2,2 , g (2k, g, j)" (1!j)(1!j)(1!j)(1!j)(1!j)
(78)
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
135
Fig. 19. Logarithm of the number of states of di!erent symmetry types as a function of the total polyad quantum number N . Numbers of states are given for four one-dimensional representations of D group. P F
j((k#1)jI!(k!1)jI>)#kjI#2jI>!kjI> g (2k, u, j)" , k"1, 2,2 , (1!j)(1!j)(1!j)(1!j)(1!j)
(79)
(1#j)((k#1)jI>#jI>!kjI>) , k"1, 2,2, a"u, g . g (2k#1, a, j)" (1!j)(1!j)(1!j)(1!j)(1!j)
(80)
Fig. 19 shows numbers of states of di!erent symmetry types as a function of N . It should be noted P that densities of states of di!erent symmetry types scale in fact by a constant factor in the high energy region (Quack, 1977). That is why the most important is the number of states of the (k"0, g,#) symmetry within the polyads. To see the organization of the energy levels within N polyads it is useful to introduce the P sub-polyad structure, namely (N , N ) polyads with N "n #n #n being the number of P Q Q stretching quanta. To construct the generating function which gives the numbers of states which belong to one sub-polyad characterized by two given quantum numbers N and N , we can P Q introduce a generating function depending on two auxiliary parameters 1 . g (j, k)" (1!kj)(1!kj)(1!j)
(81)
The additional new parameter k counts only stretching excitations and it makes no di!erence between three stretching modes. Now, the power series expansion of the function g (j, k) (82) g (j, k)" c(N , N )j,P k,Q , P Q P Q , , gives the numbers c(N , N ) of vibrational states within one polyad with two given (N , N ) P Q P Q quantum numbers. The "rst terms of this expansion read g (j, k)"1#4j#10j#(20#k)j#(35#4k)j # (56#12k)j#(84#28k#k)j#2 .
(83)
136
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
We can also expand the generating function in Eq. (81) only in one auxiliary variable k. Such an expansion has the form c(N ; j)k,Q Q . (84) g (j, k)" (1!j) Q , Individual terms of this expansion are the generating functions for the number of states in (N "const., N ) sub-polyads. Q P j#2j j#2j#3j 1 # k# k#2 . (85) g (j, k)" (1!j) (1!j) (1!j) We can equally give more detailed formulae for numbers of states of certain symmetry within the (N , N ) sub-polyad. The simplest is the generating function for D invariants (i.e. for states of P Q F k"0, g,#type): 1#kj . g (k"0, g,#; j, k)" (1!kj)(1!kj)(1!kj)(1!j)(1!j)
(86)
Naturally, generating functions for all other symmetry types can be written in a similar way. Similar multi-parameter generating functions will be relevant for many more complicated physical and chemical applications because they allow one to produce a simple model description of the numbers of states of di!erent symmetry types (density of states of certain symmetry) which are quite important for modeling thermodynamic properties of molecules and chemical reactions. 5.5. Internal structure of polyads formed by two-quasi-degenerate modes We now turn out to the description of the internal structure of vibrational polyads. The simplest case is the internal structure of polyads formed by two quasi-degenerate modes. To interpret the internal structure of vibrational polyads we use an e!ective Hamiltonian written in terms of the vibrational angular momentum operators which are the bilinear components of creation and annihilation operators (a>, a>, a , a ) for two quasi-degenerate vibrational modes: J "(J )>"a>a , > \ 1 1 J " (J !J )" (p q !p q ) , \ 2i > 2 J "(J #J )"(p p #q q ) , > \ J "(a>a !a>a )"(p !p #q !q ) , J"2N"(a>a #a>a )"(p #p #q #q ) . (87) We use indices (1,2,3) instead of (x, y, z) to avoid a confusion with the Cartesian coordinate system related to the molecular frame. Eqs. (87) give a well-known Schwinger representation (Schwinger, 1965) of the angular momentum in terms of two pairs of boson operators. As soon as the representation spanned by (J , J , J ) is known the group image for the problem > \ we consider and the group action on the polyad phase space S are completely de"ned. In many
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
137
Fig. 20. Correspondence between the quantum spectrum of the simplest D -symmetric 2D-oscillator at J"10 and the energies of stationary points on the classical vibrational energy surface. Quantum levels (full lines) and classical solutions (dashed lines) are obtained with the same model operator in Eq. (88). Quasi-degeneracy of vibrational clusters are indicated by numbers 2 and 3 on the right.
cases the symmetry of an e!ective vibrational Hamiltonian di!ers from that of the initial problem because the group image is di!erent from the initial symmetry group. All qualitatively di!erent classical vibrational Hamiltonian functions, which in the two-mode case are called vibrational energy surfaces, can be easily classi"ed by their sets of stationary points. Thus, the simplest vibrational Hamiltonians for several group images can be easily constructed in a way similar to e!ective rotational Hamiltonians with the only di!erence that the number of possible symmetry groups now is larger. The complete list of possible group images for di!erent two-dimensional vibrational problems is given in Pavlov-Verevkin and Zhilinskii (1988a) (see also Zhilinskii, 1989b). In the simplest case of a non-linear AB molecule the image of the C group in the axial vector T representation spanned by the components of the vibrational angular momentum constructed from two stretching modes, l and l , with symmetries A and B is the C group (Pavlov Verevkin and Zhilinskii 1988a; Zhilinskii, 1989b). (See Fig. 20.) An excursion to a series of works by Kellman (1990) (see also Xiao and Kellman 1989; Kellman and Chen, 1991; Kellman, 1985) provides examples of qualitatively di!erent and similar vibrational energy surfaces for the case of vibrational polyads formed by two quasi-degenerate modes of non-linear AB molecules. Several examples of the qualitative characteristics of vibrational energy surfaces are given in Table 18. When the vibrational energy surface is of the simplest Morse type with one minimum and one maximum, the molecule is described by the normal mode limit. If two additional points exist on the vibrational energy surface the internal structure of vibrational polyads is characterized by the local mode model. One should take into account the fact that at low total number of quanta 2J"v #v the number of energy levels in a polyad is too small to observe the formation of vibrational clusters due to the e!ect of the localization near the classical stationary points. That is why even when the classical limit function of the quantum Hamiltonian is
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
138
Table 18 Morse functions de"ned over an S manifold under the presence of the C symmetry: Qualitatively di!erent types of vibrational energy surfaces for two stretching modes of a C triatomic molecule T Type
C ss
C as
2C
0L 0G
min max
max min
* *
1 1
sad min
max sad
min max
1G 1G
sad max
min sad
max min
Genealogy
Molecular examples (Kellman, 1990) SO (v45)
0L C* (ss) 0L C* (as) 0G C* (ss) 0G C* (as)
H O (v45)
O (v45)
ss and as denote C -invariant orbits which correspond to symmetric and anti-symmetric stretching. `Genealogya indicates the type and the place (the orbit) of the bifurcation leading to a more complicated Morse function from the simplest one.
of type 1 the existence of two equivalent regions of localized motion becomes apparent only at higher values of 2J, as clearly indicated by the formation of doublets in the quantum spectrum. Another particular example of the bending vibration of a D symmetric X molecule is quite F simple from the theoretical point of view and serves as a concrete illustration to rather abstract results and to a somewhat new interpretation of the intra-molecular vibrational dynamics. To interpret the internal structure of vibrational polyads formed by overtones of the l (E ) mode of an E X molecule, we use an e!ective Hamiltonian written in terms of vibrational angular momentum operators which are the bilinear components of creation and annihilation operators (a>, a>, a , a ) for the doubly degenerate vibrational mode of type E (cf. Eq. (87)). The representation spanned by E (J , J , J ) is (A E ) and the group image for this problem is D . The action of D on the polyad E E phase space S is identical to its natural group action. The J component of the vibrational angular momentum has the A symmetry and corresponds to the projection of vibrational angular momentum on the C -axis of the molecule. Qualitatively di!erent types of vibrational energy surfaces are listed in Table 19. We conclude from this table that a simplest D symmetric Hamiltonian has two non-equivalent-by-symmetry localization areas: near the critical orbit with the C local symmetry, and near one of two critical orbits which have the C local symmetry. A corresponding quantum system has two- and three-fold clusters lying at the opposite ends of the vibrational energy spectrum. Non-local trajectories pass through the saddle points, and correspond to delocalized quantum states which lie at intermediate energies and do not form any cluster structure. An example of such a simplest D -symmetric operator can be easily constructed in terms of spherical tensors: H "¹ #a(¹ !¹ )#b(¹ #¹ ), 1'a
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
139
Table 19 Morse functions de"ned over an S manifold in the presence of the D symmetry: Qualitatively di!erent types of vibrational energy surfaces for doubly degenerate bending modes of a D triatomic molecule F Type
2C
3C
3 C
6C
0L(a) 0L(b) 0G(a) 0G(b)
max max min min
min sad max sad
sad min sad max
* * * *
1 1 1 1
max max max max
min min sad max
min max sad min
sad sad min sad
0L(a) 0L(a) 0L(a) 0L(b)
1G 1G 1G 1G
min min min min
max max sad min
max min sad max
sad sad max sad
0G(a) 0G(a) 0G(a) 0G(b)
Genealogy
C, C, C* C, C, C, C* C,
`Genealogya indicates the type and the place (the orbit) of the bifurcation showing how a more complicated Morse function is obtained from the simplest one. Only one example for each type of the Morse function is given.
Relative positions and numbers of clusters may vary but all these operators result in the same two types of localized motion, or of localized quantum wave functions, which correspond to the precession of the vibrational angular momentum around the C - and C -axis. The immediate consequence of this statement is the presence of three-fold clusters in the internal structure of bending overtones of the A molecule with D symmetry. This phenomenon is well F known in classical mechanics as formation of quasi-modes (Arnol'd, 1981). Such three-fold clusters really exist for the vibration of H>, one of the most fundamental molecular ions observed by spectroscopists and astronomers (Oka, 1980, 1992). This problem has a small number of particles and is quite interesting from the theoretical point of view since extensive quantum and classical calculations are possible. In the case of H> the calculations based on an ab initio potential surface proved to be very precise in describing the rovibrational structure of experimentally observed low-excited vibrational states. Table 20 shows a few lowest energy levels of a number of vibrational polyads for which the full assignment NlC, N"2J and l"2J , seems to be well established. These data clearly support the X conception of a localization of bending mode since the low-l levels tend to form triply degenerate clusters. More detailed quantum analysis of vibrational wave functions gives further evidences for such localization (Sadovskii et al., 1993). 5.6. Vibrational quantum bifurcations and normal local mode transition It is naturally possible to analyze the qualitative modi"cations of e!ective vibrational Hamiltonians for a series of polyads as the polyad quantum number varies. The formal analysis of this vibrational problem in the case of polyads formed by two quasi-degenerate modes is completely
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
140
Table 20 Bending overtones of H> according to Tennyson and Henderson (1989). Each value gives the distance from the lower level in cm\, for the lowest level of each polyad the absolute energy is given 3A 3A 3E
210 279 7003
4E 4E 4A
899 111 8996
5A 5A 5E
612 60 10853
6E 6E 6A
2 104 12363
equivalent to the qualitative analysis of the rotational structure under the variation of the J value. The qualitative modi"cations for one-parameter family of e!ective Hamiltonians are necessarily characterized by the same type of bifurcations as purely rotational problems. No new types of quantum bifurcations can appear because the stabilizers for vibrational problems are the same as for rotational ones. It is important to remark that the interpretation of bifurcations for the vibrational problem is often done in the complete phase space rather than on the polyad vibrational sphere. It is quite useful to compare the qualitative description of e!ective Hamiltonians on a reduced vibrational polyad space and the qualitative analysis of vibrational dynamics in terms of periodic trajectories in the full vibrational phase space together with the qualitative description of these trajectories in the coordinate space. The crucial point is the correspondence between stationary points on the reduced polyad space for the problem of two quasi-degenerate vibrations and the periodic trajectories on the initial phase space. Bifurcations of stationary points on the polyad sphere correspond to bifurcations of periodic trajectories. The simplest bifurcation of the critical C orbit for vibrational polyads formed by two stretching modes of AB molecules is associated with the transition between the normal mode picture and the local mode picture as discussed in the previous subsection. At the same time it is important to note that for more symmetrical molecules even the simplest Morse functions have several equivalent minima or/and maxima resulting in the appearance of quasi-modes or nonlinear normal modes in the simplest approximation for a generic Hamiltonian function (Montaldi et al., 1988; Montaldi, 1997; Montaldi and Roberts, 1999; Kozin et al., 1999). This is the case of bending overtones of the A molecule (D symmetry). In such a case the local mode model F becomes the "rst simplest approximation and further possible bifurcations can lead to even more complicated system of stationary points and periodic orbits. 5.7. Internal structure of polyads formed by N-quasi-degenerate modes. Complex projective space as classical reduced phase space In order to be able to analyze qualitatively the internal structure of vibrational polyads formed by an arbitrary number N of quasi-degenerate modes we should "rst construct the classical limit phase space for e!ective quantum Hamiltonians which describe the internal structure of polyads. The general scheme of such a construction is based on the generalized coherent state method (Perelomov, 1986; Zhang et al., 1990; Cavalli et al., 1985). We follow here some simple heuristic approach which enables one to see immediately the topological structure of the corresponding classical phase space.
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
141
To construct e!ective quantum Hamiltonians for vibrational polyads we use creation and annihilation operators a>, a and form the products of the form G G (a>)L 2(a>)L, (a )K 2(a )K, , ,
(89)
satisfying the condition n #2#n "m #2#m . , ,
(90)
Only such terms have non-zero matrix elements within vibrational states forming polyad. The action of any given operator in Eq. (89) on any wave function describing one of the states forming the polyad yields physically identical results if these wave functions di!er only by a common complex phase. This restriction is important under the transition to the classical limit. To take it into account we introduce the complex variables z instead of a pair of operators a>, a , put the G G G requirement "z "#2#"z ""1 and identify the complex vectors which di!er by a phase, i.e. , we identify (z ,2, z ) and (z e P,2, z e P). From a mathematical point of view this procedure is , , the constriction of the complex projective space CP which is locally a (2N!2)-dimensional ,\ real Euclidean space. Thus, the qualitative analysis of e!ective Hamiltonians for vibrational polyads in a general case of N quasi-degenerate modes is, in fact, the qualitative analysis of functions de"ned over a complex projective space. The Betti numbers for CP (b "1, b "0, 04i4K) tell us that in the ) G G> absence of any symmetry the minimal number of stationary points is K#1. In the presence of symmetry the analysis of the symmetry action on CP will give us information about the simplest ) Morse-type functions on the reduced classical phase space for the vibrational problem. In fact, the above-mentioned observations about the correspondence between stationary points on the reduced polyad phase space and the periodic trajectories on the initial space establishes this quite important correspondence between the minimal number of stationary points for reduced Hamiltonian for polyads and the number of non-linear normal modes. 5.8. Integrity bases for CP spaces , To realize the qualitative analysis of e!ective vibrational Hamiltonians for polyads we need the e!ective tools to work with functions on the corresponding classical phase spaces which are complex projective spaces. In this section the brief outline of the construction of integrity bases for CP will be given. , Explicit construction of the integrity basis for CP can be done in two steps. First, we L\ construct the system of invariants for n 2-D-vectors (with respect to SO(2) symmetry group). Second, we drop the hyper-radius in the 2n-D initial space from the denominator invariants. Molien function for SO(2) invariants on the 2n-D-space with the identical action of the SO(2) group on n pairs of variables (initial representation is n ) (1)#n ) (!1) ) can be written in terms of an integral over the group
dh 1 p , M1-" L (1!je\ F)L(1!je F)L 2p \p
(91)
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
142
which can be equivalently rewritten as
1 p dh M1-" . L 2p (1!2j cos h#j)L \p This integral can be generally represented as a rational function P (j) M1-" L\ L (1!j)L\ with polynomials P
L\ P(n"1; j)"1 ,
(92)
(93)
given as follows: (94)
P(n"2; j)"1#j ,
(95)
P(n"3; j)"1#4j#j ,
(96)
P(n"4; j)"1#9j#9j#j ,
(97)
P(n"5; j)"1#16j#36j#16j#j ,
(98)
P(n"6; j)"1#25j#100j#100j#25j#j ,
(99)
2 A general expression for the polynomial P
L\
(j) can be equally derived
1 (n!1)(n!2)j P "1#(n!1)j# L\ (2!) 1 (n!1)(n!2)(n!3)j#2#jL\ . # (3!)
(100)
This formula can be rewritten in terms of binomial coe$cients
L\ n!1 P " jI . (101) L\ k I It is well known (Weyl, 1939) that for the diagonal action of the SO(2) group on n-two-dimensional vectors, all invariants can be generated by n quadratic polynomials. These n polynomials are formed by n norms x#y, n(n!1)/2 non-diagonal scalar products x x #y y , (iOj), and G G G H G H n(n!1)/2 anti-symmetric products x y !x y , (iOj). In what follows, the notation with one index G H H G will be used r "x#y, i"1,2, n , G G G s "x x #y y , a"1,2, n(n!1)/2 , ? G H G H t "x y !y x , a"1,2, n(n!1)/2 . ? G H G H
Index a corresponds to lexicographical order of the natural double index.
(102) (103) (104)
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
143
Anti-symmetric products t are linearly independent but all their products t t can be expressed ? ? @ in terms of polynomials in symmetric products and norm. Thus, we can take all anti-symmetric products as numerator invariants. In the case of n"2 this remark resolves the problem of choice of numerator invariants (one numerator invariant and one anti-symmetric product). In the case of n"3 one symmetric invariant should be added to form the complete set of 4 numerator invariants. For general n'2 to form the complete set of (n!1) numerator invariants of degree 2 we should add (n!1)!n(n!1)/2"(n!1)(n!2)/2 symmetric invariants. For su$ciently large n53 we will have equally anti-symmetric numerator invariants of higher degrees 4k#2 (k51). To "nd the number of such invariants we introduce the action of the O(2) group and use it to construct the Molien functions for invariants and covariants
p
dh p dh # , (1!je\ F)L(1!je F)L (1!j)L \p \p 1 p dh p dh M-(u)" ! . L 4p (1!je\ F)L(1!je F)L (1!j)L \p \p Generating functions for invariants and covariants of the O(2) action are, respectively, 1 M-(g)" L 4p
(105) (106)
[P (j)#(1!j)L]/2 M-(g)" L\ L (1!j)L\
(107)
[P (j)!(1!j)L]/2 M-(u)" L\ , L (1!j)L\
(108)
where P (j) are given in Eq. (101). These formulae enables us to give explicit expressions for L\ Molien functions for the CP spaces with additional splitting of numerator invariants into L\ symmetric and anti-symmetric parts with respect to the O(2) action, i.e. with respect to complex conjugation or time reversal: M
P L\ !. " !.L\ (1!j)L\
(109)
with P (j)"1#+j, , !. P (j)"1#(1#+3,)j#j , !. P (j)"1#(3#+6,)j#(6#+3,)j#+j, , !. P (j)"1#(6#+10,)j#(21#+15,)j#(6#+10,)j#j , !. P (j)"1#(10#+15,)j#(55#+45,)j !. # (45#+55,)j#(15#+10,)j#+j,2 .
(110) (111) (112) (113)
(114)
Terms within +2, indicate that the corresponding polynomials are covariant with respect to the O(2) action (i.e. they change sign under time reversal). For su$ciently large n the problem of the explicit construction of the integrity basis requires "rst the splitting of all quadratic invariants into denominator and numerator groups of invariants.
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
144
Then, as soon as we know the number of algebraically independent (denominator) invariants in each degree we can in principle follow the general strategy consisting in choosing a su$cient number of appropriate polynomials with random coe$cients and to verify their algebraic independence (Sturmfels, 1993). Algorithms for such a procedure are described in particular in Chapter 2 of Sturmfels (1993) and can be realized using one of the packages for symbolic computations (for example Macaulay (Bayer and Stillman, 1982)). The algorithms are based on the GroK bner basis construction which proved itself to be extremely useful and popular now for computations in commutative algebra Cox et al., 1992, 1998; Eisenbud, 1995; Vasconcelos, 1998; Becker and Weispfenning, 1993). We remind that the subdivision of invariant polynomials into algebraically independent (denominator) and linearly independent but algebraically dependent (numerator) is called sometimes in mathematical literature the Hironaka decomposition. Su$ciently generic combinations of initial polynomials r , s will be as a rule algebraically G ? independent but for applications it is preferable to construct the integrity basis which is in some sense very close to initial generators. This will be formalized in a requirement to construct linear combinations of initial polynomials with coe$cients being one and zero and keeping as much as possible zeros among the coe$cients. Several such choices will be suggested below. For the CP space each basis can be represented by a rectangular matrix with "ve lines and six columns. The numerotation of columns is (r1, r2, r3, s1, s2, s3). Five lines correspond to algebraically independent invariants for SO(2) action on three-dimensional complex space. One of the simplest choices corresponds to the matrix
1
1
1 0 0 0
1 !1 0 0 0 0 0
0
0 1 0 0 .
0
0
0 0 1 0
0
0
0 0 0 1
(115)
In this case r and r could be taken as two invariants of the numerator. This choice of basis does not respect the intention to use only 0 and 1 as entries of the matrix but it is preferable from the point of view of applications than the next one given below, because the denominator invariant r #r #r is introduced explicitly and this is important from the point of view of going to a CP basis by "xing this denominator invariant to be constant. Slight modi"cation of the basis above gives a good basis for the O(2) group action on three 2-vectors:
1 0 1 0 0 0 1 1 0 0 0 0
0 0 0 1 0 0 . 0 0 0 0 1 0 0 0 0 0 0 1
(116)
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
145
r and r are again two invariants of the numerator. But this basis is less convenient from the point of view of the transformation to a CP basis because to impose condition r #r #r "const. one should take the combination of the denominator and numerator invariants. The most interesting choice of algebraically independent invariants which gave an idea of generalizing the integrity basis construction to CP , CP is based on taking all r polynomials as G algebraically independent and forming all the other polynomials as linear combinations of s : G 1 0 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0 .
(117)
0 0 0 1 1 0 0 0 0 1 0 1
s and s can be taken as numerator invariants in this case. Naturally, an arbitrary permutation of the last three columns gives an equivalently good basis because it corresponds simply to a permutation of numbers of variables. For CP we start by taking all four r , r , r , r norms as algebraically independent polynomials for the action of SO(2) on the four two-vectors (it is clear that in this case we can easily eliminate the hyper-radius from denominator invariants). To complete the set of denominator invariants we add three linear combinations of six linearly independent scalar products s , s , s , s , s , s . The choice of good linear combinations is given below in the form of a three by six matrix. For example, we have 1 1 0 0 0 1
0 1 1 0 1 0 .
(118)
0 0 1 1 0 1
As three second degree numerator invariants we can take s , s , s and as four six-degree numer ator invariants s , s , s , s s , s s , s s . For CP the choice becomes less evident. It is again possible to choose as algebraically independent polynomials "ve polynomials r , r , r , r , r and four linear combinations of s , ? a"1,2, 10. The matrix corresponding to a good basis with standard lexicographical order for s looks like ? 1 0 1 1 0 0 0 0 1 0 0 1 0 0 1 1 0 0 1 0 . (119) 0 0 0 1 0 1 1 1 0 0 0 1 1 0 0 0 1 0 0 1
The author does not know similar explicit solutions for the integrity bases for CP with N'4. , 5.9. Finite symmetry group action on CP The analysis of the symmetry group actions on CP space was studied by Zhilinskii (1989a). The most important results of such an analysis is a system of zero-dimensional strata leading to critical orbits.
146
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
Fig. 21. Action of the O symmetry group on CP space. There are two-dimensional strata and "ve zero-dimensional strata which are schematically shown. Filled squares are D orbits; Filled triangles are D orbits; Empty squares are C orbits; Empty triangles are C orbits; Filled ellipses are D orbits. Nine 2-D objects should be imagined to be S spheres forming in 4D-space many contact points of high symmetry. Among the nine S spheres three belong to one stratum formed by C orbits while six other S spheres form another C stratum.
For example, just on the basis of the analysis of the group action it is possible to predict the existence of 63 non-linear normal modes for AB molecule. This statement which appeared "rst in Montaldi et al. (1987) on the basis of complicated non-linear analysis of periodic trajectories near equilibrium for Hamiltonian system (Montaldi et al., 1988) was considered initially by the molecular community as an abstract curiosity. At the same time these 63 non-linear normal modes simply correspond to stationary points on reduced e!ective vibrational Hamiltonians for di!erent vibrational polyads formed by vibrational modes in the molecule. In particular, for AB there exists one non-degenerate vibration, one doubly degenerate, and two triply degenerate vibrations. For polyads formed by a non-degenerate l mode the system of stationary points on the `reduced polyad phase spacea includes one point because the reduced space is trivial and includes itself just one point. Reduced phase space for polyads formed by a doubly degenerate mode is a two-dimensional sphere. The image of the group acting on this space is D which leads to eight stationary points corresponding to three critical orbits (assuming the Hamiltonian function to be the simplest Morse-type one). At last the reduced polyad space for each triply degenerate mode is CP with the action of the group O on it. Fig. 21 shows the action of the O group on CP space (Zhilinskii, 1989a). This action leads to 27 stationary points situated on zero-dimensional strata. All that gives for the complete vibrational problem 1#8#27#27"63 stationary points corresponding to non-linear normal modes.
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
147
5.10. Continuous symmetry group action on CP The continuous symmetry group action on CP space arises naturally for linear molecules with resonances between three vibrational modes. The most natural is the resonance condition between doubly degenerate bending modes and one of non-degenerate stretching modes. This leads to the construction of an e!ective Hamiltonian de"ned over the CP space in the presence of SO(2) or O(2) symmetry. Due to the presence of continuous symmetry the space of orbits in this case has dimension three (we remind that the SO(2) group is a one-dimensional Lie group) and can be rather well visualized. Moreover, the 3D orbit space can, in fact, be sliced by surfaces corresponding to a constant value of the second integral of motion associated with continuous SO(2) symmetry. Such geometrical analysis was realized for example recently by Cushman et al. (1999). It is reasonable as well to introduce a model with even higher SO(3) symmetry. Nuclear and particle physics use naturally various Lie groups for mathematical models of physical systems but we do not touch here this enormous "eld. Nevertheless, three vibrational modes of a molecular system can be approximately considered as transforming according to an irreducible representation of the group SO(3) of weight (1). The internal structure of vibrational polyads in this case becomes in some sense trivial because the space of orbits of the SO(3) action on CP is one dimensional but this model can be served as one of possible limiting case describing the internal structure of polyads formed by triply degenerate modes. 5.11. Nontrivial n : m resonances The construction of the CP classical phase space realized in previous sections is strictly L speaking applicable in the case of 1 : 1 : 2 : 1 resonance. General n : 2 : n resonances result in I classical phase spaces with more complicated strati"cation. Corresponding dynamic problems are actually under study in the non-linear mechanics but this analysis is still far from applications to concrete molecular models. 5.12. Vibrational polyads for quasi-degenerate electronic states Vibrational structure of two or several quasi-degenerate electronic states can be analyzed in formal analogy with the rotational structure of several vibrational states. This analogy becomes especially close if the restriction to vibrational polyads is possible for coupled electronic states. This approach gives another possibility to look at such well-known phenomena as dynamical Jahn}Teller (Jahn and Teller, 1937; Jahn, 1938; Englman, 1972) and Renner (Jungen and Merer, 1976) e!ects.
6. Rovibrational problem We have partially studied the rovibrational problem by analyzing the rotational structure of N quantum states in Section 4. Now, we return to the rovibrational problem but taking the classical limit in both rotational and vibrational variables. In order to work with the compact
148
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
phase space we will assume the existence of vibrational polyads and study the rotational structure of vibrational polyads. The case of the rotational problem for vibrational polyads formed by two quasi-degenerate modes corresponds to the analysis of the classical Hamiltonian (energy function) de"ned over the classical limit phase space which is a direct product of two two-dimensional spheres S ;S . This problem is equivalent to the problem of coupling of two angular momenta which was analyzed in a semi-quantum approach in Section 4.3. Now, we study the same problem from pure classical point of view. 6.1. Model problem: coupling of two angular momenta. Quantum and classical monodromy We return to the model problem studied in Section 4.3 1!c c H" S # (N ) S), 04c41 , X "S" "N ""S"
(120)
but now we suppose "S" to be arbitrary (with the only restriction "S"("N") and treat this problem as classical. Thus, the main idea is to compare global features which are present for a one-parameter family of Hamiltonians in Eq. (120) in a completely quantum problem (both angular momenta are quantum operators), in a semi-quantum description (one angular momentum, say S, is a quantum operator while N is a classical object), and in a completely classical picture (both N and S variables are classical). We follow here the general program of comparative qualitative quantum-classical non-linear analysis (Sadovskii and Zhilinskii, 1993a; Zhilinskii, 1996; Sadovskii et al., 1996) and our goal now is to understand the e!ect of redistribution of energy levels between branches in completely classical terms. When "S"'1/2 quantum energy-level pattern shows several redistributions between (2S#1) rotational components. Comparison between semi-quantum and totally quantum picture is given in Fig. 22.
Fig. 22. Extremal points (dashed lines) on classical energy surfaces for three-level (S"1), and four-level (S"3/2) problems and quantum energy levels (full lines) for N"4 in two cases.
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
149
The classical limit has an analytical solution for all rotational energy surfaces even for arbitrary S: S E X " X ((1!c)#2c(1!c) cos(h)#c , 1 "S"
(121)
S "S, S!1,2,!S#1,!S . X The solutions for energies of stationary points are shown in Fig. 22 along with quantum energy levels. One should note that the disagreement between classical and quantum description in the region of c&1 decreases when S/NP0. It is clear (see also Fig. 22) that for high S the model considered leads to conical intersection of all rotational energy surfaces. This extremely nongeneric situation is due to a very simple form of the Hamiltonian. At the same time after deformation of the Hamiltonian we can have only conical intersections between pairs of rotational surfaces. Since "S" and "N" are conserved, the phase space of our problem is S ;S , the product of two spheres, and the number of degrees of freedom equals 2. Indeed, each sphere S is de"ned in the respective 3-space (S , S , S ) and (N , N , N ) as V W X V W X S#S#S""S", N#N#N""N" . (122) V W X V W X Furthermore, the Hamilton function in Eq. (120) is invariant with respect to the continuous symmetry CX and J "S #N , +H, J ,"0 (123) X X X X is the corresponding integral of motion. Thus, the classical problem can be reduced to J "const. X subspace but one should take into account the fact that some of J "const. subspaces are singular X and only singular reduction can be applied. To understand global behavior of the dynamical system considered we need to study the phase portrait on the complete phase space (i.e. on all regular and singular reduced phase spaces together) and moreover as a function of the external control parameter c. To represent the classical phase portrait we can use instead of the four-dimensional phase space the three-dimensional space of orbits of the symmetry group G"SO(2)Z . We realize this construction in two steps. First, we construct orbits of the SO(2) action which are in one-to-one correspondence with points of the J reduced phase space from one point of view and with di!erent X relative con"gurations of three vectors S, N, n , where n is a unit vector in the direction of the X X z-axis, from another more formal geometrical point of view. To label the SO(2) orbits we use three algebraically independent invariants S , N , and m"NS which characterize, respectively, the X X projection of S and N on the z-axis, and the angle between S and N, and as an additional algebraically dependent (but linearly independent) invariant the triple product p"(n (SN)) X (in fact just the sign of this invariant is su$cient). S , N , m, and p form the integrity basis for the X X SO(2)Z action on S ;S . In the three-dimensional space S , N , m of `denominatora invariants, X X all points corresponding to the SO(2) orbits are inside and on the boundary given by p"0: p"N S#2mN S !m!N S!SN . X X X X
(124)
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
150
Table 21 Critical orbits of the SO(2)Z action on the phase space Orbit
N /"N " X
S /"S" X
m/"N ""S"
Energy
J X
K X
A B C D
1 !1 1 !1
1 !1 !1 1
1 1 !1 !1
1 2c!1 !1 1!2c
"N "#"S" !"N "!"S" "N "!"S" !"N "#"S"
!"N "#"S" "N "!"S" !"N "!"S" "N "#"S"
Fig. 23. Space of orbits (left) of the G action on S ;S . Sections of orbifold by planes corresponding to constant J values are shown. Figure is done for N"4, S"1. Sections correspond to J "!4.5,!4,2, 4, 4.5, 4.8. On the right X X J "!"N"#"S" section is shown in K !m variables sliced by constant energy sections for the Hamiltonian in Eq. (120) X X with c"1/2.
To remove the dependence on "N " and "S" of the geometrical form of the space of orbits we can use scaled variables S /"S", N /"N ", m/("N ""S"). Four vortices of the orbifold are critical orbits which are X X points with the SO(2)Z stabilizer. All other points on the boundary of the orbifold correspond to circles (the stabilizer is Z ), and all points inside to a couple of circles (generic orbits with trivial stabilizer). Four critical orbits A, B, C, D correspond to extremal values of N and S . They are X X explicitly characterized (see Table 21) by their positions on the orbifold, their energy, and the value of the projection of the total angular momentum (second integral of motion). We remark that the space of orbits constructed here is identical with the space of orbits in the case of a Rydberg state problem of an atom in the presence of parallel electric and magnetic "elds discussed in details in Chapter III. A family of J reduced phase spaces (more strictly orbits of the Z action on reduced phase X space) is represented in Fig. 23 as sections of the orbifold by planes J "N #S with di!erent X X X J values. The most part of these sections is regular but those passing through points A, B, C, D are X singular. Fig. 23 shows as well one example of a singular J section using as coordinates m and X K "S !N . This section represents the reduced phase space as a space of Z orbits. All internal X X X regular points correspond to a pair of circles in initial phase space (characterized by opposite values
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
151
Fig. 24. Energy momentum diagrams for a Hamiltonian in Eq. (120) with di!erent values of the parameter c. (Figures are constructed for N/S"4.)
of p invariant and related through the Z action). All regular points on the boundary correspond to one circle in the initial 4-D phase space. An exceptional point on a singular section is just a point in the initial space. To see better the dynamical meaning of the reduced phase space we should consider each J section as a space of SO(2) orbits only and to use auxiliary invariant p as a third X dynamical variable which form together with m and K the Poisson algebra which can be X transformed with the appropriate change of variables to a standard form of the algebra for the three components of the angular momentum. For all regular values of J the reduced phase X space is therefore a S sphere but for the singular value of J the reduced phase space is X a topological sphere with one exceptional point. Each J section can be sliced further into levels of constant energy by "nding the intersection X with H"const. plane of constant energy. Due to an appropriate choice of variables, H"const. sections are represented in Fig. 23 as straight lines. For a given J section the energy can take values between E (J ) and E (J ). Plotting X
X
X E (J ) and E (J ) as functions of J we arrive at the energy}momentum diagram (see Fig. 24).
X
X X One point lying on the boundary of the energy}momentum diagram corresponds to one point on the space of G orbits and to one point on the space of SO(2) orbits. At the same time one point lying inside the energy}momentum diagram corresponds to an interval on the space of G orbits and to a circle on the space of SO(2) orbits. To "nd the inverse image of di!erent energy sections in the initial phase space we need to distinguish regular and singular sections. The inverse image of the regular energy and J section is either a torus (for an interval) or a circle (for a point). A singular X section corresponds to one point, or to a pinched torus. The last situation takes place only for singular J sections and the energy sections going through singular points with an intermediate X energy value (which is less than the E and more than E for given J ).
X To analyze now the global phase portrait and its variation with c we look at the evolution of the energy}momentum maps for di!erent c (see Fig. 24). For two limiting cases c"0 and 1 the energy momentum diagrams have very simple geometrical form for a Hamiltonian in Eq. (120). c"0 corresponds to parallelogram and c"1 is a slightly curved trapezoid with paraboloid lateral sides. If we represent on the classical energy momentum diagram energy levels for the quantum problem, we have in both limiting cases a regular lattice formed for c"0 by 2N#1 dots in 2S#1 horizontal lines and for c"1 the same number of lines but with a di!erent number of dots in di!erent lines. When c varies between 0 and 1, critical orbits B and D on energy}momentum diagram change their energy but keep their J values (see Table 21). X
152
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
Fig. 25. Pinched torus.
Important is the fact that all four critical orbits are on the boundary of the energy}momentum map when c is close to its limiting values. At the same time it is clear that in the intermediate case one of the critical orbits will be inside the energy}momentum map. Our analysis shows that any regular point (which is not on the boundary) on the energy} momentum diagram corresponds to a 2D torus in the initial phase space. A singular point when it is inside the energy}momentum diagram corresponds to a pinched torus (see Fig. 25). Remark that for a similar problem with higher symmetry (quadratic Zeeman e!ect in the presence of an orthogonal electric "eld (Cushman and Sadovskii, 1999) can be served as an example) the number of pinched points on the torus can be larger due to the symmetry. A regular point on the boundary of E}J diagram corresponds to a circle and a singular point on X the boundary to one point on the initial phase space. In classical mechanics it is known that the presence of an isolated pinched torus surrounded by a regular torus is related to monodromy, i.e. the obstruction to the existence of the global action-angle variables. Such situation with an exceptional pinched torus surrounded by regular torus was studied in detail recently for simple classical and quantum mechanical problems [spherical pendulum (Cushman and Duistermaat, 1988; Guillemin and Uribe, 1989), champagne bottle (Child, 1998)] and for more general case (Ngoc, 1999) and was related with classical monodromy. Recent work by Sadovskii and Zhilinskii (1999) supplies another example of this phenomenon which exists for the model Hamiltonian (120) but only for a subset of possible values of parameter c. To "nd the range of c corresponding to Hamiltonians with monodromy we need to calculate the intersection of the J section with the surface of the orbifold. X The section corresponding to J ""S"!"N " (and similar to J ""N "!"S") goes through a critiX X cal orbit (see Fig. 23). Tangent vectors to the path of intersection of an orbifold and a J section at X a critical orbit are of particular importance because they enable us to calculate the region of parameters of the Hamiltonian when the monodromy is present. As soon as a critical orbit is a singular point of the orbifold and of the J section we have two tangent vectors. The angle X a between these two tangent vectors has the form 2"S"#2"N "#"N ""S" . cos a" (4"N "#4"S"#8"N ""S"#8"N ""S"#9"N ""S"
(125)
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
153
Fig. 26. Quantum energy levels for N"16, S"4 and c"0.5 plotted in energy}momentum variables (to compare with classical energy}momentum map in Fig. 24).
It is clear that if "S"/"N "P0, this angle is going to zero. This can be easily seen in Fig. 23. J "$("N "!"S") sections in this limit are close to AC and BD straight lines which belong to the X boundary of the orbifold. Now, we can take constant energy sections of a Hamiltonian and "nd the range of values of the parameter corresponding to the existence of a monodromy. The condition is the orthogonality between the normal vector to the energy section and the tangent vector at singular point. For the Hamiltonian in Eq. (120) the region with monodromy (corresponding to critical orbit with J "!"N "#"S") is X "N " "N " 4c4 . (126) 2"N "#"S"!2("N ""S" 2"N "#"S"#2("N ""S" In the case of S/N going to zero the domain of c associated with monodromy is going to one point c"1/2. Standard manifestation of a classical monodromy in the quantum energy spectrum was given either through the analysis of the defect of the lattice of quantum levels represented on the energy}momentum diagram around the monodromy point or in terms of the logarithmic singularity of the density of states (Cushman and Duistermaat, 1988; Child, 1998; Ngoc, 1999). To see the quantum monodromy for the Hamiltonian in Eq. (120) we plot in Fig. 24 quantum energy levels on the energy}momentum map for N"16, S"4 (this implies the same ratio N/S"4 as in Fig. 24) and for the parameter c"1/2 corresponding to the presence of a classical monodromy (see Fig. 26). It is clear that in any local simply connected domain which does not include the monodromy point (shown by a white disk in Fig. 24) the quantum energy levels form a regular 2D-lattice with integer quantum numbers (m, n) which order the quantum states by J and energy. This means that X we can take an elementary cell of this lattice (four quantum states labeled each by a pair of quantum numbers (m, n), (m, n#1), (m#1, n), (m#1, n#1), shown in Fig. 24 as a shadowed cell) and to move it through the lattice without ambiguity using local parts of a regular lattice. If we "nally move the elementary cell through the closed path around the monodromy point (see path
154
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
shown in Fig. 24 left) the e!ect of monodromy on the quantum spectrum is evident. After completing a closed path around the monodromy point, the elementary cell considered as a part of a regular lattice has been subjected to the unimodular transformation over integers. This is exactly the statement about the absence of globally de"ned action-angle variables or globally de"ned sets of two quantum numbers. Namely, this interpretation of classical monodromy in quantum problem was discussed in earlier publications (Cushman and Duistermaat, 1988; Child, 1998; Ngoc, 1999). The analysis of quantum-classical correspondence realized by Sadovskii and Zhilinskii (1999) suggests another more clearly visible in physical applications manifestation of classical monodromy. Namely, the redistribution of quantum energy levels between di!erent branches in the energy spectrum, under the variation of a control parameter, should be considered as a "ngerprint of the classical monodromy. The discussed phenomenon of the redistribution of energy levels between di!erent branches can be easily located even in the case of a su$ciently low number of quantum states. This phenomenon is apparently topologically stable and should be present even after deformation to non-integrable case. 6.2. Rotational structure of bending overtones in linear molecule Qualitative analysis of vibrational polyads formed by bending modes of a linear molecule requires a natural generalization of the problem described in the previous section and a generalization of the monodromy e!ect. Let us consider just linear four-atomic molecules with two doubly degenerate bending modes. If two vibrational angular momenta associated with each bending mode can be considered as approximately good quantum numbers then for "xed value of these vibrational momenta the internal structure of corresponding group of levels is completely described by the Hamiltonian studied in the previous section. Thus, the presence of monodromy is possible. Additional degrees of freedom will ensure the presence of monodromy phenomenon for a family of additional integrals of motion. The analogy with the description of defects of crystalline structure could be useful in further analysis. 6.3. Rotational structure of triply degenerate vibrations. Complete classical analysis Qualitative analysis of the rotational structure of a triply degenerate vibrational state can be done using the semi-quantum approach, i.e. by taking the classical limit in rotational variables and analyzing the classical rotational problem for three quantum states within the e!ective 3;3 matrix symbol. Let us take as an example the e!ective Hamiltonian of an octahedral molecule in a triply degenerate vibrational state. The molecular prototype of this example is the l (F ) band of S Mo(CO) studied experimentally by Asselin et al. (2000) and analyzed further from the classical point of view by Dhont et al. (2000). The corresponding e!ective Hamiltonian can be represented as a series of coupled terms HX) LC constructed from vibrational and rotational tensor operators, <C and RX) LC, respectively, HX) LC"[<C RX) LC] E .
(127)
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
155
Vibrational operators are in turn built as symmetrized tensor products of creation and annihilation operators for a triply degenerate (F ) vibrational mode, a>$S and a$S , S
$S a$S ]C , (128) N N while rotational tensor operators are constructed from elementary angular momentum operators J , J , and J (Champion et al., 1992). Including all possible rovibrational operators up to order V W X three, we obtain the Hamiltonian H"k < E #k RE E #k HE $E #k HE E #k HE #E #k HE $E # k HE $E #k HE $E . (129) The "rst three terms in H describe the degenerated three-dimensional harmonic oscillator, the spherical top rigid rotor, and the Coriolis interaction, respectively. The notation of coe$cients, k G (i"1,2, 8), does not follow the standard tensorial notation in order to simplify expressions. Parameters k , k , and k de"ne the scalar contribution to the rotational energy 1 4 4 E (J)" k ! k J! k J , (130) Q 3 (3 (3 where (131)
J"(J(J#1)
is the amplitude of the angular momentum and J is the rotation quantum number. The term E (J) Q is common to all levels within the rotational multiplet of a triply degenerate band. Consequently, the parameters k , k , and k are of no interest to the study of the internal structure of this multiplet. Using the vibrational basis functions +"v , v , v 2,"+"1, 0, 02, "0, 1, 02, "0, 0, 12, (132) V W X and writing in explicit form the tensor components in Eq. (129) we get the following standard matrix representation of the Hamiltonian H within the semi-quantum model:
HH HH HK " H (133) H HH H H H with H being functions of J, h, . GH To construct a purely classical analog of the quantum Hamiltonian in Eq. (129) we should transform quantum operators of vibrational coordinates and conjugated momenta (q( , p( ), G G i"1, 2, 3 of the triply degenerate oscillator to classical variables. To this end it su$ces to replace (q( , p( ) for the classical variables (q , p ). Equivalently, the classical analogs of the three creation and G G G G annihilation operators are the three complex variables together with their conjugates (Bargmann, 1961, 1962) H
(a>, a )P(zH, z )"(q #ip , q !ip ), I I I I I I I I
k"1, 2, 3 .
156
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
Next step is to introduce the integral of motion (134) N"(z zH#z zH#z zH) corresponding to the polyad quantum number n (total number of quanta in the vibrational mode) of the quantum Hamiltonian. This integral is the harmonic oscillator part of the vibrational Hamiltonian, or equally the sum of actions of the three one-dimensional oscillators. The assumption that n is preserved introduces an approximate dynamical symmetry. Reduction of this oscillator symmetry (see, for example Appendix B in Cushman and Bates (1997)) leads from the initial six-dimensional Euclidean phase space R to a reduced phase space of real dimension 4 which is a complex projective space CP . This space can be described in terms of homogeneous quadratic polynomials constructed of z and zH. Analysis of the topological and group-theoretical features related with this model was initiated by Zhilinskii (1989a) (see Sections 5.7 and 5.9) and continued later in Sadovskii and Zhilinskii (1993b). The total classical phase space of our model rovibrational Hamiltonian for the triply degenerate (F ) mode is the product CP ;S (for other examples of classical phase spaces of this kind see S (Sadovskii and Zhilinskii, 1993a; Zhilinskii, 1996)) Due to the action of the symmetry group O there are several critical orbits on this space. These critical orbits are necessarily stationary F points of any smooth Morse function de"ned over CP ;S . Critical orbits are de"ned entirely by the action of the O group on the vibrational CP and rotational S spaces, respectively. More F precisely, since the spatial inversion does not modify the angular momentum components and quadratic vibrational polynomials, the action of the O group reduces to that of the O group. F (The image of O in the space of rotational and quadratic vibrational variables is O.) The action of F the O group on CP has been studied in detail in Zhilinskii (1989a) (see Section 5.9, and Fig. 21 in particular) and the action of this group on the rotational space S [on (J , J , J )] is, of course, well V W X known (see Chapter I). The seven critical orbits of the O group action on the CP ;S space are characterized in Table 22, where for each orbit we specify its stabilizer on the total space, and on the vibrational and rotational subspaces CP and S , the number of points in the orbit, and the branch assignment in the limiting case of normal Coriolis splitting. Positions of these critical orbits can be obtained directly from Table 4 of Zhilinskii's paper (1989a) and from the positions of critical orbits of the O group action on S . In Table 22 we give these positions in terms of complex vibrational coordinates z , k"1, 2, 3 and rotational coordinates J , a"x, y, z. Since the energy I ? (and everything else) is exactly the same for all points on the same critical orbit, only one point in each orbit is represented. All points in the critical orbits in Table 22 are stationary points of any Morse function (a smooth function with only non-degenerate stationary points) de"ned on our phase space CP ;S . However, using simple topology arguments we can show that such a function should also have additional stationary points. We can also suggest that an O-symmetric Morse function on CP ;S would have two additional non-critical C orbits of stationary points. Using the description of the O group action (Zhilinskii, 1989a) we can "nd that the points in question project on the C -invariant spheres S in CP . It is possible to characterize them further by taking the time-reversal symmetry of our rovibrational Hamiltonian into account. Action of time-reversal operation T on the classical variables (z , z , z , J , J , J )P(zH, zH, zH,!J ,!J ,!J ) V W X V W X
(135)
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
157
Table 22 Critical orbits of the action of the O group and stationary points of the Hamiltonian function on the classical phase space CP ;S . Reversing symmetry T "TC is used to restrict non-critical C orbits to the corresponding invariant sub-manifold S where the two orbits x and y represent the two stationary points of a generic Hamiltonian function Stabilizer of the orbit
Position on CP
Position on S
Type of orbit
Total
vib
rot
(z , z , z )
(J , J , J )
C
D
C
J(1, 0, 0)
crit
C
C
C
J(1, 0, 0)
crit
C
C
C
N(1, 0, 0) N (0, 1,#i) (2 N (0, 1,!i) (2
J(1, 0, 0)
crit
C
D
C
(3
C
C
C
1 1 N 2 ,! !i,! #i 2 (3 (3 (3
(3
C
C
C
1 1 N 2 ,! #i,! !i 2 (3 (3 (3
(3
C
D
C
(2
C T
C T
C T
C T
C T
C T
N
J
(1, 1, 1)
(3
N
(1#x N (1#y
J
J
(0, 1, 1)
N
J
(2
ix,
iy,
1
1 ,! (2 (2 1
1 ,! (2 (2
J (2 J (2
(1, 1, 1)
crit
(1, 1, 1)
crit
(1, 1, 1)
crit
(0, 1, 1)
crit
(0, 1, 1)
non-crit
(0, 1, 1)
non-crit
is equivalent to simultaneous complex conjugation on CP and inversion on S . Since the Hamiltonian itself is, of course, invariant with respect to T, we can extend its initial symmetry group from the spatial group O to O T which on CP ;S becomes OT. F F Analysis of the OT action on CP ;S indicates the presence of a one-dimensional stratum formed by 12-point orbits whose stabilizer is the four-element group C ;T "[Id, C , (C T), (C T)] . (The C subgroup of this group contains the C element and the second order-2 subgroup T contains a reversing operation T "C T; the so-called `diagonala axes C and C are orthogonal to one of the C -axis.) This stratum is a union of 12 isolated S circles. Each circle lies on the corresponding C -invariant sphere with stabilizer C . A Morse function has two stationary points on S . These points are at the same time stationary on the C -invariant sphere in CP and on the complete space CP ;S due to the OT symmetry action. Of course, the exact position of the two stationary points depends on the concrete Hamiltonian function, but restricting this
158
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
function to one of the invariant circles greatly facilitates the computation. Having done all preparatory analysis described above and having transformed our Hamiltonian in Eq. (129) to its classical analogue H(z, J) we can appreciate the results. Indeed, we obtain the energies of all relative equilibria corresponding to critical orbits (in particular the energies of all C and C symmetric relative equilibria) after a trivial substitution of classical dynamical variables z and J in the Hamiltonian function H for the coordinates of the points in Table 22. If the scaling constant corresponding to the value of the oscillator integral N is set to 1, resulting analytic expressions which are given below in Eqs. (136)}(138) are identical to those obtained from the analysis of the eigenvalues of the 3;3 matrix Hamiltonian (133) for the l "1 state (Dhont et al., 2000). To "nd explicitly the position of the two remaining C -symmetric stationary points and the energy of the corresponding relative equilibria, we restrict our classical Hamiltonian H to the S subspace of CP indicated in Table 22. The coordinates x and y of the two points are de"ned in terms of the polar coordinate on the circle S . They are obtained as solutions of a simple quadratic equation in de"ning the minimum and the maximum of H on S . Of course, positions x and y of the non-critical stationary points depend on the Hamiltonian parameters and the values of J. As before, after replacing variables z and J for the coordinates of the C symmetric points in Table 22 and using the solutions for x and y (with N"1) we obtain expressions listed together with solutions for critical orbits below in Eq. (138). The so obtained energies for nine stationary orbits have rather simple analytical expressions. For the C -axis we have 4(2 k J , E (J)"E (J)! Q 3 2 2(2 4(2 4 E (J)"E (J)# k J$ k JG k J$ k (2J#J) , ! Q 3 3 3 (15 (136)
where E is the scalar contribution in Eq. (130). The three eigenvalues for C axis are Q 8(3 k J , E (J)"E (J)! Q 9
2 4 4(2 4 4(3 k J$ k JG k JG k J!J . E (J)"E (J)# ! Q 3 9 3 (15 3
(137)
The three eigenvalues for the C -axis are (2 2 E (J)"E (J)! k J! k J , Q 3 (3 1 1 (2 k J# k J$ J(3aJ#6b E (J)"E (J)# ! Q 6 6 (3
(138)
with a"2k !(6k , 2 8 k J# k (4!2J) . b"2k ! (3 5
(139)
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
159
Quite remarkable is the coincidence of these analytical expressions obtained in complete classical analysis with analogous expressions for energies of stationary points on the rotational surfaces found in a semi-quantum approach, i.e. through the diagonalization of 3;3 matrix in Eq. (133).
7. Microscopic models of qualitative phenomena All our previous models were, in fact, phenomenological ones based on e!ective Hamiltonians depending on parameters reconstructed, in principle, from experimental data. The microscopic approach can be alternatively formulated which is oriented to the description of the same type of qualitative phenomena but starting from a non-empirical Hamiltonian or more properly speaking from the intra-molecular potential in the adiabatic approximation for individual electronic state. The main idea of such approach is to relate the qualitative features of rovibrational energy-level patterns with characteristics of the intra-molecular potential and to demonstrate the persistence of qualitative features under small (physically reasonable) deformations of the potential. 7.1. Microscopic theory of four-fold cluster formation in non-linear AB molecules The non-linear AB molecule gives probably the simplest demonstration of the qualitative predictions of modi"cations of the rotational structure based on the intra-molecular potential. The qualitative physical model is extremely simple (Zhilinskii and Pavlichenkov, 1988; Pavlichenkov, 1993; Kozin and Pavlichenkov, 1996). The three principal inertia moments of a non-linear tri-atomic molecule are generically di!erent. These molecules are normally of the asymmetric top type. Ocassionally, due to some relation between the masses and the equilibrium angle the two-in-plane inertia moments can become equal. In such a case the molecule becomes an accidental symmetric top. For non-rigid molecules the e!ective inertia moments corresponding to a rotation of the molecule around stationary axes naturally vary with angular momentum due to a centrifugal distortion e!ect. Thus, for the AB molecule with I (I (I "I #I in the equilibrium ! ! non-rotating con"guration (see Fig. 27) at low rotation (low J values) the axis A (which is orthogonal to the plane of molecule) is the axis of stable rotation corresponding to the critical orbit on the rotational phase sphere associated with the minimum on the rotational energy surface. The axis C is the axis of stable rotation corresponding to the critical orbit on the rotational phase sphere with the maximum energy and the axis B is the unstable stationary axis corresponding to a critical orbit of saddle points on the rotational energy surface. Remind that the symmetry group of the e!ective rotational Hamiltonian (or of rotational energy surface) for the AB non-linear molecule is D (see Fig. 3). F For a rotating molecule (under increase of J value) the centrifugal distortion e!ects will modify mainly the angle between the chemical bonds and to some extent the length of bonds A}B themselves. This causes the instantaneous moment of inertia I (J) to increase and it is possible that ! I becomes larger than I for some J value. The re#ection of this modi"cation on the rotational ! energy surface is the modi"cation of stability of the corresponding critical orbit. Such modi"cation is necessarily related with the bifurcation of the stationary point which generically leads to the formation of a non-critical stationary orbit formed by four points. The presence of a stable
160
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
Fig. 27. Centrifugal distortion e!ect for AB non-linear molecule in the case of classical rotation around axis C. For low J values the axis C is stable and axis B is unstable, whereas for high J values the axis C becomes unstable. The axis A which is orthogonal to the molecule plane remains always the stable rotation axis.
four-point orbit implies the existence of four-fold clusters in the rotational energy-level system. The presence of this phenomenon mainly depends on the equilibrium con"guration of the AB molecule and the atomic masses. The most favorable examples are the tri-atomic molecules with inter-bond angle slightly more than p/2 and the central atom A much heavier than atom B. That is why H Se and H Te were the two molecules for which this e!ect was observed experimentally (Kozin et al., 1992; Tretyakov et al., 1992) and a series of numerical calculations were done to con"rm the appearance of four-fold clusters (Makarewicz, 1990; Jensen and Kozin, 1993; Kozin and Jensen, 1993a,b; Coudert, 1998; Makarewicz, 1998). 7.2. Rotational structure and intramolecular potential To demonstrate another example of qualitative characterization of the rotational structure let us consider again the spherical top molecule with O symmetry of the e!ective rotational HamilF tonian. Phenomenological arguments used earlier in this chapter tell that the simplest Morse-type function which describes the rotational energy surface in the simplest approximation can be of two types with the same number of stationary points but with the interchange of positions of minima and maxima at critical C and C orbits. The transformation from one to another type corresponds just to changing the sign of the only phenomenological constant before the fourth-order rotational operator invariant with respect to the O group. Can we get a simple F answer which relates the sign of this phenomenological parameter with the adiabatic potential? Naturally, the answer can be obtained if we follow the standard procedure of calculating the parameters of e!ective rotational Hamiltonians starting from Wilson}Howard Hamiltonian, which is a function of the angular momentum J, and of the 3N!6 internal coordinates q and I conjugated momenta p , I ,\ p k(q) (J!p)# I #<(q) , (140) H "(J!p)2 2 2 I where p is a vector bilinear in Cartesian coordinates and momenta which corresponds to the angular momentum induced by the vibrations, k is the inverse matrix of the modi"ed inertia tensor, and < the inter-nuclear potential depending on the internal coordinates q"q , q ,2, q . ,\
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
161
Many formulae exist which give the relation between parameters of the Hamiltonian in Eq. (140) which are called often molecular parameters and phenomenological parameters of di!erent e!ective Hamiltonians (Amat et al., 1971; Papousek and Aliev, 1982; Aliev and Watson, 1985). We want to formulate here some conclusions which are independent in some way on the form of the potential used in the Hamiltonian in Eq. (140). Let us consider the four-atomic molecule A . If one supposes that the potential has an arbitrary pairwise form the rotational energy surface has always (in the limit of small distortion) minima at C critical orbit and maxima at C critical orbit (VanHecke et al., 1999). This answer can be obtained through the classical approximation for the rotation around stationary axes (critical orbits which are due to symmetry and independent on the form of potential). A more complicated question concerns the qualitative description of the rotational structure of excited vibrational states. We have discussed in the previous section the complete classical description of the rotational structure of the triply degenerate vibrational band based on the phenomenological Hamiltonian. It is possible to combine this approach with the classical transformation of the molecular Hamiltonian to the e!ective Hamiltonian for excited vibrational states. This procedure is based on the normal form transformation to the reduced Hamiltonians taking into account the approximate integrals of motion.
8. Conclusions and perspectives We demonstrated in this chapter the application of qualitative methods to the analysis of di!erent molecular problems. We have studied mainly examples with very low number of degrees of freedom but the symmetry analysis was not restricted to only low contributions. Our main idea was to show how one can describe the qualitative features within di!erent molecular models without making concrete numerical calculations. This qualitative language seems to be quite useful in analyzing new, not yet fully explored regions of molecular spectra and intra-molecular dynamical behavior. On the other side we wanted also to remind to theoreticians working in the "eld of dynamical (especially Hamiltonian) systems that molecular systems supply many interesting examples to test mathematical models and to make interesting physical predictions based on the general analysis of the Hamiltonian systems with symmetry.
Appendix A. Tables of the rotational energy surface types of individual vibrational components of CF4 The summary of the qualitative analysis of the rotational structure for di!erent vibrational states of the tetrahedral CF molecule is given in this appendix in the form of tables indicating for di!erent components and di!erent ranges of J values the qualitative type of corresponding rotational energy surfaces. 0L and 0G stand for the simplest Morse-type functions with minima, respectively, on the C and C strata.
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
162
RG "0G (C*>; C,>; C ; C,\; C\) stands for the sequence of bifurcations leading to the crossover for the l (#) vibrational component. R"0L (C,>; C ; C,; C,\) stands for the sequence of bifurcations leading to the crossover for the 2l (!1) vibrational component. 1min2 or 1max2 means that min or max of the vibrational component are close to max or min of another RES and form nearly a `conical intersectiona. The energy spectrum is similar to the pseudo-symmetrical with respect to the energy of the `conical intersectiona. (m/s) or (s/m) mean that m(in)(ax) and saddle points are so close in energy that, in fact, we cannot di!erentiate between them. Rz stands for the complicated inversion which goes through intermediate steps and which should be described (probably) in terms of symmetry which is higher than the cubic symmetry.
RES for the ground vibrational state Type
Numb
6C
8C
12C
24C Q
24C (a) Q
24C (b) Q
48C
0L
26
min
max
sad
*
*
*
*
`Genealogya
J-values 0}70
RES for the l (#1) vibrational component S Type
Numb
6C
8C
12C
24C Q
24C (a) Q
24C (b) Q
48C
0L
26
min
max
sad
*
*
*
*
`Genealogya
J-values
`Genealogya
J-values
`Genealogya
J-values
RES for the l (0) vibrational component S Type
Numb
6C
8C
12C
24C Q
24C (a) Q
24C (b) Q
48C
0G
26
max
min
sad
*
*
*
*
RES for the l (!1) vibrational component S Type
Numb
6C
8C
12C
24C Q
24C (a) Q
24C (b) Q
48C
0L
26
min
max
sad
*
*
*
*
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
163
RES for the l (!) vibrational component Type
Numb
6C
8C
12C
24C Q
24C (a) Q
24C (b) Q
48C
0L
26
min
max
sad
*
*
*
*
`Genealogya
J-values 0}70
RES for the l (#) vibrational component Type
Numb
6C
8C
12C
24C Q
24C (a) Q
24C (b) Q
48C
0G 2 3 3
26 74 98 98
max min min min
min min min max
sad sad min min
* sad sad sad
* * sad *
* * * *
0L
26
min
max
sad
*
* max max max #sad *
*
*
`Genealogya 0G (C*>) 0G (C*>; C,>) 0G (C*>; C,>; C ) RG
J-values 448 49}53 54}60 61 62}70
RES for the l vibrational state Type
Numb
6C
8C
12C
24C Q
24C (a) Q
24C (b) Q
48C
0L
26
min
max
sad
*
*
*
*
`Genealogya
J-values 0}70
RES for the (l #l ) (!1) vibrational component S Type
Numb
6C
8C
12C
24C Q
24C (a) Q
24C (b) Q
48C
0L
26
min
max
sad
*
*
*
*
`Genealogya
J-values
RES for the (l #l ) (#3) vibrational component E Type
Numb
6C
8C
0G
26
max
2 ??? 0L
74 26
12C
`Genealogya
24C Q
24C (a) Q
24C (b) Q
48C
1min2 sad
*
*
*
*
min
min
sad
(s/m)
(m/s)
*
*
0G (C*>)
min
max
sad
*
*
*
*
Rz
J-values 424 25}50... ??? 57
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
164
RES for the (l #l ) (!2) (#2) two vibrational component (lower in energy component) E S 24C Q
24C (a) Q
24C (b) Q
48C
min 1max2 sad max max sad
* (s/m)
* (m/s)
* *
* *
max
sad
*
*
*
*
Type
Numb
6C
0L 2 ??? 0G
26 74 26
8C
min
12C
`Genealogya
J-values
0G (C*>)
422 23 ??? 28}...
(higher in energy component) Type
Numb
6C
8C
12C
24C Q
24C (a) Q
24C (b) Q
48C
0G
26
max
min
sad
*
*
*
*
`Genealogya
J-values ...
RES for the (l #l ) (!3) vibrational component S Type
Numb
6C
8C
12C
24C Q
24C (a) Q
24C (b) Q
48C
0L
26
min
max
sad
*
*
*
*
`Genealogya
J-values
`Genealogya
J-values
0G (C*>)
RES for the (l #l ) (#1) vibrational component E Type
Numb
6C
8C
12C
24C Q
24C (a) Q
24C (b) Q
48C
0G 2 ??? 0L
26 74
max min
min min
sad sad
* (s/m)
* (m/s)
* *
* *
26
min
max
sad
*
*
*
*
Rz
436 37}50 ??? 68
`Genealogya
J-values
RES for the (2l ) (#2) vibrational component E Type
Numb
6C
8C
12C
24C Q
24C (a) Q
24C (b) Q
48C
0G
26
max
min
sad
*
*
*
*
430
1 1 1 0L
50 50 50 26
max max min min
min max max max
max min min sad
* * sad *
* sad * *
sad * * *
* * * *
0G (C,>) ?? 0G (C,>; C ) ... 0G (C,>; C ; C,) ? 538
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
165
RES for the (2l ) (#2) vibrational component E Type
Numb
6C
8C
12C
24C Q
24C (a) Q
24C (b) Q
48C
0L
26
min
max
sad
*
*
*
*
`Genealogya
J-values
RES for the (2l ) (0) vibrational component (lower in energy component) E Type
Numb
6C
8C
0G
26
max
1 1
50 50
max max
12C
24C Q
24C (a) Q
24C (b) Q
48C
(min) (sad)
*
*
*
*
(min) max (max) min
* *
* sad
sad *
* *
`Genealogya
J-values 430
0G (C,>) 0G (C,>; C )
?? ...50
RES for the (2l ) (0) vibrational component (higher in energy component) E Type
Numb
6C
8C
12C
24C Q
24C (a) Q
24C (b) Q
48C
`Genealogya
??? 2
J-values ???
74
max
max
sad
(m/s)
(s/m)
*
*
0L (C*>)
10}50
RES for the (2l ) (!1) vibrational component E Type
Numb
6C
8C
12C
24C Q
24C (a) Q
24C (b) Q
48C
`Genealogya
J-values
0L 1 1 1 0G
26 50 50 50 26
min min min max max
max max min min min
sad max max max sad
* * * sad *
* * sad * *
* sad * * *
* * * * *
0L (C,>) 0L (C,>; C ) 0L (C,>; C ; C,) R
410? ...14?? 15... ? 18}43
`Genealogya
J-values
RES for the (2l ) (!2) vibrational component E Type
Numb
6C
8C
12C
24C Q
24C (a) Q
24C (b) Q
48C
0L 1 1
26 50 50
min min max
max max max
sad min min
* sad *
* * sad
* * *
* * *
432 0L (C,>) 0L (C,>; C,)
33}35
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
166
RES for the l (!1) vibrational component S `Genealogya
J-values
Type
Numb
6C
8C
12C
24C Q
24C (a) Q
24C (b) Q
48C
0G
26
max
min
sad
*
*
*
*
1
50
max
min
max
sad
*
*
*
0G (C,>)
1 1
50 50
min min
min max
max max
* *
sad *
* sad
* *
0G (C,>; C,) 22... 0G (C,>; C,; C ) ...29...
`Genealogya
J-values
`Genealogya
J-values
`Genealogya
J-values
J-values
420 21
RES for the l (0) vibrational component S Type
Numb
6C
8C
12C
24C Q
24C (a) Q
24C (b) Q
48C
0L
26
min
max
sad
*
*
*
*
RES for the l (#1) vibrational component S Type
Numb
6C
8C
12C
24C Q
24C (a) Q
24C (b) Q
48C
0G
26
min
max
sad
*
*
*
*
RES for the (l #l ) (!) vibrational component Type
Numb
6C
8C
12C
24C Q
24C (a) Q
24C (b) Q
48C
0L
26
min
max
sad
*
*
*
*
RES for the (l #l ) (#) vibrational component Type
Numb
6C
8C
12C
24C Q
24C (a) Q
24C (b) 48C Q
`Genealogya
0G 2 3 3 0L
26 74 98 98 26
max min min min min
min min min max max
sad sad min min sad
* sad sad sad *
* max max max#sad *
* * sad * *
448 0G (C*>) 49}53 0G (C*>; C,>) 54}60 0G (C*>; C,>; C ) 61 562 RG
* * * * *
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
167
References Abraham, R., Marsden, J., 1978. Foundations of Mechanics. Addison-Wesley, Reading, MA. Aliev, M., Watson, J., 1985. Higher-order e!ects in the vibration}rotation spectra of semirigid molecules. In: Rao, K. (Ed.), Molecular Spectroscopy: Modern Research, Vol. 3. Academic Press, New York, pp. 1}67. Amat, G., Nielsen, H., Tarrago, G., 1971. Rotation}Vibrations of Polyatomic Molecules. Dekker, New York. Arnol'd, V., 1981. Mathematical Methods of Classical Mechanics. Springer, New York. Arnol'd, V., 1988. Modern Problems in Mathematics: Dynamical Systems. Springer, Berlin. Asselin, P., Soulard, P., Manceron, L., Boudon, V., Pierre, G., 2000. High resolution Fourier transform infrared spectroscopy and analysis of the l band of jet-cooled MoCO . J. Mol. Struct. 517, 145}155. Avron, J., Raveh, A., Zur, B., 1988. Adiabatic quantum transport in multiply connected systems. Rev. Mod. Phys. 60, 873}915. Bargmann, V., 1961. On a Hilbert space of analytic functions and an associated integral transform. Commun. Pure Appl. Math. 14, 187}214. Bargmann, V., 1962. On the representations of the rotation group. Rev. Mod. Phys. 34, 829}845. Bayer, D., Stillman, M., 1982. Macaulay: A System for Computation in Algebraic Geometry and Commutative Algebra. Harvard University, Cambridge. Becker, T., Weispfenning, V., 1993. Grobner Basis. Springer, New York. Biedenharn, L., Louck, J., 1981. Angular Momentum in Quantum Physics. Addison-Wesley, New York. Birkho!, G., 1966. Dynamical Systems. AMS, New York. Bordas, C., Helm, H., 1992. Stark e!ect in triatomic hydrogen Rydberg states. Phys. Rev. A 45, 387}402. Brodersen, S., Zhilinskii, B., 1995. The rotational structure of the vibrational states and substates of symmetry E in CF . J. Mol. Spectrosc. 172, 303}318. Burnside, W., 1911. Theory of Groups of Finite Order. Cambridge University Press, New York. Cavalli, A., D'Ariano, G., Michel, L., 1985. Compact manifolds of coherent states invariant by semi simple Lie groups. Ann. Inst. Henri Poincare 44, 173}193. Champion, J.-P., LoeK te, M., Pierre, G., 1992. Spherical top spectra. In: Rao, K.N., Weber, A. (Eds.), Spectroscopy of the Earth's Atmosphere and Interstellar Medium. Academic Press, San Diego, pp. 339}422. Child, M., 1998. Quantum states in a Champagne bottle. J. Phys. A. 31, 657}670. Child, M., Halonen, L., 1984. Overtone frequencies and intensities in the local mode picture. Adv. Chem. Phys. 57, 1. Coudert, L.H., 1998. Analysis of the rotational levels of water and determination of the potential energy function for the bending l mode. J. Mol. Spectrosc. 165, 406}425. Cox, D., Little, J., O'Shea, D., 1992. Ideals, Varieties, and Algorithms. Springer, Berlin. Cox, D., Little, J., O'Shea, D., 1998. Using Algebraic Geometry. Springer, Berlin. Creagh, S.C., Littlejohn, R.G., 1991. Semiclassical trace formulas in the presence of continuous symmetry. Phys. Rev. A 44, 836}850. Cushman, R., Ferrer, S., Hanssmann, H., 1999. Singular reduction of axially symmetric perturbations of the isotropic harmonic oscillator. Nonlinearity 12, 389}410. Cushman, R.H., Bates, L., 1997. Global Aspects of Classical Integrable Systems. Birkhauser, Basel. Cushman, R.H., Duistermaat, J.J., 1988. The quantum mechanical spherical pendulum. Bull. Am. Math. Soc. 19, 475}479. Cushman, R., Rod, D., 1982. Reduction of the semisimple 1:1 resonance. Physica D 6, 105}112. Cushman, R., Sadovskii, D., 1999. Monodromy in perturbed Kepler systems: hydrogen atom in crossed "elds. Europhys. Lett. 47, 1}7. Davarashvili, O., Zhilinskii, B., Krivtsun, V., Sadovskii, D., Snegirev, E., 1990. Experimental study of a sequence of quantum bifurcations. Pis'ma Zh. Eksp. Teor. Fiz. 51, 17}19. Dhont, G., Sadovskii, D., Zhilinskii, B., Boudon, V., 2000. Analysis of the `unusuala vibrational components of triply degenerate mode l of MoCO based on the classical interpretation of the e!ective rotation}vibration Hamiltonian. J. Mol. Spectrosc. 201, 95}108. Dorney, A., Watson, J., 1972. Forbidden rotational spectra of polyatomic molecules. Stark e!ect and d J "0 transitions of ¹ molecules. J. Mol. Spectrosc. 42, 135}148. B Eisenbud, D., 1995. Commutative Algebra with a view toward Algebraic Geometry. Springer, Berlin.
168
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
Englman, R., 1972. The Jahn}Teller E!ect in Molecules and Crystals. Wiley, New York. Fox, K., Galbraith, H., Krohn, B., Louck, J., 1977. Theory of level splitting: spectrum of the octahedrally invariant fourth rank tensor operator. Phys. Rev. A 15, 1363}1381. Fried, L., Ezra, G., 1987. Semiclassical quantization using classical perturbation theory: algebraic quantization of multidimensional systems. J. Chem. Phys. 86, 6270}6282. Gilmore, R., 1981. Catastrophe Theory for Scientists and Engineers. Wiley, New York. Golubitsky, M., Schae!er, D., 1985. Singularities and Groups in Bifurcation Theory: Vol. I. Springer, New York. Golubitsky, M., Stewart, I., 1987. Generic bifurcation of Hamiltonian systems with symmetry. Physica D 24, 391}405. Guillemin, V., Uribe, A., 1989. Monodromy in the quantum spherical pendulum. Commun. Math. Phys. 122, 563}574. Gutzwiller, M.C., 1990. Chaos in Classical and Quantum Mechanics. Springer, New York. Harter, W., 1988. Computer graphical and semiclassical approaches to molecular rotations and vibrations. Comput. Phys. Rep. 8, 319}394. Harter, W., 1993. Principles of Symmetry, Dynamics, and Spectroscopy. Wiley Interscience, New York. Harter, W., 1996. Molecular symmetry and dynamics. In: Drake, G. (Ed.), Atomic, Molecular & Optical Physics Handbook. American Institute of Physics. Woodbury, New York, pp. 378}393. Harter, W., Patterson, C.W., 1977. Orbital level splitting in octahedral symmetry and SF rotational spectra. I. Qualitative features of high J levels. J. Chem. Phys. 66, 4872}4885. Harter, W., Patterson, C.W., 1984. Rotational energy surfaces and high-J eigenvalue structure of polyatomic molecules. J. Chem. Phys. 80, 4241}4261. Harter, W., Patterson, C.W., Paixao, F., 1978. Frame transformation relations and multipole transitions in symmetric polyatomic molecules. Rev. Mod. Phys. 50, 37}56. Harter, W., Weeks, D., 1988. Rotation}vibration spectra of icosahedral molecules. I. Icosahedral symmetry analysis and "ne structure. J. Chem. Phys. 90, 4727}4743. Herman, M., Lievin, J., Auwera, J.V., Campargue, A., 1999. Global and accurate vibration hamiltonians from high resolution molecular spectroscopy. Adv. Chem. Phys. 108, 1}431. Herzberg, G., Longuet-Higgins, H.C., 1963. Intersection of potential surfaces in polyatomic molecules. Discuss. Faraday Soc. 35, 77}82. Ja!eH , C., 1988. Comment on `Semiclassical phase space evolution of Fermi resonance spectraa. J. Chem. Phys. 89, 3395}3396. Jahn, H.A., 1938. Stability of polyatomic molecules with degenerate electronic state. II. Spin degeneracy. Proc. Roy. Soc. A 164, 117}133. Jahn, H.A., Teller, E., 1937. Stability of polyatomic molecules with degenerate electronic state. I. Orbital degeneracy. Proc. Roy. Soc. A 161, 220}238. Jensen, P., Kozin, I., 1993. The potential energy surface for the electronic ground state of H Se derived from experiment. J. Mol. Spectrosc. 160, 39}57. Jonas, D., Solina, S., Rajaram, B., Silbey, R., Field, R., Yamanouchi, K., Tsuchiya, S., 1993. Intramolecular vibrational redistribution of energy in the stimulated emission pumping spectrum of acetylene. J. Chem. Phys. 99, 7250}7270. Jungen, C., Merer, A.J., 1976. The Renner}Teller e!ect. In: Rao, K.N. (Ed.), Molecular Spectroscopy, Vol. 2. Academic Press, New York, pp. 127}164. Kellman, M., 1985. Algebraic resonance dynamics of the normal/local transition from experimental spectra of ABA triatomics. J. Chem. Phys. 83, 3843}3858. Kellman, M., 1990. Approximate constants of motion for vibrational spectra of many-oscillator systems with multiple anharmonic resonances. J. Chem. Phys. 93, 6630}6635. Kellman, M., 1995. Dynamical analysis of highly excited vibrational spectra: progress and prospects. In: Day, H.-L., Field, R. (Eds.), Molecular Dynamics and Spectroscopy by Stimulated Emission Pumping. World Scienti"c, Singapore, pp. 1}60. Kellman, M., Chen, G., 1991. Approximate constants of motion and energy transfer pathways in highly excited acetylene. J. Chem. Phys. 95, 8671}8672. Kellman, M., Lynch, E., 1986. Fermi resonance phase space structure from experimental spectra. J. Chem. Phys. 85, 7216}7223.
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
169
Kirschner, S., Watson, J., 1973. Sextic centrifugal distortion of tetrahedral molecules. J. Mol. Spectrosc. 47, 347}350. Klein, F., 1884. Vorlesungen uK ber das Ikosaeder und die Au#oK sung der Gleichungen vom fuK nften Grade. Teubner, Leipzig. Kozin, I., Belov, S., Polyansky, O., Tretyakov, M., 1992. Submillimeter-wave spectrum of H Se. The evidence of fourfold clustering of rotational levels. J. Mol. Spectrosc. 152, 13}28. Kozin, I., Jensen, P., 1993a. Fourfold clusters of rovibrational energy levels for H S studied with a potential energy surface derived from experiment. J. Mol. Spectrosc. 163, 483}509. Kozin, I., Jensen, P., 1993b. Fourfold clusters of rovibrational energy levels in the fundamental vibrational states of H Se. J. Mol. Spectrosc. 161, 186}207. Kozin, I., Pavlichenkov, I., 1996. Bifurcation in rotational spectra of nonlinear AB molecules. J. Chem. Phys. 104, 4105}4113. Kozin, I., Roberts, R., Tennyson, J., 1999. Symmetry and structure of rotating H>. J. Chem. Phys. 111, 140}150. Kramer, P., Moshinsky, M., 1968. Group theory of harmonic oscillator and nuclear matter. In: Loebl, E.M. (Ed.), Group Theory and Its Applications, Vol. 3. Academic Press, New York, pp. 339}468. Krivtsun, V., Sadovskii, D., Snegirev, E., Shotov, A., Zasavitskii, I., 1990a. Diode laser study of the l and l bands of the SnH molecule. J. Mol. Spectrosc. 139, 107}125. Krivtsun, V., Sadovskii, D., Zhilinskii, B., 1990b. Critical phenomena and diabolic points in rovibrational energy spectra of spherical top molecules. J. Mol. Spectrosc. 139, 126}146. Landau, L., Lifshitz, E., 1965. Quantum Mechanics, Nonrelativistic Theory. Pergamon Press, Oxford. Lederman, S.M., Marcus, R.A., 1984. Densities of vibrational states of given symmetry species. Linear molecules. J. Chem. Phys. 81, 5601}5607. Lederman, S.M., Runnels, J.H., Marcus, R.A., 1983. Densities of vibrational states of given symmetry species and their use in statistical estimates involving Coriolis and anharmonic e!ects. J. Phys. Chem. 87, 4364. Levine, R., Kinsey, J., 1986. Anharmonic local-mode}normal-mode transformations: an algebraic treatment. J. Phys. Chem. 90, 3653}3656. Louck, J., 1976. Derivation of the molecular vibration}rotation Hamiltonian from the Schrodinger equation for the molecular model. J. Mol. Spectrosc. 61, 107}137. Lyapunov, A., 1892. General Problem of Movement Stability. Kharkov, Moscow. Makarewicz, J., 1990. Semiclassical and quantum mechanical pictures of the rovibrational motion of triatomic molecules. Mol. Phys. 69, 903}921. Makarewicz, J., 1998. Rovibrational energy surfaces of triatomic water-like molecules. J. Chem. Phys. 108, 469}479. Marsden, J., Ratiu, T., 1994. Introduction to Mechanics and Symmetry. Springer, New York. Michel, L., 1972. Non linear group actions; smooth actions of compact Lie groups on manifolds. In: Sen, R.N., Weil, C. (Eds.), Statistical Mechanics and Field Theory. Israel University Press, Jerusalem, pp. 133}150. Michel, L., 1979. Minima of Higgs}Landau potentials. In: Bacry, H. (Ed.), Regards sur la Physique contemporaine. CNRS, Paris, pp. 157}203. Michel, L., 1980. Symmetry defects and broken symmetry. Rev. Mod. Phys. 52, 617}650. Michelot, F., Moret-Bailly, J., 1975. Expressions algeH briques approcheH es de symboles de couplage et de fonctions de base adapteH s a` la symeH trie cubique. J. Phys. (Paris) 36, 451}460. Mills, I., Robiette, A., 1985. On the relationship of normal modes to local modes in molecular vibrations. Mol. Phys. 56, 743}765. Molien, T., 1897. Uber die Invarianten der linearen Substitutionsgruppen. Sitzungber. Konig. Preuss. Akad. Wiss. 52, 1152}1156. Montaldi, J., 1997. Persistence and stability of relative equilibria. Nonlinearity 10, 449}466. Montaldi, J., Roberts, R., 1999. Relative equilibria of molecules. J. Nonlinear Sc. 9, 53}88. Montaldi, J., Roberts, R., Stewart, I., 1987. Non-linear normal modes of symmetric hamiltonian systems. In: Guttinger, W., Dangelmayr, G. (Eds.), The Physics of Structure Formation. Springer, Berlin, pp. 354. Montaldi, J., Roberts, R., Stewart, I., 1988. Periodic solutions near equilibria of symmetric Hamiltonian systems. Philos. Trans. Roy. Soc. London A 325, 237}293. Ngoc, S.V., 1999. Quantum monodromy in integrable systems. Commun. Math. Phys. 203, 465}479. Oka, T., 1980. Observation of the infrared spectrum of H>. Phys. Rev. Lett. 45, 531}534. Oka, T., 1992. The infrared spectrum of H> in laboratory and space plasmas. Rev. Mod. Phys. 64, 1141}1149.
170
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
Papousek, D., Aliev, M., 1982. Molecular Vibrational}Rotational Spectra. Elsevier, Amsterdam. Patterson, C., 1985. Quantum and semiclassical description of triply degenerate anharmonic oscillator. J. Chem. Phys. 83, 4618}4632. Patterson, C., Harter, W., 1977. Orbital level splitting in octahedral symmetry and SF rotational spectra. II. Quantitat ive features of high J levels. J. Chem. Phys. 66, 4886}4892. Pavlichenkov, I., 1993. Bifurcations in quantum rotational spectra. Phys. Rep. 226, 175}279. Pavlichenkov, I., Zhilinskii, B., 1985. Rotation of molecules around speci"c axes: axes reorientation under rotational excitation. Chem. Phys. 100, 339}354. Pavlichenkov, I., Zhilinskii, B., 1988. Critical phenomena in rotational spectra. Ann. Phys. (N.Y.) 184, 1}32. Pavlov-Verevkin, V., Sadovskii, D., Zhilinskii, B., 1988. On the dynamical meaning of the diabolic points. Europhys. Lett. 6 (7), 573}578. Pavlov-Verevkin, V., Zhilinskii, B., 1987. Localization of vibrations under excitation for triatomic molecules. Khim. Fizika 6, 1459}1467. Pavlov-Verevkin, V., Zhilinskii, B., 1988a. E!ective Hamiltonians for vibrational polyads: integrity basis approach. Chem. Phys. 126, 243}253. Pavlov-Verevkin, V., Zhilinskii, B., 1988b. Rearrangements of the vibrational polyadic spectra with excitation: two-mode case. Chem. Phys. 128, 429}437. Perelomov, A., 1986. Generalized Coherent States and their Applications. Springer, Berlin. Pierre, G., Sadovskii, D., Zhilinskii, B., 1989. Organization of quantum bifurcations: crossover of rovibrational bands in spherical top molecules. Europhys. Lett. 10, 409}414. PoincareH , H., 1879. The`ses. Gauthier Villard, Paris. Poston, T., Stewart, I., 1978. Catastrophe Theory and Its Applications. Pitman, London. Quack, M., 1977. Detailed symmetry selection rules for reactive collisions. Mol. Phys. 34, 477}504. Quack, M., 1985. On the densities and numbers of rovibronic states of a given symmetry species: rigid and nonrigid molecules. J. Chem. Phys. 82, 3277}3283. Quack, M., 1990a. The role of quantum intramolecular dynamics in unimolecular reactions. Philos. Trans. Roy. Soc. London, Ser. A 336, 203}220. Quack, M., 1990b. Spectra and dynamics of coupled vibrations in polyatomic molecules. Annu. Rev. Phys. Chem. 41, 839}874. Robbins, J.M., 1989. Discrete symmetries in periodic orbit theory. Phys. Rev. A 40, 2128}2136. Sadovskii, D., Fulton, N., Tennyson, J., Zhilinskii, B., 1993. Nonlinear normal modes and local bending vibrations of H> and D>. J. Chem. Phys. 99, 906}918. Sadovskii, D., Zhilinskii, B., 1988. Qualitative analysis of vibration}rotation hamiltonians for spherical top molecules. Mol. Phys. 65 (1), 109}128. Sadovskii, D., Zhilinskii, B., 1993a. Group-theoretical and topological analysis of localized rotation}vibration states. Phys. Rev. A 47, 2653}2671. Sadovskii, D., Zhilinskii, B., 1993b. Qualitative study of a model three-level Hamiltonian with S;(3) dynamical symmetry. Phys. Rev. A 48 (2), 1035}1044. Sadovskii, D., Zhilinskii, B., 1995. Counting levels within vibrational polyads. Generating function approach. J. Chem. Phys. 103, 10520}10536. Sadovskii, D., Zhilinskii, B., 1999. Monodromy, diabolic points and angular momentum coupling. Phys. Lett. A 256, 235}244. Sadovskii, D., Zhilinskii, B., Champion, J., Pierre, G., 1990. Manifestation of bifurcations and diabolic points in molecular energy spectra. J. Chem. Phys. 92, 1523}1537. Sadovskii, D., Zhilinskii, B., Michel, L., 1996. Collapse of the Zeeman structure of the hydrogen atom in an external electric "eld. Phys. Rev. A 53, 4064}4067. Schwinger, J., 1965. On angular momentum. In: Biedenharn, L.C., Dam, H.V. (Eds.), Quantum Theory of Angular Momentum. Academic Press, New York, pp. 229}279. Sinha, A., Kinsey, J.L., 1984. Symmetry-speci"c densities of vibrational levels for polyatomic molecules. J. Chem. Phys. 80, 2029}2033. Smale, S., 1970a. Topology and mechanics. I. Invent. Math. 10, 305}331. Smale, S., 1970b. Topology and mechanics. II. Invent. Math. 11, 45}64.
B.I. Zhilinskin& / Physics Reports 341 (2001) 85}171
171
Soldan, P., Zhilinskii, B., 1996. Density of vibrational states of a given symmetry type for octahedral AB molecules. Chem. Phys. Lett. 258, 25}29. Springer, T.A., 1977. Invariant Theory, Lecture Notes in Mathematics, Vol. 585. Springer, Berlin. Stefanski, K., Pollak, E., 1987. An analysis of normal and local mode dynamics based on periodic orbits. I. Symmetric ABA triatomic molecules. J. Chem. Phys. 87, 1079}1088. Stein, S.E., Rabinovich, B.S., 1973. Accurate evaluation of internal energy level sums and densities including anharmonic oscillators and hindered rotors. J. Chem. Phys. 58, 2466}2473. Sturmfels, B., 1993. Algorithms in Invariant Theory. Springer, New York. Tennyson, J., Henderson, J.R., 1989. Highly excited rovibrational states using a discrete variable representation: the H> molecular ion. J. Chem. Phys. 91, 3815}3825. Thom, R., 1972. StabiliteH Structurelle et Morphogenese. Benjamin, New York. Tretyakov, M., Belov, S., Kozin, I., Polyansky, O., 1992. Submillimeter-wave spectrum of H Se. Analysis of rotational spectra of all selenium isotopomers in the ground state. J. Mol. Spectrosc. 154, 163}168. VanHecke, C., Sadovski, D., Zhilinskii, B., 1999. Qualitative analysis of molecular rotation starting from inter-nuclear potential. Eur. Phys. J. D 7, 199}209. Vasconcelos, W., 1998. Computational Methods in Commutative Algebra and Algebraic Geometry. Springer, New York. Watson, J., 1968. Simpli"cation of the molecular vibration}rotation Hamiltonian. Mol. Phys. 15, 479}490. Weeks, D., Harter, W., 1988. Rotation}vibration spectra of icosahedral molecules. II. Eigenfunctions and normal modes of buckminsterfullerene. J. Chem. Phys. 90, 4744}4771. Weidenmuller, H.A., 1993. Semiclassical peroidic-orbit theory for identical particles. Phys. Rev. A 48, 1819}1823. Weyl, H. The Classical Groups. Their Invariants and Representations, Princeton Univ., New Jersey, 1939. Whitten, G.Z., Rabinovich, B.S., 1963. Accurate and facile approximation for vibrational energy-level sums. J. Chem. Phys. 38, 2466}2473. Wilson, E., Decius, J., Cross, P., 1955. Molecular Vibrations. McGraw-Hill, New York. Xiao, L., Kellman, M., 1989. Uni"ed semiclassical dynamics for molecular resonance spectra. J. Chem. Phys. 90, 6086}6098. Zhang, W., Feng, D., Gilmore, R., 1990. Coherent states: theory and some applications. Rev. Mod. Phys. 62, 867}927. Zhilinskii, B., 1978. TheH orie qualitative des amas dans les spectres de vibration}rotation des moleH cules a symeH trie eH leveH e. C. R. Acad. Sci. (Paris) B 286, 135}138. Zhilinskii, B., 1979. Spectra of tensor operators adapted to nonstandard basis. Qualitative features of clustering. J. Mol. Spectrosc. 78, 203}228. Zhilinskii, B., 1989a. Qualitative analysis of vibrational polyads: N mode case. Chem. Phys. 137, 1}13. Zhilinskii, B., 1989b. Theory of Complex Molecular Spectra. Moscow University Press, Moscow. Zhilinskii, B., 1996. Topological and symmetry features of intramolecular dynamics through high resolution molecular spectroscopy. Spectrochim. Acta A 52, 881}900. Zhilinskii, B., Brodersen, S., 1994. The symmetry of the vibrational components in ¹ molecules. J. Mol. Spectrosc. 163, B 326}338. Zhilinskii, B., Brodersen, S., Madsen, M., 1993. The pattern of clusters in isolated vibrational components of CF and the semiclassical model. J. Mol. Spectrosc. 160, 192}216. Zhilinskii, B., Kozin, I., Petrov, S., 1999. Correlation between asymmetric and spherical top: imperfect quantum bifurcations. Spectrochim. Acta A 55, 1471}1484. Zhilinskii, B., Pavlichenkov, I., 1987. Critical phenomena in the rotational spectra. Zh. Eksp. Teor. Fiz. 92, 387}403. Zhilinskii, B., Pavlichenkov, I., 1988. Critical phenomenon in the rotational spectra of water molecule. Opt. Spektrosk. 64, 688}690.
Physics Reports 341 (2001) 173}264
Symmetry, invariants, topology. III
Rydberg states of atoms and molecules. Basic group theoretical and topological analysis L. Michel , B.I. ZhilinskimH * Institut des Hautes E! tudes Scientixques, 91440 Bures-sur-Yvette, France Universite& du Littoral, BP 5526, 59379 Dunkerque Ce& dex, France Contents 1. Introduction 1.1. Dynamical symmetry of Rydberg states 2. Groups and their actions appropriate for the Rydberg problem 2.1. Structure of O(4) 2.2. The adjoint representation of O(4); induced action on R"S ;S 2.3. Action of O(3), SO(3), O(3)?T, SO(3)?T, and SO(3)?T on R; their Q strata, orbits and invariants 2.4. Invariants of the one-dimensional Lie subgroups of O(3) acting on R 2.5. One-dimensional Lie subgroups of O(3)?T and their invariants 2.6. Orbits, strata and orbit spaces of the onedimensional Lie subgroups of O(3) acting on R 2.7. Invariants of "nite subgroups of O(3) acting on R 2.8. Orbits, strata and orbit spaces of "nite subgroups of O(3) acting on R 2.9. Orbits, strata and orbit spaces for T-dependent subgroups of O(3)?T 3. Construction and analysis of Rydberg Hamiltonians 3.1. E!ective Hamiltonians
175 178 179 179 182
185 188 191
192 196 197 198 203 203
3.2. Qualitative description of e!ective Hamiltonians invariant with respect to continuous subgroups of O(3) 3.3. Qualitative description of e!ective Hamiltonians invariant with respect to "nite subgroups of O(3) 4. Manifestation of qualitative e!ects in physical systems. Hydrogen atom in magnetic and electric "eld 4.1. Di!erent "eld con"gurations and their symmetry 4.2. Quadratic Zeeman e!ect in hydrogen atom 4.3. Hydrogen atom in parallel electric and magnetic "elds 4.4. Hydrogen atom in orthogonal electric and magnetic "elds 4.5. Where to look for bifurcations? 5. Conclusions and perspectives Appendix A. Geometrical representation A.1. O(3) or SO(3) invariant Hamiltonian A.2. C invariant Hamiltonian A.3. C invariant Hamiltonian T A.4. C invariant Hamiltonian F A.5. D invariant Hamiltonian A.6. D invariant Hamiltonian F
* Corresponding author. E-mail address: [email protected] (B.I. ZhilinskimH ). Deceased 30 December 1999. 0370-1573/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 0 0 ) 0 0 0 9 0 - 9
205
210
213 213 215 217 223 227 228 231 231 233 235 236 238 239
174
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
Appendix B. Molien functions for point group invariants B.1. C group B.2. C point group B.3. C point group G B.4. C point group Q B.5. C point group T B.6. D group F B.7. ¹ point group B
240 241 242 243 243 244 245 248
Appendix C. Strata and orbits for point groups Appendix D. Qualitative description of e!ective Hamiltonians based on equivariant Morse}Bott theory D.1. SO(3) continuous subgroup D.2. C continuous subgroup References
250
255 255 256 258
Abstract Rydberg states of atoms and molecules are studied within the qualitative approach-based primarily on topological and group theoretical analysis. The correspondence between classical and quantum mechanics is explored to apply the results of qualitative (topological) approach to classical mechanics developed by PoincareH , Lyapounov and Smale to quantum problems. The study of the action of the symmetry group of the problems considered on the classical phase space enables us to predict qualitative features of the energy level patterns for quantum Rydberg operators. 2001 Elsevier Science B.V. All rights reserved. PACS: 03.65.Fd; 31.15.Md; 32.80.Rm; 33.80.Rv Keywords: Atoms in "elds; Rydberg problem; Hydrogen atom
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
175
1. Introduction This chapter is devoted to the qualitative analysis of Rydberg states of atoms and molecules based on the extensive use of group actions combined with topological arguments introduced in Chapter I. Some results of such applications are published (Sadovskii et al., 1996; Sadovskii and Zhilinskii, 1998; Cushman and Sadovskii, 1999) but there are still many open problems to study within the formalism developed here. Rydberg states of atoms and molecules are very slightly bound quantum states of an electron and a positively charged ion. In the approximation one can neglect the excitations of the ionic core and all spin e!ects, the spectrum of states has a structure typical of that of the hydrogen atom: grouping in multiplets of n states for the large principal quantum number n. In the following, we shall restrict ourselves to the analysis of the internal structure of these large Rydberg multiplets when their splitting is small in some sense. The experimental study of Rydberg states is a very active "eld of physics (Aymar, 1984; Stebbings and Dunning, 1983). There are many works on Rydberg atoms isolated or in di!erent con"gurations of magnetic and electric "elds (Beims and Alber, 1993; Boris et al., 1993; Cacciani et al., 1988, 1989, 1992; Fabre et al., 1977, 1984; Flothmann et al., 1994; Frey et al., 1996; Fujii and Morita, 1994; Hulet and Kleppner, 1983; Jacobson et al., 1996; Jones, 1996; Konig et al., 1988; Lahaye and Hogervorst, 1989; Raithel et al., 1991, 1993a,b; Raithel and Fauth, 1995; Rinneberg et al., 1985; Rothery et al., 1995; Seipp et al., 1996; van der Veldt et al., 1993; Zimmerman et al., 1979). The study of Rydberg molecules is beginning (Bordas et al., 1985, 1991; Bordas and Helm, 1991, 1992; Broyer et al., 1986; Dabrowski et al., 1992; Dabrowski and Sadovskii, 1994; Davies et al., 1990; Dietrich et al., 1996; Dodhy et al., 1988; Helm, 1988; Herzberg, 1987; Herzberg and Jungen, 1972; Hiskes, 1996; Jungen, 1988; Jungen et al., 1989, 1990; Ketterle et al., 1989; Labastie et al., 1984; Lembo et al., 1989, 1990; Mayer and Grant, 1995; Merkt et al., 1995, 1996; Schwarz et al., 1988; Sturrus et al., 1988; Weber et al., 1996). Theoretical studies also exist for each type of experiments (Aymar et al., 1996; Bander and Itzykson, 1966; Bixon and Jortner, 1996; Braun, 1993; Braun and Solov'ev, 1984; Chiu, 1986; Clark et al., 1996; Delande and Gay, 1986, 1988, 1991; Delande et al., 1994; Delos et al., 1983; Engle"eld, 1972; Gourlay et al., 1993; Greene and Jungen, 1985; Herrick, 1982; Howard and Wilkerson, 1995; Huppner et al., 1996; Iken et al., 1994; Kalinski and Eberly 1996a,b; Kazanskii and Ostrovskii, 1989, 1990; Kelleher and Saloman, 1987; King and Morokuma, 1979; Kuwata et al., 1990; Laughlin, 1995; Lombardi et al., 1988; Lombardi and Seligman, 1993; Mao and Delos 1992; Pan and Lu, 1988; Rabani and Levine, 1996; Rau and Zhang, 1990; Remacle and Levine, 1996a,b; Robnik and Schrufer, 1985; Solov'ev, 1981; Tanner et al., 1996; Thoss and Domcke, 1995; Uzer, 1990; Zakrzewski et al., 1995). It is time to establish some general physical laws applying to the Rydberg multiplets. They can be obtained by a general qualitative analysis of the relevant problems based on general methods that we have introduced in the initial Chapter I of this issue. Further applications to atomic and molecular problems are currently in progress. Two features of Rydberg physics make a general approach attractive: First, when the external "elds are small enough or vanish, each Rydberg multiplet is labeled by the value n of the principal quantum number and the splitting inside each multiplet is small compared to the splitting between them. The dynamical system which describes the internal structure of an individual n multiplet has two degrees of freedom. For large n, the set of n quantum
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
176
levels is su$ciently large to be well studied by classical analysis, as a Hamiltonian dynamical system de"ned on a four-dimensional phase space with non-trivial topology and symmetry that we shall soon make precise. Second, the isolated hydrogen atom has a large dynamical symmetry: that of the orthogonal group O(4) and time reversal. As we shall recall, that is an exact symmetry for the quantum mechanical study, in the non-relativistic approximation, of an electron in the static Coulomb potential of a point-like nucleus; then the energy of bound states depends only on the principal quantum number n; that exceptional energy degeneracy between the states of di!erent orbital momenta of value l, 04l4n!1 contains l (2l#1)"n states whose state vectors form the space of an irreducible linear representation of O(4). For atomic or molecular ions the O(4) dynamical symmetry is only an initial approximation for the Rydberg states, which becomes exact at the asymptotic limit nPR when there are no external "elds. Their geometric symmetry is that of the positive ion: it is at most O(3) (case of the isolated hydrogenoid atom) and it is a "nite subgroup of O(3) in the case of non-aligned molecules. Several important physical examples of geometric symmetry are listed in Table 1. Further extension of geometric invariance groups to higher symmetry groups including time-reversal operation will be discussed as well (Section 2.9). As it is well known, the quantum mechanical study of the hydrogen atom, in the nonrelativistic approximation, was "rst made by Pauli (1926) just before the appearance of the ShroK dinger equation. In a convenient unit system, the Hamiltonian for a hydrogen atom can be written 1 1 1 . H" p! , E "! L r 2n 2k
(1)
Due to the speci"c Coulomb interaction there are two vector integrals of motion: J } the angular momentum vector, and X } the Laplace}Runge}Lenz vector, X"p;J!rr\ .
(2)
In other words, the Hamiltonian operator H commutes with the vector operators J and X and all their functions. After the energy-dependent scaled transformation K"X(!2E )\ "Xn , L
(3)
Table 1 Physical examples of the geometric invariance subgroups of Rydberg atoms and molecules O(3) C T C F C D F Point group G
Rydberg atoms with the closed shell core Rydberg atoms in the presence of a weak electric "eld Heteronuclear diatomic Rydberg molecule AB Rydberg atoms in the presence of a weak magnetic "eld Rydberg atoms in the presence of parallel magnetic and electric "elds Homonuclear diatomic Rydberg molecules A Polyatomic Rydberg molecules with the point group symmetry G H (D ), NH (¹ ),2 F B
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
177
it was shown by Hulthen (1933) and Fock (1936), that J and K form a Lie algebra with de"ning commutation relations: [J , J ]"ie J , (4) ? @ ?@A A [J , K ]"ie K , (5) ? @ ?@A A [K , K ]"ie J . (6) ? @ ?@A A We shall show in Section 2.1 that this Lie algebra is the one of the group O(4) or SO(4) (or several other groups). In the hydrogen atom the operators J, K satisfy moreover the two important relations: J ) K"K ) J"0 ,
(7)
J#K"n!1 ,
(8)
which give the value of the two SO(4) Casimir operators for the Hilbert space of vector states with the energy E . Instead of J and K we can introduce the two linear combinations: L J "(J#K)/2 , (9) J "(J!K)/2 . (10) Using only relations (4) and (5) one veri"es that these operators satisfy [J , J ]"ie J , k"1, 2 , (11) I? I@ ?@A IA [J , J ]"0 . (12) ? @ That shows that the Lie algebra of SO(4) is isomorphic to the Lie algebra of the direct product S;(2);S;(2) of two groups S;(2). Therefore, an irreducible representation of SO(4) can be labeled by a pair ( j , j ); its dimension is (2j #1)(2j #1). Relations (9) and (10) imply J !J "J ) K. Then (7) is equivalent to J "J . (13) This last relation proves that the relevant SO(4) irreducible representations for the Rydberg problems are of the form ( j, j) with n"2j#1; hence its dimension is n. The richness of the Rydberg problem makes its `qualitativea study very interesting and powerful (Bander and Itzykson, 1966; Boiteux, 1973, 1982; Brown and Steiner, 1966; Coulson and Joseph, 1967; Cushman and Bates, 1997; Engle"eld, 1972; Guillemin and Sternberg, 1990; Iwai, 1981a,b; Iwai and Uwano, 1986; McIntosh and Cisneros, 1970; Stiefel and Scheifele, 1971). We recall that many fascinating basic tools needed for the general approach are summarized in Chapter I and used for molecular problems in Chapter II where we introduce the general approach by using examples from molecular physics which are less complicated than Rydberg state problem from the point of view of the mathematical technique and give to the reader the opportunity to get the uni"ed qualitative approach developed recently for molecular rotation, vibration, and rovibration problems (Pavlichenkov and Zhilinskii, 1988, 1993a,b; Sadovskii and Zhilinskii, 1988, 1993a,b; Zhilinskii, 1989a,b, 1996; Zhilinskii et al., 1993).
178
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
Section 2 will give a more thorough study of the symmetry we have to use. The main result of this section is the description of the symmetry group action on the phase space, construction of the rings of the polynomial invariant functions and the orbifolds for all cases of the continuous invariant symmetry groups of the Rydberg problem and for "nite point symmetry groups. Section 3 gives the construction and properties of phenomenological Rydberg Hamiltonians based on the integrity bases introduced in Section 2. Section 4 will study some examples of qualitative e!ects. We restrict ourselves with the application of the technique developed to the problem of the hydrogen atom in external electric and magnetic "elds. In spite of the fact that this problem seems to be well understood [see, for example, recent reviews by Friedrich and Wintgen (1989), Hasegawa, Robnik and Wunner (1989), and Braun (1993)] it is still possible to "nd new features and to give di!erent (and we hope useful) interpretations of some qualitative modi"cations of the dynamical behavior. In particular, the simple geometrical description of the collapse phenomenon is given recently by Sadovskii et al. (1996). It is based on the orbifold construction discussed in detail in this chapter. The conclusion summarizes brie#y further steps to apply the technique developed in this chapter to a much wider class of molecular problems. The appendices give more details about the geometrical representation of Rydberg orbifolds, explain more technical mathematical constructions like Molien functions, and collect some auxiliary tables needed for further application of the qualitative approach to molecular problems with "nite point symmetry group. 1.1. Dynamical symmetry of Rydberg states Now, we can formulate the classical limit construction for the Rydberg problem studied. General scheme of the classical limit construction is based on the method of generalized coherent states (Perelomov, 1986; Cavalli et al., 1985; Zhang et al., 1990). This formal construction starts with introducing a dynamical algebra g whose generators play the role of the dynamic variables of the problem. The Hamiltonian itself in this case is considered as an operator in the enveloping algebra. For the Rydberg problem the dynamic algebra is the so(4) algebra with J and K generators (see Section 1, Eqs. (4)}(6)). An equivalent way to represent the same algebra is to use two commuting vector operators J and J . In any of these representations two important relations in Eqs. (7) and (8) restrict the space of the dynamic variables variation to four-dimensional space. The geometrical signi"cance of this space is clearly seen in the J , J representation. Di!erent points of the classical limit phase space are in one-to-one correspondence with orientations of two vectors J , J . This space is the topological product of two two-dimensional spheres S : S ;S . We will denote this space as R. It plays the essential role in all the subsequent analysis. An arbitrary e!ective Hamiltonian which gives the description of the internal structure of Rydberg multiplets may be written in the classical limit as a function de"ned over R (the classical phase space for the Rydberg problem). Following steps of the qualitative analysis of the Rydberg problem may be formulated now as follows: (i) Study of the action of the invariance group on R, classi"cation of orbits and strata. Finding the critical orbits (see Section 2). (ii) Application of Morse theory to the complexity classi"cation of functions de"ned over R in the presence of the invariance symmetry group (see Section 3).
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
179
(iii) Construction of an arbitrary e!ective Hamiltonian in terms of expansions over integrity basis. Representation of classical Hamiltonians near critical manifolds and the qualitative description of the corresponding quantum energy level patterns and wave functions (see Sections 3 and 4).
2. Groups and their actions appropriate for the Rydberg problem The dynamical symmetry group of hydrogen atom Rydberg states is O(4)T, where T is the time reversal. For other physical systems this group can be used as "rst approximation: only a subgroup of it is the exact symmetry. In the beginning of this section we build the group O(4)T and the phase space R of Rydberg problems. The rest of the section is devoted to the study of the action of the O(4)T subgroups on R. 2.1. Structure of O(4) As we have seen in the introduction, the quantum Hamiltonian of the hydrogen atom commutes with the six-dimensional Lie algebra described in Eqs. (4)}(6) and recognized under the form (11) and (12) as the Lie algebra of S;(2);S;(2). But many non-isomorphic Lie groups have the same Lie algebra, e.g. SO(3);SO(3), O(3);O(3), S;(2)!2 (de"ned in Eq. (26)), SO(4) and O(4) as shown in Eqs. (22), (23) and (28). There is some ambiguity in choosing between these di!erent groups; but a non-connected one is de"nitely richer and better adapted to the physics, so we choose here O(4) as an approximate symmetry of Rydberg states (time reversal will be added in Section 2.2.1). We begin by studying SO(4). The connected orthogonal groups SO(n) for n'2 have a double covering called Spin(n). For n"3, SO(3) is the group of rotations in the three-dimensional space and its spin group Spin(3)"S;(2), the group of 2;2 unitary matrices with determinant 1. The N SO(3). The image of this relation between these two groups is the homomorphism S;(2)P homomorphism is the full SO(3) group and its kernel is the center +I ,!I , of S;(2).We denote it by Z (!I ). This whole information can be expressed in one-line way: N SO(3)P1 . 1PZ (!I )PS;(2)P (14) This notation is an exact sequence. A less explicit notation is SO(3)"S;(2)/Z (!I ). To understand better the interrelations between di!erent Lie groups having the same su(2);su(2) algebra we give below their realization as groups of transformations of the quaternions. The elements q30, the quaternion "eld, can be represented by the 2;2 matrices (e.g. Duval, 1964): q"q I#iq p #iq p #iq p ,
(15)
The symbol is a shorthand for de"ning the groups generated by the groups or group element written on each side of this symbol. The exact sequences (see Appendix A to Chapter I) used here are all of the type 1PAPBPCP1. They mean A is the invariant subgroup of B and C"B/A. This notation is now currently used even in undergraduate studies. For more details see e.g. Michel (1980).
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
180
q 3R, a"1, 2, 3, 4, where the three Hermitian matrices p "pH are the Pauli matrices: ? I I 0 1 0 !i 1 0 , p " , p " . p " 1 0 i 0 0 !1
(16)
Notice that q is not Hermitian. Indeed qH"q with q "q , q "!q , q "!q , q "!q . We verify
(17)
qqH"qHq""q"I with "q"" q (18) ? ? where "q"""qH" is called the norm of the quaternion q. Notice that "q"'0 when qO0 and "qq """q" "q ". That last property can be checked from Eq. (18) or from "q""det q .
(19)
Eq. (18) shows that the set of quaternions forms a dimension 4 orthogonal space with the scalar product: 1 1 (q, q )" tr qq H" tr q qH" q q . (20) ? ? 2 2 ? By de"nition, the group S;(2) is the multiplicative group of 2;2 unitary matrices of determinant 1. That group is realized by the multiplication of the quaternions of norm "q""1; indeed Eq. (19) shows that det q"1 and, from Eq. (18), q is unitary. Moreover, the manifold of S;(2) elements, i.e. the quaternions of norm 1, is S , the unit sphere in the dimension 4 orthogonal space of 0. The group S;(2);S;(2) acts linearly on the quaternions by uqvH . (u, v)3S;(2);S;(2), (u, v) ) q"
(21)
This action preserves the quaternion norm. Moreover, it is transitive on the set of quaternions of a given norm: indeed the quaternion "q"I is transformed into the arbitrary quaternion q of the same norm by u"q"q"\, v"I. This shows the existence of a group homomorphism F O(4) , S;(2);S;(2)P
(22)
Im h"SO(4), Ker h"Z (I,!I) . (23) Indeed, h is continuous so its image is connected; the transitivity property we have just established requires Im h"SO(4). By de"nition Ker h is the set of group elements acting trivially on 0; they are u"v"$I. The stabilizer of the quaternions of the form q I is made of the elements v"u; one calls it the diagonal subgroup S;(2)BLS;(2);S;(2). Moreover, S;(2)B transforms into itself the threedimensional subspace of quaternions orthogonal to I (their trace vanishes); that establishes the well-known homomorphism S;(2)PSO(3), [S;(2)/Z "SO(3)].
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
181
We shall consider the 4;4 orthogonal matrix
!I
s"
0
0 1
, s"I, det s"!1 .
(24)
It is the inversion through the origin for the subgroup O(3)LO(4) which leaves "xed the coordinate q ; as Eq. (17) shows, it transforms q into qH. Every element of O(4) not in SO(4) is of the form sh(u, v). Let us study the induced action of these elements on S; ;S; from their known action on every quaternion q: sh(u, v)s\ ) q"sh(u, v) ) qH"s ) (uqHvH)"vquH ,
(25)
it is the permutation of the two factors in S; ;S; . So we are led to consider the wreath product S; !2 which is de"ned as the semi-direct product: S; !2&(S; ;S; ) ) Z (P) , (26) where Z (P) denotes the group of permutations of the two factors of S; ;S; . To summarize, we have established the two exact sequences (see Appendix A to Chapter I) F SO(4)P1 , 1PZB (!I ,!I )PS; ;S; P (27) FY O(4)P1 . 1PZB (!I ,!I )PS; !2P (28) Notice that SO(4) and O(4) have a two-element center generated by the matrix !I . Eq. (27) shows that S;(2);S;(2)"Spin(4). Its elements can be labeled by a pair of indices u , u 3S;(2). Then h(u , u ) de"nes an orthogonal matrix g3SO(4) SO(4) U g&h(u , u ) . (29) The elements of O(4) are either of the form g or sg with s de"ned in Eq. (24). The irreducible representations of a direct product of groups are the tensor product of their irreducible representations. It is customary to label the irreducible representations of S;(2) by j, 042j3Z; so the irreducible representations of S;(2);S;(2) are usually labeled by the pair ( j , j ). They have the dimension (2j #1)(2j #1). The representations of S;(2)!2 are labeled by ( j , j ) ( j , j ) (of dimension 2(2j #1)(2j #1)) when j Oj . When j "j this representation becomes reducible into the direct sum of two reducible representations ( j, j)! which are, respectively, symmetric and antisymmetric for the permutation of the two S;(2) factors. Among all representations those with j $j 3Z form the set of irreducible representations of O(4). They are faithful when j , j are both half-integers; when j , j are both integers their image is that of the `adjointa group of O(4), that is the quotient O(4)/Z (!I ) of O(4) by its center. It is easy to see that it is isomorphic to SO(3)!2. This can be summarized by 0 SO(3)!2P1 . (30) 1PZ (!I )PO(4)P Eq. (13) shows that the only irreducible representations carried by the n-dimensional space of state vectors of a Rydberg multiplet have j "j . As we shall show they are ( j, j)! with n"2j#1. It is why we say that it is O(4) and not S;(2)!2 which is the symmetry group of the Rydberg problem.
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
182
Remark that these representations are faithful only when j is half integer. That is the case of (, )>, the vector representation (i.e. that of dimension 4) of O(4) that we have studied in this subsection. 2.2. The adjoint representation of O(4); induced action on R"S ;S The adjoint representation of O(4) is (1, 0) (0, 1); indeed, by de"nition, it is the representation of O(4) on the six-dimensional vector space of its Lie algebra. We denote this space by <"< < direct sum of two three-dimensional vector spaces. The operators J , J de"ned in Eqs. (9) and (10) act e!ectively on their respective space < , < and trivially on the other one; that yields the representation of Eqs. (11) and (12). A vector of the six-dimensional space < < is naturally decomposed in a direct sum that we denote by (V). Moreover, < < is an orthogonal space with W the scalar product obtained as the matrix product
(x y )
x y
"x ) x #y ) y .
(31)
Each three-dimensional subspace is invariant by a SO(3). For the group SO(4), its matrix g&h(u , u ) [see Eq. (29)] is represented by the matrix A(g) (written in 3;3 blocks):
p(u ) 0 , 0 p(u )
A(g)"
(32)
where p has been de"ned in Eq. (14). In the preceding section we have shown that s3O(4) (de"ned in Eq. (24)) corresponds to the permutation of the two three-dimensional subspaces (see Eq. (25)):
A(s)"
0
I
I
. 0
(33)
So the adjoint representation of the group O(4) is irreducible. Instead of using for the vector space < of the Lie algebra of O(4) the decomposition <"< < corresponding to the operators J , J , it is interesting to use the decomposition <"< < corresponding to the e!ective action of the axial vector operator J and the polar vector operator K. This can be done by the matrix
I 1 I . 2 I !I The conjugation by S diagonalizes A(s): S"
I SA(s)S\"A (s)" 0
0 !I
(34)
,
(35)
1 p(u )#p(u ) p(u )!p(u ) SA(g)S\"A (g)" . 2 p(u )!p(u ) p(u )#p(u )
(36)
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
183
In this new decomposition we write the component vectors as
j
x
x#y x!y , i.e. j" , k" . y (2 (2
"S
k
(37)
We denote by gB the matrices of SO(4) which correspond through Eq. (57) to the pairs u"u "u . The matrices A (gB) which represent them in the adjoint representation are block diagonal
A (gB)"
p(u)
0
0
p(u)
.
(38)
They are the matrices of SO(4) which commute with A (s). They form the subgroup SO(3)B acting e!ectively on the axes 1, 2, 3 of the vector representation of O(4) and leaving the axis 4 "x. The orthogonal group O(3)B"SO(3)Bs is the orthogonal group of the physical space; it transforms j3< , k3< , respectively, as axial and polar vectors. 2.2.1. Extension of O(4) by including the time reversal In classical mechanics of point particles, the operation T which leaves the position of the particles "xed and reverses their velocity is called time reversal. So T also changes the sign of angular momenta. More generally in physics, T changes the sign of axial vectors (e.g. the magnetic induction) and leaves the polar vectors "xed (electric "eld, Laplace}Runge}Lenz vector, etc.) (Wigner, 1959). Wigner has shown that in quantum physics T is represented by an anti-unitary operator. T is an exact symmetry of atomic and molecular physics (except for tiny e!ects induced by weak interaction!) (Sakurai, 1964). We denote by O(4)T the group generated by O(4) and T; it is the full approximate dynamical symmetry group for the Rydberg states. The action of T on <, the space of the adjoint representation, is represented by the matrix !A (s) where A (s) (generally called P, the parity operator) is de"ned in Eq. (35). Remark that O(4)T is not the direct product of O(4) by Z (T) since T does not commute with the elements of O(4) not in the subgroup O(3)B;Z(!I ) (the last factor is the center of O(4)); it is a semi-direct product. The image of the adjoint representation of O(4)T is isomorphic to O(4). 2.2.2. The action of O(4)T on the vector space < of the adjoint representation and on the phase space R We begin by studing the action of SO(4) on <. It is de"ned by the image 0[SO(4)]" SO(3);SO(3) (30) of SO(4) in the adjoint representation. There are two strata in the action of O(3): the origin with the stabilizer O(3), and the rest of space with the stabilizer O(2). There is a unique invariant, the vector norm x ) x50 which de"nes the orbit space; the two strata are de"ned by x ) x"0 and x ) x'0. The invariant, the orbit space, and the orbits in < of index 2 subgroup SO(3) are identical. The corresponding stabilizers are, respectively, SO(3)5O(2)"SO(2), which are index 2 subgroups of the stabilizers of O(3). Note that the vector representation of SO(3) coincides with its adjoint representation. The adjoint representation of SO(3);SO(3) is the direct sum of two adjoint representations of SO(3). We obtain the strata on the space <"< < by using the general fact (see Chapter I): In a reducible representation of a compact group, the stabilizer of a vector is the intersection of the stabilizers of its components in the irreducible subspaces.
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
184
Table 2 Strata of the SO(4) adjoint representation Stablizer
Invariants
dim.str.
Topology
SO(3);SO(3) SO(2);SO(3) SO(3);SO(2) SO(2);SO(2)
x ) x"0"y ) y x ) x'0"y ) y x ) x"0(y ) y x ) x'0(y ) y
0 3 3 6
One point S S S ;S
We give the image by 0 from the stabilizer. dim.str. means dimension of the stratum. The topological structure of each orbit is given.
The two invariants x ) x and y ) y label the orbits. The orbit space is the semi-algebraic domain x ) x50, y ) y50. Its inside corresponds to the generic stratum. Its boundary is the union of the three strata x ) x'0"y ) y, x ) x"0(y ) y, and x ) x"0"y ) y. The results are given in Table 2. To obtain the stabilizers in SO(4) one must take the inverse image 0\ of those in SO(3);SO(3). The group O(4)"SO(4)s is generated by SO(4) and s whose action on < is given in Eq. (33): it exchanges the vectors x and y. To write the invariants of O(4) (or of its image SO(3)!2) we use the symmetric and anti-symmetric combinations of those of SO(4) x ) x#y ) y"j ) j#k ) k, i'0 , i"
(39)
x ) x!y ) y"2j ) k , o"
(40)
!i4o4i50
(41)
(the second form in j, k is obtained from Eq. (37)). Since o changes sign by the exchange x y the invariants of the O(4) action on < are i, o and the orbit space is de"ned by restrictions i50, i5o50. The inside of the orbit space (obtained by replacing 5 by ') corresponds to the generic stratum. The three non-generic strata form the boundary. Their equations are, respectively, i'0, o"0; i'0, i"o'0; and i"0(No"0). The corresponding orbits in the image SO(3)!2 are SO(2)!2, SO(3);SO(2), SO(3)!2. Since i and o are also T invariant, O(4)T has same orbits and same orbit space as O(4). The corresponding stabilizers [for the image (SO(3)!2)T] are given in Table 3 where T is the Q product Ts"sT. Taking into account the physical relation in Eq. (13) we have found that the phase space of Rydberg problem has the topology S ;S and we have denoted it by R. In < this phase space is de"ned by the equations R&S ;S :
i"1, o"0 .
(42)
It is an orbit of O(4)T [and even of SO(4)] belonging to the stratum o"0 of stabilizer (SO(2)!2)T.
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
185
Table 3 Strata of the O(4)T and O(4) adjoint representation. Both groups have the same set of strata but their stabilizers are di!erent Stablizer
Invariants
dim.str.
Topology
(SO(3)!2)T (SO(2);SO(3))T Q (SO(2)!2)T (SO(2);SO(2))T Q
i"0"o i"o'0 iOo"0 i!o'0
0 3 3 6
One point 2S S ;S 2(S ;S )
The image of the stabilizer is given. For the O(4) adjoint representation T or T should be omitted. Q i50 is implicit. dim.str. means the dimension of the stratum. The topological structure of each orbit is given.
2.3. Action of O(3), SO(3), O(3)T, SO(3)T, and SO(3)T on R; their strata, Q orbits and invariants In the four-dimensional space, the subgroup SO(3) leaving invariant the fourth coordinate is characterized in Eqs. (32) and (36) r3SO(3) u "u SO(3)BLSO(3);SO(3) .
(43)
In Eq. (24) we have de"ned s, the space inversion of O(3)"SO(3);Z (s). The restriction of the adjoint representation of O(4) to O(3) is reducible. The matrix S reduces this representation (the reduction is given explicitly in Section 2.2, see Eqs. (34)}(37)). This reducible representation of O(3) can be denoted by 1> 1\ where 1! is, respectively, the axial and polar vector representation. In our problem, the axial and polar vectors j, k correspond, respectively, to the angular momentum and the Runge}Lenz vector. They satisfy the relations in Eqs. (39) and (40) which completely characterize the manifold R. We need also the new invariant m of SO(3) which we de"ne as j ) j!k ) k41 . !14m"
(44)
The stabilizers of the action of O(3) on R are the intersection of O(3) with the stabilizers of O(4) that we have determined in Table 3. This gives immediately the classi"cation of the strata in the action of O(3) on R. Indeed, the stabilizers in the three-dimensional subspaces of j and k are, respectively, O(3) for the null vector, C ( j) for jO0 and C (k) for kO0. So for the points of R, when both j, k F T are non-zero vectors (they satisfy j ) k"0) the stabilizer is C ( j), the group generated by the Q re#ection through the plane orthogonal to j. Table 4 summarizes the data of the three strata which appear in the action of O(3) on R. The stabilizers appearing in the action on R of the subgroup SO(3) are obtained by taking the intersection of SO(3) with the stabilizers of the O(3) action. This shows that on R the orbits are the same for O(3) and SO(3). However, the SO(3) orbits form only two strata: the same open dense stratum with trivial symmetry and one stratum with symmetry C formed by two orbits m"$1. Any SO(3)-invariant function on R is invariant by O(3) (see Table 4).
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
186
Table 4 Strata and orbits in the actions of O(3) and SO(3) on R. The orbits of shape S are critical G
Stabilizer
dim.
Equation
n.o.
topol.
O(3)
C (k) T C ( j) F C ( j) Q C 1
2 2 4
m"!1 m"1 !1(m(1
1 1 R
S S RP
2 4
"m""1 !1(m(1
2 R
S RP
SO(3)
dim."dimension of the stratum. n.o."number of orbits. topol."topological structure of each orbit.
It is a general theorem of Weyl (1939) that the invariants of a direct sum of representations of SO(n) (n53) are made from all distinct scalar products and determinants. Its application to the action of SO(3) on the six-dimensional space < proves that i, m, o form a minimal set of generators of the invariant polynomials since they are algebraically independent. As a scalar product of an axial vector j and a polar vector k, o is a pseudo-scalar for O(3). For this group, i, m, o form an integrity basis. There is no di!erence in R between the P- and P1- invariant polynomial rings since o"0 (and i"1). Taking into account the fact that the values of polynomials of an integrity basis label the orbits, we can conclude that O(3) and SO(3) have the same orbits on R. So on <, when oO0, the O(3) orbits split into two orbits of SO(3) with opposite values of o. Since i, m, o are algebraically independent polynomials on the six-dimensional space <, the ring of invariant polynomials of SO(3) and O(3) are the polynomial rings: P1-"P[i, m, o] ,
(45)
P-"P[i, m, o] .
(46)
Table 4 shows that any function on R, invariant under O(3) or SO(3), depends only on the variable m de"ned in Eq. (44); indeed this parameter labels completely the orbits (common to O(3) and SO(3)). The geometrical representation of the orbifold (which is a 1D-segment) is given in Appendix A. Table 4 also shows that the two orbits de"ned by m"$1 are isolated in their strata; therefore they are critical: these two orbits are orbits of extrema of any O or SO(3) invariant real function de"ned on R. We give a direct and explicit proof of this property: Lemma. Any O(3)-invariant function f (m) on R has at least two orbits of extrema, dexned by m"$1. Indeed, at the point
j
k
3R, m"2
j
!k
.
(47)
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
187
For a function on R, we have to project m on the tangent plane to R. Two normal vectors at (H )3R are obtained by di!erentiating the equations i"1, o"0 de"ning R, i.e. I Ri/Rj j Ro/Rj k " , v " " , v v "d . (48) v " G H GH Ri/Rk k Ro/Rk j
The rank one orthogonal projectors on these two vectors are, respectively:
j21 j j21k k21k k21 j , P " . P " k21 j k21k j21k j21 j
(49)
Since P P "0"P P the gradient of m on the tangent plane at the point (H )3R is I (1!m)j .
R m"(I !P !P ) m" !(1#m)k
(50)
To end the proof of this lemma, we simply verify that this vector vanishes for m"1 (so k"0) and for m"!1 (so j"0). Remark that this vector never vanishes on the interval !1(m(1. So the only other orbits of extrema of f (m) occur for the values of m on this interval, satisfying f (m),df/dm"0. The extension to the case of three di!erent two-dimensional Lie groups including T-dependent symmetry transformations can be easily done because the space of orbits remains the same and the same invariant m can be used to distinguish orbits and strata. These groups are SO(3)T, O(3)T, and SO(3)T . Special care should be taken only to specify the stabilizers of di!erent Q strata. They are given in Table 5. As explained in Section 2.2.1 the action of T on the adjoint space < is represented by !A (s), so T "sT is represented by !I . Hence T leaves i, m, o invariant and the orbits and strata of Q Q Table 5 Strata and orbits in the actions of O(3)T and SO(3)T on R. The orbits of shape S are critical G
Stabilizer
Equation
O(3)T
C (k)T T C ( j)T F Q C ( j)T Q QY C T C T T C T Q T Q
m"!1 m"1 !1(m(1
SO(3)T
SO(3)T Q
m"!1 m"1 !1(m(1 "m""1 !1(m(1
T is the product of time reversal and re#ection in plane orthogonal to j. Q This group is generated by the re#ection in plane orthogonal to the vector j and by the product Ts of time reversal T and the re#ection s in plane of j and k. T is the product of time reversal T by C rotation around axis orthogonal to vector j. T is the product of time reversal T by C rotation around k vector.
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
188
SO(3)T are those of SO(3). Similarly, the orbits and strata of O(3), O(3)T, and SO(3)T are Q identical. Invariants i, m, o characterize orbits unambiguously. So we have P1-"P1-;TQ , (51) P-"P-;T"P1-;T . (52) An orbit, common to di!erent groups, has di!erent conjugacy classes of stabilizers for each group. The stabilizers on R for these "ve groups are given in Tables 4 and 5. 2.4. Invariants of the one-dimensional Lie subgroups of O(3) acting on R Up to a conjugation, there are "ve one-dimensional Lie subgroups of O(3) whose traditional notations in molecular physics are C ,C ,C ,D ,D . (53) F T F We denote unit vectors by a hat, e.g. n( , and by C (n( ) the group of rotations around the axis de"ned by n( . The stabilizers for the action of these groups on R and on the six-dimensional space < are the intersections with the stabilizers of O(3). The generic orbits are one dimensional, so we need at least "ve polynomial invariants to label the orbits on R, the vector space of the O(4) adjoint representation. With the two coordinate vectors j, k and the vector n( "xing the C -axis, we can form "ve scalar products which are "ve algebraically independent invariant homogeneous polynomial for C ; we label them by Greek letters (i, m, o have already been de"ned): i"j ) j#k ) k , (54) m"j ) j!k ) k ,
(55)
o"2 j ) k,
(56)
k"n( ) j ,
(57)
l"n( ) k ,
(58)
the "rst three and the last two are, respectively, of degree two, one in the coordinates. However, this is not enough for generating the ring P! of C polynomial invariants and for labeling all the C -orbits. Indeed, the Molien function is dh 1 p M (j)" ! (1!j)(1!2j cos h#j) 2p 1#j " . (59) (1!j)(1!j)
The degrees of j in the "ve factors of the denominator are equal to the degrees of the "ve homogeneous polynomials of variables j , k of Eqs. (54)}(58). But the numerator requires a sixth G H We could have chosen for the direction n( the third coordinate axis; then k"j , l"k .
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
189
generator of P! of degree 2. The square of this invariant should be a polynomial function of six basic invariant polynomials. This statement can also be veri"ed geometrically: indeed any set of values satisfying Schwarz inequalities for the scalar products: o4i!m ,
(60)
k4(i#m) , l4(i!m) de"nes two circles"C -orbits which can be distinguished by the sign of p"(n( , j, k),n( ) ( j;k) .
(61) (62)
(63)
Moreover,
1 i 1 1 p"(i!m)#okl! o# (k#l)! mk# ml 50 . 4 2 2 2
(64)
So the module of C -invariant functions on <, the six-dimensional vector space of the O(4) adjoint representation may be represented as P! "P[i, m, o, k, l]䢇(1, p) . (65) To each point of the "ve-dimensional domain de"ned by the inequalities in Eqs. (60)}(62) and (64) corresponds a unique C orbit and conversely: this is the orbit space <"C . The four other one-dimensional subgroups have C as invariant subgroup. We can build their orbit spaces from the one of C . Let G be any of the three groups C , C , D . Then G is the F T union of two C cosets: G"C 6C C where C denotes, respectively, the re#ections C , C , 6 6 T F and C a rotation by n around an axis (in the plane h) orthogonal to the rotation axis of C . The action of the quotient group G/C on the C orbit space can be obtained by computing the action of these three operations C on the C invariants o, k, l, p (the O -invariants i, m are invariants for 6 its "ve one-dimensional subgroups). The result is given in Table 6. We call those pseudo-invariants which are multiplied by !1 or invariant under the action of the symmetry elements (more properly speaking pseudo-invariants transform according to onedimensional real representation of the symmetry group). Their squares are invariants and we will obtain the set of denominator invariants of G by replacing the pseudo-invariants among o, k, l by their squares. The product of any two pseudo-invariants transforming according to the same one-dimensional representation of the symmetry group is an invariant which will be a numerator invariant. We can treat similarly the group D which is made of the four cosets of C which are in F the union of the three groups G's. So its denominator invariants are i, m, k, l, o. The "rst three lines of Table 6 show that lp is the only non-trivial numerator invariant. The structure we found for the module of the invariant polynomials on < of these four other one-dimensional subgroups of O(3), can be compared with their Molien functions. To compute We use the property that the square of the determinant on m m-component vectors v ,14a4m is the determinant of ? the matrix of elements v .v . ? @
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
190
Table 6 Action of C and discrete symmetry generators of the group O(3)T on a set of C -invariants. i and m are invariant of the total group O(3)T. They are omitted from the table. The ! sign indicates the multiplication by !1 Element
o
k
l
p
C C T C F C
# ! ! #
# ! # !
# # ! !
# # ! !
T T T T F T
! # # !
! # ! #
# # ! !
! ! # #
This line is introduced to show explicitly that we deal with invariants of C
group.
them it is useful to compute the (incomplete) Molien functions M of the C cosets not containing % 1 of the "rst three groups. We obtain
1 p dh 1 " , M T " ! 2p (1!j) (1!j)
(66)
dh 1 p M F " ! (1!j)((1!j)!4j(cos h)) 2p 1 " , (1!j)(1!j)
(67)
dh 1 p 1 M " " . " (1#j)(1!j) (1#j)(1!j) 2p
(68)
We can express the complete Molien function for the invariants in terms of these M : M
!T
"(M #M T ) , ! !
(69)
M
!F
"(M #M F ) , ! !
(70)
M
"
"(M #M ) , ! "
(71)
M
"(M #M T #M F #M ) . ! ! " "F ! Then we obtain for the expression of the Molien functions: M
1#j#j#j " !T (1!j)(1!j)(1!j)
(72)
1#j " , (1!j)(1!j)
(73)
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
191
M
!F
1#2j#j " , (1!j)(1!j)(1!j)
(74)
M
"
1#j#2j " , (1!j)
(75)
M
1#j#j#j " . "F (1!j)(1!j)
(76)
We verify again that Molien functions are rational fractions. However, for Eq. (73) the form we have used does not correspond to the reduced one (which is given between brackets). The situation is di!erent from the Sloane counter example referred to in Chapter I (Sloane, 1977). The reduced form of the C Molien function does correspond to a module given in Eq. (78) while the module T corresponding to nonreduced form (given in Eq. (73) without parenthesis) is given in Eq. (77): (77) P!T "P[i, m, o, k, l]䢇(1, p, ok, pok) , P!T "P[i, m, p, k, l]䢇(1, ok) . (78) To prove the equivalence of these two modules we replace p by its square in the denominator invariants of Eq. (78) and add to the numerator invariants the numerator invariants multiplied by p; the two modules have same basis. Eq. (64) shows that p is a function of o and of i, m, k, l, ok; since the latter are other invariants of Eq. (78), we can replace p by o; that ends the transformation of Eq. (78) into Eq. (77). We prefer to use the "rst form because o"0 in one of the equations de"ning the manifold R and our future qualitative analysis uses basically G-invariant functions on R (de"ned by i"1, o"0). Table 7 enables us to write explicitly the structure of the "ve modules of G-invariant functions on R: P! "R "P[m, k, l]䢇(1, p) , P!T "R "P[m, k, l]䢇(1, p) , P!F "R "P[m, k, l]䢇(1, lp) , P" "R "P[m, k, l]䢇(1, kl, kp, lp) , P"F "R "P[m, k, l]䢇(1, lp) .
(79) (80) (81) (82) (83)
2.5. One-dimensional Lie subgroups of O(3)T and their invariants Extension to groups including time-reversal symmetry operation was discussed in Section 7 of Chapter I. We summarize here in Table 7 the system of invariants of all 16 one-dimensional subgroups of the group O(3)T. The restriction of the system of invariant polynomials on R becomes especially simple for four symmetry groups C T , C T, C T , and D T. For all these groups we have on T T F F R only three denominator invariants: P! ;TT "R "P[m, k, l] ,
(84)
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
192
Table 7 Denominator and numerator invariants in the action on R of the 16 one-dimensional Lie subgroups of O(3)T. h "i and h "m are present for all subgroups and therefore omitted from the table. Invariants which have a "xed value on R,S ;S (due to o"0) are between ( ). Four parts of the table correspond, respectively, to connected subgroup C , to seven subgroups with two connected components, to seven subgroups with four connected components and to the D T group with eight connected components F G
h
h
h
u
u
u
C C T C F D C T C T T C T F C T D F C T T C T F D T C T T F C T F T D T T D T F
(o)
k
l
p
(o) (o) (o) (o) (o) (o) (o)
k k k k k k k
l l l l l l l
p lp kl kp
(ok) (ol) kp (ok)
(okp) (op) lp (op)
p p
kl (lo)
(klp) (lop)
(o) (o) (o) (o) (o) (o) (o)
k k k k k k k
l l l l l l l
lp (ko) klp kp p (lo) kl
(kop)
(klo)
(po) (lko) (lko)
(klo) (lpo) (lkpo)
(o)
k
l
(okl)
P!T ;T"R "P[m, k, l] ,
(85)
P!F ;T "R "P[m, k, l] ,
(86)
P"F ;T"R "P[m, k, l] .
(87)
It is curious to note that especially these groups have the most natural physical interpretation. C T is the symmetry group for the atom in a static electric "eld or for Rydberg states of T a heteronuclear diatomic molecule. C T is the symmetry group of an atom in a constant magnetic "eld (Zeeman e!ect). F T C T is the maximal common subgroup of C T and C T . Thus, it corresponds to T T F T the symmetry of an atom in the simultaneous presence of two parallel magnetic and electric "elds. The D T group is the symmetry group for Rydberg states of a homonuclear diatomic F molecule, or of the quadratic Zeeman e!ect. 2.6. Orbits, strata and orbit spaces of the one-dimensional Lie subgroups of O(3) acting on R The stabilizers of O(3) have been determined in Table 4; they belong to three O(3) conjugation classes: those of C (k), C ( j), C ( j). The stabilizers of the "ve groups G of Eq. (53) are their T F Q
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
193
Table 8 Strata and orbits in the action of the "ve one-dimensional Lie subgroups of O(3) on R G R"G
Stabilizer
gc
dim. strat.
Strata equations and inequalities
Number orbits
Nature orbits
cr?
C S
C 1
c g
0 4
k"1, l"1 k(1, l(1
4 R
A, B, C, D S
cr
C T
C T C C ( j ) Q 1
c c
l"1 k"1 k"0, l(1 0(k(1Nl(1
2 1 R R
C, D AB S 2S
cr cr
g
0 0 3 4
C F
C F C
c c
0 0
k"1 l"1
2 1
A, B CD
cr cr
S
C (n) Q C G 1
R
g
2 4
1#m 04 "k(1Nl"0 2 m"1, k(1 m(1, 0(k(1Nl(1
R R
S S 2S
D Susp. RP
C C 1
c c g
0 1 4
k#l"1, kl"0 m"1, k"l"0 (m"1, 0(k(1, 0(l(1)
2 2 R
AB, CD C ,C 2S
cr cr
D F
C F C T C ( j) F C (k) T
c c c c
0 0 1 1
k"1 l"1 m"1, k"l"0 m"!1, k"l"0
1 1 1 1
AB CD C C
cr cr cr cr
B
C (n) Q C G C ( j ) Q
R
2S 2S 2S
B
1
2
2 2 3 g
4
1#m 04 "k(1Nl"0 2 m"1, 0(k(1 k"0, l(1, m(1#l 1!m 1#m (1, l( (1, 0(k( 2 2 (m!1)#k#l'0
R R R
4S
Column 1: Below the group G is given the topological nature of the orbit space R"G (Susp. is for suspension). Column 2: Stabilizer of the stratum. C ( j )"C ( j) except when j"0; then it is C (n( ;k). Q Q Q Column 3: c is for closed stratum and g for generic stratum. Column 4: Dimension of the stratum. Column 5: The stratum is a semi-algebraic set; its de"ning polynomial equations and inequalities are given here. Column 6: Number of orbits in the stratum. Column 7: Geometry of each orbit in the stratum. Column 8: Critical orbits have a `cra in this column.
intersections with the stabilizers of O(3). So, for these group G, we can directly make the list of strata, and give the invariant equations (for closed strata) and/or inequalities which de"ne them. We are interested only about the orbits and strata on R. This information is given in Table 8.
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
194
First, let us write again all equations on the C invariants when they are restricted on the manifold R de"ned by i"1 and o"0. This corresponds to the study of the three-dimensional orbifold R"C . Eqs. (60)}(62) and (64) become (88) k4(1#m) , l4(1!m) , (89) (90) !4p4 and 044p"(1!m)!2(1!m)k!2(1#m)l41 .
(91)
This last equation shows explicitly that p is determined up to a sign by m, k, l. From the two previous equations we deduce k"$1Nm"1, l"0"p ,
(92)
l"$1Nm"!1, k"0"p .
(93)
This means that for a given n( there are four points A, B, C, D of R "xed by C (n( ): A: j"n( , k"0 , ,k"1, m"1, l"0 ,
(94)
B: j"!n( , k"0 , ,k"!1, m"1, l"0 ,
(95)
C: k"n( , j"0 , , l"1, m"!1, k"0 ,
(96)
D: j"0, k"!n( , ,l"!1, m"!1, k"0 .
(97)
On S ;S the four points correspond to the pairs of poles: NN, SS, NS, SN, respectively. As we shall see any function invariant by one of the "ve one-dimensional Lie subgroups of O(3), has an extremum on each of these four points of R. Similarly, we have two circles on R: C : k"0, n ) j"0Nm"1, k"l"0 , (98) C : j"0, n ) k"0Nm"!1, k"l"0 . (99) They correspond, on the two equators of S ;S , to pairs of identical points, diametrically opposed points, respectively. Each of these two circles C , C form a critical orbit of D or D . F Finally, we should consider the particular cases where one of the vectors j, k vanishes, i.e. m"$1. When m"1 then k"0; since any axial vector j is invariant by the symmetry through the origin (while any non-trivial polar vector changes of sign), m"1 contains a stratum with stabilizer
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
195
C (the group generated by the symmetry through the origin) for the groups C and D . When G F F m"!1 then j"0 and the polar vector k is invariant by the symmetry through any plane containing it. The symmetry through the plane which contains also n( belongs to a stabilizer of C and D and this is still true when jO0 and collinear to n( ;k, see Table 8. T F The "ve orbit spaces R"G, are of dimension three. We will "rst make a direct study of their topology. We shall "rst prove that topologically R"C &S . Then, as suggested by the } signs in the columns of k, l, p of Table 6, the orbit spaces of C , C , D are obtained by identifying the T F points symmetrical through, respectively, a three-, two-, one-dimensional linear manifold containing the center of the sphere S . In the four-dimensional space of parameters m, k, l, p, when !1(m(1, Eq. (91) shows that the section of R"C &S by a hyper-plane m"constant is an ellipsoid &S : !1(m(1 , (100) 2p#(1!m)k#(1#m)l"(1!m) . Indeed this is true because each coe$cient (function of m) is strictly positive. Furthermore, this equation implies Eqs. (88)}(90). We are left to study the particularly two cases: m"$1. m"1Nl"p"0
(101)
and from Eq. (88): !14k41. Similarly, m"!1Nk"p"0
(102)
and from Eq. (89): !14l41. These two segments of line are the limits when mP$1 of the ellipsoids de"ned by Eq. (100). This proves that the orbifold for the C action on R is the 3D-sphere R"C &S (103) with four marked points. Its four points A, B, C, D represent four critical orbits of one point each, and the complement is the image of the generic stratum. Remark that in the invariant space of `orthogonal coordinatesa p, m, k, l, the orbit space R"C has three symmetry hyper-planes p"0, k"0, l"0, mutually orthogonal. Table 7 shows that R"C is obtained from R"C by identifying its points of coordinates T m, p, l, $k. We can represent it by the intersection of R"C with the closed half-space k50. This T orbit space R"C is therefore topologically equivalent to a hemisphere, i.e. a ball B : T R"C 5(k50)"R"C &B . (104) T This is also the topological nature of the projection of R"C on its symmetry hyper-plane k"0; T indeed from (1!m)k50 we deduce from Eq. (91) that 2p#m#(1#m)l4 ,
(105)
Some general results are known on the topology of three-manifolds (a short for three-dimensional manifolds): (Fomenko, 1983; Thurston, 1969). They may be used to "nd the topology of the orbifolds on the basis of the geometrical representation (see Appendix A).
196
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
but one must remark that there is a unique point of this projection which does not correspond to a unique point of the orbit space R"C : indeed m"1, p"k"l"0 if the projection of the whole T segment m"1, p"l"0, !14k41. To summarize, Table 8 shows that the orbit space R"C T contains four strata represented by (1) two points C, D de"ned in Eqs. (97) and (98): each corresponds to a one point C -orbit; T (2) one point X: m"1"k, p"0"l, representing a two point C -orbit, +A, B, (see Eqs. (95) and (96)); (3) the boundary (RB &S ) minus the two points C, D representing the C stratum; Q (4) the interior of B minus the point X representing the generic stratum. We now prove R"C &S . Indeed, from Table 7 in the parameter space one obtains R"C from F F R"C by identifying the points symmetrical through the two-plane P(m, k) of the coordinates m, k (it is the intersection of the two symmetry hyper-planes l"0, p"0). Consider the sphere S : m#k#l#p"1 topologically equivalent to R"C . By identi"cation of its points in the pair s "(m, k, $l, $p (which are not in P(m, k))), one obtains a topological space &R"C . The ! F projection on the two-plane P(m, k) is the disk B : m#k41. Note that RB "S 5P(m, k) while each point p(m, k) of the interior of B is the projection of a circle C of equation N l#p"1!m!k in the two-plane perpendicular to P(m, k). By the identi"cation of its points symmetric through its center the circle C is transformed into a circle C ; this holds for each p in the N N interior of B . Correspondingly this point identi"cation transforms S into S . We note from Table 7 that one transforms R"C into R"D by identifying the points of F F opposite k coordinate. As for the transformation of R"C into R"C study above, we obtain that T R"D &B . F We are left with the study of R"D . It is obtained from R"C by identifying the points in each pair s "(m, $k, $l, $p), i.e. the points symmetric through the intersection axis of the three ! symmetry hyper-planes k"0, l"0, p"0. This axis is the normal to the hyper-plane H: m"0. We remark that R"C &S is topologically equivalent to the double cone of vertices $1, 0, 0, 0 and basis B"(R"C )5H&S ; this is also called a suspension of B. The point identi"cation transforms B into B &RP. So: R"D &suspension (RP) . (106) Table 8 gives in its "rst column the topological nature of the "ve orbit spaces studied in this section. The geometrical representation of the orbit spaces for one-dimensional Lie subgroups studied in this section is discussed in more detail in Appendix A. 2.7. Invariants of xnite subgroups of O(3) acting on R The construction of invariant functions on the R manifold is based on the preliminary construction of the integrity basis on the six-dimensional space where the action of the symmetry group of the problem is linear. The Molien function and the invariants themselves for the six-dimensional space x y or k j may be found from known expressions for Molien functions and integrity bases for irreducible representations. Next step includes the restriction of the polynomial algebra on the sub-manifold R of the 6D-space. The general procedure of the restriction of the polynomial ring
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
197
de"ned on the manifold to the sub-manifold was outlined in Section 5.2 of Chapter I. The sub-manifold R is de"ned in the six-dimensional space x y by the polynomial equations x", y". If the point symmetry group does not include improper rotations (inversion or re#ections) these polynomials may always be considered as denominator invariants. In such a case we just eliminate them from the integrity basis constructed for the 6D-space and the resulting integrity basis gives the basis for the 4D sub-manifold. For point groups which are not the subgroups of SO(3) we start with the consideration of the similar problem for the proper rotation subgroup and after that take into account the e!ect of improper symmetry elements working directly on the 4D-sub-manifold R. The detailed realization of this procedure is given in Appendix B on several examples. Below we just summarize the results for the case of the C point group as the invariance group of the problem. Let us take x and y to coincide with the C -axis. In such a case x , x , y , y transform according to the anti-symmetric representation B and x , y according to the totally symmetric representation A of the C point group. We are interested in the module of invariant functions de"ned on the sub-manifold R given by two polynomial equations: x #x #x " , y #y #y " . After the restriction on R the Molien function for invariants has the form 1#6j#j . M "R " ! (1!j)(1!j)
(107) (108)
(109)
The explicit form of the module of invariant functions on R may be easily given as well: P! "R "P[x , x , y , y ]䢇(1, u , u , u , u , u , u , u ) , where
(110)
u "x y , u "x y , u "x y , u "x y , u "x x , u "y y , u "x x y y . More condensed notation for the set of numerator invariants may be used
(111) (112)
P! "R "P[x , x , y , y ]䢇((1, x x )(1, y y ), (x , x )(y , y )) . (113) Instead of listing explicitly [as in Eq. (110)] all eight numerator invariants, we show in Eq. (113) that they may be reconstructed as products of slightly simpler monomials. One should remark that the choice of denominator and numerator invariants is not unique and the choice proposed here is just one of the possible ones. Much more information may be found in Appendix B. 2.8. Orbits, strata and orbit spaces of xnite subgroups of O(3) acting on R The strata for any "nite subgroup G of O(3) for its six-dimensional representation (k j which is the sum of polar and axial vector representations) follow immediately from the well-known results
198
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
Table 9 Strata and orbits in the action of C
T
on R
Stabilizer
gc
dim.
Number
Nature
C T C CT Q CB Q 1
c c
0 0 2 2 4
2 1 R R R
1 2 2 2 4
g
g is for generic stratum; c is for closed stratum. Dimension of the stratum. Number of orbits in the stratum. RL stands for the n-dimensional stratum. Number of points in each orbit.
for polar and axial vector representations. We list below strata for group actions on R manifold for several point groups which are the most interesting from the point of view of possible applications. These are examples of molecules with the C , D , and ¹ point symmetry. Similar tables for all T F B other possible group symmetries are given in Appendix C. Table 9 listed in this subsection show in column 1, the stabilizer of the stratum. In column 2 closed and generic strata are indicated. We remark once more that some strata are neither closed nor generic. In column 3 the dimension of the stratum is given. For the "nite group action on R the dimension of the generic stratum is always 4. Column 4 gives the number of orbits in the stratum. If the dimension of the stratum is zero the number of orbits in the stratum is "nite. If the dimension of the stratum is positive the number of orbits in the stratum is in"nite. We denote it as RL with n equal to the dimension of the stratum. Column 5 shows the number of points in each orbit. For "nite group actions on R this number is always "nite and for the generic stratum it is equal to the number of elements of the group. 2.9. Orbits, strata and orbit spaces for T-dependent subgroups of O(3)T We have shown that there are 16 one-dimensional Lie subgroups of the complete symmetry group O(3)T of the problem. There are naturally many "nite subgroups which can be obtained in a way similar to our construction of one-dimensional subgroups. Extension of point subgroups of O(3) to points subgroups of O(3)T is analogous in some sense to the construction of antisymmetry (Shubnikov, 1951) or magnetic symmetry groups (color groups) well known in solid-state physics (Hamermesh, 1964; Shubnikov and Belov, 1964). We give in Tables 10, 11 and 12 the analysis of the orbits, strata and orbit spaces for some groups including time reversal. We have chosen those groups which have simple physical realization as symmetry groups of the hydrogen atom in the presence of di!erent external "elds. This simple quantum system enables one to study quite a lot of di!erent invariance symmetry groups. Let us consider several physical situations together with more or less detailed description of corresponding symmetry groups: E Hydrogen atom without any external "elds. The symmetry group includes the full orthogonal group O(3) and the time reversal. We can write the group as the direct product O(3)T where
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264 Table 10 Strata and orbits in the action of D
F
199
on R
Stabilizer
gc
dim.
Number
Nature
C F C T C T C C F C Q 1
c c c c
0 0 0 0 2 2 4
1 1 2 1 R R R
2 2 3 6 6 6 12
g
g is for generic stratum; c is for closed stratum. Dimension of the stratum. Number of orbits in the stratum. RL stands for the n-dimensional stratum. Number of points in each orbit.
Table 11 Strata and orbits in the action of ¹ on R B Stabilizer
gc
dim.
Number
Nature
C T C S C T C Q 1
c c c c
0 0 0 0 2 4
2 1 1 1 R R
4 8 6 6 12 24
g
g is for generic stratum; c is for closed stratum. Dimension of the stratum. Number of orbits in the stratum. RL stands for the n-dimensional stratum. Number of points in each orbit. Table 12 Invariant manifolds for the C
F
T symmetry group action on R (Hydrogen atom in magnetic "eld.) T
Stabilizer
dim.
Equations
C T F T C T T [C T ] F T [C T ] G T T T T C
0 0 2 2 3 3 4
k"$1 l"1 l"0; m"2k!1 l"0; m"1 p"0 l"0 R
This group includes four elements: E, p , T , T . F T This group includes four elements: E, i } inversion, T , T . T
200
E
E
E
E
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
the two element group Z (T) is formed by the identity E and time reversal T. In fact, the dynamic symmetry group for hydrogen atom in the absence of external "elds is higher. We use this higher group to introduce the approximate quantum number corresponding to an individual n-shell of the Rydberg atom even in the case of external "elds (assuming that the splitting of the n-shell is small as compared to the inter-n-shell gap). Di!erent external "elds break this symmetry till various subgroups. Hydrogen atom in the presence of constant magnetic "eld. This case corresponds to the Zeeman e!ect for the hydrogen atom. C group is the invariance group of spatial geometric transformaF tions. The complete symmetry group including time-reversal operations is C T . It is the F T semi-direct product of C and two element group T "+E, T , including two elements: the F T T identity E and the product of time reversal and space re#ection T in the plane of symmetry axis. T The group C T includes the following classes of conjugated elements: +E, is the identity, F T +p , the re#ection in the plane orthogonal to the magnetic "eld axis, +R , the rotation on angle F (
around magnetic "eld, +S , the rotation-re#ection on angle around magnetic "eld, +T ( ), ( T the time reversal followed by re#ection in a plane passing through the magnetic "eld, +T ( ), the product of time reversal and rotation by n around an axis orthogonal to the "eld axis. An operation T can be equivalently described as a product of T and S operations. Table 12 gives T ( invariant manifolds for the C T group action on R. We give here invariant manifolds F T instead of strata to simplify the de"ning equations. Hydrogen atom in the presence of constant electric "eld. This is the case of the Stark e!ect for the hydrogen atom. C group is the invariance group of spatial geometric transformations. The T invariance under the time reversal takes place in the absence of a magnetic "eld as well. Therefore, the complete invariance group can be written as C T. This group includes the T following symmetry operations: E is the identity, T the time reversal, R the rotation on angle (
around electric "eld, p the re#ection in plane passing through the "eld axis, TR the time ( ( reversal followed by rotation, T ( ) the time reversal followed by re#ection. The system of Q invariant manifolds of the C T action on R is given in Table 14. T Collinear electric and magnetic "elds. The symmetry group includes the C subgroup of rotations around the common direction of the electric and magnetic "eld. The complete invariance group can be written as C T . Naturally, this group is the subgroup of the T symmetry group appropriate for the case of only electric or only magnetic "eld. The system of invariant manifolds of the C T action on R is given in Table 13. T Orthogonal electric and magnetic "elds. The symmetry group is "nite in this case and includes only four symmetry elements: identity E, re#ection in the plane orthogonal to the magnetic "eld
Table 13 Invariant manifolds for the C T symmetry group action on R (Hydrogen atom in parallel magnetic and electric T "elds.) Stabilizer
dim.
Equations
C T T T T C
0 3 4
l"1; k"1 p"0 R
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264 Table 14 Invariant manifolds for C
T
201
T symmetry group action on R (Hydrogen atom in electric "eld.)
Stabilizer
dim.
Equations
C T T C T T [C T ] Q QY [C T] Q C Q T T C
0 0 2 2 3 3 4
l"$1 k"1 k"0; m"1!2l k"0; m"!1 k"0 p"0 R
This group includes four elements: E is the identity, p the re#ection in the plane including the "eld axis and a vector orthogonal to the vector L, T the time reversal followed by re#ection in the plane formed by "eld axis and the vector L, QY T the time reversal followed by rotation over p around "eld axis.
p , product Tp of time reversal T and re#ection p in plane formed by two orthogonal "elds, F T T and the symmetry operation TC which is the product of the time reversal T and the C rotation around the electric "eld direction. We will use the notation G for this symmetry group. This group has three subgroups of order two which will be useful below. The group T includes as non-trivial symmetry operation the product Tp of time reversal T and T T re#ection p in plane formed by two orthogonal "elds. The group C includes the re#ection in the T Q plane orthogonal to the magnetic "eld (this plane includes the electric "eld vector). The group T includes as a non-trivial symmetry operation, the product of time reversal and the C rotation around the electric "eld direction. E Generic non-orthogonal non-collinear con"guration of two "elds. The symmetry group T has Q order two and includes one non-trivial symmetry operation Tp which is the product of the time reversal and the re#ection in plane formed by the two "elds. T is the subgroup of the symmetry Q group G for orthogonal "elds and the subgroup of the symmetry group C T for collinear T Q "elds. It is the only common subgroup for these two important limiting cases of parallel and orthogonal "elds. E To complete the list of interesting symmetry groups we add here the symmetry group of the quadratic Zeeman e!ect or of Rydberg states of homonuclear diatomic molecules. This is the D T group which is the maximal one-dimensional subgroup of the O(3)T symmetry F group. The set of invariant manifolds for this group is given in Table 15. Inclusion of additional symmetry operations enables simpli"cations in such physically important cases as the Zeeman e!ect, Stark e!ect, hydrogen atom in parallel "elds or quadratic Zeeman e!ect the integrity basis and the geometrical representation of the orbifold. Going from the C to C T , from C to C T, from C to C T , and from D to D T leads to an Q T T F F Q F F integrity basis consisting of only basic `denominatora invariants. The space of orbits for extended groups possesses richer strati"cation (see Tables 12}15) but the geometrical form of the orbifold becomes simpler (see Figs. 1}3) as compared to the geometrical form of the orbifold for purely geometrical point group symmetry studied in Appendix A.
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
202
Table 15 Invariant manifolds for the D T symmetry group action on R (Complete symmetry group of the quadratic Zeeman F e!ect.) Stabilizer
dim.
Equations
C T F Q C T T C T F Q C T T C (n)T Q Q C T G Q [C T] Q [C T ] Q QY C ( j)T Q C Q T T Q C
0 0 1 1 2 2 2 2 2 3 3 3 4
k"1 l"1 m"1, k"l"0 m"!1, k"l"0 1#m"2k m"1 k"0; m"!1 k"0; m"1!2l k"0, l"0 k"0 p"0 l"0 R
C subgroup includes the symmetry plane orthogonal to the vector L; T is the time reversal followed by re#ection in F Q plane formed by L and magnetic "eld. C is the re#ection in plane orthogonal to the C axis; T is the time reversal followed by re#ection in plane of K and L. Q Q This group includes four elements: E is the identity, p the re#ection in the plane including the "eld axis and a vector orthogonal to the vector L, T is the time reversal followed by re#ection in the plane formed by "eld axis and the vector QY L, T the time reversal followed by rotation over p around "eld axis. T is the time reversal followed by C rotation around K.
Fig. 1. Space of orbits for the C
F
T action on R. (H atom in the presence of magnetic "eld.) Q
Fig. 2. Space of orbits for the C T action on R. (H atom in the presence of parallel magnetic and electric "elds.) Q
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
203
Fig. 3. Space of orbits for the C T action on R. (H atom in the presence of electric "eld.) Q Fig. 4. Three slices of D T orbifolds for di!erent k values. Di!erent strata are denoted by letters as follows: F a!C T ; b!C ( j)T ; c!C T; d![C T]; e!C T; f![C T ]; g!C ; h!C T ; i!T ; F Q Q T Q T Q QY Q G Q j!C (n)T ; k!T ; l!C ; m!C T . De"ning equation for the strata are given in Table 15. Q Q Q F Q
To see better the strati"cation of the R under the action of D T symmetry group which F includes 13 di!erent strata we give in Fig. 4 the schematic representation of the space of orbits through three sections corresponding to k"0, 0(k(1, and k"1. Similar extension of the geometrical point group symmetry can be done in cases of "nite symmetry groups; e.g. see the example of the G symmetry group for the hydrogen atom in two orthogonal "elds mentioned a little earlier in this section is analyzed in more details in Section 4.4 and in a separate publication (Sadovskii and Zhilinskii, 1998).
3. Construction and analysis of Rydberg Hamiltonians 3.1. Ewective Hamiltonians Let us discuss now the simplest e!ective Hamiltonians invariant with respect to subgroups of the O(4) group and their relation with invariant functions de"ned on the R manifold. We start with the Hamiltonian for a hydrogen atom which is O(4) invariant. It was introduced in Eq. (1) and rewritten in terms of the angular momentum vector J and the transformed Laplace}Runge}Lenz vector, K Eq. (3) which satis"ed the commutation relations (4)}(6) and two relations (7) and (8). Further we want to study a slightly perturbed Hamiltonian which conserves essentially the presence of n multiplets typical for the hydrogen atom (this is precisely the situation with Rydberg atoms and molecules in the limit of small splitting of n multiplets). An e!ective Hamiltonian for a given n shell may be written as a phenomenological e!ective Hamiltonian constructed from J and K operators. Relations (7) and (8) are important to reach the correspondence between the quantum
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
204
operators J and K and their classical analogs, because the classical variables j, k satisfy the normalization condition (40), (39), (42) which is independent of the particular realization of the quantum operator for any concrete n shell. As a consequence, the correspondence between quantum operators and classical dynamic variables has the form J (n!1j , ? ?
(114)
K (n!1k . (115) ? ? Using the integrity basis for the classical representation we can now construct easily the basic polynomial invariants formed by the quantum operators and to indicate their classical analogs J!K"(n!1)m ,
(116)
J "(n!1k , X
(117)
(118) K "(n!1l . X For higher-order invariants one needs to make the anti-symmetrization of the operators which do not commute. For example, the operators representing the auxiliary numerator invariants have the form +J , K ,!+J , K ,"(n!1)p , V W W V +K , (+J , K ,!+J , K ,),"(n!1) lp , X V W W V +J , K ,"(n!1)kl , X X +J , (+J , K ,!+J , K ,),"(n!1) kp , X V W W V where
(119) (120) (121) (122)
(123) +A, B,"(AB#BA) . Sometimes it is useful to work in the x, y representation (40), (39) instead of j, k. We denote the corresponding quantum operators by J "(J!K)/2 , (124) J "(J#K)/2 . (125) The Hilbert space of wave functions associated with the n multiplet may in such a case be formed by the basis "J "J "(n!1)/2; M , M 2 , (126) M "!(n!1)/2,2, (n!1)/2 . (127) G The correspondence between quantum operators J , J and classical dynamic variables follows directly from Eqs. (116)}(122): J ) J "(n!1)m/4 ,
(128)
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
J #J "2(n!1k , X X
205
(129)
J !J "(n!1l , (130) X X J J !J J "(n!1)p/2 , (131) W V V W +(J !J ), (J J !J J ),"(n!1) lp , (132) X X W V V W J !J "(n!1)kl , (133) X X +(J #J ), (J J !J J ),"(n!1) kp . (134) X X W V V W There are two ways leading to the construction of the e!ective operators. One is purely a phenomenological approach based on the utilization of all operators allowed by symmetry with their coe$cients being the adjustable parameters of the model. The other one uses the transformation of the initial operator to the e!ective one by applying some kind of perturbation theory or contact transformation, etc. This second procedure results in the e!ective operator with "xed coe$cients which depend on the initial Hamiltonian. Both approaches result in the Hamiltonian which may be written in terms of integrity basis. We remind that as soon as invariant polynomials are known, an arbitrary Hamiltonian may be written in the form of expansions (properly symmetrized to take into account the non-commutativity of quantum operators) in terms of denominator invariants h and G numerator invariants u Q (135) C 2 I u hL hL 2hLI . H" I L L L _Q Q 2 L L LI Q An explicit form of invariants for all continuous subgroups of O(3) was found in Section 2.4 (see Table 7) and for some point groups in Section 2.7 and in Appendix B. The purely phenomenological way to represent e!ective Hamiltonians supposes that all the C 2 I coe$cients are L L L _Q adjustable parameters, whereas perturbation treatment gives the C 2 I coe$cients in L L L _Q Eq. (135) as explicit functions of the parameters of the initial Hamiltonian. We will start by analyzing "rst the phenomenological approach to the construction of e!ective Hamiltonians taking into account their symmetry properties. The main idea of our approach is to introduce along with the phenomenological construction some kind of complexity classi"cation of e!ective Hamiltonians which is based on the quantitative measure } the number of orbits of extrema. We also call them the stationary orbits (points, or "nite number of points, or manifolds). 3.2. Qualitative description of ewective Hamiltonians invariant with respect to continuous subgroups of O(3) We discuss here the qualitative classi"cation of e!ective Hamiltonians invariant with respect to di!erent symmetry groups. By qualitatively di!erent Hamiltonians we mean those operators which are characterized by di!erent sets of stationary orbits with each stationary orbit characterized by its stabilizer and Morse index. The number of stationary orbits may be used as a measure of the complexity of the Hamiltonian. The zero level of complexity corresponds to a set of Hamiltonians with the minimal possible
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
206
number of stationary orbits (compatible with the topological structure and the symmetry group action according to the equivariant Morse}Bott theory). We will list below qualitatively di!erent Morse}Bott Hamiltonians of some low level of complexity, whereas in Appendix D more general results are outlined. 3.2.1. O(3) invariant Hamiltonian Any O(3) invariant Hamiltonian may be written in the form H"H(m) ,
(136)
where H(m) is an arbitrary (su$ciently good) function of one variable which is de"ned for !14m41. m is the only invariant polynomial which is not constant on the R manifold (see Section 2.3). For quantum operators instead of m we should use equivalent operators given by Eq. (116) (or (128)) in the J, K (or J , J ) representation. So any e!ective Hamiltonian may be written as a power series in m or more generally as an arbitrary function of m as in Eq. (136). Any O(3) invariant classical Hamiltonian possesses two critical manifolds due to the presence of the two critical orbits. These two critical orbits have very simple physical meaning. C critical F orbit corresponds on the orbifold to the point m"1 ( j ) j"1, k ) k"0), i.e. the quantum state localized near this point has maximal (possible for a given n) value of the orbital momentum l"n!1. Another critical orbit corresponds to the point m"!1& ( j ) j"0, k ) k"1) i.e. for the quantum problem the quantum state localized in the phase space near this point has minimum possible value of the orbital momentum l"0. More complicated Hamiltonians can be classi"ed according to the number of stationary orbits (the RP manifolds) which belong to the generic C stratum. Qualitatively di!erent generic O(3) Q invariant Hamiltonians are given in Table 16. It is clear that any O(3) invariant Hamiltonian results in a system of energy levels which is completely characterized by one quantum number l, the weight of the irreducible representation of the group O(3). This one regular sequence may be rather complicated in the general case but for the simplest Morse}Bott-type Hamiltonian (p"0 in Table 16) the sequence should be monotonic. It is reasonable to assume that the number of extrema of the H"H(m) function in a general case is much smaller than the number n imposing limit on possible l values, l4n!1. In such a case the energy spectrum explicitly shows the regular behavior. In contrast, if the number of extrema on H(m) is larger or of the same order as n (assuming n is large), the energy spectrum may seem to be erratic. Near an extremum, corresponding to the C critical orbit (m"1&( j ) j"1, k ) k"0)) for any F (su$ciently good) Hamiltonian, there is a system of regular energy levels, E(l)"const. l(l#1)
(137)
characterized by l"(n!1), (n!2),2, whereas near an extremum, corresponding to the C critical orbit (m"!1&( j ) j"0, k ) k"1)) there is a regular system of energy levels, T E(l)"const. l(l#1)
(138)
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
207
Table 16 Classi"cation of qualitatively di!erent O(3) invariant Hamiltonians due to the complexity level Level of complexity
C T stratum
C F stratum
N with index 0
N with index 1
0 0 1 1 2 2
max min max min max min
min max max min min max
no no no 1 1 1
no no 1 no 1 1
2p 2p
max min
min max
p p
p p
2p#1 2p#1
max min
max min
p p#1
p#1 p
N is the number of stationary RP manifolds on C stratum. Q p is a non-negative integer.
which is characterized by l"0, 1, 2,2 . Near an extremum corresponding to generic C orbit the Q system of energy levels may be described as a bent series of the form E(l)"const.[l(l#1)!l (l #1)] with the l value varying around some generally non-integer l value.
(139)
3.2.2. C invariant Hamiltonian The C invariant phenomenological Hamiltonian may be written as (140) H " C mL kL lL # CN pmL kL lL L L L L L L ! with m, k, l, p given in Eqs. (116)}(119), or in (128)}(131) with all necessary symmetrization. Any Hamiltonian (140) has four critical orbits (stationary points A, B, C, D, see Table 8 and Eq. (101)). There is only one type of the simplest C invariant Hamiltonian. It is characterized by the absence of critical manifolds of non-zero dimension (see Table 17). The simplest classical Hamiltonian possesses one minimum and one maximum on two critical orbits and two saddle points (with Morse index 2) on the other pairs of critical orbits. The description of qualitatively di!erent Hamiltonians of the "rst and second level of complexity is given in Table 17 as well. We use in Table 17 Morse counting polynomials to represent the system of stationary orbits for a given set of the qualitatively similar Hamiltonians. To reduce the number of di!erent classes we neglect in Table 17 all Hamiltonians which may be obtained by a simple transformation (H) (!H). It is clear that the inversion of the sign of the Hamiltonian is associated with the transformation of the Morse indices of stationary points (k (4!k)) and of one-dimensional stationary manifolds (k (3!k)). For several classes both Hamiltonians H and (!H) are described by the same Morse counting polynomial (we add to the level of complexity the index
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
208
Table 17 Classi"cation of qualitatively di!erent C
invariant Hamiltonians
Level of complexity
C stratum
C stratum
0!
1#2t#t
no
1 1
1#3t 1#t#2t
t t
2! 2 2! 2! 2 2
1#2t#t 1#2t#t 4t 2#2t 1#3t 2#2t
t#t 1#t 1#t t#t 2t t#t
Morse counting polynomial is given in this column to characterize the system of stationary orbits. Morse counting polynomial for the S stationary orbits is given. The superscript $ in the "rst column indicates those classes which are invariant under the sign inversion of the Hamiltonian.
$ in such a case, see Table 17). For all other classes the change of the sign of the Hamiltonian will modify the redistribution of the stationary orbits over Morse indices. For example, there are two extra classes of the Hamiltonians of the "rst level of complexity and three additional classes for the second level of complexity, but we omit them from the table. Near minimum and maximum the simplest Morse-type C -invariant Hamiltonian (zero level of complexity) may be approximated as 2D-isotropic harmonic oscillator. So it is characterized by a qualitative energy level pattern shown in Fig. 5. Lower and upper parts of the multiplet are formed by a sequence of polyads typical for 2D-isotropic harmonic oscillator. 3.2.3. C invariant Hamiltonian T The C invariant phenomenological Hamiltonian may be written as T H T " C mL (k)L lL # CN pmL (k)L lL L L L L L L !
(141)
with m, (k), l as denominator invariants and p as one numerator invariant given in Eqs. (116)}(119) or in Eqs. (128)}(131) with all necessary symmetrization. Any Hamiltonian (141) has three critical orbits: one C orbit consisting of two stationary points (A, B) and two C orbits consisting of one T stationary point (C, D) each (stationary points A, B, C, D, see Table 8 and Eq. (101)). The most natural physical application of the C symmetry concerns the Stark e!ect. In this case T the symmetry group should be extended to include the time reversal as a symmetry operation. Taking into account the invariance of m, k, l with respect to time reversal and the alternation of the sign of p under the same operation the general form of Hamiltonian (141) becomes simpler. In the case of Stark e!ect the e!ective Hamiltonian (141) is independent of p.
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
209
Fig. 5. Qualitative energy level pattern for e!ective C -invariant Hamiltonians of the simplest Morse}Bott type. In the neighborhood above the minimum and below the maximum the energy level pattern is similar to that of a twodimensional isotropic harmonic oscillator.
3.2.4. C invariant Hamiltonian F The C invariant phenomenological Hamiltonian may be written as F H F " C mL kL (l)L # CJN (lp)mL kL (l)L (142) ! L L L L L L with m, k, (l) as denominator invariants and (lp) as one numerator invariant as given in Eqs. (116)}(118), (120), or in (128)}(130), (132) with all necessary symmetrization. Any Hamiltonian (142) has three critical orbits: one C orbit consisting of two stationary points (C, D) and two C orbits consisting of one stationary point (A, B) each (stationary points A, B, C, D, see F Table 8 and Eq. (101)). In the case of the Zeeman e!ect after extending the symmetry group from C to C T (see F F Q Section 2.9) the general form of the e!ective Hamiltonian becomes independent of (lp). 3.2.5. D invariant Hamiltonian The D invariant phenomenological Hamiltonian may be written as H " C mL (k)L (l)L # CJN (lp)mL (k)L (l)L " L L L L L L # CIN (kp)mL (k)L (l)L # CJI (lk)mL (k)L (l)L (143) L L L L L L with m, (k), (l) as denominator invariants and (lp), (kp), and (lk) as three numerator invariants given in Eqs. (120)}(122), or (132)}(134) with all necessary symmetrization. Any Hamiltonian (143) has four critical orbits: two C orbits consisting each of two stationary points (A, B and C, D) and two C orbits (both being S stationary manifold C , C ) (stationary points A, B, C, D, and stationary circles C , C see Table 8 and Eqs. (101) and (102)). 3.2.6. D invariant Hamiltonian F The D invariant phenomenological Hamiltonian may be written as F H F " C mL (k)L (l)L # CJN (lp)mL (k)L (l)L " L L L L L L
(144)
210
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
with m, (k), (l) as denominator invariants and (lp) as one numerator invariant given in Eqs. (120) or (132) with all necessary symmetrization. Any Hamiltonian (144) has four critical orbits: one C orbit consisting of two stationary points (A, B), one C orbit consisting of two stationary F T points (C, D), one C orbit consisting of a S stationary manifold C , and one C orbit consisting F T of a S stationary manifold C (stationary points A, B, C, D, and stationary circles C , C are given in Table 8 and in Eqs. (101) and (102)). 3.3. Qualitative description of ewective Hamiltonians invariant with respect to xnite subgroups of O(3) We characterize below in Tables 18 and 19 the simplest Morse-type functions for all "nite point group symmetries. For "nite groups, orbits include a "nite number of isolated points and any Morse-type function possesses only isolated extrema. We can work within the initial Morse theory (without extension to the Morse}Bott approach). For each point group the possible sets of stationary points are indicated up to trivial change of Morse indices c c . Stationary points I \I for each group are split into columns according to their Morse index k. The number of stationary points c with the Morse index k within each orbit and their symmetry types are given. If there are I several orbits of stationary points with the same Morse index it is indicated explicitly as a sum. It should be noted that for many point groups the stationary points are situated on zero-dimensional strata only and their positions are "xed (i.e. only critical orbits are present) (see Section 2.8 and Appendix C for the description of all strata for the "nite group action on R). For some low symmetry groups (C , C , S ,C ) there are no orbits isolated within the stratum and the positions Q G of stationary points are not "xed by symmetry. At the same time the presence of a closed 2D-stratum for the C and S groups indicates that a number of stationary points should lie on Q the closed stratum. For these groups it is necessary to verify further Morse inequalities for the restriction of the complete initial function on the close stratum. For O and > groups, only part of the simplest Morse functions is given in Table 19. To obtain F F the complete list it is necessary to interchange in all possible manners the C and C critical LT LF orbits. Near minima or maxima any Hamiltonian may be approximately represented as 2D-harmonic oscillator. Taking into account the presence of several, say k (k is the dimension of the orbit of stationary points) equivalent by symmetry minima the model problem appropriate for the description of internal dynamics near the extrema is the motion of a particle in a 2D-potential with k equivalent extrema (the potential can be anisotropic or isotropic and slightly anharmonic near each extremum) assuming small tunneling between di!erent extrema. We can introduce three characteristic parameters for such a problem: (i) anisotropy of the 2D-harmonic oscillator, (ii) anharmonicity of the 2D-oscillator, and (iii) splitting due to tunneling. Let us consider several simple limiting situations from the point of view of the energy level patterns for quantum problems near extrema. Let d be the anisotropy of the model Hamiltonian near an extremum d &h"l !l "/(l #l ) , (145) where l , i"1, 2, are two harmonic frequencies of the model operator, d be the characteristic G energy splitting due to tunneling, and d be the anharmonicity correction.
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
211
Table 18 Simplest Morse type functions de"ned on the R manifold in the presence of point group symmetry (lower symmetry groups) Point group
c (c )
c (c )
c
c (c )
c (c )
C C Q
1(C ) 1(C ) Q 1(C ) Q 1(S ) 1(S ) 1(S ) L
no
2(C ) 2(C ) 1(C )#1(C ) Q Q 2(C ) 1(S )#1(S ) 2(C ) L
no
1(C ) 1(C ) Q 1(C ) Q 1(S ) 1(S ) 1(S ) L
C L n52
1(C ) L
no
1(C )#1(C ) L L
no
1(C ) L
C LF n52
1(C ) LF
no
2(C ) L
no
1(C ) LF
C LT n52
1(C ) LT
no
2(C ) L
no
1(C ) LT
D D L n53
2(C ) 2(C ) L 2(C ) L n(C ) 2(C )#2(C ) L L 2(C ) LF
2C n(C ) n(C ) n(C ) n(C ) n(C ) T
2(C )#2(C ) n(C )#n(C ) n(C )#2(C ) L 2(C )#2(C ) L L n(C ) 2n(C )
2(C ) n(C ) n(C ) n(C ) n(C ) n(C ) T
2(C ) 2(C ) L n(C ) n(C ) n(C ) 2(C ) LT
2(C ) LF 2(C ) LF 2(C ) LF 2(C ) LF 2(C )#2(C ) F T 2(C )#2(C ) F T 2(C )#2(C ) F T 2(C )#2(C ) F T 2(C )#2(C ) F T 2(C )#2(C ) F T
n(C ) F n(C ) F n(C ) T n(C ) T 4(C ) T 4(C ) T 4(C ) T 4(C ) F 4(C ) F 4(C ) F
n(C )#n(C ) T F n(C )#n(C ) T T n(C )#n(C ) T F n(C )#n(C ) F F 4(C ) F 4(C ) F 4(C ) T 4(C ) F 4(C ) T 4(C ) T
n(C ) T n(C ) F n(C ) F n(C ) T 4(C ) T 4(C ) F 4(C ) F 4(C ) T 4(C ) T 4(C ) F
2(C ) LT 2(C ) LT 2(C ) LT 2(C ) LT 4(C ) F 4(C ) T 4(C ) F 4(C ) T 4(C ) F 4(C ) T
S S L n52
n"3, 5 only D (D ) LF LB n53, odd D (D ) LF LB n52, even n"4 n"4 n"4 n"4 n"4 n"4
only only only only only only
no no no no no
no no no no no
There are three di!erent C strata for the D group. Each stratum includes two orbits. To reach complete description of qualitatively di!erent Morse-type functions it is necessary to specify the stratum for all stationary points. There are two di!erent C strata for even n54, whereas there is only one stratum for odd n. Further speci"cation of strata is needed. There are three di!erent C and three di!erent C strata for the D (D ) T F F B group. There are two di!erent C and two di!erent C strata for the D (D ) group for n54, even. T F LF LB
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
212
Table 19 Simplest Morse-type functions de"ned on the R manifold in the presence of point group symmetry (higher symmetry groups). To complete the list of Morse functions for O and > groups it is necessary to add other sets of stationary points F F resulting from di!erent permutations within pairs of C and C strata LT LF Point group
c (c )
c (c )
c
c (c )
c (c )
¹
4(C ) 4(C ) F 4(C ) T 6(C ) 6(C ) 8(C ) 6(C )#6(C ) 6(C ) F 6(C ) F 8(C ) F 6(C )#6(C ) F T 12(C ) 12(C ) 20(C ) 12(C )#12(C ) 12(C ) F 12(C ) F 20(C ) F 12(C )#12(C ) F T
6(C ) 6(C ) F 6(S ) 12(C ) 12(C ) 12(C ) 12(C ) 12(C ) F 12(C ) F 12(C ) F 12(C ) F 30(C ) 30(C ) 30(C ) 30(C ) 30(C ) F 30(C ) F 30(C ) F 30(C ) F
4(C )#4(C ) 8(C ) 8(C ) 6(C )#8(C ) 8(C )#8(C ) 6(C )#6(C ) 8(C ) 6(C )#8(C ) T F 8(C )#8(C ) F T 6(C )#6(C ) F T 8(C ) F 12(C )#20(C ) 20(C )#20(C ) 12(C )#12(C ) 20(C ) 12(C )#20(C ) T F 20(C )#20(C ) F T 12(C )#12(C ) F T 20(C ) F
6(C ) 6(C ) T 6(C ) T 12(C ) 12(C ) 12(C ) 12(C ) 12(C ) T 12(C ) T 12(C ) T 12(C ) T 30(C ) 30(C ) 30(C ) 30(C ) 30(C ) T 30(C ) T 30(C ) T 30(C ) T
4(C ) 4(C ) F 4(C ) T 8(C ) 6(C ) 8(C ) 8(C ) 8(C ) T 6(C ) T 8(C ) T 8(C ) T 20(C ) 12(C ) 20(C ) 20(C ) 20(C ) T 12(C ) T 20(C ) T 20(C ) T
¹ F ¹ B O
O F (?)
>
> F
In some cases the harmonic approximation of the Hamiltonian near an extremal orbit should be isotropic due to symmetry (extremal orbit is a critical one with its stabilizer being a group with high symmetry). In such a case we have only the anharmonicity parameter d and the tunneling splitting parameter d. Typically d is su$ciently small but nevertheless at the same time d 'd. In such a case the energy spectrum of the quantum problem near the extremum may be represented as a spectrum of a k-fold 2D-dimensional isotropic (slightly anharmonic) oscillator. It means that each polyad of a 2D-isotropic (slightly anharmonic) harmonic oscillator is replaced by a k-fold cluster of similar polyads. Typically for the model problem with an anisotropic harmonic oscillator we can neglect the anharmonicity correction and assume equally that d(d . In such a case the energy spectrum of the quantum problem near the minimum or maximum may be represented as a spectrum of a k-fold 2D anisotropic harmonic oscillator. It means that each non-degenerate level of the 2D-anisotropic harmonic oscillator is replaced by a k-fold cluster of energy levels. If both d and d are small and have the same order of magnitude, then polyads are formed by energy levels of each almost isotropic harmonic oscillator and the total energy level pattern is a system of k-fold clusters of vibrational polyads. Internal structure of each cluster depends on the relation between d and
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
213
d and may be complicated and vary signi"cantly from one polyad to another because the ratio between anisotropic and tunneling splittings changes with the vibrational energy increase.
4. Manifestation of qualitative e4ects in physical systems. Hydrogen atom in magnetic and electric 5eld Hydrogen atom in magnetic and electric "elds gives us an opportunity to study the dependence of the Rydberg n multiplets on variable parameters such as strength of electric and magnetic "elds. To apply directly our approach we should remind once more that external "elds are su$ciently low to ensure the small splitting of the n multiplets with respect to the splitting between multiplets. In the case of non-zero electric "eld the ionization is always possible and consequently the strict non-relativistic Hamiltonian of the hydrogen atom in an external electric "eld possesses everywhere the continuous spectrum. We will neglect here the e!ect of ionization and restrict ourselves with the analysis of e!ective Hamiltonians diagonal in n. To see the limits of the applicability of the present treatment in the case of a Rydberg atom in an external magnetic "eld, for example, we "nd here conditions which enable one to treat n-multiplets separately. It is su$cient that the diamagnetic shift which roughly varies as Gn (G is a magnetic "eld strength in atomic units, 1 a.u."2.35;10 T) remains smaller than the energy di!erence between two consecutive multiplets (*E"n!(n#1)&1/n), i.e. Gn (1. This estimation shows that for a su$ciently low magnetic "eld G we can always "nd a relatively high n shell (to ensure the applicability of the classical approach) which can be analyzed in terms of e!ective Hamiltonians for an isolated n. 4.1. Diwerent xeld conxgurations and their symmetry The qualitative description of the hydrogen atom in two external "elds depends strongly on the symmetry of the problem created by the two "elds. To specify the symmetry we de"ne "rst the absolute con"guration of two "elds in the laboratory "xed frame. It is given by two vectors: F is the electric "eld (polar) vector and G the magnetic "eld (axial) vector. Any two absolute con"gurations which can be mutually transformed by some rotation of the laboratory frame should be considered as physically equivalent and form one (relative) con"guration of two "elds. This means that the classi"cation of di!erent relative con"gurations of two "elds is equivalent to the classi"cation of orbits of the O(3) group on the six-dimensional space generated by one polar and one axial vector. Three parameters, F, G, and the scalar product (FG) with a natural relation between them (FG)4FG, are needed to characterize completely all relative con"gurations of two "elds. Thus, a one-to-one correspondence exists between di!erent relative con"gurations of two "elds and points of the "lled cone in the 3D-space (see Fig. 6). The cone shown in Fig. 6 is the orbifold of the O(3) group action on the space of absolute "eld con"gurations. Qualitatively di!erent relative con"gurations are characterized by di!erent symmetry groups. There are six di!erent types of "eld con"gurations which are listed in Table 20 together with their symmetry groups. Let us specify now the symmetry groups introduced in Table 20 to characterize di!erent types of "eld con"gurations. In each case the complete symmetry group includes two kinds of symmetry
214
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
Fig. 6. Geometrical representations of di!erent "eld con"gurations. F is the electric "eld vector and G the magnetic "eld vector.
Table 20 Symmetry classi"cation of relative con"gurations of the electric "eld F and the magnetic "eld G De"ning equations
Geometrical description
Symmetry group
Physical problem
G"F"0 G"0, F'0 G'0, F"0 G'0, F'0, (GF)"GF G'0, F'0, (GF)"0 G'0, F'0, (GF)(GF, (GF)O0
Point O Ray OF Ray OG Conical surface Plane GOF Interior of the cone
O(3)T C T T C T F Q C T Q G T Q
No "elds Stark e!ect Zeeman e!ect Parallel "elds Orthogonal "elds Generic con"guration
operations: purely geometrical spatial transformations forming one of the standard point symmetry groups and symmetry operations related to time reversal. We use the following notation. The group T includes two elements, identity E and time reversal ¹. The group T includes also two elements, identity E and the symmetry operation (¹p) which is Q the product of the time reversal and the re#ection in a plane including both "elds. Group G includes four symmetry elements: identity E, re#ection in the plane orthogonal to the magnetic "eld p , product ¹p of time reversal ¹ and re#ection p in plane formed by two orthogonal "elds, F and the symmetry operation ¹C which is the product of the time reversal ¹ and the C rotation around the electric "eld direction. G has three subgroups of order two: C "(E, p ), T "(E, ¹p), Q F Q and T "(E, ¹C ). We can study now the dependence of the dynamics on the ratio of the "eld strengths assuming that the total e!ect of both "elds is kept to be more or less the same even if we change the "eld con"guration from that corresponding to pure Zeeman e!ect till that of pure Stark e!ect. These "eld con"gurations lie in some section of the cone in Fig. 6. To make an interesting section of it we take into account that the energy correction to the hydrogen atom in the linear Zeeman limit is
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
215
*E +$Gn and in the linear Stark limit is *E +$3Fn where n is the principal quantum L L number for a perturbed hydrogen atom n+(!1/(2E). Thus for given n it is natural to "x S"((Gn)#(3Fn) and to vary only the relative strength of the two "elds: Gn , G" Q ((Gn)#(3Fn)
(146)
3Fn F" . Q ((Gn)#(3Fn)
(147)
We consider the case of small S but we allow large variations of the relative strengths of the two "elds 04F , G 41 with the restriction F#G"1. Q Q Q Q A very interesting question concerns the qualitative behavior of the n-shell dynamics under the variation of "eld parameters (assuming S to be small enough). Are e!ective Hamiltonians for the n-shell always of the simplest Morse type? Or there are some regions in the space of parameters where the e!ective Hamiltonian should possess more than a minimal number of stationary points. If such a region exists it means that under the variation of "eld parameters (even for a low "eld limit) some bifurcations are present and the qualitative modi"cations of the dynamics take place. Two important cases to study are the case of two parallel "elds (con"gurations of "elds represented by points on the surface of cone in Fig. 6) and the case of orthogonal "elds (con"gurations represented by points on the bisectral plane of the cone in Fig. 6). But before looking on these cases we will brie#y apply the qualitative analysis to the limiting case of the Zeeman e!ect. 4.2. Quadratic Zeeman ewect in hydrogen atom Let us consider the Hamiltonian for the quadratic Zeeman e!ect p 1 G H" ! # (x#y) . r 8 2
(148)
This Hamiltonian is very popular from the point of view of theoretical investigations of di!erent dynamical regimes in quasi-regular and chaotic regions (Braun, 1993; Delande and Gay, 1986; Delos et al., 1983; Fano and Sidky, 1992; Farrelly and Krantzman, 1991, Farrelly and Milligan, 1992; Friedrich and Wintgen, 1989; Herrick, 1982; Huppner et al., 1996; Krantzman et al., 1992, Kuwata et al., 1990; Liu et al., 1996; Mao and Delos, 1992; Robnik and Schrufer, 1985; Sadovskii et al., 1995; Solov'ev, 1981,1982; Tanner et al., 1996; Uzer, 1990). The Hamiltonian in Eq. (148) is D T invariant. Remark that its symmetry is higher than that F of an atom in the presence of magnetic "eld (which is C T ) because we neglect the terms linear F Q in angular momentum. Consequently any e!ective diagonal in n operator can be rewritten in terms of operators corresponding to the D polynomial invariants, m, k, l (see Eqs. (116)}(118)). The F auxiliary invariant lp does not enter in Hamiltonian (148) due to additional time-reversal invariance. We replace Hamiltonian (148) by the diagonal in n e!ective operator just by projecting it on the manifold of the non-perturbed hydrogen atom wave functions with a given n quantum number. Such procedure is physically meaningful in the low magnetic "eld limit.
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
216
The explicit expressions for the diagonal in n matrix elements of the diamagnetic terms are either diagonal in l or non-diagonal in l with *l"$2 (they are diagonal in m). It is necessary to "nd such combinations of diagonal in n operators which give the same matrix elements as the diamagnetic perturbation. The diagonal in n perturbation of the hydrogen atom may be equivalently represented in the form of the e!ective operator identical to that used by Solov'ev (1981), Herrick (1982), Delande and Gay (1986) and many others. In terms of invariant polynomials the Hamiltonian for the n shell has the following form: H "const.#G L
n(n!1) [k!2m!5l] , 16
(149)
where the const. is n-dependent. This Hamiltonian in Eq. (149) is invariant with respect to the D T group. Its action on the phase space of the reduced problem (n-shell e!ective HamilF tonian) is summarized in Table 15. Energy levels of this Hamiltonian can be trivially visualized on the D T orbifold because the energy function is linear in invariant polynomials used to F construct the orbifold. Let us analyze now the essential part of the Hamiltonian: k!2m!5l .
(150)
There are four critical orbits and the topological structure of the energy levels varies by passing through the critical orbits only. Corresponding energies are (in increased order) E"!3, C T T } critical orbit; E"!2, C T } critical orbit (one-dimensional manifold); E"!1, C T F F } critical orbit; E"2, C T } critical orbit (one-dimensional manifold). So, the Rydberg T multiplet in relative units lies between !3(E(2. It is split into three di!erent regions. The lowest one occupies 1/5 of the multiplet width. The highest one occupies 3/5 of the multiplet width (see Fig. 7). Let us consider the lowest part of the multiplet. The energy surface near the minimum may be represented as the energy surface for two equivalent 2D slightly anharmonic oscillators. Energy levels form polyads. The nth polyad consists of 2n levels. There are two e!ects which lead to the splitting of the energy levels within a polyad: anharmonicity of the isotropic oscillator and the tunneling between two equivalent wells. The splitting of polyad due to anharmonicity e!ects
Fig. 7. Qualitative description of critical orbits versus energy for e!ective D -invariant Hamiltonian for quadratic F Zeeman e!ect.
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
217
preserves the C symmetry. As a consequence, the degeneracy of energy levels with the same "m", the projection of the angular momentum is present. If we neglect the tunneling splitting, for any "m"O0 there are four degenerate energy levels. The numerical results for the quantum problem clearly show this patterns. We can estimate the position of the classical energy minimum for the C orbit from quantum results by assuming the harmonic T oscillator model and taking the energy of the zero level equal to the fundamental frequency (this is the linear approximation). Thus for example, for n"45 the harmonic frequency (the fundamental transition) is 0.09442 and the estimated energy of the classical energy minimum found from the entirely quantum calculations is E (C )"!2.99797 as compared to (!3) for the purely T classical model. At the other end of the energy multiplet (near C orbit) the energy level pattern is similar to that T of a one-dimensional rotator plus one-dimensional oscillator. The one-dimensional harmonic oscillator describes energy levels with m"0. The harmonic frequency (for n"45) can be again estimated from the quantum calculations as a fundamental transition. It is equal to 0.194414. Adding the half quanta of the corresponding harmonic frequency to the highest (extremal) quantum energy level gives the estimation for the classical energy of the (C ) critical orbit: T E (C )"1.99987 as compared with E"2 in the classical limit. This numerical comparison is T completely satisfactory taking into account the extreme simplicity of the classical model. It is clearly seen from the numerical results that the m"0 energy levels form the regular sequence of non-degenerate energy levels within the energy interval 25E5!1 and the regular sequence of doublets within the energy interval !25E5!3. This fact well correlates with the energy of the critical orbit C characterizing by the energy E(C )"!2. F F It is important to note that the existence of four critical orbits is the consequence of the symmetry of the Hamiltonian and does not depend on the concrete form of the Hamiltonian. At the same time the relative positions in energy of critical orbits strictly depend on the concrete form of the operator. 4.3. Hydrogen atom in parallel electric and magnetic xelds We demonstrate shortly in this section one particular application of the qualitative analysis of Rydberg states by studying the transition from a Zeeman to a Stark structure of a weakly split Rydberg n-multiplet of the H atom in parallel magnetic and electric "elds (Sadovskii et al., 1996). The geometrical approach clearly shows the origin of the new phenomenon, the collapse of the energy levels. The use of classical mechanics, topology, and group theory provides detailed description of the modi"cations of dynamics due to the variation of the electric "eld. We focus on the point where the collapse of the Zeeman structure occurs, give the sequence of classical bifurcations responsible for the transition between di!erent dynamic regimes, and compare it to the quantum energy-level structure. In fact, Rydberg atoms in parallel magnetic and electric "elds have been extensively studied both theoretically and experimentally during the last decade. In particular, many studies have focused on the situation where the "elds are (relatively) weak and the dynamics can be analyzed in terms of additional approximate integrals of motion (Braun, 1993; Braun and Solov'ev, 1984; Cacciani et al., 1988, 1989, 1992; Delande and Gay, 1986; Farrelly et al., 1992; Iken et al., 1994; Seipp et al., 1996; van der Veldt et al., 1993). We use a similar idea to analyze several dynamic regimes that exist for
218
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
Fig. 8. Collapse of the Zeeman structure for the magnetic "eld G"0.06 due to the increasing electric "eld. Quantum levels are calculated for the n"10 multiplet of the hydrogen atom. Scaled electric "eld strength is given in units FI "GI /3 [see Eq. (153)].
di!erent strengths of the electric (F) and magnetic (G) "elds. These regimes clearly manifest themselves in the energy level pattern in Fig. 8. At very weak electric "eld, where most of the studies were done, the energy levels are grouped according to the value of ¸ , the projection of the angular X momentum on the "eld axis. The internal structure in this region is mainly due to quadratic Zeeman e!ect (see Fig. 9 and discussion in the Section 4.2). When F increases this structure quickly disappears. Instead we observe regular `resonancea structures at certain values of F (see Fig. 8). This culminates in an almost complete collapse of the internal structure. Surprisingly, and contrary to the F&0 case the dynamics near this collapse has not been earlier analyzed in detail. Neglecting the spin e!ects the Hamiltonian for the hydrogen atom in constant parallel magnetic G and electric F "elds (along the z-axis) has the form (in atomic units) G p 1 G H" ! # ¸ # (x#y)!Fz r 2 X 8 2
(151)
with G and F in units of 2.35;10 T and 5.14;10 V/cm. We restrict ourselves to the low "eld case where the splitting of an n-shell caused by both "elds is small compared to the splitting between neighboring n-shells (see Fig. 10). As is well known, in the absence of electric "eld low-m submanifolds of the n-shell show characteristic pattern of the second order Zeeman e!ect. When the electric "eld e!ect is of the same order as the quadratic Zeeman e!ect (see Fig. 9), this pattern disappears and turns into a Stark structure for each m sub-manifold (Braun and Solov'ev, 1984; Delande and Gay, 1986). Much lesser attention has been paid to the region where the Stark splitting of the n-shell (J3Fn) is of the same order as the n-shell splitting due to magnetic "eld
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
219
Fig. 9. Deformation of the QZE structure of the n"10, m"0 multiplet of the hydrogen atom. Dashed lines show the energy in stationary points of the classical Hamiltonian restricted on k"0.
Fig. 10. Neighboring Rydberg multiplets n"9, 10, 11 of the hydrogen atom.
220
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
(JGn) (Fig. 8), in other words when F "G/(3n) . (152) Eq. (152) gives the collapse condition for shell n. To ensure that this collapse happens when the n-shell splitting remains small compared to the gap between n-shells, we take G(1/n. Under this assumption we can study an isolated n-shell and can use scaling FI "Fn, GI "Gn, EI "2nE #1 , (153) L L to remove n from the e!ective n-shell Hamiltonian [Eq. (157) below]. Our purpose is to study the dynamics under the variation of the electric "eld F in the neighborhood of its critical value. The natural parameter for this study is d"3Fn/G!1. The analysis is based on the transformation of the initial Hamiltonian (151) into an e!ective one for an individual n-shell. This can be done either by quantum or by classical perturbation theory (SoloveH v, 1981; Delande and Gay, 1986; Farrelly et al., 1992; Stiefel and Scheifele, 1971). An e!ective n-shell Hamiltonian can be expressed in terms of angular momentum L and Runge}Lenz vector A"p;L!r/r, or, alternatively, in terms of their linear combinations J "(L#A)/2 and J "(L!A)/2. For the linear Stark}Zeeman e!ect in parallel "elds the e!ective Hamiltonian is 1 H" (!1#Gn¸ #3FnA ) . X X 2n
(154)
If we impose the relation between "eld strengths (152) this Hamiltonian becomes 1 H" (!1#3F n(J ) ) . X 2n
(155)
The n energy levels in the n-shell described by Eq. (155) form n-fold degenerate groups. The levels in each group are labeled by the same value of (J ) and by di!erent values of (J ) . Fig. 8 shows X X how the Zeeman structure of the n-shell at f"0 transforms into this highly degenerate structure at F"F (FI "GI /3). We call this e!ect the collapse of the Zeeman structure caused by electric "eld. To describe the "ne structure of each (J ) manifold of states the second-order e!ects should be X taken into account. To develop the e!ective n-shell Hamiltonian to higher orders we consider n as an integral of motion and use the perturbation theory to reduce the initial problem (151) to two degrees of freedom. Naturally, the pair (¸ , X ) describes one of these degrees; the other degree can be X * described by A and X (Farrelly et al., 1992). Of course, for Hamiltonian (151) ¸ is strictly X X conserved and the n-shell Hamiltonian does not depend on X . However, to study the collapse we * should consider the energy level structure of the n-shell as a whole, and therefore, we should keep ¸ as a dynamical variable. Hence our n-shell Hamiltonian is a function of dynamical variables X (¸ , A , X ) and parameters (n, F, G). X X The classical phase space R for e!ective n-shell Hamiltonian is a 4D space with topology S ;S . Its parametrization can be done either using the L, A variables with L#A"n, and L ) A"0, or using the J , J variables with J "J "n/4. (In the classical limit n is su$ciently large and n+n!1.)
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
221
For the qualitative analysis of the n-shell dynamics we use invariant polynomials l"A /n; k"¸ /n; m"(¸!A)/n (156) X X forming integrity basis which is used both to label the points of the phase space and to expand the Hamilton function as explained in Section 5.6 of Chapter I (see also Appendix A). Furthermore, the proper scaling in Eqs. (156) and (153) results in equations which do not depend on n. The symmetry group G"C T of the problem and its action on R are described in detail in Q Section 2.9. Consecutive steps in the qualitative analysis of an e!ective Hamiltonian in the presence of the symmetry group (Sadovskii and Zhilinskii, 1993a) which are summarized brie#y in Section 2 of Chapter II include the study of the action of the symmetry group on the classical phase space, construction of the space of orbits (orbifold), and the analysis of the system of stationary points (orbits) of the Hamilton function using the topological and group theoretical information about the phase space. The strati"cation of the R phase space under the action of the symmetry group C T is given in Tables 8 and 13. Q The Hamilton function can be expressed as a polynomial H"H(l, k, m) of invariant polynomials l, k, m. Up to quadratic in F and G terms the scaled energy EI has the form (157) EI "GI k#3FI l!GI m#(9FI #GI )k#(3FI !5GI )l#(3GI !17FI ) . To qualitatively characterize classical and quantum dynamics we "nd the system of stationary points (manifolds) of the energy function on the phase space. Group theory asserts that four points A, B, C, D (critical orbits) must be stationary for any smooth function de"ned over the phase space (see Section 4 of Chapter I). Energy values (157) at these points are shown in Figs. 8 and 10. Morse inequalities con"rm that the simplest Morse-type functions possessing stationary points only on the four critical orbits really exist on R and have one minimum, one maximum, and two saddle points. For more complicated functions any other stationary points can be found by looking for those energy sections of the orbifold which correspond to the modi"cation of the topology of the energy section. Simple geometrical analysis shows that in the linear (in F and G) approximation for F(F the energy function is of the simplest type with minimum in B, maximum in A, and two saddle points in C and D. For F'F the energy function is again of the simplest type with minimum in D, maximum in C, and two saddle points in A and B. Sudden transition from one simplest type of the energy function to another one in the linear model occurs due to the formation of the degenerate stationary manifold at value F"F corresponding to Hamiltonian (155). In the linear model the energy surface touches the orbifold through the whole interval [C, A] or [B, D]. Introduction of the F and G terms into the energy function removes this degeneracy. The energy surface (157) is the second-order surface in m, k, l variables. It can touch the orbifold O at some isolated points on the p"0 surface which are di!erent from the critical orbits A, B, C, D. If this happens, additional stationary orbits are present. The detailed analysis of a system of stationary points as a function of F near the collapse value F shows how the transformation from the Zeeman-type energy function (with only four stationary critical orbits having minimum and maximum in B and A) to the Stark-type energy function (with only four stationary critical orbits having minimum and maximum in D and C) occurs. Two sequences of bifurcations are present with two bifurcations in each sequence. As F increases, one sequence begins with a bifurcation at point B which creates a new
222
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
stationary S orbit of EI on R. The corresponding point on the surface of the orbifold moves from B to D and disappears at D after the second bifurcation. Another sequence of bifurcations proceeds in the similar way with two bifurcations at C and A and the new additional stationary orbit moving from C to A. Positions of all stationary orbits can be found by solving the Hamiltonian equations on R. Alternative way is to use the geometrical representation of the orbifold and of the energy surface. To "nd non-critical stationary orbits we "nd points where the energy surface touches the orbifold. In other words, we "nd points where the normal vector to the p"0 surface and the normal vector to the energy surface k"(k , k , k ) are collinear. This geometric view gives us extremely simple J I K conditions for bifurcations at points A, B, C, D: A: 4k k "k!k , d +!GI /8!GI /16 , (158) K I J I C: 4k k "k!k, d +2GI /3#GI /72 , (159) K J J I ! B: 4k k "k!k, d +!GI /8#GI /16 , (160) K I I J D: 4k k "k!k, d +!2GI /3#GI /72 . (161) K J I J " When d varies between d and d an additional stationary orbit exists on the surface of the orbifold ! and moves from the point A to the point C. Similarly, for d between d and d another additional " stationary orbit moves from D to B. The energies of all stationary orbits near the bifurcation points and the quantum energy levels are shown in Fig. 11.
Fig. 11. Bifurcation diagram near the collapse region. Classical (left) versus quantum (right) representation.
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
223
Simple quantum mechanical interpretation of the e!ect of the transformation of the Zeeman-type structure into Stark-type one can be done by looking at the two extremal states (with minimal and maximal energy) of the same n multiplet (Fig. 11). We can characterize each extremal state by two average values 1¸ 2 and 1A 2. From the positions of stationary points on X X the orbifold it follows immediately that for the state with maximal energy 1¸ 2+n for d(d , X 1¸ 2+0 for d'd , whereas 1¸ 2 varies almost linearly with d for d (d(d . For the same X ! X ! state 1A 2+0 for d(d , 1A 2+n for d'd , whereas 1A 2 varies almost linearly with d for X X ! X d (d(d . ! We conclude that important qualitative modi"cations of dynamics take place in the collapse region. This suggests new experimental investigations which can use the detailed information on many energy-level crossings in the collapse region to obtain quantum states with desired properties by "ne tuning of the "eld parameters. Existence of collapsed levels with di!erent projections of the orbital momentum m can be used in the experiment to selectively produce states with any possible m using adiabatic change of "eld parameters. This is specially important for formation of so-called `circulara states with very high m&n values (Delande and Gay, 1988; Germann et al., 1995; Hulet and Kleppner, 1983; Kalinski and Eberly, 1996a,b). 4.4. Hydrogen atom in orthogonal electric and magnetic xelds Hydrogen atom in crossed (orthogonal) electric and magnetic "elds was the subject of many experimental (Flothmann et al., 1994; Raithel and Fauth, 1995; Raithel et al., 1993a,b,1991; Raithel, Fauth and Walther, 1993; Rinneberg et al., 1985) and theoretical (Farrelly, 1994; Gourlay et al., 1993; von Milczewski et al., 1994,1996) studies. The purpose of this section is to show what kind of qualitative information can be obtained taking into account only general topology and symmetry information. The hydrogen atom in the presence of two orthogonal "elds gives us an example of a system with the "nite symmetry group G (see Sections 2.9 and 4.1). The space of orbits in this case is four dimensional and it is naturally more di$cult to visualize its strati"cation and to represent the orbifold in a geometrical way. Nevertheless, very useful topological and group-theoretical information can be found by studying the invariant subspaces of di!erent symmetry. We remind on this example again major steps of the qualitative analysis realized in Chapter II for di!erent rovibrational problems. More details can be found in Sadovskii and Zhilinskii (1998). First, we construct the Molien function and the integrity basis for the ring of G invariant polynomials on R. The procedure we follow here is formally the same as that used in Appendix B for molecular point group symmetry. The simplest way to write the Molien function is to work in the J , J representation, to consider the ring of all invariant functions constructed from six dynamic variables and to restrict this ring to 4D classical phase space, R. We start with the situation without any additional symmetry (the symmetry group is trivial, C ). On the six-dimensional space (J ) , (J ) , (a, b"x, y, z) the ? @ Molien function for invariants has a trivial form 1 . M " ! (1!j)
(162)
224
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
The ring of the invariant polynomials on the 6D-space P! is the ring of all polynomials P! "P[(J ) , (J ) , (J ) , (J ) , (J ) , (J ) ] . (163) V W X V W X To restrict the polynomial ring on the sub-manifold h "(J )#(J )#(J )"const. , (164) V W X h "(J )#(J )#(J )"const. , (165) V W X we are obliged to introduce two second-order polynomials, h , h given by Eqs. (164) and (165) as denominator invariants. This may be done by multiplying both the numerator and denominator of the Molien function (162) by (1#j). The new form of the Molien function 1#2j#j M " ! (1!j)(1!j)
(166)
corresponds now to another description of the ring of invariants P! which is considered as a free module P! "P[(J ) , (J ) , h , (J ) , (J ) , h ]䢇(1, (J ) )(1, (J ) ) (167) V W V W X X with four auxiliary invariants u "1, u "(J ) , u "(J ) , u "(J ) (J ) . We use the nota X X X X tion (a, b)(c, d)"(ac, ad, bc, bd) to show that four numerator invariants are represented as products of more simple terms. Having the form (166) of the Molien function and the form (167) of the module of invariant functions it is easy to make the restriction to the sub-manifold R. We just eliminate two denominator invariants, h , h , corresponding to the equations de"ning R. The resulting Molien function and the module of invariants on R are written as follows: 1#2j#j , M "R " ! (1!j)
(168)
(169) P! "R "P[(J ) , (J ) , (J ) , (J ) ]䢇(1, (J ) )(1, (J ) ) . V W V W X X The choice of basic and auxiliary invariants is ambiguous and the ones proposed here is just an example. The important point is that the integrity basis may be constructed which includes four numerator invariants (including 1) and four denominator invariants. To "nd now the integrity basis in the A, L representation we can simply transform invariants from J , J to A, L representation. Let us now decrease slightly the symmetry and consider two non-parallel and non-orthogonal "elds. The "nite symmetry group in this case has order 2 and includes one non-trivial operation: product of time reversal and space re#ection in the plane de"ned by two "elds. To specify the action of the symmetry group on basis polynomials we "x the coordinate frame in such a way that the x-axis coincides with the electric "eld vector and the y-axis belongs to the plane of two "elds and has the positive projection of the magnetic "eld on it. Let Tp be the symmetry operation for the generic con"guration of two "elds considered. It follows immediately that A , A , ¸ , ¸ are invariant with respect to this symmetry operation, V W V W whereas A , ¸ are pseudo-invariant (change the sign). We see as well, that all basic invariants of X X the ring of polynomials (167), and (169) are invariant with respect to the Tp operation. At the same
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
225
time between three non-trivial numerator (auxiliary) invariants only, one is invariant with respect to the Tp operation, whereas two others change sign under this operation. This means that the Molien function and the module of invariants on R in the case of a generic con"guration of two "elds has the form 1#j , M$ "R " (1!j)
(170)
(171) P$ "R "P[(J ) , (J ) , (J ) , (J ) ]䢇(1, (J ) (J ) ) . V W V W X X The symmetry group for the case of two orthogonal "elds is higher. It includes four symmetry elements E is the identity, Tp, introduced just above, p the re#ection in the plane orthogonal to the magnetic "eld B (this plane includes the electric "eld vector), and TC the product of the time reversal and the C rotation around the electric "eld. For the particular case of orthogonal "elds it is useful to change the notation of axes in order to show explicitly their orientation with respect to external "elds. We use below in this section the coordinate frame +e, b, p, with vector e along the electric "eld vector, vector b along the magnetic "eld vector, and vector p chosen to form the right-hand frame. Symmetry properties of A , ¸ and of all basic and auxiliary invariants of module (167) and (171) ? @ are summarized in the Table 21. The "rst consequence is the necessity to change the two basic invariants which de"ne the sub-manifold R. Instead of J "J "const. we can use J #J "const. and J !J "0. The "rst transformed equation is invariant with respect to the symmetry group. The second is pseudo-invariant. To deal with invariants only on the 6D-space we should "rst change the integrity Table 21 Transformation properties of dynamic variables for two orthogonal "elds under the action of G group
A C A @ A N ¸ C ¸ @ ¸ N (J ) C (J ) @ (J ) C (J ) @ (J ) N (J ) N (J ) (J ) N N (J ) (J )
E
TpC@
pCN
TCC
# # #
# # !
# ! #
# ! !
# # #
# # !
! # !
! # #
# # # #
# # # #
# #
! !
#
#
!(J ) C (J ) @ !(J ) C (J ) @ !(J ) N !(J ) N #
!(J ) C (J ) @ !(J ) C (J ) @ (J ) N (J ) N #
# #
# #
(J ) (J )
(J ) (J )
226
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
basis by introducing as basic invariants h , h , and as an auxiliary invariant u and instead of ? ? @ (J ) , (J ) , (J ) , (J ) their linear combinations which have well-de"ned symmetry properties C @ C @ (irreducible tensors) with respect to the symmetry group h "(J #J ) , h "(J !J ), u "(J !J ) , (172) ? ? @ h "(J ) !(J ) , h "(J ) #(J ) , (173) C C @ @ h "(J ) #(J ) , h "(J ) !(J ) , (174) C C @ @ u "(J ) !(J ) , u "(J ) #(J ) , u "(J ) (J ) . (175) N N N N N N Now, we have the description of the ring of C invariant polynomial on the 6D-space in terms of the integrity basis which includes invariants and pseudo-invariants of the symmetry group for two orthogonal "elds. The Molien function and the ring of invariants P! can be written as (1#2j#j)(1#j) , (176) M! " (1!j)(1!j)(1!j) (177) P! " "P[h , h , h , h , h , h ]䢇(1, u , u , u )(1, u ) . ? ? @ Reduction on R should be done now by eliminating two basic invariants, h , h and one auxiliary ? ? invariant u which corresponds to zero on R. Among basic denominator invariants we have two, @ (h , h ), which are pseudo-invariants with respect to the total symmetry group of the problem. To insure that all basic numerator polynomials are invariants of the total symmetry group we can change the integrity basis by introducing instead of h , h two new basic invariants h "h , ? h "h and three auxiliary polynomials u "h , u "h , u "h h . ? ? @ A After such a modi"cation we can rewrite the Molien function on R and the module of polynomials on R in terms of an integrity basis including as basic polynomials only invariant polynomials with respect to the total symmetry group, and as auxiliary polynomials both invariants and pseudo-invariants of the total symmetry group: (1#2j#j)(1#j) , M "R " ! (1!j)(1!j)
(178)
P! "R "P[h , h , h , h ]䢇(1, u , u , u )(1, u , u , u ) . ? ? ? @ A The list of polynomials forming the integrity basis is as follows:
(179)
h "A , h "¸ , h "¸, h "A , C @ ? C ? @ u "A , u "¸ , u "(J ) (J ) , N N N N u "¸ , u "A , u "¸ A , ? C @ @ A C @ u u , u u , u u , u u , u u , u u , ? @ A ? @ A u u , u u , u u . ? @ A
(180) (181) (182) (183) (184)
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
227
Table 22 Invariant manifolds for the G symmetry group action on R (Hydrogen atom in perpendicular electric and magnetic "elds.) Stabilizer
dim.
Topology
Equations
G C Q T T Q C
1 2 2 2 4
S S S S ;S S;S
¸#A"1 @ C ¸"A"0, ¸ !A40 C @ N N ¸"A"0, ¸ !A50 C @ N N ¸#¸#A#A"1 C @ C @ R (L#A"1; (L ) A)"0)
Now, we can simply eliminate all auxiliary polynomials which are not invariant with respect to the total symmetry group for two orthogonal "elds. This gives the following Molien function and the module of invariant on R: (1#2j#j) , M $"R " (1!j)(1!j)
(185)
(186) P $"R "P[A , ¸ , ¸, A]䢇(1, ¸ A , (J ) (J ) , (J ) (J ) ¸ A ) . C @ C @ C @ N N N N C @ Remark that we can replace (J ) (J ) by (¸!A). N N N N Next step of the qualitative analysis is the strati"cation of the phase space R under the action of the G symmetry group. Invariant manifolds are listed in Table 22. Keeping in mind this information we apply the Morse theory arguments to get restrictions on the number and location of stationary points of the Hamilton function. As long as there are no zero-dimensional strata of the group action, there are no critical orbits. Nevertheless, we can say that there should be at least two stationary orbits on the G invariant subspace and at least one additional stationary T invariant orbit formed by two equivalent points. Under the variation of the strength of two Q "elds these stationary points move but they are obliged to be always on these invariant subspaces. If we compare results of the qualitative analysis of the hydrogen atom in parallel "elds with that for hydrogen atom in orthogonal "elds it becomes clear that for su$ciently low "elds the evolution of the Zeeman multiplet into the Stark multiplet goes through a sequence of bifurcations for parallel "elds, whereas for orthogonal "elds no generic bifurcations are present. Immediately a natural question arises. What can be said about generic "eld con"gurations? Is it reasonable to expect the presence of bifurcations under some variation of relative "elds and their orientation? 4.5. Where to look for bifurcations? To answer this question we represent in Fig. 12 the space of relative con"gurations of two "elds F, G imposing the restriction of the type F#aG"S, where S is supposed to be su$ciently small and the positive parameter a can be chosen in such a way that the splitting of the Rydberg multiplet in Zeeman and Stark limits are approximately the same. If electric and magnetic "elds are non-collinear and non-orthogonal, the only non-trivial symmetry operation is the composition of time reversal and the space re#ection in the plane
228
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
Fig. 12. Schematic representation of qualitatively di!erent Morse-type Hamiltonians for a hydrogen atom in two external "elds (electric and magnetic). Two parameter family of Hamiltonians is split into regions corresponding to the simplest and non-simplest Hamiltonians (as a Morse-type functions).
including two "elds. So the symmetry group for such generic orientation is the T group Q introduced earlier. The only non-trivial invariant subspace is the T invariant torus. On this torus Q four stationary points are generically present. The same four stationary points should always be present for the R phase space. The appearance of additional stationary points should be veri"ed "rst of all near the collinear con"guration of two "elds corresponding to the non-simplest Morse-type Hamiltonian. Near this point the Hamiltonian is not of the simplest Morse type as we show below. Let us consider the collinear con"guration of two "elds corresponding to the range of F, G parameters such that additional extrema on R exist. These additional extrema correspond to points on p"0. After a small deformation of the con"guration of "elds ("elds become non-orthogonal after an arbitrarily small perturbation) the symmetry is broken but each stationary orbit leads to one stationary orbit of the lower symmetry group. As soon as for parallel "elds on a T invariant Q torus (with one particular orientation of symmetry plane) there are more than four stationary points, their number should be conserved after the symmetry is broken by small perturbation. So near the collapse region for parallel "elds there should be the region where the Hamiltonian is a non-simplest Morse-type function even for non-parallel "elds. Fig. 12 schematically demonstrates this fact.
5. Conclusions and perspectives The main idea of this chapter was to demonstrate how general group-theoretical and topological methods work in a particular physical problem, Rydberg states of atoms and molecules. As compared to the similar analysis realized earlier for molecular rotations and vibrations, the mathematical di$culty was overcome in this study. This is the presence of continuous symmetry related with the generalization from standard Morse theory of functions with isolated nondegenerated stationary points to Morse}Bott theory of functions with non-degenerate stationary
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
229
manifolds. At the same time the general scheme of the qualitative analysis follows the way developed earlier for molecular rotations and vibrations (Zhilinskii and Pavlichenkov, 1987, Pavlichenkov and Zhilinskii, 1988; Zhilinskii, 1989a,b; Sadovskii and Zhilinskii, 1993a; Zhilinskii, 1996) and brie#y summarized in Chapters I and II. In contrast to many speci"c theoretical molecular analyses, we do not impose at the beginning the concrete form of the Hamiltonian. This allows us to "nd some features of the energy spectrum (or of the classical dynamical behavior) which are common for a relatively large set of possible Hamiltonian functions. The Hamiltonian which is interesting from the point of view of physical applications is sometimes just a single particular operator from the wide set of objects considered. General statements can be surely applied to one particular example but the information obtained in such way is certainly more restrictive than the result of concrete numerical analysis of the particular problem. This is a disadvantage of the generic qualitative analysis. Otherwise, conceptually it is extremely useful to understand the presence of features which are independent of the precise form of the Hamiltonian, especially taking into account the fact that any Hamiltonian used for a concrete physical application is always an approximation. For Rydberg problems we have studied very simple case of small splitting of n-shell. It is certainly possible to "nd some physical situations when such a model is rather accurate and reasonable. At the same time it is clear that for real molecular and atomic systems there are many interactions (and additional degrees of freedom) which become important and often can modify considerably even the qualitative behavior. Among the simplest but essential corrections are those due to spin, the motion of the center of mass in the presence of external "elds (Johnson et al., 1983; Farrelly, 1994). Highly excited states of the hydrogen atom occupy rather special place among Rydberg states of atomic systems with one excited electron. The origin of this is the additional dynamical O(4) symmetry. Many di!erent experimental and theoretical analyses of the non-hydrogenic atom Rydberg states were done. The quantum defect theory is the most popular and powerfool tool of the theoretical analysis and interpretation of experimental data. Aymar et al. (1996) reviewed recently this subject. At the same time the systematic qualitative analysis of e!ective n-shell Rydberg Hamiltonians for free non-hydrogenic atoms or atoms in external "elds has yet not been done. Doubly excited Rydberg states of atoms give much more complicated examples with rich qualitative structure which was pointed out initially by Herrick and Sinanoglu (1975) on the basis of comparison of the approximate O(4) dynamical symmetry and extensive numerical calculations. To understand better the qualitative features of electronic excited states and especially their localization properties [see for example papers by Goodson and Watson (1993), Dunn et al. (1994) and references therein] it would be interesting to perform the topological and symmetry analysis of a two-electron problem on the same basis as it is done in the present paper for one-electron Rydberg problem. We have not practically touched in the present review the concrete applications of the qualitative theory to Rydberg states of diatomic or polyatomic molecules except for the general symmetry analysis. It should be noted that the application of a qualitative analysis should begin with the reanalysis of the excited states of the simplest one-electron diatomic molecule H>. In spite of its apparent simplicity the description of the "ne structure of highly excited states of H> still remains a question of interest (Brown and Steiner, 1966; Coulson and Joseph, 1967; Grozdanov and
230
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
Solov'ev, 1995). Diatomic molecules are of particular theoretical interest due to the presence of continuous symmetry groups which are di!erent for homonuclear and heteronuclear molecules. Due to the variety of diatomic molecules it is possible to "nd examples of molecules with slight or strong breaking of the interchange symmetry of two nuclei. Moreover, for a number of diatomic molecules experimental or numerical data exist (Bordas et al., 1985; Dabrowski et al., 1992; Dabrowski and Sadovskii, 1994; Davies et al., 1990; Fujii and Morita, 1994; Howard and Wilkerson, 1995; Jacobson et al., 1996; Jungen, 1988; Jungen et al., 1989, 1990; Greene and Jungen, 1985; Kim and Mazur, 1995; Merkt et al., 1995, 1996, Michels and Harris, 1963; Watson, 1994) and experimental data or results of alternative numerical modelization are ready to be compared through the qualitative analysis in order to reveal some universal features of the system of Rydberg states of diatomics. Among polyatomics, the H molecule is surely the most popular as an object to study Rydberg states (Bordas and Helm, 1991; Bordas et al., 1991; Bordas and Helm, 1992; Dodhy et al., 1988; Helm, 1988; Lembo et al., 1989, 1990; Herzberg, 1981; Ketterle et al., 1989; King and Morokuma, 1979; Pan and Lu, 1988; Stephens and Greene, 1995). In fact, this molecule belongs to the class of so-called Rydberg molecules for which the chemical bonding is formed due to the Rydberg electron (Herzberg, 1987). One can imagine the Rydberg molecule as a stable molecular ion plus an electron on a high Rydberg orbit. Typically, Rydberg molecules are bound only in excited electronic states and their predissociation becomes more pronounced under electronic desexcitation. H and NH are typical Rydberg molecules (Herzberg, 1981). More exotic examples of Rydberg dimers like (H ) or (NH ) , etc., are discussed by Boldyrev and Simons (1992a), Boldyrev and Simons (1992b) and Wright (1994). Chemical processes related with Rydberg electrons were studied even for such big objects as fullerenes (Weber et al., 1996) or metal surfaces (Ganesan and Taylor, 1996). Comparison of simple model electronic Rydberg calculations with concrete molecular experiments is naturally much more complicated due to the presence of additional degrees of freedom (vibration and rotation). At the same time this gives possibility for new qualitative e!ects like, for example, the stabilization of unstable rotational axes of an asymmetrical top molecule due to the interaction with a Rydberg electron as proposed by Basov and Pavlichenkov (1994) or core induced stabilization of molecular Rydberg states discussed by Lee et al. (1994). Monitoring of intramolecular dynamics through preparation of special Rydberg electron wave packets is no longer a science "ction but the subject of current interest (Beims and Alber, 1993; Boris et al., 1993; Dietrich et al., 1996; Frey et al., 1996; Jones, 1996; Rabani and Levine, 1996; Remacle and Levine, 1996a,b; Thoss and Domcke, 1995). Naturally, it is impossible to expect that a simple qualitative model based on the n-shell approximation can be used in the region where n is no longer an approximate integral of motion. So a further natural question arises: How to extend the approach presented in this paper to the case of `overlappinga and to the case of more serious `interactionsa of di!erent n-shells? This problem can be attacked from two opposite sides. One possibility is to start with the model of `complete classical chaosa and to study peculiarities of quantum systems through some kind of statistical or some other `chaologicala methods (Bixon and Jortner, 1996; Friedrich and Wintgen, 1989; Hasegawa et al., 1989; Lombardi et al., 1988; Lombardi and Seligman, 1993; von Milczewski et al., 1994, 1996; Zakrzewski et al., 1995). An alternative approach consists in studying qualitative e!ects related to the violation of the individual n-shell approximation. Such an extension of the qualitative approach to the qualitative theory allowing the `couplinga of di!erent n-shells should be extremely
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
231
useful for the Rydberg problem in the intermediate region between quasiregular and completely chaotic motion. The analog of such an extension was proposed earlier in the qualitative analysis of molecular rovibrational structure. Rotational multiplets of di!erent individual vibrational components can be considered as formal analogs of di!erent n-multiplets. In the classical limit the violation of individual components (vibrational components for rovibrational problems or n-shells for Rydberg problems) is associated with the formation of `diabolica points (conical intersection points) between di!erent energy surfaces. As was shown on simple initial example by Pavlov-Verevkin et al. (1988) the new qualitative phenomenon, namely the redistribution of energy levels between di!erent branches in the energy spectrum typically appears in such a case under variation of the integral of the motion. Each elementary qualitative phenomenon can be characterized by a topological invariant which is stable under small deformation of the Hamiltonian. Recent theoretical analysis (Zhilinskii and Brodersen, 1994; Brodersen and Zhilinskii, 1995b; Brodersen and Zhilinskii, 1995a; Zhilinskii, 1996) supports this point of view and makes some interesting relations with rather di!erent physical phenomena like recoupling of angular momenta and quantum Hall e!ect (Avron et al., 1983, 1988; Bellissard, 1989; Leboeuf et al., 1992; Niu et al., 1985; Simon, 1983) or purely mathematical questions like topological obstructions to integrability (Nekhoroshev, 1972; Duistermaat, 1980; Cushman and Bates, 1997). Complete classical analysis of the redistribution phenomenon discussed in Section 6.1 of Chapter II (Sadovskii and Zhilinskii, 1999) has given an interesting mathematical relation with classical monodromy. Further study of this phenomenon (Cushman and Sadovskii, 1999) shows the presence of classical monodromy for the hydrogen atom in orthogonal electric and magnetic "elds. The redistribution of energy levels between di!erent n-shells in quantum Rydberg problems and corresponding analysis of associated classical models will surely become a new subject of further theoretical and experimental study.
Appendix A. Geometrical representation We consider in this appendix the geometrical representation of orbifolds which is the initial step for the geometrical representation of qualitatively di!erent types of the Morse}Bott-type functions de"ned over classical phase space for various invariance groups. As long as we are working with functions which are supposed to be the classical analog for quantum e!ective Hamiltonians, we will use the notion `energy levela for a solution of the equation H"const. In our particular case the Hamiltonian function is de"ned over a four-dimensional space (R"S ;S ), so the energy level is normally a three-dimensional region of the phase space which may be characterized by its topology. Taking into account the action of the symmetry group we can reach more detailed information about the structure of each energy level by indicating the topological structure of orbits from one side and the topological structure of the orbifold section from another side. A.1. O(3) or SO(3) invariant Hamiltonian The case of SO(3) invariant operator is not very interesting in our particular problem because any SO(3) invariant operator is also invariant with respect to the O(3) group on the R"S ;S
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
232
Fig. 13. Orbifold for the O(3) group action on R"S;S. Orbits are parametrized by the value of one invariant polynomial, m.
Table A.1 Topological description of energy levels for the simplest Morse}Bott-type Hamiltonian (O(3) invariant). `noa in the Morse index column means absence of stationary points Morse index
Dim of orbit
Topological structure of the energy level
Local symmetry
0(2) no 2(0)
2 3 2
e;S e;RP e;S
C T C Q C F
manifold. So if one is restricted to the consideration of the diagonal in `na e!ective operators, there is no di!erence between SO(3) and O(3) symmetry groups. The orbifold for the O(3) action on the R"S ;S manifold is shown in Fig. 13. From the topological point of view it is a 1D-ball (R"O(3)&B ). Orbits are parametrized by the value of one invariant polynomial, m. The detailed description of orbits and strata is given in Table 4 of this paper. The simplest Morse}Bott-type function de"ned on R and which is O(3) invariant possesses two critical manifolds coinciding with two critical orbits of the O(3) group action. Di!erent energy levels of such a simplest function are characterized by the topological structure in Table A.1. There are, in fact, two possible choices of the simplest Morse}Bott function di!ering in interchanging the position of minimum and maximum critical manifolds. As soon as the dimension of critical orbits is two, the Morse indices for minimal or maximal critical manifolds are equal to 0 or 2 correspondingly. Any function H"H(m) which has RH/RmO0 for !14m41 may be used as an example of the simplest Morse}Bott-type function. If we have an e!ective Hamiltonian which possesses an extremum within the interval !1(m(#1, then the system of di!erent energy levels has more complicated topological properties. For example, the e!ective Hamiltonian of the form H"c(m#)
(A.1)
has "ve topologically di!erent energy levels given in Table A.2. The Morse index for the RP stationary manifold may be either 0 or 1 (because the dimension of the RP stationary manifold is 3). One can easily see that the total number of critical manifolds may be arbitrary but the numbers of critical manifolds with given indices are related among themselves.
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
233
Table A.2 Topological description of energy levels for the (A.1) type Hamiltonian (O(3) invariant). Morse indices are indicated for the c'0 (c(0) choice of the parameter. The disconnected components of the energy level are represented within the curly brackets Morse index
Topological structure of the energy level
Local symmetry
0(1) no
e;RP 2+e;RP,
2(0) no
+e;S ,# +e;RP,
no 2(0)
e;RP e;S
C Q C Q C T C Q C Q C F
If the total number of critical manifolds is even there are one maximum and one minimum which are situated on critical orbits (C and C ) and equal number of minima and maxima on generic T F (C ) orbits. If the total number of critical manifolds is odd, there are two maxima (or two minima) Q on critical orbits (C and C ) and odd, 2p#1, (even, 2p) number of minima and even, 2p, T F (odd, 2p#1) number of maxima on generic (C ) orbits (p50 being nonnegative integer). Q A.2. C
invariant Hamiltonian
Orbits for the C action on R are parametrized by values of three denominator invariants (m, k, l) and one numerator invariant, p. So, we can represent the orbifold in m, k, l variables taking into account that for p"0 there is one-to-one correspondence between points of the orbifold and m, k, l values satisfying the equalities in Eq. (16). If p'0 there are two orbits with the same m, k, l values but with the di!erent signs of p (A.2) p"$[(1!m)!(1!m)k!(1#m)l] . We represent an orbifold as consisting of two parts, one corresponding to p50 and other corresponding to p40. Both these parts are shown in Fig. 14. They are identical in 3D-(m, k, l)space. For m"1 we have l"0 and k41. For m"!1 we have k"0 and l41. For any m(1, putting p"0 we can "nd the geometrical form of the boundary of each part of the orbifold in the 3D-(m, k, l)-space. The equation for the boundary has the form 1 1 1 k# l" . 1!m 2 1#m
(A.3)
So any m"const. section of one part of the orbifold is an ellipsoid. The complete orbifold may be constructed by identifying the surfaces of two 3D-bodies corresponding to p50 and p40 parts of the orbifold. To specify the topological structure of the orbifold it is necessary to use some additional information about topology of three-dimensional manifolds (Fomenko, 1983; Thurston, 1969).
234
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
Fig. 14. Orbifold for the C group action on R"S;S. Each part of the orbifold is parametrized by three invariant polynomials, m, k, l. Two parts corresponding to di!erent signs of the auxiliary polynomial p should be glued together through the identi"cation of respective points on the boundary p"0. Fig. 15. Representation of the orbifold for the C group action on R"S;S in non-polynomial variables e "arccosm, e "arccos(k#l), e "arccos(k!l). Only one part of the orbifold is shown.
Every three-manifold can be obtained from two handle-bodies (of some genus) by gluing their boundaries together. Such a representation is called a Heegard decomposition. It is not unique and de"ned by number g (the genus of the boundary of two auxiliary manifolds) and by mapping of the boundaries. It is known that the only three-manifold, possessing the Heegard decomposition of genus 0 is the 3D sphere. The above constructed orbifold is given just in the form of the Heegard decomposition of genus 0, thus the topological structure of the orbifold is the three-dimensional sphere S (R"C &S ) with four marked points corresponding to the 0-D stratum. This result agrees perfectly with the result of the analytical treatment made in Section 2.6. Using a non-polynomial transformation of the variables m, k, l to new ones of the type arccos m, arccos(k#l), arccos(k!l) ,
(A.4)
we can represent the orbifold in a more simple geometrical way (as two tetrahedra with identi"ed surfaces) but with the same topological properties (see Fig. 15). These non-polynomial variables may be interpreted as angles in the (x"j#k, y"j!k) representation characterizing angles between x and y and the symmetry axis. For the group C an example of the simplest e!ective invariant operator may be written in the form of the linear combination of the invariant polynomials H"c m#c k#c l . (A.5) In fact, if the auxiliary numerator invariant p does not enter in Hamiltonian (A.5) the complete symmetry of this Hamiltonian is higher, namely it is C T . To get the generic Hamiltonian it is Q necessary to choose the coe$cients c in such a way that any section of the orbifold includes at most G
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
235
one critical orbit. Let us choose one particular form H"2k!l .
(A.6)
We have four sections of the orbifold which go through the critical orbits and three connected components of the generic sections. They are denoted by letters according to their energy: a, E"!2; b, !1'E'!2; c, E"!1; d, 1'E'!1; e, E"1; f, 2'E'1; g, E"2. Four sections a, c, e, g corresponding to energies of critical orbits are shown in the Fig. 16. If we vary the coe$cients in Eq. (A.5) the orientation of the constant energy level planes with respect to the orbifold changes. This means that critical orbits which are stationary for any choice of Hamiltonian can change its stability. One particular simple physical example of a hydrogen atom in parallel electric and magnetic "elds corresponds to Hamiltonian (A.5) in the simplest approximation (see Section 4.3). A.3. C
T
invariant Hamiltonian
Orbifolds for other one-dimensional Lie subgroups of O(3), G"C , C , D , D , can be T F F constructed from that for the C group. It is su$cient to take into account the action of G/C on invariant polynomials given by Table 6 and to note that the action on orbits is equivalent to the action on invariant polynomials. The orbifold for the C subgroup results from that of a C one by noting that the p operation T T relates two orbits which belong to the same part of the C orbifold characterized by a given sign of the p invariant. So it is su$cient to take the k'0 parts of both 3D-bodies corresponding to di!erent signs of p. To properly represent the C orbifold we use the m, l, k variables which are T the invariant polynomials for the C group. In such variables each m"const., mO$1, section of T the orbifold is a parabola. The orbifold is shown in Fig. 17.
Fig. 16. Representation of energy levels of the Hamiltonian H"2k!l on the C orbifold. Only one part (p50) of the orbifold is shown. Fig. 17. Orbifold for C group action on R. Two parts corresponding to di!erent signs of the auxiliary polynomial T p should be glued together through the identi"cation of respective points on the boundary p"0.
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
236
It should be noted that the identi"cation of points on the surfaces of C orbifold results in the identi"cation of the C points on the surfaces of two parts of the C orbifold. All points on T the surfaces which do not belong to the C stratum should be identi"ed by pairs. At the same Q time the C invariant orbits lying on p50 and on p40 parts of the C orbifold at Q T k"0, pO0 are di!erent. This is due to the fact that they correspond to pO0. We can again use non-linear transformation of variables to reach a more simple geometrical form of the orbifold. In new variables arccos m, arccos((k#l), arccos((k!l) ,
(A.7)
each part of the orbifold is a tetrahedron shown in Fig. 18. These non-polynomial variables may be again interpreted as angles in the (x"j#k, y"j!k) representation characterizing angles between x and y and the symmetry axis. From the topological point of view the orbifold is a 3D-ball, B , with one marked point (C orbit) inside and with the S surface which includes 2D-stratum (C ) plus two isolated points (C orbits). The schematic topological structure of the Q T orbifold is shown in Fig. 19. It is useful to note that the sub-manifold formed by both C and Q C strata is a closed 3D-manifold invariant with respect to the C subgroup of the C invariance T Q T group. Its topological structure is the suspension of the 2D-torus (see Section 2.6 for discussion of the topology of this closed sub-manifold). This fact is important for a detailed classi"cation of the C -invariant Morse}Bott functions. T A.4. C
F
invariant Hamiltonian
For the C subgroup the p operation relates C orbits with opposite p and l values. In order F F to have the one-to-one correspondence between orbits and invariant polynomial values, we must unify at one point of the C orbifold pairs of orbits of the C orbifold according to the rule F (m, k, l)"(m, k,!l) . (A.8)
Fig. 18. Orbifold for the C group action on R represented in non-polynomial variables k "arccos m, k "arT ccos("k"#l), k "arccos("k"!l). Only one part of the orbifold is shown. Fig. 19. Schematic topological structure of the orbifold for C
T
group action on R.
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
237
Fig. 20. Orbifold for the C group action on R. Two parts corresponding to di!erent signs of auxiliary polynomial lp F should be glued together through the identi"cation of respective points on the boundary lp"0. Fig. 21. Orbifold for the C group action on R represented in non-polynomial variables t "arccos m, F t "arccos(k#"l"), t "arccos(k!"l"). Only one part of the orbifold is shown.
Orbits of C having l"0, p"0 are invariant with respect to the C group. Thus, the orbifold F for the C group may be constructed from a C one by taking only l50 parts of two bodies with F the identi"cation of all corresponding points on the surfaces. This orbifold is shown in Fig. 20. As soon as invariant polynomial for C has the form l rather than l, it is more meaningful to F change the variables and to give the orbifold in m, k, l variables. In these variables each m"const., mO0, section has the form of a parabola 1!m 1!m k# . l"! 2 1#m
(A.9)
One should remark that the geometrical form of the C orbifold in the m, k, l variables is the F same as the geometrical form of the C orbifold in the m, k, l variables (whereas the system of T strata is completely di!erent in the two cases). We can use more complicated non-polynomial variables arccos m, arccos(k#(l), arccos(k!(l)
(A.10)
to reach the simpler geometrical form of the orbifold. It is shown in Fig. 21. These non-polynomial variables are again the angles characterizing the mutual positions of x, y and the symmetry axis in the (x, y) representation. From the topological point of view this orbifold is given in the form of the Heegard decomposition of genus 0. So the C orbifold is a S manifold with one S marked circle and three marked F points: one isolated (C orbit) point and two (C orbits) lying on the S marked circle (formed by F C , C and C strata). Q G F We remark again that there are two closed sub-manifolds formed each by two di!erent strata. One is formed by C and C strata and another by C and C strata. These both manifolds are Q F G F S spheres from the topological point of view.
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
238
A.5. D
invariant Hamiltonian
In the case of the D symmetry the action of the G/C group on the C orbifold form larger orbits by relating orbits (m, k, l) and (m,!k,!l) with opposite p values. To construct the orbifold for the D group we "rst subdivide the C orbifold into parts with semide"nite signs of the D group numerator invariants (lp, kp, kl). Such a splitting of the m, k, l space of the C invariant polynomials into parts with speci"c signs of the D numerator invariants is shown in Fig. 22. Taking into account the action of the C operation on C orbits, the D orbifold may be represented as four 3D-bodies shown in Fig. 23 in D invariant polynomial variables m, k, l with the following identi"cation of faces, edges and vortexes: A B C D "a b c d , A B C D "a b c d , A B C "A B C , A C D "a c d , A C D "a c d , a b c "a b c , A C "a c "A C "a c , A B "A B "a b "a b , B C "B C "b c "b c , A D "a d "A D "a d , C D "c d "C D "c d , A "A "a "a , B "B "b "b , C "C "c "c , D "D "d "d .
(A.11)
(A.12)
(A.13)
(A.14)
Fig. 22. Splitting of the space of the C invariant polynomial variables m, k, l into parts with particular signs of D auxiliary (numerator) invariants.
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
239
Fig. 23. Orbifold for the D group action on R represented in polynomial variables m, k, l. The identi"cation of faces, edges, and vortexes of four bodies is given in Eqs. (A.11)}(A.14).
In fact, we have constructed the representation of the orbifold as a simplicial complex. It consists of four 3D-simplexes, six 2D-simplexes, "ve 1D-simplexes and four 0D-simplexes. The EulerPoincareH characteristics of this complex di!ers from zero (!1)NaN"4!5#6!4"1 .
(A.15)
Here aN is the number or p-dimensional simplexes. As it is shown in Section 2.6, the orbifold is the suspension (RP ). A.6. D
F
invariant Hamiltonian
To go from the D orbifold to a D one, it is su$cient to consider the action of the p operation F F on the D orbifold. As soon as the p operation changes, simultaneously signs of l and p it relates F orbits characterized by (lp'0, kp'0, kl'0) and (lp'0, kp(0, kl(0) and in a similar way orbits characterized by (lp(0, kp(0, kl'0) and (lp(0, kp'0, kl(0). As a consequence, for the D orbifold we have only one body instead of each pair of bodies with the same F geometrical form. The D orbifold is shown in Fig. 24. F It consists of two bodies. The points on two pairs of faces must be identi"ed whereas the third pair of faces (formed by the C stratum) must not be identi"ed. From the topological point of view Q
240
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
Fig. 24. Orbifold for the D group action on R represented in polynomial variables m, k, l. Two parts corresponding F to di!erent signs of the auxiliary polynomial lp should be glued together through the identi"cation of respective points on the boundary lp"0.
Fig. 25. Schematic topological representation of the orbifold for D
F
group action on R.
the D orbifold is equivalent to a 3D-ball. The boundary is formed by the C , C , C , C F Q T F T strata. Inside the ball there are C , C , C , and the generic C strata. The schematic topological G Q F view of the D orbifold is given in Fig. 25. Several closed subspaces formed by di!erent strata are F clearly seen in Fig. 25. It is important to verify that the Morse}Bott inequalities are satis"ed on all these subspaces.
Appendix B. Molien functions for point group invariants This appendix deals with a technical question: How to describe the system of invariant polynomials on the R manifold in the presence of a non-linear action of the symmetry group. The schematic answer to this question was initially formulated in Section 4 of Chapter I. Some applications have been discussed in Chapter II and in Section 2.7 of the present chapter. Particular
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
241
example of the symmetry group G for the hydrogen atom on orthogonal electric and magnetic "elds was treated in more details in Section 4.4. The group G has an interesting physical meaning because it includes symmetry transformations which contain both spatial and time reversal operations but it is Abelian and this simpli"es signi"cantly the construction of the Molien function for invariants. Below we explain on several examples of point group symmetry with increasing complexity the procedure of the Molien function construction. The construction of invariant functions on the R manifold is based on the preliminary construction of the integrity basis on the six-dimensional space where the action of the symmetry group of the problem is linear. The Molien function and the invariants themselves for the six-dimensional space x y or k j may be found from known expressions for Molien functions and integrity bases for irreducible representations. Next step includes the restriction of the polynomial algebra on the sub-manifold R of the 6D-space. The general procedure of the restriction of the polynomial ring de"ned on the manifold to the sub-manifold was outlined in Chapter I and realized in several examples in Section 2.7. The sub-manifold R is de"ned in the six-dimensional space x y by the polynomial equations x", y". If the point group symmetry does not include improper rotations (inversion or re#ections) these polynomials may always be considered as denominator invariants. In such a case we just eliminate them from the integrity basis constructed for the 6D-space and the resulting integrity basis gives the basis for the 4D sub-manifold. For point groups which are not the subgroups of SO(3) we start with the consideration of the similar problem for the proper rotation subgroup and after that take into account the e!ect of improper symmetry elements working directly on the 4D-sub-manifold R. B.1. C group This trivial case is useful to demonstrate how to take into account the restriction of the polynomial ring on the sub-manifold. We work in x y representation. On the 6D-space there are six basic invariants x , y (i"1, 2, 3) G G and the Molien function for invariants has a trivial form 1 . M " ! (1!j) The ring of the invariant polynomials on the 6D-space P! is the ring of all polynomials P! "P[x , x , x , y , y , y ] . To restrict the polynomial ring on the sub-manifold
(B.1)
(B.2)
(B.3) h "x #x #x " , (B.4) h "y #y #y " , we are obliged to introduce two second-order polynomials, h , h given by Eqs. (B.3) and (B.4) as denominator invariants. This may be done by multiplying both the numerator and denominator of the Molien function in Eq. (B.1) by (1#j). The new form of the Molien function 1#2j#j M " ! (1!j)(1!j)
(B.5)
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
242
corresponds now to another description of the ring of invariants P! which is considered as a free module: (B.6) P! "P[x , x ,h , y , y ,h ]䢇(1, x )(1, y ) with four auxiliary invariants u "1, u "x , u "y , u "x y . We use the notation (a, b)(c, d)"(ac, ad, bc, bd) to show that four numerator invariants are represented as product of more simple terms. Having the form (B.5) of the Molien function and the form (B.6) of the module of invariant functions it is easy to make the restriction to the sub-manifold R. We just eliminate two denominator invariants, h , h , corresponding to the equation de"ning R. The resulting Molien function and the module of invariants on R are written as follows: 1#2j#j , M "R " ! (1!j)
(B.7)
(B.8) P! "R "P[x , x , y , y ]䢇(1, x )(1, y ) . The choice of basic and auxiliary invariants is ambiguous and the one proposed here is just an example. The important point is that the integrity basis which includes four numerator invariants (including 1) and four denominator invariants may be constructed. To "nd now the integrity basis in the j k representation we can simply transform invariants from x, y to j, k representation. B.2. C point group Let us take x and y to coincide with the C -axis. In such a case x , x , y , y transform according to the B representation and x , y according to the A representation of the C point group. The Molien function for invariants constructed from 6D reducible representation 4B#2A has the form 1#6j#j . M " ! (1!j)(1!j)
(B.9)
The explicit form of the module of invariant functions on 6D-space may be easily given as well: (B.10) P! "P[x , x , x , y , y , y ]䢇((1, x x )(1, y y ), (x , x )(y , y )) . There are six denominator invariants and eight numerator invariants. We have in Eq. (B.10) second degree denominator invariants which may be replaced by invariants de"ning the sub-manifold R in Eqs. (B.3) and (B.4). After such a substitution, to make the restriction on R it is su$cient to omit two denominator invariants corresponding to equations de"ning R. The resulting Molien function for invariant polynomials on R and the description of the ring of invariant function as a module are as follows: 1#6j#j M "R " , ! (1!j)(1!j)
(B.11)
P! "R "P[x , x , y , y ]䢇((1, x x )(1, y y ), (x , x )(y , y )) .
(B.12)
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
243
To "nd the integrity basis in the j k representation we can use again the transformation from x, y to j, k representation. B.3. C point group G For point groups which are not the subgroups of SO(3) the procedure of constructing the Molien function for invariants on R and the integrity basis is slightly di!erent. We start with the consideration of the similar problem for the proper rotation subgroup of the point group. There are two groups C (or in other notation S ) and C which have no non-trivial rotational subgroups. G Q So we use them to illustrate the construction of invariant functions for these cases. The C group G includes only one non-trivial symmetry operation, inversion. Its action on x, y variables corresponds to interchange xy. Taking this action into account it is easy to make the transformation of the C invariant denominator and numerator polynomials (forming the module of invariant polynomials on R) into form with irreducible transformation properties with respect to the C group: G P!G "R "P[x #y , x !y , x #y , x !y ]䢇((1, x #y )(1, x !y )) .
(B.13)
This form of the module of the C invariant function on R is well adapted to transformation to the module of C invariant functions. First, we change the denominator invariants (x !y ) and G (x !y ) into (x !y )&x y and (x !y )&x y which are both C and C invariants. This G may be achieved by multiplying the numerator and denominator of the Molien function by (1#j). The module of the C invariant function on R in such a case will include 16 auxiliary (numerator) invariants but among them only 8 are C invariants. The resulting Molien function G and the module of the C invariants on R are written as follows: G 1#j#4j#j#j , M G "R " ! (1!j)(1!j)
(B.14)
P!G "R "P[x #y , x #y , x y , x y ]䢇(1, u , u , u , u , u , u , u ) , u "x #y , u "x y , u "(x !y )(x !y ) ,
(B.15)
u "u u ,
(B.16)
u "u u , u "(x !y )(x !y ) ,
u "(x !y )(x !y ) .
(B.17)
B.4. C point group Q The construction of the module of the C invariant functions on R is very similar to the realized Q above construction for the C invariants. The only di!erence is that the action of the C non-trivial G Q operation (re#ection in the symmetry plane) is now di!erent from the action of the inversion for the C group. Let us suppose the symmetry plane to be orthogonal to x (y ) axes. In such a case G (x !y ) and (x !y ) are symmetrical with respect to the C group whereas (x #y ) and Q
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
244
(x #y ) are anti-symmetrical. Taking this into account the Molien function and the module of C invariants on R are written immediately as follows: Q 1#j#4j#j#j , (B.18) M Q "R " ! (1!j)(1!j) P!Q "R "P[x !y , x !y , x y , x y ]䢇(1, u , u , u , u , u , u , u ) , u "x #y , u "x y , u "(x #y )(x #y ) , u "u u , u "u u , u "(x !y )(x #y ) , u "(x !y )(x #y ) . B.5. C
T
(B.19) (B.20) (B.21) (B.22)
point group
For the point group C which is not a subgroup of SO(3), we start again with the consideration T of the similar problem for the proper rotation subgroup C . We can take the Molien function for the C invariant and integrity basis for the C group as the initial point. All invariants of C span invariants and pseudo-invariants of C (A A representations). So we "rst make the linear T transformation of numerator and denominator invariants for the C group resulting in a new set of C invariants which are at the same time either of A or of A type with respect to the C group. T This linear transformation does not change the form of the Molien function. It is of the form (B.11) found for the C invariants on R. However, this transformation changes the basis of the module of C invariant functions on R. To "nd proper linear combinations which accordingly transform irreducible representation of the C group we take into account the fact that two symmetry operations (p , p ) which belong T to the C group but do not belong to the C group may be represented as products of inversion T and the C rotation around the axis orthogonal to the re#ection plane. We note again that the action of the inversion results in the interchange of x with y and the action of p , p on x , y G G G G may be represented as follows: p x !y , ( j"1, 3); p x y , (B.23) H H p x !y , ( j"2, 3); p x y . (B.24) H H So, instead of four denominator invariants (x , y , x , y ) we should take two linear combinations h "(x #y ) and h "(x !y ) which are invariant with respect to C and two combinations Q Q T h "(x !y ) and h "(x #y ) which are pseudo-invariants of type A with respect to C . ? ? T In a similar way, we form new numerator invariants. Six numerator invariants are of type A with respect to C : 1, u "x x !y y , u "x y , u "x y , u "x y !x y , T Q Q Q V u "x x y y . There are equally two numerator C invariants which are pseudo-invariants of Q the type A with respect to C : u "x y #x y , u "x x #y y . The module of T ? ? C invariant functions of R is represented now as (B.25) P! "R "P[h , h , h , h ]䢇(1, u , u , u , u , u , u , u ) . Q Q ? ? Q Q Q Q Q ? ? Now, we can transform the basis of the module of the C invariant functions in such a way that new denominator invariants become C invariants rather than pseudo-invariants. To do that it is T
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
245
su$cient to multiply the numerator and denominator of the Molien function by (1#j)(1#j). This action corresponds simply to the substitution of two C denominator invariants h and ? h by their squares which are C invariants. The new representation of the module of the ? T C invariants on R takes the form P! "R "P[h , h , (h ), (h )]䢇(1, u , u , u , u , u , u , u )(1, h )(1, h ) . Q Q ? ? Q Q Q Q Q ? ? ? ?
(B.26)
To make the restriction to C invariant functions it is su$cient now to take only those numerator T invariants which are C invariants. The representation of the module of C invariant functions on T T R takes the form P!T "R "P[h , h , h , h ]䢇((1, h h )(1, u ), (h , h )( u )) Q Q ? ? ? ? GQ ? ? G?
(B.27)
including 16 numerator invariants. The corresponding Molien function has the form (B.28) M
!T
1#4j#3j#3j#4j#j "R " (1!j)(1!j)(1!j) 1#3j#3j#j , " (1!j)(1!j)
(B.28)
(B.29)
which turns out to be reduced to a more simpler one in Eq. (B.29) which includes only eight numerator invariants. Construction of the integrity basis corresponding to the reduced form of the Molien function (B.29) requires additional analysis because we should "nd three secondorder algebraically independent invariants. At the same time the straightforward procedure realized above gives one of the possible basis of the module of invariant functions although not a minimal one. B.6. D
F
group
We begin by constructing the Molien function of D invariants over the 6D space x y. This 6D vector space is the six-dimensional reducible representation of the form (E A ) (E A ). The corresponding Molien function is 1#2j#4j#10j#4j#2j#j . M " " (1!j)(1!j)
(B.30)
Its restriction on the R subspace has the form 1#2j#4j#10j#4j#2j#j . M "R " " (1!j)(1!j)
(B.31)
To go now to the D group we "rst rewrite the Molien function for the D invariants on R using F two auxiliary variables j , j instead of one j. We use subscript to distinguish the behavior of Q ? D invariants with respect to re#ection.
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
246
First, the Molien function for the D invariants on R with two parameters has the form 1#2j#2j#6j#2j#2j#j#(2j#4j#2j) Q Q Q Q Q Q ? ? ? . (B.32) M "R " " (1!j)(1!j)(1!j)(1!j) Q ? Q ? We transform it to new denominator invariants which are at the same time the D invariants: F (1#2j#2j#6j#2j#2j#j)#(2j#4j#2j) Q Q Q Q Q Q ? ? ? (1#j)(1#j) . M "R " ? ? " (1!j)(1!(j))(1!j)(1!(j)) Q ? Q ? (B.33) To pass to D invariants one must take only those numerator D invariants which are F D invariants. The Molien function for D invariants is written in the form with the only F F parameter j M
"F
(1#j)[1#2j#2j#6j#2j#2j#j] "R " (1!j)(1!j)(1!j)(1!j) #
(j#j)[2j#4j#2j] , (1!j)(1!j)(1!j)(1!j)
(B.34)
which is equivalent to M
"F
1#2j#2j#6j#5j#8j#8j#5j#6j#2j#2j#j "R " . (1!j)(1!j)(1!j)(1!j) (B.35)
There are four D denominator invariants (degree 2, 3, 4 and 6) and 48 numerator invariants. F Remark that this is not the simplest form of the Molien function because the numerator may be factorized and the (1#j) factor may be simultaneously eliminated from numerator and denominator resulting in the reduced form of the Molien function M
1#j#2j#5j#3j#3j#5j#2j#j#j " . R" "F (1!j)(1!j)(1!j)
(B.36)
Although the question is open how to construct integrity basis corresponding to this simpli"ed Molien function in Eq. (B.36), the construction of the non-minimal integrity basis corresponding to the form in Eq. (B.35) of the D Molien function is straightforward. F To give the explicit form of the integrity basis over the six-dimensional space we need the basis for all invariants and covariants over three-dimensional space. Instead of x and y we can use G G irreducible tensors with respect to the D group on each 3D-subspace to construct the integrity basis on R of functions invariant with respect to the D group action on the 6D-space. First of all we give the explicit form of D invariants and covariants on three-dimensional space which form the basis of the module of polynomials. There are three basic (denominator) invariants: x #x , x ,
c (x)"x !3x x
(B.37)
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
247
and one auxiliary (numerator) invariant x p (x), with p (x)"3x x !x .
(B.38)
There are two auxiliary covariants of type A (degree 1 and 3), namely x and p (x), and four pairs of auxiliary covariants of type E (degree 1, 2, 2 and 3)
x , x
x !x , !2x x
x x , !x x
2x x x . (x !x )x
(B.39)
Taking into account the form of these invariants and covariants for 3D x and 3D y spaces and the expression for the Molien function in Eq. (B.31) for D invariants on R we can write the following representation for the module of D invariant functions of R: P" "R "P[x , c (x), y , c (y)]䢇(1, u ,2, u ) .
(B.40)
There are 24 (including 1) numerator invariants. They are listed below in the form which shows clearly that they are simply invariants produced by coupling x and y covariants: u "x p (x), u "y p (y), u "x y p (x)p (y) ,
(B.41)
u "x y , u "x y p(y), u "y x p(x) ,
(B.42)
u "x p(x)y p(y) ,
(B.43)
x u " x
x u " x
y x , u " y x
y !y , !2y y
y y x , u " !y y x
2y y y , (y !y )y
x !x u " !2x x
y x !x , u " y !2x x
x !x u " !2x x
y y x !x , u " !y y !2x x
x x u " !x x
y x x , u " y !x x
x x u " !x x
y y , !y y
2x x x u " (x !x )x
y !y !2y y
2y y y , (y !y )y
y !y !2y y
x x u " !x x
y 2x x x , u " y (x !x )x
2y y y (y !y )y
y !y , !2y y
(B.44)
(B.45)
(B.46)
(B.47)
(B.48)
(B.49)
(B.50)
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
248
2x x x y y u " , (B.51) (x !x )x !y y 2x x x 2y y y . (B.52) u " (x !x )x (y !y )y We change the basis of the module of D invariant functions in such a way that all invariants become invariants or pseudo-invariants for the D group. To do this it is su$cient to form simple F linear combinations of denominator and numerator invariants used in Eq. (B.40). We remark that the action of the p operation which should be added to go from D group to a D one is given by F F p x !y , ( j"1, 2) and p x y . The new representation of the module of D invariant F H H F functions corresponding to the Molien function in Eq. (B.32) with two parameters is as follows:
(B.53) P" "R "P[h , h , h , h ]䢇(1, u , 2, u , u , 2, u ) , Q Q ? ? Q Q ? ? h "x #y , h "c (x)!c (y) , (B.54) Q Q h "x !y , h "c (x)#c (y) , (B.55) ? ? u "u !u , u "u #u , u "u !u , (B.56) Q Q Q u "u !u , u "u #u , u "u #u , (B.57) Q Q Q u "u !u , u "u !u , u "u , u "u , (B.58) Q Q Q Q u "u , u "u , u "u , u "u , u "u , (B.59) Q Q Q Q Q u "u #u , u "u !u , u "u #u , (B.60) ? ? ? u "u #u , u "u !u , u "u !u , (B.61) ? ? ? u "u #u , u "u #u . (B.62) ? ? We change now two denominator invariants h , h into (h ), (h ) which are D invariants. ? ? ? ? F This results in increasing the number of D numerator invariants four times. But we are interested only in those numerator invariants which are D invariants as well. There are 48 such invariants F which correspond to the form (B.35) of the Molien function. The structure of the module of D invariant functions on R may be now given explicitly F P" "R "P[h , h , (h ), (h )]䢇((1, h h )(1, u ,2, u ), (h , h )(u , 2, u )) . (B.63) Q Q ? ? ? ? Q Q ? ? ? ? There are four denominator invariants and 48 numerator invariants which may be reconstructed from Eqs. (B.41)}(B.52), (B.53)}(B.62). Apparently, the smaller integrity basis with only 24 numerator invariants may be found as the form (B.36) of the Molien function indicates. B.7. ¹ point group B To construct the Molien function for the ¹ group invariants on R we use exactly the same B procedure as for the D group but we start now from the ¹ group in (x y) representation. The F
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
249
Molien function for the invariants of the ¹ group on R has the form (1#j)#2(j#j)#(j#j#2j#j#j) . M "R " 2 (1!j)(1!j)
(B.64)
More detailed form of the Molien function with two parameters may be given after the transformation of the ¹ invariants into the form irreducible with respect to ¹ (A or A symmetry types with B respect to ¹ ). We use below j for those ¹-invariants which are at the same time the ¹ invariants B Q B and j for those ¹-invariants which are pseudo-invariants (of the A type) of the ¹ group. ? B (1#j#j#5j#3j#8j#3j#5j#j#j#j) Q Q Q Q Q Q Q Q Q Q M "R " 2 (1!j)(1!j)(1!j)(1!j) Q ? Q ? (j#2j#3j#6j#3j#2j#j) ? ? ? ? ? ? . # ? (1!j)(1!j)(1!j)(1!j) Q ? Q ?
(B.65)
Now to form the Molien function for the ¹ invariant on R we multiply both numerator B and denominator by (1#j) (1#j) and take in the numerator only those terms which are ? ? ¹ invariant B (1#(jj))(1#j#j#5j#3j#8j#3j#5j#j#j#j) ? ? Q Q Q Q Q Q Q Q Q Q M B "R " 2 (1!j)(1!(j))(1!j)(1!(j)) Q ? Q ? (j#j)(j#2j#3j#6j#3j#2j#j) ? ? ? ? ? ? ? ? . # ? (1!j)(1!(j))(1!j)(1!(j)) Q ? Q ?
(B.66)
The total number of numerator invariants now is 96. If we use only one auxiliary parameter j we can simplify considerably the Molien function for ¹ invariants. Formula (B.66) becomes B (1#j)(1#j)(1#j)(1#4j#2j#4j#j) M B "R " 2 (1!j)(1!j)(1!j)(1!j) 1#4j#2j#4j#j . " (1!j)(1!j)(1!j)
(B.67)
(B.68)
The last simpli"ed form in Eq. (B.68) of the Molien function for the ¹ invariants on R seems to be B rather reasonable. Probably, it gives the minimal basis of denominator and numerator invariants on R. If the integrity basis corresponding to this simpli"ed form (B.68) exists it may be used in some applications. Number (12) of auxiliary invariants is not enormous. The initial form (B.67) of the integrity basis including 96 numerator invariants is not encouraging at all. To conclude this appendix we give the generating Molien functions for icosahedral symmetry > and > : F N(> ) N(>) F , g(> )" , g(>)" F (1!x)(1!x) (1!x)(1!x)
(B.69)
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
250
where N(>)"1#x#x#6x#6x#6x#6x#15x # 16x#15x#16x#15x#32x#15x # 16x#15x#16x#15x#6x#6x #6x#6x#x#x#x
(B.70)
and N(> )"1#x#x#3x#3x#3x#3x F # 7x#8x#7x#8x#7x#16x#7x # 8x#7x#8x#7x#3x #3x#3x#3x#x#x#x .
(B.71)
Appendix C. Strata and orbits for point groups We gave in Section 2.8, strata and orbits in action on R of three point groups C , D and ¹ . T F B This choice is due to the fact that among polyatomic Rydberg molecules studied experimentally or theoritically molecules with such symmetry groups are most typical. We can cite examples of H (D symmetry) (Bordas and Helm, 1991, 1992; Dodhy et al., 1988; Helm, 1988; Lembo et al., 1989; F Herzberg, 1981, 1987; Ketterle et al., 1989; King and Morokuma, 1979; Lembo et al., 1990; Pan and Lu, 1988; Stephens and Greene, 1995), Na (D symmetry) (Broyer et al., 1986), H O F (C symmetry) (Petfalakis et al., 1995), H F (C symmetry) (Bordas et al., 1985) and NH (¹ T T B symmetry) (Herzberg, 1981, 1987; Herzberg and Hougen, 1983; Watson, 1984). At the same time many other molecules are potentially interesting from the point of view of their excited Rydberg states or even Rydberg character of the ground state (Basov and Pavlichenkov, 1994; Boldyrev and Simons, 1992a,b; Chiu, 1986; Mayer and Grant, 1995; Wang and Boyd, 1994; Weber et al., 1996; Wright, 1994). Their symmetry groups vary from very simple C for HCO (Mayer and Grant, 1995) Q till the highest icosahedral symmetry I for C (Weber et al., 1996). That is why we give in this appendix orbits and strata for all possible "nite symmetry groups. The list of critical orbits for each symmetry group enable us to give the description of simplest Morse-type functions (see Tables 18 and 19 in Section 3.3). Tables C.1}C.8 listed in this appendix show in column 1 the stabilizer of the stratum. In column 2 closed and generic strata are indicated. We remark once more that some strata are neither closed nor generic. In column 3 the dimension of the stratum is given. For the "nite group action on R, the dimension of the generic stratum is always 4. Column 4 gives the number of orbits in the stratum. If the dimension of the stratum is zero the number of orbits in the stratum is "nite. If the dimension of the stratum is positive the number of orbits in the stratum is in"nite. We denote it RL with n equal to the dimension of the stratum. Column 5 shows the number of points in each orbit. For "nite group actions on R this number is always "nite.
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
251
Table C.1 Strata and orbits in the action of low symmetry groups C , C , C , C , S , S (n52) on R. Column `gca indicates L Q LF L generic (g) and closed (c) strata Group
Stabilizer
gc
dim.
Number
Nature
C C L
1
g
4
R
1
C L 1
c g
0 4
4 R
1 n
C Q
C Q 1
c g
2 4
R R
1 2
C LF
C LF C L C Q 1
c c g
0 0 2 4
2 1 R R
1 2 n 2n
S
S 1
c g
2 4
R R
1 2
S L
S L C L S 1
c c
0 0 2 4
2 1 R R
1 2 n 2n
g
Table C.2 Strata and orbits in the action of C on R. Strati"cation is di!erent for n even and n odd. Column `gca indicates generic LT (g) and closed (c) strata Group
Stabilizer
gc
dim.
Number
Nature
C n even LT
C LT C L CT Q CB Q 1
c c
0 0 2 2 4
2 1 R R R
1 2 n n 2n
0 0 2 4
2 1 R R
1 2 n 2n
C n odd LT
C LT C L C Q 1
g c c g
Along with tables of strata and orbits given in this appendix we give below strata equations for critical strata for some point groups. The form of the equation de"ning strata depends on the choice of variables (representation). At the same time the information concerning strata and orbits listed in Tables in this appendix does not depend on the representation. Throughout this paper we use either x y or j k representation of the R manifold.
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
252
Table C.3 Strata and orbits in the action of D on R. Strati"cation is di!erent for n even and n odd. Column `gca indicates generic L (g) and closed (c) strata Group
Stabilizer
gc
dim.
Number
Nature
D n even L
C L C C 1
c c c g
0 0 0 4
2 2 2 R
2 n n 2n
D n odd L
C L C 1
c c g
0 0 4
2 4 R
2 n 2n
Table C.4 Strata and orbits in the action of D
LF
on R. Column `gca indicates generic (g) and closed (c) strata
Group
Stabilizer
gc
dim.
Number
Nature
D n odd LF
C LF C LT C T C C F C Q 1
c c c c
0 0 0 0 2 2 4
1 1 2 1 R R R
2 2 n 2n 2n 2n 4n
C LF C LT C T C T C F C T C F C Q C Q 1
c c c c c c
0 0 0 0 0 0 2 2 2 4
1 1 1 1 1 1 R R R R
2 2 n n n n 2n 2n 2n 4n
D n even LF
g
g
For the C group action on R there is one closed zero-dimensional stratum with the stabilizer C . L L Equations de"ning this stratum in x, y representation are x ", y " . In the j, k representation the same Eqs. (C.1) are j "1, k "1 .
(C.1)
(C.2)
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264 Table C.5 Strata and orbits in the action of D
LB
253
(n52, even) on R. Column `gca indicates generic (g) and closed (c) strata
Group
Stabilizer
gc
dim.
Number
Nature
D n even LF
S L C LT C T C T C F C T C F C Q C Q 1
c c c c c c
0 0 0 0 0 0 2 2 2 4
1 1 1 1 1 1 R R R R
2 2 n n n n 2n 2n 2n 4n
S L C LT C F C C F C Q 1
c c c c
0 0 0 0 2 2 4
1 1 2 1 R R R
2 2 n 2n 2n 2n 4n
D n odd LF
g
g
Table C.6 Strata and orbits in the action of ¹, ¹ on R. Column `gca indicates generic (g) and closed (c) strata F Group
Stabilizer
gc
dim.
Number
Nature
¹
C C 1
c c g
0 0 4
4 2 R
4 6 12
¹ F
C F C C F C T CF Q CT Q 1
c c c c
0 0 0 0 2 2 4
2 1 1 1 R R R
4 8 6 6 12 12 24
g
For the C group action on R the only closed stratum with the stabilizer C has dimension two. Q Q In x, y and j, k representations the same stratum is given, respectively, by x "!y , x "!y , x "!y ,
(C.3)
j "j "0, k "0 .
(C.4)
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
254
Table C.7 Strata and orbits in the action of O and O on R. Column `gca indicates generic (g) and closed (c) strata F Group
Stabilizer
gc
dim.
Number
Nature
O
C C C 1
c c c g
0 0 0 4
2 2 2 R
6 8 12 24
O F
C T C F C T C F C T C F C Q CB Q 1
c c c c c c
0 0 0 0 0 0 2 2 4
1 1 1 1 1 1 R R R
6 6 8 8 12 12 24 24 48
g
Table C.8 Strata and orbits in the action of > and > on R. Column `gca indicates generic (g) and closed (c) strata F Group
Stabilizer
gc
dim.
Number
>
C C C 1
c c c g
0 0 0 4
2 2 2 R
12 20 30 60
> F
C T C F C T C F C T C F C Q 1
c c c c c c
0 0 0 0 0 0 2 4
1 1 1 1 1 1 R R
12 12 20 20 30 30 60 120
g
Nature
The C group action on R produces two closed strata. The C stratum is de"ned in x, y, ( j, k) LF LF representation by x "y "$ , j "1 . The C stratum is de"ned in x, y, ( j, k) representation by L x "!y "$, k "1 .
(C.5) (C.6)
(C.7)
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
255
For the S group action on R the S closed stratum is given by the following equations in x, y and j, k (C.8) representations: x "y , x "y , x "y , k"0 .
(C.8)
Appendix D. Qualitative description of e4ective Hamiltonians based on equivariant Morse}Bott theory D.1. SO(3) continuous subgroup To "nd possible Morse}Bott-type functions we use here the equivariant Morse inequalities. Strata and orbits of the SO(3) group action on R are listed in Table 4. There are two Morse counting polynomials. One for critical orbits (which are S manifolds), and another for generic orbits (PR manifolds with the SO(3) group acting freely on this manifold). We can write the Morse inequalities on the p manifold in the form (1#t)M (t) 1#2t#t ! "(1#t)Q(t), Q(t)50 . M (t)# 1!t 1!t
(D.1)
We remind that Q(t)50 in Eq. (D.1) means that all coe$cients of the Q(t) polynomial are not negative. Here the (1!t)\ stands for the Poincare` polynomial for the universal classifying space of the SO(3) group (see Appendix B of Chapter I) P
1-
1 (t)" . 1!t
(D.2)
The (1#2t#t) is the ordinary Poincare` polynomial for R, and the (1#t) the ordinary PoincareH polynomial for the S sphere. The form of two Morse counting polynomials is M (t)"n #n t , (D.3) M (t)"c #c t#c t, c 50, c #c #c "2 . (D.4) G In fact, it is easy to see that c "0, otherwise the left side of the Morse inequalities is not a "nite polynomial. To simplify the analysis we can just look for three di!erent cases corresponding to three di!erent polynomials M (t)"1#t ,
(D.5)
M (t)"2t ,
(D.6)
M (t)"2 .
(D.7)
Three associated Morse inequalities are written in the form n #n t"(1#t)Q(t) , n #1#n t"(1#t)Q(t) , n !1#n t"(1#t)Q(t) .
(D.8) (D.9) (D.10)
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
256
So, we have an obvious answer. When two critical orbits are minimum and maximum, the number of generic maxima is equal to the number of generic minima. When both critical orbits are minima, the number of generic maxima is larger by one than the number of generic minima. When both critical orbits are maxima, the number of generic maxima is smaller by one than the number of generic minima. D.2. C
continuous subgroup
There are four isolated one point orbits with C stabilizer and a generic stratum of C orbits. So any C invariant function on R possesses at least four stationary points coinciding with critical orbits and probably some extra S critical manifolds corresponding to generic C orbits. We give below the complete list of qualitatively di!erent C invariant functions. This means that we characterize each function by several numbers giving the numbers of critical orbits of each Morse index over each stratum. As long as for this concrete example four C orbits are critical and they should have even Morse index, there are 15 possible classes of functions which may be distinguished by their behavior on the C stratum. Within each such class further classi"cation of generic functions should take into account the number of S critical manifolds of each Morse index. We will use Morse counting polynomials separately for C and C strata. For the C stratum this polynomial has the form (D.11) M! (t)"n #n t#n t, n #n #n "4, n 50 . G It simply indicates that there are n C -orbits with Morse index 0, n C -orbits with Morse index 2, and n C -orbits with Morse index 4. It may be veri"ed that there are 15 di!erent possibilities to choose n , n , n to be non-negative integers and to satisfy n #n #n "4. A similar counting polynomial for critical C orbits is a polynomial of the third degree (D.12) M! (t)"k #k t#k t#k t , because any C orbit is a S manifold. It is clear that k should be non-negative integers but further G restrictions on k should follow from the Morse theory. G If we apply the approach based on ordinary homology, the Morse inequalities take the form (1#t)M! (t)#M! (t)!(1#2t#t)"(1#t)Q(t), Q(t)50 .
(D.13)
Unfortunately, this form of Morse inequalities is insu$cient in several cases. For example, it gives no restrictions on the number and type of C stationary orbits for the class of C invariant functions characterized by M! (t)"1#2t#t .
(D.14)
Better results can be obtained by using the equivariant version of Morse inequalities. In such a case contributions from C and C stationary orbits count di!erently and the equivariant Morse inequalities take the form M! (t) 1#2t#t M! (t)# ! "(1#t)Q(t), Q(t)50 . 1!t 1!t
(D.15)
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
257
Now, we can even forget our previous statement about the existence of 15 classes of functions with respect to their behavior on C stratum. These 15 classes may be rediscovered from the equivariant Morse inequalities. The simple requirement is that the left side of the last equation should be a polynomial. Further step is to "nd restrictions on M! (t) for each of the 15 di!erent classes of functions. These restrictions follow from the fact that the left part should be a polynomial of degree three and it should be divisible by (1#t). So, the M! (t) being the polynomial of degree three should depend for each class on three integer numbers. We summarize results in Table D.1. There is only one type of simplest C invariant Hamiltonian. It is characterized by the absence of critical manifolds of non-zero dimension. The next level of complexity includes Hamiltonians with one critical manifold on the C stratum. There are four such Hamiltonians which belong to di!erent classes with respect to their behavior on the C stratum. To give the list of qualitatively di!erent Hamiltonians of the next level of complexity (those possessing two C stationary orbits) it should be noted that within each class all Hamiltonians have the same parity of the number of C stationary orbits. This is due to the fact that increasing any of the a, b, c coe$cients by 1 results in the increase of the number of stationary C orbits by 2. So, the set of Hamiltonians of the second level of complexity includes three qualitatively di!erent Hamiltonians from the class 1#2t#t and six other Hamiltonians (each from di!erent class). The number of qualitatively di!erent Hamiltonians increases rapidly with the increase of complexity. (See Table D.2 showing numbers of qualitatively di!erent Hamiltonians for several low level of complexity.) It is interesting to note that among qualitatively di!erent functions constructed above there are 7 which are perfect in the equivariant sense (i.e. Q(t) is identically zero in the equivariant Morse
Table D.1 Classi"cation of qualitatively di!erent C
invariant Hamiltonians
M! (t)
M! (t)
Coe$cients
Simplest M! (t)
1#2t#t
a#(a#b)t#(b#c)t#ct
a, b, c50
0
3t#t 2#t#t 1#t#2t 1#3t
(a#1)#(a#b)t#(b#c)t#ct a#(a#1#b)t#(b#c)t#ct a#(a#b)t#(b#c#1)t#ct a#(a#b)t#(b#c)t#(c#1)t
a, b, c50 a, b, c50 a, b, c50 a, b, c50
1 t t t
3#t 4t 2#2t 2#2t 2t#2t 1#3t
a#(a#2#b)t#(b#c)t#ct (a#1)#(a#b)t#(b#c)t#(c#1)t a#(a#1#b)t#(b#c)t#(c#1)t a#(a#1#b)t#(b#c#1)t#ct (a#1)#(a#b)t#(b#c#1)t#ct a#(a#b)t#(b#c#2)t#ct
a, b, c50 a, b, c50 a, b, c50 a, b, c50 a, b, c50 a, b, c50
2t 1#t t#t t#t 1#t 2t
3#t t#3t
a#(a#b#2)t#(b#c)t#(c#1)t (a#1)#(a#b)t#(b#c#2)t#ct
a, b, c50 a, b, c50
2t#t 1#2t
4 4t
a#(a#3#b)t#(b#c)t#(c#1)t (a#1)#(a#b)t#(b#c#3)t#ct
a, b, c50 a, b, c50
3t#t 1#3t
258
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
Table D.2 Classi"cation of qualitatively di!erent C
invariant Hamiltonians by their complexity
Level of complexity
Numbers of qualitatively di!erent C invariant Hamiltonians
0 1 2 3 4 5 6 7
1 4 9 14 26 24 52 42
inequalities). All these perfect in the equivariant sense functions belong to di!erent classes with respect to their behavior on the C stratum. It is evident that they are the most simple functions within their class but they can have several C critical orbits. There is one perfect function with zero level of complexity (it belongs to class (1#2t#t)), two perfect functions characterized by the "rst level of complexity (they are from classes (3t#t) and (1#t#2t) correspondingly), two perfect functions characterized by the second level of complexity (from classes (2t#2t) and (1#3t)), one perfect function with the third level of complexity (class (t#3t)), and one perfect function with the fourth level of complexity (class (4t)).
References Avron, J., Raveh, A., Zur, B., 1988. Adiabatic quantum transport in multiply connected systems. Rev. Mod. Phys. 60, 873}915. Avron, J., Seiler, R., Simon, B., 1983. Homotopy and quantization in condensed matter physics. Phys. Rev. Lett. 51, 51}53. Aymar, M., 1984. Rydberg series of alkaline-earth atoms Ca through Ba. The interplay of laser spectroscopy and multichannel quantum defect analysis. Phys. Rep. 110, 163}200. Aymar, M., Greene, C., Luc-Koenig, E., 1996. Multichannel Rydberg spectroscopy of complex atoms. Rev. Mod. Phys. 68, 1015}1123. Bander, M., Itzykson, C., 1966. Group theory and the hydrogen atom, I, II. Rev. Mod. Phys. 38, 330}358. Basov, S., Pavlichenkov, I.M., 1994. Precessional motion around intermediate inertia axis: molecules and deformed nuclei. Phys. Rev. Lett. 72, 3953}3956. Beims, M., Alber, G., 1993. Bifurcations of electronic trajectories and dynamics of electronic Rydberg wave packets. Phys. Rev. A 48, 3123}3129. Bellissard, J., 1989. Gap labelling theorems for Schrodinger operators.. In: Waldschmidt, M., Moussa, P., Luck, J.-M., Itzykson, C. (Eds.), From Number Theory to Physics. Springer, Berlin, pp. 538}630. Bixon, M., Jortner, J., 1996. The electronic quasicontinuum. J. Phys. Chem. 100, 11914}11919. Boiteux, M., 1973. The three-dimensional hydrogen atom as a restricted four-dimensional harmonic oscillator. Physica 65, 381}395. Boiteux, M., 1982. Theory of nonbijective canonical transformations in mechanics: application to the Coulomb problem. J. Math. Phys. 23, 1311}1314. Boldyrev, A., Simons, J., 1992a. Rydberg bonding in (NH ) . J. Phys. Chem. 96, 8840}8843.
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
259
Boldyrev, A., Simons, J., 1992b. Theoretical search for large Rydberg molecules NH CH , NH (CH ) , NH(CH ) , N(CH ) . J. Chem. Phys. 97, 6621}6627. Bordas, C., Broyer, M., Chevaleyre, J., Labastie, P., Martin, S., 1985. Multichannel quantum defect theory analysis of the Rydberg spectrum of Na . J. Phys. (Paris) 46, 27}38. Bordas, C., Helm, H., 1991. Predissociation of H near its ionization threshold induced by very weak electric "elds. Phys. Rev. A 43, 3645}3652. Bordas, C., Helm, H., 1992. Stark e!ect in triatomic hydrogen Rydberg states. Phys. Rev. A 45, 387}402. Bordas, C., Lembo, L.J., Helm, H., 1991. Spectroscopy and multichannel quantum-defect theory analysis of the np Rydberg series of H . Phys. Rev. A 44, 1817}1827. Boris, S., Brandt, S., Dahmen, H., Stroh, T., Larsen, M.L., 1993. Quantum mechanical motion of Gaussian wave packets on elliptical and hyperbolic Kepler orbits in the Coulomb "eld. Phys. Rev. A 48, 2574}2585. Braun, P., 1993. Discrete semiclassical methods in the theory of Rydberg atoms in external "elds. Rev. Mod. Phys. 65, 115}161. Braun, P., Solov'ev, E.A., 1984. The Stark e!ect for the hydrogen atom in a magnetic "eld. Zh. Eksp. Teor. Fiz. 86, 68}83. Brodersen, S., Zhilinskii, B., 1995a. The rotational structure of the vibrational states and substates of symmetry E in CF . J. Mol. Spectrosc. 172, 303}318. Brodersen, S., Zhilinskii, B., 1995b. Transfer of clusters between the vibrational components of CF . J. Mol. Spectrosc. 169, 1}17. Brown, W.B., Steiner, E., 1966. On the electronic energy of a one-electron diatomic molecule near the united atom. J. Chem. Phys. 44, 3934}3940. Broyer, M., Delacretaz, G., Labastie, P., Whetten, R.L., Wolf, J., Woste, L., 1986. Spectroscopy of Na . Z. Phys. D 3, 131}136. Cacciani, P., Delsart, C., Luc-Koenig, E., Pinard, J., 1992. Laser spectroscopic studies of the Stark structure of the n"35 hydrogen diamagnetic manifold in parallel "elds. J. Phys. B 25, 1991}2000. Cacciani, P., Liberman, S., Luc-Koenig, E., Pinard, J., Thomas, C., 1989. Anticrossing e!ects in Rydberg states of lithium in the presence of parallel magnetic and electric "elds. Phys. Rev. A 40, 3026}3035. Cacciani, P., Luc-Koenig, E., Pinard, J., Thomas, C., Liberman, S., 1988. Rydberg atoms in parallel magnetic and electric "elds. J. Phys. B 21, 3473}3545. Cavalli, A., D'Ariano, G., Michel, L., 1985. Compact manifolds of coherent states invariant by semi simple Lie groups. Ann. Inst. Henri Poincare 44, 173}193. Chiu, Y.-N., 1986. Crystal-"eld theory for the Rydberg states of polyatomic molecules. Can. J. Phys. 64, 782}795. Clark, W., Greene, C., Miecznik, G., 1996. Anisotropic interaction potential between a Rydberg electron and an open-shell ion. Phys. Rev. A 53, 2248}2261. Coulson, C., Joseph, A., 1967. A constant of the motion for the two-centre Kepler problem. Int. J. Quant. Chem. 1, 337}347. Cushman, R.H., Bates, L., 1997. Global Aspects of Classical Integrable Systems. Birkhauser, Basel. Cushman, R., Sadovskii, D., 1999. Monodromy in perturbed Kepler systems: hydrogen atom in crossed "elds. Europhys. Lett 47, 1}7. Dabrowski, I., DiLonardo, G., Herzberg, G., Johns, J., Sadovskii, D., Vervloet, M., 1992. Spectra of the rare gas hydrides. IV. Three new bands of argon deuteride involving a low-lying `pa Rydberg state. J. Chem. Phys. 97, 7093}7110. Dabrowski, I., Sadovskii, D., 1994. Spectra of the rare gas hydrides. V. The np Rydberg series of KrD. Mol. Phys. 81, 291}326. Davies, P., Guest, M., Stickland, R.J., 1990. Infrared laser spectroscopy of H and D Rydberg states. I. Application of the polarization model. J. Chem. Phys. 93, 5408}5416. Delande, D., Gay, J., 1986. The hydrogen atom in a magnetic "eld. Spectrum from the Coulomb dynamical group approach. J. Phys. B 19, L173}L178. Delande, D., Gay, J., 1988. A new method for producting circular Rydberg states. Europhys. Lett. 5, 303}308. Delande, D., Gay, J., 1991. Supersymmetric factorization for Rydberg atoms in parallel electric and magnetic "elds. Phys. Rev. Lett. 66, 3237}3240. Delande, D., Taylor, K., Halley, M., van der Veldt, T., Vassen, W., Hogervorst, W., 1994. Scaled energy spectra of non-hydrogenic Rydberg atoms in a magnetic "eld. J. Phys. B 27, 2771}2784.
260
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
Delos, J., Knudson, S., Noid, D., 1983. Highly excited states of a hydrogen atom in a strong magnetic "eld. Phys. Rev. A 28, 7}21. Dietrich, H.-J., Muller-Dethlefs, K., Baranov, L., 1996. Fractional Stark state selective electric "eld ionization of very high-n Rydberg states of molecules. Phys. Rev. Lett. 76, 3530}3533. Dodhy, A., Ketterle, W., Messmer, H.-P., Walther, H., 1988. A zero Kelvin Rydberg spectrum of H . Chem. Phys. Lett. 151, 133}137. Duistermaat, J., 1980. On global action-angle coordinates. Commun. Pure Appl. Math. 33, 687}706. Dunn, M., Germann, T., Goodson, D., Traynor, C., Morgan III, J., Watson, D.K., Herschbach, D.R., 1994. A linear algebraic method for exact computation of the coe$cients of the 1/D expansion of the Schrodinger equation. J. Chem. Phys. 101, 5987}6004. Duval, P., 1964. Homographies, Quaternions and Rotations.. Oxford University, Oxford. Engle"eld, M., 1972. Group Theory and the Coulomb Problem. Wiley, New York. Fabre, C., Goy, P., Haroche, S., 1977. Millimetre resonance in Na Rydberg levels detected by "eld ionization quantum defects and Stark e!ect studies. J. Phys. B 10, L183}L189. Fabre, C., Kaluzny, Y., Calabrese, R., Jun, L., Goy, P., Haroche, S., 1984. Study of non-hydrogenic near degeneracies between m sublevels in the linear Stark e!ect of sodium Rydberg states. J. Phys. B 17, 3217}3228. Fano, U., Sidky, E., 1992. Semianalytic theory of Rydberg electron diamagnetism. Phys. Rev. A 45, 4776}4791. Farrelly, D., 1994. Motional Stark e!ect on Rydberg states in crossed electric and magnetic "elds. Phys. Lett. A 191, 265}274. Farrelly, D., Krantzman, K., 1991. Dynamic symmetry of the quadratic Zeeman e!ect in hydrogen: semiclassical quantization. Phys. Rev. A 43, 1666}1668. Farrelly, D., Milligan, J., 1992. Action-angle variables for the diamagnetic Kepler problem. Phys. Rev. A 45, 8277}8279. Farrelly, D., Uzer, T., Raines, P., Skelton, J.P., Milligan, J., 1992. Electronic structure of Rydberg atoms in parallel electric and magnetic "elds. Phys. Rev. A 45, 4738}4751. Flothmann, E., Main, J., Welge, K.H., 1994. The Kepler ellipses of the hydrogen atom in crossed electric and magnetic "elds. J. Phys. B 27, 2821}2833. Fock, V., 1936. Zur Theorie des Wassersto!atoms. Z. Phys. 98, 145}154. Fomenko, A., 1983. Di!erential Geometry and Topology. Additional Chapters. Moscow State University Press, Moscow. Frey, M., Dunning, F., Reinhold, C., Burgdorfer, J., 1996. Ionization of very-high-n Rydberg atoms by half-cycle pulses in the short pulse regime. Phys. Rev. A 53, R2929}R2932. Friedrich, H., Wintgen, D., 1989. The hydrogen atom in a uniform magnetic "eld } an example of chaos. Phys. Rep. 183, 37}79. Fujii, A., Morita, N., 1994. Laser spectroscopic investigation of high Rydberg states of NO: decay dynamics near the "rst ionization threshold. Laser Chem. 13, 259}271. Ganesan, K., Taylor, K., 1996. Rydberg states of the hydrogen atom near a metal surface. J. Phys. B 29, 1293}1306. Germann, T., Herschbach, D., Dunn, M., Watson, D., 1995. Circular Rydberg states of the hydrogen atom in a magnetic "eld. Phys. Rev. Lett. 74, 658}661. Goodson, D., Watson, D., 1993. Dimensional perturbation theory for excited states of two-electron atoms. Phys. Rev. A 48, 2668}2678. Gourlay, M., Uzer, T., Farrelly, D., 1993. Asymmetric-top description of Rydberg-electron dynamics in crossed external "elds. Phys. Rev. A 47, 3113}3117. Greene, C., Jungen, C., 1985. Molecular applications of quantum defect theory. Adv. At. Mol. Phys. 21, 51}121. Grozdanov, T., Solov'ev, E., 1995. Hidden crossings and the separation constant of a hydrogenlike atom in spheroidal coordinates. Phys. Rev. A 51, 2630}2633. Guillemin, V., Sternberg, S., 1990. Variations on a Theme by Kepler. American Mathematical Society, Providence, RI. Hamermesh, M., 1964. Group Theory and its Application to Physical Problems. Addison-Wesley, Reading, MA. Hasegawa, H., Robnik, M., Wunner, G., 1989. Classical and quantum chaos in the diamagnetic Kepler problem. Prog. Theor. Phys. 98 (Suppl.), 198}286. Helm, H., 1988. Measurement of the ionization potential of triatomic hydrogen. Phys. Rev. A 38, 3425}3429. Herrick, D., 1982. Symmetry of the quadratic Zeeman e!ect for hydrogen. Phys. Rev. A 26, 323}329.
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
261
Herrick, D., Sinanoglu, O., 1975. Comparison of doubly-excited helium energy levels, isoelectronic series, autoionization lifetimes, and group-theoretical con"guration } mixing predictions with large-con"guration } interaction calculations and experimental spectra. Phys. Rev. A 11, 97}110. Herzberg, G., 1981. Rydberg spectra of triatomic hydrogen and of the ammonium radical. Discuss. Faraday Soc. 71, 165}173. Herzberg, G., 1987. Rydberg molecules. Ann. Rev. Phys. Chem. 38, 27}56. Herzberg, G., Hougen, J.T., 1983. Spectra of the ammonium radical: the Schuster band of ND . J. Mol. Spectrosc. 97, 430}440. Herzberg, G., Jungen, C., 1972. Rydberg series and ionization potential of the H molecule. J. Mol. Spectrosc. 41, 425}486. Hiskes, J., 1996. Molecular Rydberg states in hydrogen negative ion discharge. Appl. Phys. Lett. 69, 755}757. Howard, J., Wilkerson, T., 1995. Problem of two "xed centers and a "nite dipole: a uni"ed treatment. Phys. Rev. A 52, 4471}4492. Hulet, R., Kleppner, D., 1983. Rydberg atoms in `circulara states. Phys. Rev. Lett. 51, 1430}1433. Hulthen, L., 1933. Uber die quantenmechanische Herleitung der Balmerterme. Z. Phys. 86, 21. Huppner, B., Main, J., Wunner, G., 1996. Nonhydrogenic Rydberg atoms in a magnetic "eld: a rigorous semiclassical approach. Phys. Rev. A 53, 744}759. Iken, M., Borondo, F., Benito, R., Uzer, T., 1994. Periodic-orbit spectroscopy of the hydrogen atom in parallel electric and magnetic "elds. Phys. Rev. A 49, 2734}2747. Iwai, T., 1981a. On a `conformala Kepler problem and its reduction. J. Math. Phys. 22, 1633}1639. Iwai, T., 1981b. On reduction of the four-dimensional harmonic oscillator. J. Math. Phys. 22, 1628}1632. Iwai, T., Uwano, Y., 1986. The four dimensional Kepler problem reduces to the three-dimensional Kepler problem with a centrifugal potential and Dirac's monopole "eld. Classical theory. J. Math. Phys. 27, 1523}1529. Jacobson, P., Labelle, R., Sturrus, W., Ward Jr., R., Lundeen, S., 1996. Optical spectroscopy of high ¸ n"10 Rydberg states of nitrogen. Phys. Rev. A 54, 314}319. Johnson, B., Hirschfelder, J., Yan, K., 1983. Interaction of atoms, molecules, and ions with constant electric and magnetic "elds. Rev. Mod. Phys. 55, 109}153. Jones, R., 1996. Creating and probing electronic wave packets using half-cycle pulses. Phys. Rev. Lett. 76, 3927}3930. Jungen, C., 1988. Quantum defect theory for molecules, in: Briggs, J., Kleinpoppen, H., Lutz, H. (Eds.), Fundamental Processes of Atomic Dynamics, Plenum, New York, pp. 79}103. Jungen, C., Dabrowski, I., Herzberg, G., Kendall, D., 1989. High orbital angular momentum states in molecular hydrogen and molecular deuterium. II. J. Chem. Phys. 91, 3926}3933. Jungen, C., Dabrowski, I., Herzberg, G., Vervloet, M., 1990. High orbital angular momentum states in molecular hydrogen and molecular deuterium. III. J. Chem. Phys. 93, 2289}2298. Kalinski, M., Eberly, J., 1996a. New states of hydrogen in a circularly polarized electromagnetic "eld. Phys. Rev. Lett. 77, 2420}2423. Kalinski, M., Eberly, J., 1996b. Trojan wave packets. Mathieu theory and generation from circular states. Phys. Rev. A 53, 1715}1724. Kazanskii, A., Ostrovskii, V., 1989. Equivalent operators for doubly excited states of the helium atom. J. Phys. B 22, 3829}3836. Kazanskii, A., Ostrovskii, V., 1990. Rydberg electron in a multipole "eld. Zh. Exp. Teor. Fiz. 97, 78}86. Kelleher, D., Saloman, E., 1987. Rydberg states with anisotropic ion cores: Stark e!ect. Phys. Rev. A 35, 3327}3338. Ketterle, W., Messner, H.-P., Walther, H., 1989. The l vibration of H> and autoionizing Rydberg states of H . Europhys. Lett. 8, 333}338. Kim, S., Mazur, E., 1995. Rydberg states of HD. Phys. Rev. A 52, 992}1004. King, H., Morokuma, K., 1979. Theory of the Rydberg spectrum of triatomic hydrogen. J. Chem. Phys. 71, 3213}3220. Konig, A., Neukammer, J., Vietzke, K., Grabka, M.K.H.-J., Hieronymus, H., Rinneberg, H., 1988. Transition from magnetic to electric-"eld F""B dominated spectra of barium at threshold. Phys. Rev. A 38, 547}550. Krantzman, K., Milligan, A., Farrelly, D., 1992. Semiclassical mechanics of the quadratic Zeeman e!ect. Phys. Rev. A 45, 3093}3103.
262
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
Kuwata, M., Harada, A., Hasegawa, H., 1990. Derivation and quantisation of Solov'ev's constant for the diamagnetic Kepler motion. J. Phys. A 23, 3227}3244. Labastie, P., Bordas, M., Tribollet, B., Broyer, M., 1984. Stroboscopic e!ect between electronic and nuclear motion in highly excited molecular Rydberg states. Phys. Rev. Lett. 52, 1681}1684. Lahaye, C., Hogervorst, W., 1989. Stark manifolds and electric-"eld-induced avoided level crossings in helium Rydberg states. Phys. Rev. A 39, 5658}5665. Laughlin, C., 1995. An asymptotic-potential method for high-l Rydberg levels of calcium. J. Phys. B 19, 2787}2801. Leboeuf, P., Kurchan, J., Feingold, M., Arovas, D., 1992. Topological aspects of quantum chaos. Chaos 2, 125}130. Lee, E., Farrelly, D., Uzer, T., 1994. Core induced stabilization and autoionization of molecular Rydberg state. Chem. Phys. Lett. 231, 241}245. Lembo, L., Bordas, C., Helm, H., 1990. f-electron Rydberg series of triatomic hydrogen. Phys. Rev. A 44, 1817}1827. Lembo, L., Helm, H., Huestis, D., 1989. Measurement of vibrational frequencies of the H molecule using two-step photoionization. J. Chem. Phys. 90, 5299}5308. Liu, W., Gu, S., Li, B., 1996. Fine structure in a strong magnetic "eld: Pashen-Back e!ect reconsidered in Rydberg atoms. Phys. Rev. A 53, 3044}3049. Lombardi, M., Labastie, P., Bordas, M.C., Broyer, M., 1988. Molecular Rydberg states: classical chaos and its correspondence in quantum mechanics. J. Chem. Phys. 89, 3479}3490. Lombardi, M., Seligman, T.H., 1993. Universal and nonuniversal statistical properties of levels and intensities for chaotic Rydberg molecules. Phys. Rev. A 47, 3571}3586. Mao, J.-M., Delos, J., 1992. Hamiltonian bifurcation theory of closed orbits in the diamagnetic Kepler problem. Phys. Rev. A 45, 1746}1761. Mayer, E., Grant, E.R., 1995. Double resonance spectroscopy of the high Rydberg states of HCO. I. A precise determination of the adiabatic ionization potential. J. Chem. Phys. 103, 10513}15519. McIntosh, H., Cisneros, A., 1970. Degeneracy in the presence of a magnetic monopole. J. Math. Phys. 11, 896}916. Merkt, F., Mackenzie, S., Softley, T., 1995. Rotational autoionization dynamics in high Rydberg states of nitrogen. J. Chem. Phys. 103, 4509}4518. Merkt, F., Xu, H., Zare, R.N., 1996. Preparation and characterization of long-lived molecular Rydberg states: application to HD. J. Chem. Phys. 104, 950}961. Michel, L., 1980. Symmetry defects and broken symmetry. Rev. Mod. Phys. 52, 617}650. Michels, H., Harris, F., 1963. Molecular orbital studies of excited states of HeH. J. Chem. Phys. 39, 1464}1469. Nekhoroshev, N., 1972. Action-angle variables and their generalization. Trans. Mosc. Math. Soc. 26, 181}198. Niu, Q., Thoulees, D.J., Wu, Y.-S., 1985. Quantized Hall conductance as a topological invariant. Phys. Rev. B 31, 3372}3377. Pan, S., Lu, K.T., 1988. Triatomic molecular H frame transformation theory of the l-uncoupling e!ect. Phys. Rev. A 37, 299}302. Pauli, W., 1926. Uber das Wassersto!spektrum vom Standpunkt der neuen Quantenmechanik. Z. Physik 36, 336}363. Pavlichenkov, I., Zhilinskii, B., 1988. Critical phenomena in rotational spectra. Ann. Phys. (N.Y.) 184, 1}32. Pavlov-Verevkin, V., Sadovskii, D., Zhilinskii, B., 1988. On the dynamical meaning of the diabolic points. Europhys. Lett. 6 (7), 573}578. Perelomov, A., 1986. Generalized Coherent States and their Applications. Springer, Berlin. Petfalakis, I., Theodorakopoulos, G., Child, M., 1995. Ab initio multichannel quantum defects for the A Rydberg states of H O. J. Phys. B 28, 5179}5192. Rabani, E., Levine, R., 1996. The dynamics of Rydberg states of molecules in the intermediate regime: the role of vibrations. J. Chem. Phys. 109, 1937}1952. Raithel, G., Fauth, M., 1995. Observation of high angular momentum states of Rydberg atoms in strong magnetic and weak crossed electric "elds. J. Phys. B 28, 1687}1706. Raithel, G., Fauth, M., Walther, H., 1991. Quasi-Landau resonances in the spectra of rubidium Rydberg atoms in crossed electric and magnetic "elds. Phys. Rev. A 44, 1898}1909. Raithel, G., Fauth, M., Walther, H., 1993a. Atoms in strong crossed electric and magnetic "elds: evidence for states with large electric-dipole moments. Phys. Rev. A 47, 419}440. Raithel, G., Held, H., Marmet, L., Walther, H., 1993b. Rubidium Rdberg atom in strong static "eld. J. Phys. B 27, 2849}2866.
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
263
Rau, A., Zhang, L., 1990. Mapping degenerate perturbation in atoms onto an asymmetric rotor. Phys. Rev. A 42, 6342}6353. Remacle, F., Levine, R., 1996a. Decay of high Rydberg states: a paradigm for the intramolecular dynamics in a congested bound level structure coupled to a continuum. J. Chem. Phys. 104, 1399}1414. Remacle, F., Levine, R., 1996b. Intramolecular dynamics in a congested set of states: a reduced description of the quantal long time evolution with application to high Rydberg states. Chem. Phys. Lett. 257, 111}118. Rinneberg, H., Neukammer, J., Jonsson, G.G., Hieronymus, H., Konig, A., Vietzke, K., 1985. High-n Rydberg atoms and external "elds. Phys. Rev. Lett. 55, 382}385. Robnik, M., Schrufer, E., 1985. Hydrogen atom in a strong magnetic "eld: calculation of the energy levels by quantising the normal form of the regularised Kepler Hamiltonian. J. Phys. A 18, L853}L859. Rothery, N., Storry, C., Hessels, E.A., 1995. Precision radio-frequency measurements of high-¸ Rydberg states of lithium. Phys. Rev. A 51, 2919}2925. Sadovskii, D., Shaw, J., Delos, D., 1995. Organization of sequences of bifurcations of periodic orbits. Phys. Rev. Lett. 75, 2120}2123. Sadovskii, D., Zhilinskii, B., 1988. Qualitative analysis of vibration}rotation hamiltonians for spherical top molecules. Mol. Phys. 65 (1), 109}128. Sadovskii, D., Zhilinskii, B., 1993a. Group-theoretical and topological analysis of localized rotation}vibration states. Phys. Rev. A 47, 2653}2671. Sadovskii, D., Zhilinskii, B., 1993b. Qualitative study of a model three-level Hamiltonian with S;(3) dynamical symmetry. Phys. Rev. A 48 (2), 1035}1044. Sadovskii, D., Zhilinskii, B., 1998. Tuning the hydrogen atom in crossed "eld between the Zeeman and Stark limit. Phys. Rev. A 57, 2867}2884. Sadovskii, D., Zhilinskii, B., 1999. Monodromy, diabolic points and angular momentum coupling. Phys. Lett. A 256, 235}244. Sadovskii, D., Zhilinskii, B., Michel, L., 1996. Collapse of the Zeeman structure of the hydrogen atom in an external electric "eld. Phys. Rev. A 53, 4064}4067. Sakurai, J., 1964. Invariance Principles and Elementary Particles. Princeton University Press, Princeton, NJ. Schwarz, M., Duchowicz, R., Demtroder, W., Jungen, C., 1988. Autoionizing Rydberg states of electronic-rotational interactions. J. Chem. Phys. 89, 5460}5472. Seipp, I., Taylor, K., Schweizer, W., 1996. Atomic resonances in parallel electric and magnetic "elds. J. Phys. B 29, 1}13. Shubnikov, A., 1951. Symmetry and Antisymmetry of Finite Figures. AN SSSR, Moscow. Shubnikov, A., Belov, N., 1964. Coloured Symmetry. Pergamon Press, London. Simon, B., 1983. Holonomy, the quantum adiabatic theorem, and Berry's phase. Phys. Rev. Lett. 51, 2167}2170. Sloane, N., 1977. Error correcting codes and invariant theory: new applications of a nineteenth-century technique. Am. Math. Mon. 84, 82}107. Solov'ev, E., 1981. Approximate integral of motion of H atom in a magnetic "eld. Pis'ma Zh. Eksp. Teor. Fiz. 34, 278}281. Solov'ev, E., 1982. Hydrogen atom in a weak magnetic "eld. Zh. Eksp. Teor. Fiz. 82, 1762}1771. Stebbings, R., Dunning, F., 1983. Rydberg States of Atoms and Molecules. Cambridge University Press, Cambridge, England. Stephens, J., Greene, C., 1995. Rydberg state dynamics of rotating vibrating H and the Jahn}Teller e!ect. J. Chem. Phys. 102, 1579}1591. Stiefel, E., Scheifele, G., 1971. Linear and Regular Celestial Mechanics. Springer, Berlin. Sturrus, W., Hessels, E., Arcuni, P., Lundeen, S.R., 1988. Laser spectroscopy of (l "0, R"1)10F and (l "0, R"1)10G states of H : A test of polarization model. Phys. Rev. A 38, 135}151. Tanner, G., Hansen, T., Main, J., 1996. The semiclassical resonance spectrum of hydrogen in a constant magnetic "eld. Nonlinearity 9, 1641}1670. Thoss, M., Domcke, W., 1995. A model for the quantum dynamics of Rydberg states of large molecules. Chem. Phys. Lett. 245, 364}370. Thurston, W., 1969. The Geometry and Topology of Three-Manifolds. Princeton University Press, Princeton, NJ. Uzer, T., 1990. Zeeman e!ect as an asymmetric top. Phys. Rev. A 42, 787}5790.
264
L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264
van der Veldt, T., Vassen, W., Hogervorst, W., 1993. Helium Rydberg states in parallel electric and magnetic "elds. J. Phys. B 26, 1945}1955. von Milczewski, J., Diercksen, G., Uzer, T., 1994. Classical dynamics of Rydberg electrons in crossed "elds: the structure of phase space and chaos-order alternations. Int. J. Bifurc. Chaos 4, 905}920. von Milczewski, J., Diercksen, G., Uzer, T., 1996. Computation of the Arnol'd web for the hydrogen atom in crossed electric and magnetic "elds. Phys. Rev. Lett. 76, 2890}2893. Wang, J., Boyd, R., 1994. Electronic structure and bonding in Rydberg molecules: NH , H O, and related molecules. Can. J. Phys. 72, 851}855. Watson, J., 1984. Assignment of the SchuH ler band of the ammonium radical to the 3pF !3sA electronic transition. J. Mol. Spectrosc. 107, 124}132. Watson, J., 1994. E!ects of a core electric dipole moment on Rydberg states. Mol. Phys. 81, 277}289. Weber, J., Ruf, M., Hotop, H., 1996. Rydberg electron transfer to C and C . Z. Phys. D 37, 351}357. Weyl, H., 1939. The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, NJ. Wigner, E., 1959. Group Theory and its Application to the Quantum Mechanics of Atomic Spectra (1st Edition 1931). Academic, New York. Wright, J., 1994. Theoretical search for the Rydberg dimer of H . Can. J. Phys. 72, 866}870. Zakrzewski, J., Dupret, K., Delande, D., 1995. Statistical properties of energy levels of chaotic systems: Wigner or non-Wigner? Phys. Rev. Lett. 74, 522}525. Zhang, W., Feng, D., Gilmore, R., 1990. Coherent states: theory and some applications. Rev. Mod. Phys. 62, 867}927. Zhilinskii, B., 1989a. Qualitative analysis of vibrational polyads: N mode case. Chem. Phys. 137, 1}13. Zhilinskii, B., 1989b. Theory of Complex Molecular Spectra. Moscow University Press, Moscow. Zhilinskii, B., 1996. Topological and symmetry features of intramolecular dynamics through high resolution molecular spectroscopy. Spectrochim. Acta A 52, 881}900. Zhilinskii, B., Brodersen, S., 1994. The symmetry of the vibrational components in ¹ molecules. J. Mol. Spectrosc. 163, B 326}338. Zhilinskii, B., Brodersen, S., Madsen, M., 1993. The pattern of clusters in isolated vibrational components of CF and the semiclassical model. J. Mol. Spectrosc. 160, 192}216. Zhilinskii, B., Pavlichenkov, I., 1987. Critical phenomena in the rotational spectra. Zh. Eks. Teor. Fiz. 92, 387}403. Zimmerman, M., Littman, M., Kash, M., Kleppner, D., 1979. Stark structure of the Rydberg states of alkali-metal atoms. Phys. Rev. A 20, 2251}2275.
Physics Reports 341 (2001) 265}336
Symmetry, invariants, topology. IV
Fundamental concepts for the study of crystal symmetry L. Michel* Institut des Hautes E! tudes Scientixques, 91440 Bures-sur-Yuette, France
Contents 1. Introduction 2. The Delone sets S(r, R) of points 3. Lattice symmetry: crystallographic systems, Bravais classes 3.1. General de"nitions. Intrinsic lattices 3.2. Euclidean geometry and Euclidean lattices 3.3. Two-dimensional crystallographic systems and Bravais classes 3.4. Three-dimensional Bravais crystallographic systems and classes 4. Geometric and arithmetic classes. Brillouin zone. Time reversal 4.1. Two maps on the set +AC, of arithmetic B classes 4.2. Geometric and arithmetic elements and classes in dimension 2 4.3. Geometric and arithmetic elements and classes in dimension 3 4.4. Brillouin zone, its high symmetry points 4.5. Time reversal T 5. VoronomK cells and Brillouin cells 5.1. VoronomK cells, their faces and corona vectors
267 269 272 272 274 276 279 281 281 282 284 290 296 296 296
5.2. Delone cells. Primitive lattices 5.3. The primitive principal VoronomK cells of type I 5.4. VoronomK cells for d"2, 3 5.5. High-symmetry points of the Brillouin cells 6. The positions and nature of extrema of invariant functions on the Brillouin zone 7. Classi"cation of space groups from their nonsymmorphic elements 7.1. Action of the Euclidean group on its space 7.2. Space group stabilizers and their strata ("Wycko! positions) 7.3. Non-symmorphic elements of space groups. Their classi"cation 7.4. Some statistics on space groups 8. The unirreps of G and G and their I corepresentations with T 8.1. The unitary irreducible representations of G I 8.2. The irreducible corepresentations of G[ I 8.3. The irreducible representations of a space group G References
* Deceased 30 December 1999. 0370-1573/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 0 0 ) 0 0 0 9 1 - 0
300 303 305 309 312 316 316 317 319 323 326 326 331 332 335
266
L. Michel / Physics Reports 341 (2001) 265}336
Abstract Fundamental concepts for the study of crystal symmetry are systematically introduced on the basis of group action analysis. 2001 Elsevier Science B.V. All rights reserved. PACS: 61.50.Ah Keywords: Periodic crystals; Arithmetic classes; Invariant functions; Critical orbits
L. Michel / Physics Reports 341 (2001) 265}336
267
1. Introduction In solids, every atom vibrates around an average position; the amplitude of the vibrations is related to the temperature. To a good approximation these average positions are "xed. We denote by S the set of points in the d-dimensional Euclidean space E which represent these positions. In B condensed matter the interatomic distances are of the same order of magnitude as the size of the atoms, so we cannot neglect their sizes; it is therefore natural to assume that there is a minimum distance d"2r between any pair of points of S. In the study of solids in bulk, surface e!ects are neglected; the natural idealization is to suppress the surface by the assumption that the set S extends to in"nity. However this is not enough since it leaves the possibility to have internal holes with a well-de"ned surface; there must be some upper bound for the distance between `neighboringa points which is of the order of magnitude of the atomic size. This can be obtained by the following condition: R is the upper bound of the radii of the `empty spheresa, those which do not contain points of S in their interior. The mathematical theory of these (r, R) sets has been elaborated by Delone and its school, mainly in Soviet Union (Delaunay, 1932a, b; Delone et al., 1974). Here we call them Delone sets. Each such set de"nes two dual orthogonal tessellations of the Euclidean space. We summarize these mathematical results in Section 2. Of course it can be skipped since we are interested only in periodic crystals. However, the discovery of aperiodic crystals (Shechtman et al., 1984) requires a broader mathematical frame for a uni"ed study of crystals. This frame seems to be provided by the theory of Delone sets; we wanted to mention it to the reader and show how classical crystallography is the study of a very rich particular case. The other sections of this chapter study periodic crystals, i.e. those whose symmetry contains a lattice of translations ¸. These lattices and their symmetry are studied in Section 3. It will lead to the de"nition of basic concepts: the crystallographic systems and the Bravais classes; they are both examples of strata. It will also lead to the concept of Brillouin zone, fundamental for the study of many types of experiments. Translation lattices ¸ de"ne pavings of space by a `fundamental domaina repeated by the translations. Among the possible domains, one is intrinsic (i.e. basis independent): it is studied later, in Section 5. The crystal structure is de"ned by the positions of the atoms in a fundamental domain. Only the crystals of chemical elements can have a unique atom in a fundamental domain. Such type of space groups are semi-direct products of Bravais groups of lattices by the translation lattice itself. There are 14 of them among the 230 space groups of dimension three; every other space group is one of their subgroups. The list of the 230 space groups has been determined in 1892 by a mineralogist, Fedorov (1885), and a mathematician, Schoen#ies (1891), working independently. Presently the `International Tables of Crystallographya (ITC, 1996) (abbreviated hereafter as ITC) is the standard reference for the data on the structure of these groups. They are labelled in ITC by a symbol (using at most seven letters or digits and two typographic characters M , / ) which gives the
As usual, when statements do not depend on the value of the dimension, we call it d. That is not the general case. Diamond and graphite, two crystallographic states of carbon, have respectively 2 and 6 atoms per fundamental domain. By comparing their results before publication, each one made very few corrections to the results of the other one.
268
L. Michel / Physics Reports 341 (2001) 265}336
structure of each space group! This remarkable notation system is universally used by crystallographers but not yet by some physicists. The translation lattice ¸ is an in"nite Abelian group which is an invariant subgroup of any space group G. The quotient group P"G/¸ is called the point group; it is a "nite group. Many physical phenomena depend on a less-re"ned classi"cation of the symmetry of crystal. For example, most of their macroscopic properties depend only on the 32 geometrical classes, i.e. the conjugacy classes in O of the point groups P. These classes have been listed by Frankenheim (1826a, b, 1842) and Hessel (1830) just before the introduction of the word group by Galois (1846). Mechanical, optical, electric and magnetic properties of the crystals are described by tensors invariant by P. These tensors are given by the invariants of the `vectora representation of P; the rings of these invariants have been given in terms of 3}6 generators in Chapter I, Tables 4, 5. Some microscopic properties of crystals may depend only on P, but in a more re"ned manner which takes into account the action of P on the translation lattice ¸; this action can be written in terms of matrices whose elements are integers. Such phenomena depend only on the 73 arithmetic classes, i.e. the distinct conjugacy classes of the point groups as subgroups of G¸(3, Z); they correspond to 73 di!erent actions of the point groups on the Brillouin zone. Arithmetic classes are studied in Section 4. Their ring of invariants on the Brillouin zone is the main subject of Chapter V. In Section 4 we also show that for stationary states (i.e. invariant by time translations) of these microscopic phenomena, time reversal T reduces the number of relevant arithmetic classes to 24. Section 5 gives some insight into the structure of the VoronomK cell in d-dimension before describing the two and "ve types occurring for d"2, 3, respectively. In the reciprocal space, this is the Brillouin cell: it is a geometrical realization of the Brillouin zone. We transpose in this geometrical description the results obtained in the preceeding section on the high symmetry points of the Brillouin zone. Section 6 gives in a one page Table 7 a very useful application of the general methods given in Chapter I, to these 24 classes. Indeed Table 7 gives the positions on BZ of the extrema common to all invariant continuous functions on the Brillouin zone and indicates the nature of these extrema for `Morse-simple functionsa (the function with the minimum possible number of extrema); these functions occur most often in the simple physical models. Of course many physical properties of crystals depend on the more re"ned classi"cation of their symmetry by the 230 space groups. Is it possible to predict for these types of properties some general results by brute force, i.e. verifying them for the 230 space groups! Another method, (which we prefer for the sake of culture) is to prove such result as a mathematical theorem, whose proof is based on properties speci"c to space groups; more often the proof will have to consider several cases, each containing only a subclass of space groups. This is illustrated in Chapter VI for the study of the symmetry and topology of the electron energy bands. In English one sometimes uses the expression `factora group. This is misleading because P is generally not a subgroup of G. Moreover quotient structure are obtained by a very general construction in mathematics that we use here for the particular case of groups. The linear representation of P on the physical space. Of course this very fundamental concept is de"ned in ITC, p. 719. It is strangely absent from most physics text books. We will quote some classic papers where it is used. These recents results have been published (Michel, 1996; Michel, 1997b) but not in a physical journal.
L. Michel / Physics Reports 341 (2001) 265}336
269
Section 7 gives an elementary and original approach of the important concepts relevant to the rich diversity of space groups. This section will also help the understanding and use of the international tables (ITC, 1996). Section 8 recalls the structure of the unirreps of the space groups G; they are built by induction from the `alloweda unirreps of the stabilizers G of the action of G on I the Brillouin zone. The images of these allowed G unirreps are also those of the group P (k) I X (Herring, 1942), whose structure is much simpler. There are many introductory books on crystallography, e.g. Buerger (1956); shorter introductions are also included in most solid state books and in some books of applications of group theory to physics. None overlaps very much with this chapter. Our aim is to emphasize fundamental concepts used little or not at all in these introductions, but so necessary for physics of crystals; these concepts are introduced and explained in a direct and completly original method. Moreover this chapter prepares the reader to use e!ectively ITC; using tables can fully satisfy technicians (they will soon be replaced by softwares). But would physicists accept to use tables of trigonometric functions without knowing their analytic properties and the geometry; Chapter IV tries to give them the deeper knowledge they need on crystallography. Most of the concepts studied in this chapter are necessary for understanding Chapters V and VI which contain essentially original results; but the reader can skip details and come back to them when they are referred to in the last two chapters or when he needs them later in his own work!
2. The Delone sets S(r, R) of points De5nition. A Delone set of points S(r, R) in a d-dimensional Euclidean space E is de"ned by two B real numbers: 2r the lower bound of the distance between two points of the set and R the upper bound of the radius of the spherical balls of E which contain no point of S in their interior. B The existence of the bound R implies that the set is in"nite. The existence of the bound 2r implies that the number of points in any bounded domain is "nite (otherwise there would be an accumulation point). From now on we use the shorthand `balla for a spherical ball; its surface is a (d!1)-dimensional sphere. It is well known that in E one needs d#1 points in general B position for determining a (d!1)-sphere.
The overlap is much greater with Burckhardt (1966), Schwarzenberger (1980), Engel (1998), and the elegant and more elementary Senechal (1991). This method does not require more sophisticated mathematics as, for instance, cohomology theory, which is of course very useful for a deeper knowledge. Strangely, in the action of the space groups on the Euclidean space, these tables give only the geometric class of the stabilizer and not their arithmetic class. However they give very useful information on the subgroups of space groups. They do not study the Brillouin zone. There exists for each dimension d a lower bound for R/r; e.g. it is 1 in the trivial case d"1. For d"2 it is probably obtained by the two-dimensional hexagonal lattice R/r"(2/(3)"1.133972 I do not know what is known in higher dimensions. There are many results for the special case of lattices; many references are in Conway and Sloane (1988). That means that any subset of k#1 points, 1(k4d, is not contained in a linear manifold of dimension (k. We use the shorthand `k-planea for a k-dimensional linear submanifold of E . We also recall that k#1 points in general B position are the vertices of a simplex.
270
L. Michel / Physics Reports 341 (2001) 265}336
De5nition. A hole of S is a d-dimensional ball which contains no point of S in its interior and has on its surface at least d#1 points; among them there must be a subset of d#1 points in general position. Of course, in a Delone set S, any subset of d#1 points in general position does not determine a hole (the sphere it de"nes may contain set points in its interior). Let us consider the complete set of holes of S. We denote by V the set of their centers (two centers cannot coincide) and for a given v3V, we denote by D the convex hull of the set points on the surface of the T corresponding sphere; this d-dimensional convex polyhedron is called a Delone cell. Beware that D may not contain v (this does not happen for d"2 but it is easy to make counter-examples in T dimension 3). Proposition 2a. In the Euclidean space E the Delone cells of a Delone set S form a tesselation ("a B facet-to-facet paving) denoted by D . 1 Indeed, consider a facet (i.e. a (d!1)-face) U of D . We denote by HU the (d!1)-plane T containing U. It determines two half-spaces. We denote by H> U the one containing D and by T H\ U the other one. We denote by C the (d!2)-dimensional sphere intersection of the (d!1)dimensional sphere of the hole with HU . The set of (d!1)-spheres containing C is called a linear sheave of spheres. The centers of its spheres are on the straight line K perpendicular to HU at the center of C. Let us consider the continuous family of spheres S of the sheave whose center x3K V moves continuously from v in the direction of H\ U . The vertices of the D 's in the interior of H> U do T not belong to these spheres; so we can "nd a new hole whose sphere center is denoted by v . Then D 5D "U. Continuing this construction for all facets of all Delone cells, we obtain the full T TY tessellation. De5nition. A Delone set is primitive when all its Delone cells are simplexes (" convex hull of d#1 points in general position). The set of balls of radius r centered at the points of the Delone set S is called the sphere packing on S. The only possible intersection between these balls are contact points between two tangent balls at the middle of the segment formed by two set points at the minimum distance 2r. We denote by d(p, q) the distance between two points of E . B De5nition. The Voronon( cell D (p) at the point p3S is the set of points of E whose distance to p is 1 B smaller or equal to the distance to any other point of S: p3S, D (p)"+x3E ; ∀q3S, d(p, x)4d(q, x), . 1 B
(1)
Proposition 2b. The Voronon( cells of a Delone set are convex polyhedrons; each D (p) is contained in 1 the ball, centered in p, of radius R. The bisector plane of the segment pq de"nes two-half spaces and D (p) is in the one containing p. 1 So D (p) is the intersection of all these half-spaces made for all set points qOp. As an intersection 1 of convex domain, it is convex. It contains the sphere of radius r centered at p. We now show that it is contained in the sphere of radius R centered at p. Indeed, assume that x3D (p) is outside this 1
L. Michel / Physics Reports 341 (2001) 265}336
271
sphere (i.e. xp'H); since, by de"nition of D (p), x is nearer to p than any other points of S, the 1 spherical ball of radius R centered at x does not contain points of S (inside or on its surface). It is absurd to "nd a hole at x of radius 'R. We have already remarked that the number of points of S inside any bounded domain is "nite; that is the case of the sphere of radius 2R centered at p: so the number of points q whose bisector pq intervenes in the constructions of D (p) is "nite. This ends 1 the proof of Proposition 2b. Corollary 2b. The Voronon( cells of a Delone set form a tessellation of the Euclidean space. The intersection of two VoronomK cells D (p)5D (p ) is in the bisector plane of pp . Either it is 1 1 empty or it is a common facet. Indeed, any point of E is either in the interior of a VoronomK cell or on B the boundary of k52 VoronomK cells; when k"2 it is the common face of the two cells. Let us consider a k -dimensional face U of the Delone cell D , with 04k 4d (when k "d, IY T U is the Delone cell). Let F , k#k "d be the k-plane containing v and perpendicular to the IY I supporting k -plane of U . The points of F are equidistant of the vertices of U , so F contains the IY I IY I common k-dimensional face F of the VoronomK cells at the vertices of U . In particular, when I IY k "d, then v is the vertex common to the VoronomK cells de"ned by the vertices of D (which form T a hole). Remark that "U ", the number of vertices of U satisfy "U "5k #1; the equality occurs for IY IY IY all k when the Delone set is primitive. Conversely, by construction of the VoronomK tessellation, the points of F , the k-plane supporting a k-face F , are equidistant from at least d#1!k points of I I S and the convex hull of these points (which form a face U of D ) is orthogonal to F . IY 1 I To summarize: the vertices of the Delone cells are points of S, i.e. constructing centers of the VoronomK cells of S and their vertices are the centers of the spheres circumscribed to the Delone cells; more generally in the tessellations D and D there is an orthogonal duality between their 1 1 faces. These two tessellations are said to be dual orthogonal. We leave to the reader the straightforward proof of Proposition 2c. If the holes of a Delone set S(2r, R) have the same radius (which has to be R), the Delone tessellation of S is the Voronon( tessellation of V the set of centers of the holes"the set of vertices of the Voronon( cells of S. As we shall see in Section 4 this is the case of all two-dimensional lattices and of the three-dimensional lattices belonging to 7 (out of 14) Bravais classes. It seems that the concept of a Delone set is a good tool for unifying the study of periodic and aperiodic crystals (see e.g. Dolbilin et al., 1998). The local symmetry can be studied with the following tools: De5nition. The star, respectively (l-star) of a point p3S is the set of straight line segments joining p to all points of the Delone sets, respectively (all points whose distance to p is 4l). Stars (l-stars) are congruent when they can be transformed into each other by an Euclidean transformation. Let us denote by St(S) (St(l,S)) the set of congruence classes of the stars (l-stars) of S. When St(l,S) is "nite (as is the case for Penrose tilings) there is some order in S.
272
L. Michel / Physics Reports 341 (2001) 265}336
The crystal structure is due to some order correlation which does not vanish at in"nite distance. Delone sets with their classes of stars St(S) are a possible frame for a comprehensive study of crystals and their symmetry when St(l,S) is "nite. When all stars form one class, S is called a regular Delone set. Theorem 2a. A regular Delone set is a periodic crystal. Equivalently: S is the union of orbits of a translation lattice. The statement equivalent to this theorem was proven by Schoen#ies (1891) in dimension 3 and extended by Bieberbach (1910, 1912) to an arbitrary dimension. It proves that periodic crystals are a particular (and very important) case of Delone sets. There is also an important mathematics literature on classi"cation of tessellations, also called tilings with, eventually, matching rules, and their possible local or global symmetries: see e.g. Moody (1995). The other sections deal only with periodic crystals.
3. Lattice symmetry: crystallographic systems, Bravais classes 3.1. General dexnitions. Intrinsic lattices We recall that an orthogonal d-dimensional vector space E contains the Abelian group RB of B addition of its vectors and an orthogonal scalar product that we denote by (x, y); the norm of a vector is N(*)"(*, *). De5nition. A lattice in E is the subgroup of RB generated by a basis +b , of E ; i.e. the vectors B H B of ¸ are those with integer coordinates on +b ,. Hence the group isomorphism ¸&ZB. Let +b , H H be another basis of ¸; the new basis vectors must have integer coordinates in the old basis and vice versa, so b " m b , m3G¸(d, Z) , (2) G GH H G indeed the matrix m and its inverse must have integer elements, so their determinants have same GH value which is $1. Eq. (2) shows that the set of bases of a lattice ¸ is a principal orbit of G¸(d, Z). In order to deal later with the orthogonal group, we choose an orthonormal basis (e , e )"d of G H GH E . For any basis +b , of E the matrix bI of elements bI "(b , e ) is the matrix of the d components B G B GH G H of its d vectors. It is invertible since the basis vectors are linearly independent, so bI 3G¸(d, R). Conversely, given a matrix of G¸(d, R) the elements of its d lines can be considered as the d components of the d vectors of a basis of E . So the correspondance +b ,bI is bijective and it B G gives a bijective map between the set B of bases of E and the set of elements of the group G¸(d, R) B B (notice that this identi"cation gives to B a structure of a manifold). Using Eq. (2), we can identify B the set of bases of ¸ as the right coset G¸(d, Z)bI of G¸(d, Z) in G¸(d, R) and we can identify L , the B
L. Michel / Physics Reports 341 (2001) 265}336
273
set of d-dimensional lattices, as the set of right cosets of G¸(d, Z) in G¸(d, R). In Chapter I Section 3.1, we have shown that this set of cosets can also be interpreted as the orbit space of the action of G¸(d, Z) on B "G¸(d, R), i.e. B L "B " G¸(d, Z)"G¸(d, Z) : G¸(d, R), (3) B B where, as sets, B "G¸(d, R). The orthogonal group O is the group of automorphisms of E . The B B B elements of O transform the lattice ¸ in a family of lattices which are simply di!erent orientations B in E of the same `intrinsica lattice ¸. For instance, given r3O , if bI is the matrix representing B B a generating basis of the lattice ¸, bI r\"bI r? represents a basis generating r ) ¸, the lattice ¸ transformed by r. An intrinsic lattice is de"ned by the Gram matrix of its generating basis; its elements are (4) (b , b )"(bI bI ? ) . GH G H We can also identify an intrinsic lattice of ¸ to the orbit O ) ¸. So ¸G , the set of intrinsic lattices, is B B the set of such orbits. Since the action of r3O is bI C bI r\, from Eq. (3) we can also identify LG with B B the double cosets of G¸(d, Z) and O in G¸(d, R). To summarize B LG "L " O "G¸(d, Z) : G¸(d, R) : O , +BCS, "L ""O . (5) B B B B B B B Bravais (1850) did consider the types of intrinsic lattices and the strata of O on L are called B B Bravais crystallographic systems in ITC p. 722. The stabilizer (O ) is the point group of ¸ and its conjugacy class in O is called the holohedry of B* B ¸ that we shall denote by P (or P ). The orthogonal transformation r is a symmetry of ¸ if bI r\ is * another basis of ¸; then, from Eq. (2), there exists m3G¸(d, Z), mbI "bI r\0 m"bI r\bI \0 r"bI \m\bI .
(6)
Equivalently (we use the lower index b to remind the dependence on the basis) PX "bI P bI \LG¸(d, Z), P "O 5bI \G¸(d, Z)bI LO . (7) @ @ @ B B As the intersection of a compact and a discrete subgroups of G¸(d, R), the point group P is a "nite @ group. PX is called the Bravais group of ¸ and its conjugacy class PX in G¸(d, Z) is the Bravais class @ of ¸. In other words one can say that the Bravais group PX gives the action of the point group on the lattice while the holohedry gives simply the action of the point group on the space. This (Y +BCS, . suggests that there is a natural surjective map +BC, P B B We also remark that !I , the matrix representing the symmetry through the origin, is an B element of P and PX (in every basis); indeed if l3¸, then !l3¸. The matrix q "bI bI ? de"ned in Eq. (4) is a symmetric positive matrix. We denoted it by @ q because it represents a positive quadratic form q (x)" x (q ) x . The set Q of d;d symmetric @ GH G @ GH H B real matrices ("d variable quadratic forms) forms a N"d(d#1)/2-dimensional orthogonal vector space E whose scalar product is (q , q )"tr q q . The positive quadratic forms form , The concept of strata is not explicitly mentioned in ITC.
274
L. Michel / Physics Reports 341 (2001) 265}336
a convex cone that we denote by C (Q ). From (2) we can de"ne the action of G¸(d, Z) on Q : > B B K mqm? . m3G¸(d, Z), q3Q , qP B
(8)
This action transforms C (Q ) into itself. So the de"nitions of intrinsic lattices and Bravais > B classes can be reformulated as orbit and stratum spaces LG "C (Q ) " G¸(d, Z), +BC, "C (Q ) "" G¸(d, Z) . (9) B > B B > B As we have seen, the Bravais groups are "nite. Since they are the stabilizers of the action of G¸(d, Z) on C (Q ), the theorem of Palais (1961) proves the `gooda strati"cation of this action: so there is > B a unique minimal symmetry stratum (the group Z (!I )), it is open dense and its boundary is the B set of strata with larger symmetry. A lattice is an integral lattice when its quadratic form has integer coe$cients and they are relatively prime; as Eq. (8) shows, this property is basis independent. When the basis +b , generates G ¸, for any positive real number j we denote by j¸ the lattice generated by the basis of vectors +jb ,; G it is enough to consider j'0 since !I3P. All lattices j¸ have the same symmetry. De5nition. The dual lattice ¸H of ¸ is the set of vectors whose scalar product with every vector of ¸ is an integer. To prove that this set of vectors is a lattice, let +b , a basis generating ¸; we de"ne H the dual basis +bH, by G (bH, b )"d ; x" mbH, y" g b , then (x, y)" m g (10) G H GH G H H G G G H G with this simple form of the scalar product one veri"es that the dual basis generates the dual lattice. It is also straightforward to verify that if the Bravais group of ¸ is represented by the matrices g's, in the dual basis, PX H is represented by the matrices * ? g3PX g " (g )\"(g\)?3PX H . (11) * * These two representations of the holohedry P are equivalent on the real, but they may be inequivalent on Z. The de"nition of dual basis in Eq. (10) shows that (¸H)H"¸. Note that an integral lattice is a sublattice of its dual. 3.2. Euclidean geometry and Euclidean lattices The Euclidean group is the automorphism group of the Euclidean space E ; it is the semi-direct B product Eu "RB)O where RB is the invariant subgroup of translations. Instead of y"t ) x, we B B denote by y"t#x the transformed one of the point x3E by the translation t. This formalism has B been established in the last quarter of the XIXth century: for instance we can also write Beware that this action (8) preserves det q but not the scalar product tr q q . Also explained in Chapter I at the end of Section 4, before Section 4.1.
L. Michel / Physics Reports 341 (2001) 265}336
275
t"xy"y!x. For the mid-point o of the segment ab in E we can write o"a#ab"(a#b); B we denote by p the symmetry through the point o, so M b"p a"a!2oa"2o!a 0 o"(a#b) . (12) M More generally a point of E can be written as a linear combination of points with the sum of B coe$cients "1. Let x be a set of n points indexed by the values of k, 14k4n; with real I coe$cients j : I j x with j "1 is the linear manifold de"ned by the x 's, I I I I I I j x with j 50, j "1 is the convex hull of the x 's, I I I I I I I (1/n) x is the barycenter of the x 's. I I I To write explicitly the group law of the Euclidean group Eu we choose an origin o on the B Euclidean space E . Then every element of Eu can be written as the product of "rst, an orthogonal B B transformation A3O and second, of a translation t3RB. We write such an element +t, A,. Its B action on the point x3E and its group law are B +t, A, ) x"Ax#t; +s, A,+t, B,"+s#At, AB, , +s, A,\"+!A\s, A\, .
(13)
Since a lattice ¸ is a subgroup of the translation group RB, De5nition. An Euclidean lattice ¸M LE is an orbit ¸#x of a d-dimensional lattice of translaB tions. As is well known, by choosing a point o3E we have an identi"cation xox of this B Euclidean space with a vector space of origin o. Given an Euclidean lattice ¸M , by choosing any point o3¸M , we obtain its lattice (of translations). The symmetry group G of ¸M is the stabilizer of * ¸M in Eu : B G "¸ ) PXLEu , elements: +l, A,, l3¸, A3PX . (14) * B It is easy to verify that it is the Bravais group of ¸ which enters the semi-direct product (indeed to build it one must know the action of P on ¸); hence we obtain a new de"nition of the Bravais class from the structure of the stabilizer (Eu ) of ¸ in the Euclidean group B* PX "(Eu ) /¸ . (15) * B* Let l, l , l 3¸; We know that !I 3P and from Eqs. (14) and (13) we obtain B +l #l ,!I, ) l"l #l !l . (16) The comparison with Eq. (12) shows that the element of G in Eq. (16) is the symmetry through the * middle of the segment l l ; if we consider also the case l "l we have proven: In the solid-state literature, as here, the usual convention for product of operators or group elements is adopted: g g means "rst g then g . However, it seems to be a tradition to write the elements of the Euclidean group +A, t,, with the operation A performed before the translation t. Here we cannot adopt this incoherence. The original paper of Bravais (1850) on Euclidean lattices is very interesting to read.
276
L. Michel / Physics Reports 341 (2001) 265}336
Proposition 3a. An Euclidean lattice ¸M has an inxnity of symmetry centers: they are its points and the middle of the segments formed by any pair of its points. With the notation introduced between Eqs. (9) and (10). Corollary 3a. The symmetry centers of ¸M form an Euclidean lattice that we denote ¸M . We are interested by intrinsic Euclidean lattices: they are de"ned modulo their position in E ; so B the set of intrinsic lattices can be identi"ed with the set +Eu ) ¸M , of Euclidean lattice orbits. B With the choice of a basis one easily de"nes a fundamental domain of ¸M , that is a domain in E which contains one, and one only, point of each orbit of the translation lattice ¸. Let us give B examples of fundamental domains, de"ning them by the coordinates x of their points x: G (17) P "+x; 04x (1,, PM "P !w; w"([ ]B) , @ G @ @ the last expression means that every coordinate of w is , so PM is the same parallelepiped as P , but @ @ it is centered at the origin. Remark that the closure of P (replace (1 by 41 in the de"nition) @ contains 2B points of the Euclidean lattice containing o. We denote by ¸ the set of lattice vectors of norm a. For the shortest vectors of ¸ we also write ? S"¸ . The minimum distance for two points of the Euclidean lattice ¸M is d"(s. As we have seen Q for Delone sets, the number of points of ¸M in any bounded domain is "nite (otherwise there would exist an accumulation point). So all ¸ 's are "nite sets. The set N of values of the norm of the ? vectors of ¸ is countable and discrete; it contains 0, the other values are positive and unbounded and s is the smallest positive value of N. Then ¸" N ¸ . ?Z ? 3.3. Two-dimensional crystallographic systems and Bravais classes From Eq. (7) we know that the point group is a subgroup of O and is conjugate to the Bravais B group PX, a subgroup of G¸(n, Z). This implies the necessary condition that the trace of its matrices must be integers; this condition is su$cient for d"2. As we saw in Chapter I, Section 3, Eqs. (7)}(8), the matrices of O represent either rotations or re#ections and their respective traces are 2 cos h and 0. So the rotations must be of order 1, 2, 3, 4, 6. That gives ten geometric classes: 1"c , 2"c , 3"c , 4"c , 6"c , m"c , 2mm"c , 3m"c , 4mm"c , 6mm"c . Q T T T T As we remark after Eq. (7), !I is an element of every holohedry P. For d"2 this matrix represents also the rotation by p. The groups of six geometric classes contain it. It is easy to show that 4"c and 6"c cannot be the full symmetry group of a lattice. Let us prove it for 4"c . Let S be the set of short vectors. They form one orbit of four vectors $s , (s , s )"sd , i"1, 2; indeed G G H GH if there were another c orbit (of the same norm) we can verify that these two orbits of vectors would generate a dense subgroup of R. We prove that S generates the full lattice. Assume that it is not correct, i.e. S generates only a sublattice ¸ whose vectors are those with integer coordinates in the basis +s ,; so there should be in ¸, a vector *" j s ,¸ whose coordinates j are not both G G G G G integers. By translation in ¸ we transform it to a vector w of coordinates l satisfying 04l 41; G G
L. Michel / Physics Reports 341 (2001) 265}336
277
Fig. 1. Euclidean lattices illustrating the two orthorhombic Bravais classes. They have exactly the same symmetry re#ection axes; the intersections of these symmetry axes are symmetry centers for both lattices. c2mm is obtained from p2mm by adding as lattice points the symmetry centers of the rectangles (" body centring). Then new symmetry centers appear for c2mm: they are the small dots. The Bravais groups of these two lattices are conjugated in O but not in G¸(2, Z). Their space groups, also denoted by p2mm, c2mm, are not isomorphic.
then the norm of at least two of the four vectors w, w!s , w!s !s is smaller than the norm of G the s : that is absurd. Since S has the symmetry group 4mm"c , that is also the holohedry of the G T lattice it generates. By a similar proof we obtain that 6"c is not a holohedry. So there exists four crystallographic systems in dimension 2; we list their names and their symmetry group +CS, "diclinic: 2"c , orthorhombic: 2mm"c , T quadratic: 4mm"c , hexagonal: 6mm"c . T T The group 2"c is the center of G¸(2, Z) so the diclinic system has a unique Bravais class. We prove that the orthorhombic system has two Bravais classes. Fig. 1 shows two Euclidean lattices which have the same holohedry but belong to two di!erent Bravais classes p2mm, c2mm (p is for principal and c for centered). These two Euclidean lattices have exactly the same symmetry re#ection axes m forming two families of parallel axes in orthogonal directions and whose intersections are symmetry centers. Following the de"nition of P in Eq. (17), p2mm has rectangles as fundamental domains and its @ points are the vertices of the rectangles. c2mm is obtained from p2mm by adding the rectangle centers as lattice points; crystallographers say `c2mm is obtained by centering p2mma. So p2mm is a sublattice of c2mm and, as Abelian groups, their quotient is Z . From Corollary 3a, c2mm has also a family of isolated symmetry centers obtained by centring the lattice ¸ of the symmetry centers of ¸"p2mm. So the symmetry groups (" stabilizers in Eu ) of the two lattices are di!erent; their quotient is also Z . The Bravais groups that we obtain from Eq. (15) are di!erent as we will show. The fundamental domain P of c2mm is a rhomb. Let +b ,, (b , b )"b, (b , b )O0 be the @ G G G corresponding basis of c2mm; then a "b $b is an (orthogonal) basis of p2mm. The two !
278
L. Michel / Physics Reports 341 (2001) 265}336
di!erent representations of the holohedry are generated by the matrices
1 0 p2mm: $p "$ , 0 !1
0 1 c2mm: $p "$ , 1 0
(18)
where p , p are Pauli matrices given previously in Eqs. (16) of Chapter III. These two representa tions are equivalent on the complex, on the real and even on the rational numbers; but they are inequivalent on Z, the ring of integers. That means that the two (isomorphic) groups p2mm and c2mm are non-conjugate subgroups of G¸(2, Z). The proof is very simple: matrices conjugated in G¸(d, Z) have the same g.c.d.("greatest common divisor) of their elements; the g.c.d. of the elements of the matrices I #p and I #p are, respectively, 2 and 1. Note that this di!erence comes from the existence of two inequivalent integral representations pm and cm of the re#ections for d"2. We leave to the reader to show that there are only two such representations and to verify that the general integral matrix representing a re#ection is
a
b
c
!a
, a#bc"1, a#b#c odd for pm, even for cm ,
(19)
(hint: verify that (a#b#c) mod 2 is invariant by a conjugation in G¸(2, Z); by such a conjugation one can transform an element of this group into an upper triangular matrix, so c"0, and for a re#ection a"$1 and b3Z; then try to conjugate in G¸(2, Z) the matrices of this form). Note that the lattice c2mm can be de"ned, in an orthogonal basis +a , as G
(a , a )"ad , a Oa , cmm" a a G H G GH G G G with a both 3Z or both 3Z# . G The dual basis +aH, is also orthogonal. In this basis, the dual lattice is de"ned by G
(20)
(21) (cmm)H" a aH , G G G with a 3Z, a #a 32ZN(cmm)H&2(cmm) . Indeed, although these two lattices are di!erent, their fundamental domains are rhombs and their Bravais groups are Z-equivalent. Remark that if the centring of p2mm is done by atoms > di!erent from the atoms X of this lattice, the symmetry is not changed and the space group of the new crystal is again p2mm. There is the same density of the two kinds of atoms, so the crystal has X> for its chemical formula; it is a pure convention to say that > is a centring of the rectangles of X, the other convention exchanging X> is just as good. We leave to the reader to check that when the rectangles of p2mm become squares (the Bravais group is p4mm), the rhombs which de"ne the fundamental domains of c2mm, also become squares; that is why there exists a unique Bravais class in the square crystallographic system. When it is It is also true in any dimension d'1.
L. Michel / Physics Reports 341 (2001) 265}336
279
Fig. 2. A fundamental domain of the action of G¸(2, Z) on C (Q ), the cone of positive quadratic forms q. With the > parametrization tr q"q #q '0, m"(q !q )(tr q)\, g"2q (tr q)\, q"(tr q)(I #mp #gp ) and the pos tivity implies m#g(1, the fundamental domain is the triangle HQC minus the vertex C(1,0) on the surface of the cone. H(0, ) represents the p6mm lattices, Q(0, 0) the p4mm lattices, the side QC the p2mm lattices, the two sides QH and HC the cmm lattices with, respectively, four and two shortest vectors; the interior of the triangle represents the diclinic lattices (generic case).
Table 1 The seven Bravais crystallographic systems and the geometrical class of their holohedry in the Schoen#ies and the ITC notations Bravais CS
Triclinic
Monoclinic
Orthorhombic
Tetragonal
Rhombohedral
Hexagonal
Cubic
Schoen#ies ITC
C G 1
C F 2/m
D F mmm
D F 4/mmm
D B 3 m
D F 6/mmm
O F m3 m
possible, ITC prefer to use orthogonal bases (as we have done in the last two equations) although they are not generating bases of the lattice. It is very important to remark that when a "a (square lattice), the centring yields an equivalent lattice. So the quadratic system has only one Bravais class. It is interesting to de"ne a fundamental domain for the action of G¸(2, Z) on the cone C (Q ). > That was "rst done by Lagrange (1773) under the heading `reduction of quadratic forma. Following Michel (1995), we present the same result di!erently, in the basis of C (Q ) which is > obtained by tr q"1. The result is given by triangle HQC of Fig. 2. 3.4. Three-dimensional Bravais crystallographic systems and classes In Chapter I, Table 1 (in Section 3) we introduced the 32 geometric classes (conjugacy classes of point groups in O ), essential for the classi"cation of the symmetry of the macroscopic physics of crystalline states, and gave their Schoen#ies and ITC notations. Eleven classes of point groups contain !I , the symmetry through the origin. By a proof similar to that we have done for 4"c , Note for reading the literature: as we have done in the last two equations, the orthogonal bases used in the ITC are not normalized (i.e. the basis vectors do not have unit length). On the contrary, solid state physicists prefer, in general, to use orthornormal bases. Here, in all chapters, we follow the ITC tradition.
280
L. Michel / Physics Reports 341 (2001) 265}336
we can show that four of them are not holohedry: they are 4/m"C , F 3 "C , 6/m"C , m3 "¹ . The seven others are the holohedries de"ning the seven Bravais G F F crystallographic systems: We now established that these seven crystallographic systems contain, respectively: 1, 2, 4, 2, 1, 1, 3 Bravais classes. We shall essentially follow Bravais (1850); it is the standard approach in crystallography (Table 1). Group 1 has only the representation $I ; its Bravais group is denoted by P1 . Group 2/m contains a re#ection; so it has two inequivalent integral representations P2/m, C2/m obtained by an obvious extension of Eq. (18). Similarly to the two-dimensional case, for the orthorhombic group lattices we denote by Pmmm the Bravais group of the lattice with a rectangular parallelepiped domain. The other possible Bravais classes are obtained by (i) centring this domain with the vector w "(, , ), which gives the Bravais class Immm (the I comes from `innera centring), (ii) centring the three faces with the vectors f "(0, , ), f "(, 0, ), f "(, , 0) which gives the Bravais class Fmmm (the F comes from faces), (iii) centring one face only: that is very similar to d"2; which gives the Bravais class Cmmm. The lattices Fmmm and Immm are dual of each other. When the symmetry becomes tetragonal, as for the two-dimensional case, the P and C lattices becomes equivalent; it can be shown that the same occurs for the F and I lattices. So the tetragonal system has two Bravais classes P4/mmm and I4/mmm; the last choice is a convention. However, for the cubic system, the equivalence between the tetragonal I and F centring no longer holds because the equivalence of representations does not extend from the group 4/mmm"D to the larger group m3 m"O . So the cubic system has three F F Bravais classes: Pm3 m, Fm3 m, Im3 m. Finally the rhombohedral and hexagonal systems have only one Bravais class. The corresponding Bravais groups are, respectively, denoted by R3 m and P6/mmm. Fig. 3 gives the partially ordered set of conjugacy classes of Bravais groups as subgroups of G¸(3, Z) and the map on the partially ordered set of holohedries (conjugacy classes in O ); this map is order preserving.
Their de"nition is given on p. 722 of ITC and they are called Bravais systems; while ITC calls crystal systems (p. 721) those introduced earlier by Weiss (1816); there are also seven of them with "ve which coincide with "ve of the Bravais systems. The union of the Bravais (rhombohedral6hexagonal) systems coincide with the union of the Weiss (trigonal6hexagonal) systems and it is called `hexagonal familya in the international tables. Strangely, these tables refuse to make a choice between the two partitions of the `familya. The Weiss classi"cation was remarkable but empirical; trigonal is for the crystals which have a 3 or 3 symmetry and hexagonal is for the crystals with a 6 or 6 symmetry. The latter belong also to the Bravais hexagonal system. The international tables distinguish among the space groups of the trigonal crystal systems by their "rst letter R, P those who belong, respectively, to the rhombohedral and hexagonal Bravais systems. More generally, in ITC, the `international symbolsa for space groups follow the Bravais classi"cation. Here we use the Bravais classi"cation as the natural and fruitful one for the study of crystal symmetry. The "rst classi"cation was made by Frankenheim (1842): he found 15 classes. Bravais (1850) not only corrected the error, but made a deep mathematical analysis of the two- and three-dimensional lattices (e.g. he introduced the concept of dual lattices) extending the work of Gauss (1805) that he quotes several times. The determination of the set of strata: C (Q )""G¸(3, Z) has been made by Schwarzenberger (1980). > Indeed the ITC could have chosen F4/mmm (they do it for listing some subgroups of space groups); probably the inner centring was considered simpler than the 3-face centring.
L. Michel / Physics Reports 341 (2001) 265}336
281
Fig. 3. For three dimensional crystallography, the left diagram shows the partial order on +BC, , the set of the 14 Bravais classes and the right one shows the partial order on +CS, , the set of the 7 Bravais crystallographic systems. The dotted horizontal lines give the order preserving map de"ned after Eq. (7).
4. Geometric and arithmetic classes. Brillouin zone. Time reversal 4.1. Two maps on the set +AC, of arithmetic classes B As we recall in the introduction, crystals have a translation lattice of symmetry ¸ and the crystal structure is de"ned by the positions of the atoms in a fundamental domain of ¸. In the next section we will show the existence of a fundamental domain D invariant by the Bravais group PX of ¸. * * In D , if the atom positions, for each kind of atom, are not a union of orbits of PX, the point * symmetry of the crystal is smaller and it corresponds to an arithmetic class, i.e. the conjugacy class of a "nite group of G¸(d, Z) that we also denote by PX (Bravais groups are particular examples of arithmetic classes). Since G¸(d, Z)(G¸(d, R), the conjugacy class [PX] de"nes the conjugacy %*B 8 class [PX] , i.e. a real linear representation of PX up to an equivalence. Since PX is "nite, it is %*B 0 well known that such a representation is equivalent to an orthogonal one, which de"nes the geometric class P"[PX] B of PX, i.e. its conjugacy class in O . In other words, we have constructed B the natural map ( +GC, +AC, P B B
(22)
between the set of arithmetic and geometric classes in d-dimension. As we already explained, in general the symmetry of the macroscopic physical properties of the crystal depends only on its geometric class. This invariant domain is called the Wigner}Seitz domain by the physicists. It was known much before: see the beginning of Section 5.
282
L. Michel / Physics Reports 341 (2001) 265}336
The Bravais groups form a subset of the arithmetic classes. There is also a natural surjective map from the arithmetic classes to the Bravais classes (see e.g., Michel and Mozrzymas, 1989) ? +BC, . +AC, P B B
(23)
This map is not obvious. Notice that, as all other symmetries, the crystal symmetry must be de"ned up to conjugacy class of the space group. Which one? It is more than the conjugacy class in Eu , the B Euclidean group, but less that the conjugacy class in A+ , the a$ne group. It is only part of the B latter: it is Eu 5[G] B , the intersection of the Euclidean group and of the conjugacy class of G in B DD the a$ne group. Indeed, as long as no phase transition occurs, the variation of external parameters (e.g. temperature, pressure) does not change the symmetry of the crystal, but only conjugate the space group in A+ but with the condition that it remains a subgroup of Eu . The lattice of B B translations is modi"ed and, when its symmetry is not that of a maximal Bravais class, it may increase and belong to every larger Bravais class (think, as an example, to the space group P1 for which the whole class [G] B is inside Eu ). That larger symmetry of ¸, the translation lattice, B DD has to be considered as accidental. We can make a mathematical description of this phenomenon by using the theorem of Palais (1961) of good strati"cation when all the stabilizers (here for the action on G¸(3, Z) on C (Q )) are "nite. Indeed given a group K belonging to an arithmetic > class PX, we consider C (Q )), the set of positive quadratic form invariant by K. It is the > intersection of the cone C (Q ) with the vector subspace in E (the vector space of quadratic > , forms) which carries the trivial representation of K. The strati"cation in Bravais classes on C (Q) (" strata of the action of G¸(d, Z)) induces by restriction on C (Q)) a strati"cation with > > an open dense stratum; the corresponding Bravais class de"nes the value of the map a of Eq. (23) for the arithmetic class of K. As a preliminary study of arithmetic and geometric classes, one can study the conjugacy classes of "nite-order elements in G¸(3, Z) and G¸(3, R). We call these classes arithmetic elements and geometric elements. As we know, for d"2, 3 their order is 1, 2, 3, 4, 6. To label the arithmetic and geometric elements we use the notation of ITC for the cyclic group they generate. These elements are the building bricks of the arithmetic and geometric classes. Their numbers are, respectively, 7 and 6 for d"2, and, respectively, 16 and 10 for d"3. 4.2. Geometric and arithmetic elements and classes in dimension 2 We have already listed in Section 3.3 the 10 geometric classes for d"2. As we have seen, they are formed from six geometric elements: "ve rotations 1, 2, 3, 4, 6 in SO and m, the conjugacy class of the re#ections through an axis; it contains all the elements of determinant !1 of O . It requires some arguments of number theory to prove that each of the geometric classes corresponding to the rotation groups 3, 4, 6 has a unique arithmetic class. We have shown in Eq. (19) that there are two arithmetic classes corresponding to m and given their notation pm, cm. To summarize: The ITC de"ne the geometric elements and give their list in p. 6. The de"nition of the arithmetic elements is only implicit.
L. Michel / Physics Reports 341 (2001) 265}336
283
For d"2, the arithmetic elements are: p1, p2, p3, p4, p6, pm, cm. A choice of matrices representing them is: (p1)"I "!(p2), 1 0 0 1 0 !1 , (cm)" , (p4)" , (pm)" 0 !1 1 0 1 0
(p3)"
0 !1
1 !1
, (p6)"
1 !1
1
0
.
(24)
((pm) and (cm) were given in Eq. (18)). This simple choice of matrices in Eq. (24) requires the following restrictions on the vectors of the basis: their angle is p/2 for (pm) and (p4), arbitrary for (cm) and 2p/3 for (p3), (p6); except for (pm), the two vectors have the same norm. The maximal geometric classes are 4mm"c , 6mm"c . From the fact that the arithmetic T T classes of 4 and 6 are unique it is easy to show that each one of these two maximal geometric classes have unique arithmetic classes p4mm, p6mm. The conjugacy classes of their subgroups are given in Chapter I, Fig. 1. For these two "nite groups, when the same geometric class appears for di!erent conjugacy classes, one has to check if those become conjugate in G¸(2, Z); if the answer is negative, these conjugacy classes are distinct arithmetic classes and we need some notations for distinguishing them. For 4mm"c that is the case for c which yields pm, cm and c which yields T Q T p2mm, c2mm. For 6mm"c , it is the case for c ; ITC denote the two corresponding arithmetic T T classes p3m1 and p31m and distinguish them in a given coordinate system. Let us follow this pedestrian argument before giving one which is coordinate independent. In the system of coordinates corresponding to Eq. (24) the representations of p3m1 and p31m are, respectively, generated by the pairs of matrices (p3), (cm) and (p3),!(cm); the two $(cm) correspond to the orthogonal duality between the sets of three symmetry axes in the two groups. These two representations are conjugated in G¸(2, R); the matrices x which conjugate them must satisfy the linear homogeneous system of equations x(p3)"(p3)x, x(cm)"!(cm)x ,
solutions x"j
1 !2
!2
1
, det x"3j .
(25)
The value of the determinant shows that if j3Z the corresponding matrix cannot be in G¸(2, Z); for j"((3)\, x3SO . It is a rotation by p/2 since x"!I ; indeed Eq. (25) requires that x commutes with the rotations (so it is one of them) and transforms a symmetry axis into an orthogonal one. We could have made the same proof without a choice of coordinate: (12)mm"c 3O is the normalizer of 6mm"c in O and is a realization of the automorphism T T group Aut 6mm. The outer automorphisms exchange the two conjugacy classes of the subgroups 3m"c ; but we know that (12)mm has no two-dimensional integral representations. T The arithmetic classes p3m1 and p31m are distinct and the space groups (semi-direct products with the hexagonal lattice of translation) that they de"ne are non-isomorphic. Let S be the set of the six shortest vectors of this lattice: the 3 symmetry axes of p3m1 contain the S vectors while the 3 symmetry axes of p31m are bisectors of the p/3 angles formed by the S vectors. The six endpoints of the S vectors de"ne with the origin 6 equilateral triangles which are Delone cells of the hexagonal
284
L. Michel / Physics Reports 341 (2001) 265}336
lattice. For the group p3m1 the sides of the Delone cells are carried by the symmetry axes; therefore it is a group generated by re#ection. Indeed it is the group de"ned by the triangular kaleidoscope; this kaleidoscope is popular as a toy and we strongly advise the reader to play with it if he has not yet done it! Fig. 4 shows the two diagrams representing the partial order, the left one on +AC, the set of the 13 arithmetic classes, the right one on +GC, , the set of the 10 geometric classes. The dotted horizontal lines give the order preserving map de"ned in Eq. (22). We give in the next equation the correspondence a de"ned in Eq. (23), mapping the 13 arithmetic classes on the "ve Bravais classes; the value of a is the last element of each subset: a: p1, p2, pm, p2mm, cm, c2mm, p4, p4mm, p3, p3m1, p31m, p6, p6mm .
(26)
The "rst eight listed arithmetic classes are orthogonal (i.e. 4p4mm"O(2, Z)) and therefore self-dual. The "ve arithmetic classes belonging to the hexagonal Bravais class cannot be represented by orthogonal integral matrices; among them, the two arithmetic classes p3m1, p31m which correspond to the same geometric class, form (up to an equivalence on Z) a pair of dual arithmetic classes. Indeed, by conjugation by the matrix
0 !1 !ip " 1 0
the representation (given before Eq. (25)) of each one is transformed into the other representation. So, among the 13 arithmetic classes in dimension 2, there is a unique pair of dual arithmetic classes duality
p3m1 p31m .
(27)
4.3. Geometric and arithmetic elements and classes in dimension 3 As we showed in Chapter I, Table 1 there are two maximal geometric classes: m3 m"O and F 6/mmm"D . From the previous section (see Fig. 3), we know that the second one has a unique F Bravais class, P6/mmm while the "rst one has three. One of them has been implicitly studied in Chapter I, Fig. 2; it corresponds to O(3, Z)"Pm3 m. We now built the two others and verify their properties:
(e , e )"d , Pm3 m" k e , k 3Z . G G G G H GH G
(28)
Delone cells have been de"ned in Section 1. By the translations we obtain the Delone cells of any points; they are triangles so the hexagonal lattice is primitive (this concept is also de"ned in Section 1). The other two-dimensional space groups generated by re#ection are p6mm, p4mm, p2mm. The kaleidoscope is the smaller polygon formed by the in"nite set of symmetry axes; it is, respectively, a triangle of angles p/2,p/3,p/6, an isocele triangle of angles p/2,p/4,p/4, a rectangle (see Fig. 1); only in the last case the kaleidoscope is also a Delone cell of the lattice. There exists a classi"cation of the re#ection space groups in any dimension d; they are labelled by an extended Dynkin diagram and they are the Weyl groups of some Kac-Moody algebras. This beautiful theory is outside the scope of this monography, but it is used in other "elds of physics.
L. Michel / Physics Reports 341 (2001) 265}336
285
Fig. 4. For d"2 the two diagrams show the partial order, the left one on +AC, , the set of the 13 arithmetic classes, the right one on +GC, , the set of the 10 geometric classes. The underlined classes in a, b are, respectively, the Bravais classes and the crystallograpic systems. The dotted horizontal lines give the order preserving map de"ned in Eq. (22).
Let us write the I-cubic lattice as the centering of the P-cubic one, with w "(, , ): Im3 m"Pm3 m6(w #Pm3 m)
0Im3 m" k e , either k 3Z, or k 3Z# . G G G G G Let us de"ne:
Fm3 m"(Im3 m)H0 Fm3 m" k e , k 3Z, k 32Z . G G G G G G We may choose for the basis of the last two lattices, with the notation of Eq. (4):
0 1 1
bI " 1 0 1 , $ 1 1 0
!1
b " '
1
1
(30)
1
1 , Nb b ?"I . $ ' 1 !1
1 !1
(29)
(31)
The basis bI shows explicitly that Fm3 m is obtained from Pm3 m by centering the three faces of the $ unit cube and the last relation of Eq. (31) proves the duality in Eq. (30). The corresponding quadratic forms are q ,q(Fm3 m)"I #J , q ,q(Im3 m)"I !J with (J ) "1, q q "I . (32) $ ' GH $ ' Instead of the cubic system we could have worked in the orthorhombic system. In Eq. (28) we start not from an orthonormal basis but from an orthogonal one: (e , e )"ad . Eqs. (28)}(31) are G H G GH unchanged but in Eq. (31) the basis bI is expressed in the dual orthogonal basis (e , e )"a\d . ' G H G GH Then, from the matrices representing the re#ection (Pm ) (given in Chapter I, Eq. (16)) we can G
286
L. Michel / Physics Reports 341 (2001) 265}336
compute those representing the re#ections (Fm )"bI \(Pm )bI . We obtain G $ G $ 1 1 1 0 0 !1
0 !1 , (Fm )" 1 1 (Fm )" 0 0 !1 0 !1 0
1 , 0
0 !1 0
(Fm )" !1 1
0 0 .
(33)
1 1.
We verify the relations:
(Fm )"I, i, j, k permutation of 1, 2, 3, G (Fm ) (Fm )"(Fm ) (Fm )"!(Fm ) . (34) G H H G I So the (Fm )'s generate Fmmm, the !(Fm )'s generate F222 and (Fm ), (Fm ) generate (Fmm2) . The G G G H I re#ections (Im ) are represented by the dual matrices: H (35) (Im )"(Fm )? . H H The three F (respectively, I) re#ections can be transformed into each other by conjugation by the matrices of the arithmetic class R3 which permute the coordinates circularly. To verify that these six matrices belong to the arithmetic class Cm [which we have represented, in Chapter I, Eq. (17) by Z (Cm ), where (Cm ) is the permutation matrix permuting the coordinates 1, 2], it is su$cient to show it explicitly on one of them for each family: (Fm )"X(Cm )X\, (Im )"X\?(Cm )X? , 0 !1 0
X" !1 1
We will later need
0 1 .
(36)
0 0
(Fm )"X(Im )X\ , (Fm )"!(Fm )(Fm )"!X(Cm )(Im )X\ . (37) Notice that (Im ) and (Cm ) commute. To summarize we have introduced another (i.e. Z equivalent) representative of the Fmmm arithmetic class: X\(Fm )X"(Fm ) "+(!(Cm )(Im ), (Cm ), (Im ), . (38) G G We now show that the two integral representations Fmmm and Immm of the geometric class mmm are inequivalent on Z. For that we consider the most general 3;3 real matrix S whose elements satisfy the system of linear equations: (Im )S"S(Fm ), (Im )S"S(Fm ); using Eq. (34) we verify
L. Michel / Physics Reports 341 (2001) 265}336
287
Fig. 5. Diagram of the partially ordered set of the conjugacy classes of subgroups of the maximal Bravais groups Fm3 m and Im3 m; there are four pairs of classes whose elements belong to the same arithmetic classes: they have a"sign between them.
that this implies (Im )S"S(Fm ). The most general solution in integers is k#l l k
S"
l
l#j
j
k
j
j#k
, det S"4jkl, j, k, l3Z .
(39)
This shows that the two representations are equivalent on the real and the rational but are not equivalent on the integers: indeed the determinant of S cannot be $1 when j, k, l are integers. We verify in Chapter I, Fig. 2 that each of the "ve cubic group is generated by 222"D and one of the "ve rhombohedral groups. So the inequivalence on Z of F222 and I222 can be extended to the representations Fm3 m and Im3 m or to any other pair of corresponding cubic arithmetic class. The partially ordered set of conjugacy classes of the subgroups of Fm3 m, Im3 m are shown in Fig. 5 (We recall that those of Pm3 m, P6/mmm are shown in Chapter I, Figs. 2 and 3). We have seen in the previous section (after Eq. (21)) that for the square crystallographic system (" geometric class 4mmm) there is a unique Bravais class whose Bravais group is p4mm&O(2, Z). The last expressions of Eqs. (29)}(30) de"ne cubic lattices in any dimension d and it is not di$cult to prove for d'4 the existence of three maximal Bravais classes P, F, I for the geometric class (called also the Coxeter group B ) of symmetry of the d-dimensional cube (one can use w with its B B d coordinates" in the orthonormal basis (e , e )"d ). For dimension d"4, (w , w )" G H GH 1"(e , e ); these vectors belong to the orbit of a larger Coxeter group F (with "F : B ""3) which G G
288
L. Michel / Physics Reports 341 (2001) 265}336
has some deep relations with the quaternions and is the symmetry group of a regular self-dual polytope which exists only in d"4. For this dimension the lattices F and I are equivalent and distinct from the P lattice of Bravais group O(4, Z). Another obvious generalization to any dimension is to extend the de"nition in Eq. (32) of the matrix J to J (i.e. all its elements are 1); verify that J"dJ . Then the quadratic forms B B B q "I #J , q"I !(d#1)\J , q q"I , (40) B B B B B B B B B de"ne two dual lattices denoted as A and A since they are the root and weight lattices of the B B simple Lie algebra A &S;(d#1). We have already seen that for d"2 they are in the same B Bravais class: the hexagonal one. The lattice A has been studied in arbitrary dimensions, "rst by B VoronomK (1908, 1909); it is studied here in Section 5.3. We still have to study the tetragonal system. As we have done in Eq. (33) we compute
(F4)"bI \(P4)bI " $ $
1 1
1
0 0 !1
!1 0
0
"(Cm ) (Fm )"(Fm ) (Cm ) and "nd the equivalence with its dual arithmetic class:
(I4)"(F 4)"
0 1
(41)
0
0 1 !1 "X\(F4)X
!1 1
0
"(Cm ) (Im )"(Im ) (Cm ) , (42) where X has been de"ned in Eq. (36). So the tetragonal system has only two Bravais classes of lattices: P and F&I. In ITC, the notation I has been chosen; naturally we follow the notations of ITC (e.g. in Fig. 5). To see more explicitly the self-duality of the I-tetragonal Bravais class, we study the arithmetic class I422. First we notice that the square of the I-rotation by p/2 is I(4)"!(Im ), the so-called `verticala I-rotation by p. So I422 is generated by (I4) and Im , Im , the two `horizontala rotations (one is enough) by p. The two other `horizontala rotations by p are (use the third equality in Eq. (41)) : !(I4) (Im )"!(Cm ), !(I4) (Im )"(Cm ) (Im ). From Eq. (38), we see that these rota tions are !(Fm ) and !(Fm ) and moreover !(Fm ) "!(Im ). To summarize (and to be used in the future) the three re#ections of F mmm are: (Cm ), (Im ), (Cm ) (Im ) . We have also given a matrix realization of the arithmetic class I422; from it we "nd that the maximal subgroups of I422 are I4, I222, F mmm .
(43)
(44)
Although in several tables of tetragonal I space groups they use also F presentation in the subtables of maximal subgroups.
L. Michel / Physics Reports 341 (2001) 265}336
289
That can be read in Fig. 4, but only in terms of arithmetic class (hence independently of their realization). In this "gure the duality corresponds to the exchange FI in the cubic and orthorhombic system but not in the I tetragonal system except for the exchange of the two arithmetic classes I4 m2I4 2m which belong to the same geometric class: we leave to the reader, with the use of Eq. (36), to check that the contragradient representation of one of them is Z-equivalent to the representation of the other. To help the reader we list here the 15 pairs of dual arithmetic classes in 3D. By de"nition the orthogonal arithmetic classes are self-dual. We "rst recall the duality FI between the elements of three pairs in the orthorhombic system and "ve pairs in the cubic system orthorhombic F222I222, Fmm2Imm2, FmmmImmm .
(45)
cubic F23I23, Fm3 Im3 , F432I432, F4 3mI4 3m, Fm3 mIm3 m .
(46)
There are "ve other pairs of dual arithmetic classes belonging to the same geometric class and, by the map a of Eq. (23), to the same Bravais class. One veri"es that the contragradient representation of one arithmetic group of such a pair is equivalent on Z to the representation of the other group of the pair; for the hexagonal system we use a generalization of the two-dimensional case (27) I4 m2I4 2m, P321P312, P3m1P31m, P3 m1P3 1m, P6 m2P6 2m .
(47)
We have shown in Chapter I, Figs. 2, 3 that all 33 conjugacy classes of subgroups of O(3, Z)"Pm3 m and the 16 conjugacy classes of subgroups of P6/mmm which are not 4Cmmm are distinct arithmetic classes. The set of conjugacy classes of subgroups that Fm3 m and Im3 m have in common with Pm3 m are the 10 arithmetic classes 4R3 m. The set of other subgroup conjugacy classes among these two cubic groups only, de"nes 14 other arithmetic classes; their images by a of Eq. (23) are the I4/mmm, Immm, Fmmm Bravais classes. Finally each of these two cubic Bravais groups have a speci"c set of "ve cubic arithmetic classes. That gives a total of 73 arithmetic classes. The statistics according to the "rst letter is P : 37, R : 5, C : 6, A : 1, F : 8, I : 16 . It is also interesting to look at the distribution of the arithmetic classes among the geometric classes AC per GC: 1 2
3
4
5,
GC:
8 12 8
3
1
total"32 ,
AC:
8 24 24 12 5
total"73 .
The cyclic point groups belong to 10 of the 32 geometric classes and de"ne the 10 geometric elements. They are denoted in ITC: 1, 2, 3, 4, 6, 1 , m, 3 , 4 , 6 , (m replaces 2 ); a matrix with a\ is obtained by multiplying by !I the corresponding rotation matrix. In his beautiful elementary book H. Weyl (1952) gives 70 for the number of arithmetic classes in three dimensions, reproducing an error which appeared and still appears in most mathematic books dealing with this topics.
290
L. Michel / Physics Reports 341 (2001) 265}336
For each geometric element there is an arithmetic element labelled P ("principal). There are six other arithmetic elements, and in total the labels for 16 arithmetic elements are P1
P2 C2
P3 R3
P4 I4
P6
P1
Pm Cm
P3 R3
P4 I4 .
P6 ,
We have shown that the re#ections of Fmmm and Immm are all equivalent to Cm (see Eq. (39)): similarly the rotations of these two groups are equivalent to C2. 4.4. Brillouin zone, its high symmetry points For periodic crystals, all functions describing their physical properties have the periods of the crystal. Most experiments measure the Fourier transforms of these functions, i.e. functions on the Brillouin zone. In this subsection we de"ne the Brillouin zone and begin the study of the space group actions on it. In Section 3.1 we have de"ned ¸H, the dual lattice of the lattice ¸, as the set of vectors whose scalar products with all l3¸ are integers. Physicists also consider the reciprocal lattice which is 2p¸H. Indeed it is the one which is obtained in di!raction experiments (with X-rays, neutrons, electrons) by a crystal of translation lattice ¸. It corresponds to Fourier transform; the momentum variable is usually denoted by k and the vector space of the k's is called the momentum or the reciprocal space. A unitary irreducible representation of the translation group is given by k(x)"exp[i(k ) x)]. Here we are interested in the subgroup of the translation group RB de"ned by the lattice of translations ¸. By restriction to ¸, two unirreps k and k of RB such that k !k32p¸H, yield the same unirrep of ¸. So the set ¸K of inequivalent unirreps is ¸ U lCk(l)"e k l, ¸K "+k mod 2p¸H, .
(48)
Equivalently, with a choice of dual bases (see Eq. (10)) l" k b , k" i bH, l C e H GH IH , k 3Z, i mod 2p . H H H H H H H H The set ¸K of the unirreps has the structure of a group, with the group law
(49)
k,k#k mod 2p¸H0i ,i#i mod 2p . (50) H H H This group is called the dual group of ¸ by the mathematicians and the Brillouin zone ("BZ) by the physicists. It is isomorphic to the group ¸K "BZ&;B . (51) We denote by kK the elements of BZ in order to distinguish clearly between k and kK ": k mod 2p¸H. ? The Bravais group PX of ¸ acts on BZ through its contragradient representation PI X ": (PX )\ . * * * More generally, since by de"nition of BZ the translation group acts trivially, a space group G acts through its quotient F G/¸"PX . GP
(52)
L. Michel / Physics Reports 341 (2001) 265}336
291
So the space groups belonging to the same arithmetic class PX have the same action. As usual, we denote by G the stabilizer in G of kK 3BZ and PX the stabilizer in PX. The latter stabilizer depends I I only on the arithmetic class; beware that for a given kK the stabilizers G "h\(PX ) for the I I di!erent space groups of the arithmetic class PX are, in general, non-isomorphic. Notice that the G 's are also space groups. I The action of the space groups on BZ preserves its group structure, i.e. ∀g3G, g ) kK #g ) kK "g ) (kK #kK ) .
(53)
As a consequence we have the obvious proposition, that we write for any arithmetic class. Proposition 4. In the action of PX on BZ, the xxed elements form a group, generally denoted by (BZ).X; the elements of a coset of this group have the same stabilizer. We also remark that an orbit contains only elements of BZ of the same order l (we recall that l is the smallest integer such that lkK "0). A d-dimensional torus is the topological product of d circles; this is the topology of the group ;B . So there is a natural, global system of coordinates on it made of d angles u , 14i4d, with all u "0 for the 0 element of this Abelian group; the group G G law is the addition modulo 2p of each coordinate u . For instance the elements of order 2 have G 04m(d coordinates 0 and d!m coordinates p. So BZ has 2B!1 elements of order 2. In this section we only study for d"2,3 the strata of dimension 0 which appear in the action on BZ of the 5, 14 Bravais groups (listed in Figs. 4 and 3, respectively). We point out that, even for this non-linear action, the stabilizers of these strata cannot belong to an axial or planar (i.e. leaving "xed an axis or a plane) geometrical class. The smallest Bravais group, in any dimension, is Z (!I ) (where !I is the symmetry through the origin). Since !p,p mod 2, this group has B B the 2B "xed points: those who satisfy 2kK "0, i.e. the elements of order 2 and the origin. Since this group is a subgroup of any other Bravais group, for these groups the set of order 2 elements of BZ is partitioning into orbits belonging to strata of dimension 0. For d"2. The matrices generating the "ve Bravais groups are given in Eq. (24). The matrices of p2 and p2mm are diagonal so the four points satisfying 2kK "0, i.e. (0, 0), (p, 0), (0, p), (p, p), are "xed points. The re#ection (cm) exchanges (p, 0) and (0, p) and leaves (p, p) "xed; since the orthogonal Bravais group p4mm is generated by p2mm and c2mm it has the same action on the 2kK "0 points as c2mm. To study the hexagonal Bravais group p6mm, let us compute the contragradient of (p3) and study its action
(p 3)"
!1 !1 1
0
, (p 3)
i
" i
!i !i . i
(54)
In the action of p6mm on BZ, the element (p3) transforms (p, 0) into (p, p) and this point into (0, p). So the three elements of order 2 of BZ form a unique orbit. Eq. (54) tells us that the coordinates of The map h is not invertible, so h\ alone has no meaning; but it is an accepted tradition to denote by h\(P ) the I counter image of P by h, i.e. the unique subgroup of G such that h(G )"PX . I I I We use the additive notation for the group BZ.
292
L. Michel / Physics Reports 341 (2001) 265}336
Table 2 Strata of zero-dimension in the action of Bravais groups on BZ for d"2. The points are grouped into orbits and the stabilizer is given Points of BZ Their order
() 1
(p ) 2
() p 2
(p) p 2
Diclinic Orthorhombic
p2 p2mm
p2 p2mm
p2 p2mm
p2 p2mm
Orthorhombic Square
c2mm p4mm
Hexagonal
p6mm
p2 p2mm
(p ), (p ) p p 3
c2mm p4mm p2mm
p3m1
the "xed points of the group p3 satisfy: 2i #i ,0 mod 2p, i ,i mod 2p ,
2p 2p 4p 4p , , kK " , , Nsolutions, (0, 0), kK " ! ! 3 3 3 3
3kK "3kK "0 . ! !
(55)
The points kK , kK are also invariant by (cm) and are exchanged by (p2)"!I ; hence their ! ! stabilizer is p3m1 (whose representation was given just before Eq. (25)). Table 2 gathers the information we have obtained on the strata of dimension zero in the action of the "ve two-dimensional Bravais groups on BZ. For d"3. Triclinic, monoclinic systems and P-orthorhombic Bravais class: The extension of the results obtained for d"2 is straightforward. The representations of the Bravais groups P1 , P2/m, Pmmm can be made diagonal (on Z), so they produce on BZ a unique 0-dimensional stratum, that of the eight "xed points 2kK "0. For the group C2/m, the zero-dimensional strata contain only the eight points 2kK "0; the seven elements of order 2 are partitioned into "ve orbits of 1, 1, 1, 2, 2 elements; those of two elements are (p, 0, 0), (0, p, 0) and (p, 0, p), (0, p, p). C-orthorhombic and P-tetragonal Bravais class: The same results are obtained for the groups Cmmm, P4/mmm since these groups are generated by C2/m and, respectively, by P2/m and Pmmm. Hexagonal Bravais class: For the Bravais group P6/mmm, (0, 0, p) is a "xed point; the six other elements of order 2 are partitioned into two orbits: (p, 0, 0), (0, p, 0), (p, p, 0) and (p, 0, p), (0, p, p), (p, p, p) whose stabilizers are P2/m. Moreover, from Eq. (55), we obtain two orbits of two points: (2p/3, 2p/3, 0), (4p/3, 4p/3, 0) and (2p/3, 2p/3, p), (4p/3, 4p/3, p), whose common stabilizer is P6 2m (do not forget the duality given in Eq. (47)). The elements of these two orbits satisfy, respectively: 3kK "0 and 6kK "0. Rhombohedral Bravais class: The Bravais group R3 m is generated by the groups P1 and R3m which is the permutation group of the three coordinates. So the zero-dimensional strata contains only 0 and the seven elements of order 2; those are partitioned into three orbits of 3, 3, 1 elements which contain respectively 1, 2, 3 coordinates p (and the others are 0).
L. Michel / Physics Reports 341 (2001) 265}336
293
P-cubic Bravais class: We obtain the same results as the rhombohedral one since Pm3 m is generated by its subgroups Pmmm and R3 m. F-orthorhombic Bravais class: For the group Fmmm, we study "rst the action of the three rotations of F222 on BZ; their action is represented by the matrices !(Im ). One obtains, H respectively, for j"1, 2, 3:
i !i i C !i #i , i !i #i
!i #i , !i !i #i
!i #i !i #i . !i
(56)
We verify that among the seven elements of order 2 (which are the elements invariant by !I , the symmetry through the origin), three of them, B "(0, p, p), B "(p, 0, p), B "(p, p, 0) are "xed points and the four others form one orbit whose stabilizer is P1 . There are no other points in zero-dimensional strata. For the equivalent realization F mmm given in Eq. (43), one obtains for the three "xed points: (0, 0, p),(p, p, 0),(p, p, p) and the four other points of order 2 form one orbit (with stabilizer P1 ). It will also be useful later to know the one-dimensional stratum with (Fmm2) as stabilizer. From Eq. (56) we "nd that the elements "xed by the rotation !Im satisfy: 2i ,0 mod 2p, i !i "$i . That de"nes three loops: (i , i , 0), (i , i #p, p), (i , i !p, p) .
(57)
These three loops are the intersection of the two two-dimensional tori, submanifolds of BZ, i !i "!i , i !i "i of "xed points by the matrices !Im and !Im , respectively. So the stabilizer of the points of these three loops is (Fmm2) except at the points B . G I-orthorhombic Bravais class: Similarly, for the group Immm, we study the action of the three rotations; their action is represented by the matrices !(Fm ). One obtains, respectively, for H j"1, 2, 3:
i !i !i !i , i C i i i
i !i !i !i , i
i . i !i !i !i
(58)
We verify that among the seven elements of order 2, there is a "xed point: (p, p, p) and the six others fall into three orbits of two points: (p, 0, 0), (0, p, p),
(0, p, 0), (p, 0, p);
(0, 0, p), (p, p, 0) .
(59)
Their stabilizers are, respectively, the three C2/m subgroups. Moreover there are two points invariant by the three rotations of Eq. (58):
p p p , , 2 2 2
and
3p 3p 3p , , ; they satisfy 4kK "0 . 2 2 2
The stabiliser is I222; obviously these two elements are exchanged by the space inversion.
(60)
294
L. Michel / Physics Reports 341 (2001) 265}336
I-tetragonal Bravais class: The group I4/mmm is generated by its subgroups Immm and (I4) whose action on BZ is
i
i #i #i . (61) (I 4) ) i " !i i !i This matrix leaves invariant the point (p, p, p) and exchanges the two points of the third orbit of Eq. (59); so the stabilizer is I422 (see Fig. 5). The matrix (I 4) mixes the "rst two orbits, so the partition into I4/mmm-orbit of the set of points of order two is 1, 2, 4 with (C2/m) as stabilizers. This matrix exchanges the two points of Eq. (60); hence I4 lets them "xed. So these two points of order 4 form an orbit of I4/mmm with stabilizer I4 2m (see e.g. Table 3). I-cubic Bravais class: The group Im3 m is generated by its subgroups Immm and R3m, the permutation group of the three coordinates; so it leaves "xed (p, p, p) and it merges the three two-element orbits of Eq. (59) into a six-element orbit. So the stabilizers have eight elements and form a conjugacy class of Im3 m. Let us study the stabilizer of the orbit element: B "(p, p, 0); it is invariant by the matrices (Cm ) (the permutation matrix of the coordinates 1, 2) and by (Im )"(Fm )? (de"ned in Eq. (33)). We have shown in Eq. (38) that these two matrices generate the group F mmm of the arithmetic class Fmmm. R3m leaves also "xed the two points of Eq. (60), so they form a two-element orbit of Im3 m whose stabiliser is I4 3m (see Table 3). F-cubic Bravais class: The Bravais group Fm3 m is generated by the two Bravais groups Fmmm and R3 m. From the orbits of these two groups on the set of elements of order 2, we deduce that Fm3 m has two orbits on this set: one has three elements, (de"ned as B when we studied Fmmm, see G also Table 3), the other has four other elements (they form a unique orbit for Fmmm and two orbits for R3 m). The stabilizers are, respectively, F4/mmm&I4/mmm and R3 m as the only Fm3 m subgroups of index 3 and 4; for a direct veri"cation, check that B is "xed by both matrices (F 4) (given in Eq. (42)) and !(F 4) while R3 m leaves "xed (p, p, p). In general, when the group acting on BZ is enlarged some stratum of dimension 0 can appear if the stabilizer of strata of higher dimension is enlarged, at some isolated points, with an element whose "xed points are isolated. Let us start from the one-dimensional stratum of Fmmm that we have studied (under the heading of this group) and whose stabilizer is (Fmm2) ; Fig. 5 shows that it is a maximal subgroup of I4mm, I4 2m and Fmmm (irrelevant for our study). So we have to study the "xed points of the matrices $(F 4) (de"ned in Eq. (42)) which generate the representation of F4&I4 and F4 &I4 on the reciprocal space. The "xed points of !(F 4) satisfy the equations
i #i "0, 2i "i , i "i #i . (62) The coordinates of the "xed points of the group I4 2m (generated by I4 and Fmm2) satisfy Eqs. (62) and (57). They are: (p/2, 3p/2, p) and (3p/2, p/2, p). The subgroup R3m of Fm3 m permutes the components of these BZ elements, making a six-element orbit with stabilizer I4 2m. Table 3 presents the obtained results on the zero-dimensional strata in the action of the Bravais groups on BZ. For each Bravais group, the set X of the points in these orbits has a number of elements 64"X"414. It is easy to decompose this set into orbits for any arithmetic class PX: the set X is that of the Bravais group a(PX) (this map is de"ned in Eq. (23)) and the stabilizers are the intersection with PX of the stabilizers of the Table 3; in a few cases an orbit splits into two (with the
L. Michel / Physics Reports 341 (2001) 265}336
295
Table 3 Zero-dimensional strata in the action of Bravais groups on BZ. For each group, the table gives the corresponding orbits and their stabilizers. The number of elements in one orbit and the common order of the elements are (independently) 1, 2, 3, 4, 6. When the order of the orbit elements is '2, only one element kK is given for the orbit; the others are given in Eq. (63)
kK 3BZ
0
p
p
0
0
0
p
p
0
p
0
p
0
p
0
p
0
p
0
0
p
p
p
0
0 1
R 2
A 2
A 2
A 2
B 2
B 2
B 2
P1 P2/m Pmmm
# # #
# # #
# # #
# # #
# # #
# # #
# # #
# # #
C2/m Cmmm P4/mmm
# # #
# # #
R3 m Pm3 m
# #
# #
Fmmm
#
Their label Their order
P1 P2/m Pmmm
P1 P2/m Pmmm
# # #
C2/m G P4/mmm
C2/m G P4/mmm
P1
No. orbit
1
Variable
kK
0
R, A , A , A , B , B , B
Order
1
2
P6/mmm Immm
0 0
A R
I4/mmm
0
R
Im3 m
0
R
Fm3 m
0
A B : F mmm
A A B B : C2/m
A A A B B B : F mmm RA A A : R3 m
#
2
B B B : F4/mmm
#
6
#
2
2
p/2
p/2
2p/3
2p/3
p/2
3p/2
2p/3
2p/3
p/2
p
0
p
4
A A B , B B R: Cmmm A B , A B , A B , : C /m G
# # #
4
I222 I4 2m I4 3m I4 2m
3
6
P6 2m
P6 2m
296
L. Michel / Physics Reports 341 (2001) 265}336
same number of elements) when one passes from the Bravais group PX to a subgroup PX. When the * order of the orbit elements is '2, only one element kK is given for the orbit in Table 3; we complete here the 2-element orbits: 4kK "0,
3kK "0,
6kK "0,
p/2
3p/2
2p/3
4p/3
2p/3
4p/3
p/2 ,
3p/2 ,
2p/3 ,
4p/3 ,
2p/3 ,
4p/3 .
p/2
3p/2
0
0
p
(63)
p
The six elements of the order 4 orbits of Fm3 m are obtained by the permutations of the components: (p/2, 3p/2, p). 4.5. Time reversal T In classical Hamiltonian mechanics if, at a given instant one reverses the momenta, the trajectories are unchanged but they are followed in the reverse direction. That symmetry has been called (a little abusively) time reversal; we denote it by T. The fundamental paper on T in quantum mechanics is (Wigner, 1932); it has been extended to quantum "eld theory. In both frames, T is represented by an anti-unitary operator. In Section 8 we will study the merging of space group symmetry and time reversal. Here we study only the modi"cation of the space group action on BZ. The change of sign of momenta transforms a unirrep of the group ¸ into its complex conjugate. This corresponds on BZ (" the set of inequivalent unirreps of ¸) to the transformation kK !kK . For simplicity, we study here T only when the spin coordinates do not intervene explicitly. Time reversal invariance is a symmetry of many equilibrium states; the real functions on BZ describing their physical properties, e.g. the energy function, must satisfy the relation E(kK )"E(!kK ). The e!ect of this symmetry can be obtained by enlarging PX, the group acting e!ectively on BZ, with !I , when B PX does not already contain the symmetry through the origin. We denote this enlarged group by P[ X; this is simply PX for the 7, 24 arithmetic classes (for d"2,3) which contain the symmetry through the origin. Table 4 gives the list of PXOP[ X's which correspond to the same P[ X. 5. VoronommK cells and Brillouin cells The next two sections are based on some common work and many discussions with Peter Engel and Marjorie Senechal. 5.1. VorononK cells, their faces and corona vectors We have mentioned that one can built a fundamental domain of a lattice independently of the choice of a coordinate system. It was introduced for d"2 by Dirichlet (1850) and also Hermite We give this table for the convenience of the reader because we have not seen it in textbooks.
L. Michel / Physics Reports 341 (2001) 265}336
297
Table 4 Arithmetic classes P[ X obtained by enlarging PX with !I P[ X
PX
P[ X
PX
(p2) (p2mm) (c2mm)
p1; pm; cm;
p4 (p4mm) p6 (p6mm)
p3; p3m1, p31m;
(P1 ) (P2/m) (C2/m) (Pmmm) (Cmmm) (Fmmm) (Immm) P4/m (P4/mmm) I4/m (I4/mmm) R3
P1; P2, Pm; C2, Cm; P222, Pmm2; C222, Cmm2, Amm2; F222, Fmm2; I222, Imm2; P4, P4 ; P422, P4mm, P4 2m, P4 m2; I4, I4 ; I422, I4mm, I4 m2, I4 2m; R3;
(R3 m) P3 P3 1m P3 m1 P6/m (P6/mmm) Pm3 (Pm3 m) Fm3 (Fm3 m) Im3 (Im3 m)
R32, R3m; P3; P312, P31m; P321, P3m1; P6, P6 ; P622, P6mm, P6 m2, P6 2m; P23; P432, P4 3m; F23; F432, F4 3m; I23; I432, I4 3m;
This enlargement includes the e!ect of time reversal on BZ. In dimension d"2, 3, the 13, 73 arithmetic classes PX and P[ X are mapped onto the 7, 24 ones containing !I and denoted by P[ X. The 5, 14 arithmetic classes of the Bravais groups are between ( ). For the labels of the arithmetic classes and for the order in their listing, we follow the `International Tables for Crystallographya [ITC].
(1850); these domains are hexagons or rectangles (with squares as a particular case). Fedorov (1885) wrote a book on the d"3 case; he found the "ve combinatorial types of polytopes realizing these domains. The "rst general study in arbitrary dimension d was done by VoronomK (1908). He obtained remarkable results and these domains are called VoronomK cells in mathematics. The 3D-domains were introduced in physics much later: by Brillouin (1930) in the reciprocal space, by Wigner and Seitz (1933) in the direct space. We have already introduced VoronomK cells in Section 2, in the more general setting of Delone sets of points. We study them here; we begin to prove results in d-dimension because the proofs are exactly the same for d"3. Let us "rst prove that Euclidean lattices are a particular case:
E.S. Fedorov wrote this book between the age of 16 and 26, while serving in the army, or studying medicine, chemistry and physics. Then he became a mineralogist and six years later his book was accepted for publication in a crystallography series. No translation in a western language is known. A detailed analysis of it has been made in English by Senechal and Galiulin (1984). As we shall point out some results were "rst found by Minkowski (1897). These authors do not quote earlier references. Seitz told me that he learned Brillouin's use of the cells in the Brillouin (1931) book. The "gures are in p.304 in the Chapter added to the original French edition.
298
L. Michel / Physics Reports 341 (2001) 265}336
Proposition 5a. Euclidean lattices are Delone sets of points. There is a minimum distance between the points of a Euclidean lattice (see the end of Section 3.2). To complete the proof we have to give an upper limit of R, the largest radius of the holes. Let 2R @ the longest diagonal of the parallelepiped P (the fundamental domain de"ned in Eq. (17)) and @ S the sphere which has this diagonal as diameter; by construction this sphere has a radius 5 that @ of any hole. When we consider the G¸(d, Z) orbit of bases, R (which has no upper bound) has @ a lower bound R 5R, the radius of the largest hole. The equality occurs when P is a hypercube, @ @ so bI bI ?"I : it de"nes the simplest cubic lattice. B A fundamental domain of an Euclidean lattice ¸M is its VoronomK cell de"ned in Eq. (1); it is a polyhedron, de"ned at each point l3¸M , independent of the choice of basis. We denote it by D (l); * by the translations of ¸ one obtains all of them from any chosen one. We now study D at the * origin o; it can be de"ned directly from the translation lattice ¸LE , the vector space it spans. B Eq. (1) becomes D "+x3RB, ∀l3¸, N(x)4N(x!l), * "+x3RB, ∀l3¸, (x, l)4(l, l), .
(64)
Since the lattice ¸ is a subgroup of RB, if we consider the coset x#¸,x!¸"+x!l, l3¸,, we can interpret Eq. (64) as x3D 0 x is shortest in its coset RB/¸ . (65) * x unique shortest vector if x3 interior of D . * (66) x not unique shortest vector if x3*D the boundary . * So D is the fundamental domain of the translation group ¸. Since D has been de"ned in terms of * norm or scalar product of vectors, the symmetry group PX of the lattice is a symmetry group of the * VoronomK cell D . * We call corona the set of VoronomK cells which surround the one at the origin, i.e. they have with D (o) a non-empty intersection. J De5nition (Corona vectors). We say that 0Oc3¸ is a corona vector if the VoronomK cell centered at c"o#cOo has common points with the VoronomK cell at the origin; then c "o#c, the middle of oc is one of these common points; moreover c is the symmetry center of D (o)5D (c) . * * as 2D , we can give two equivalent de"nitions of C, the set of corona vectors of ¸: * * C"+c3¸, D (o)5D (c)O,0C"¸5*2D . (67) * * * From Eq. (66) we deduce: Denoting D
Proposition 5b. The corona vectors are the shortest lattice vectors in their ¸/2¸ cosets.
L. Michel / Physics Reports 341 (2001) 265}336
299
c3C0∀04l3¸, N(c#2l)!N(c)50 0 c ) l#N(l)50 .
(68)
Equivalently:
Remark c3CN!c3C. So, for the number of corona vectors, we have the inequality 2(2B!1)4"C" .
(69)
In Eq. (68), let us replace 2 by m'2: c3C, m'2, ∀04l3¸, m\(N(c#ml)!N(c))"2(c ) l#N(l))#(m!2)N(l)'0 .
(70)
This shows that a corona vector is the shortest vector in its coset ¸/m¸, m'2. So for m"3 and cO0, we obtain: "C"43B!1 .
(71)
De5nition (Facet vectors). When the bisector plane of the corona vector c supports a facet ("a face of dimension d!1) of D , we say that c is a facet vector. We denote by F the set of facet * vectors: F"+ f3C, dim(D (o)5D ( f ))"d!1,LC . (72) * * In the Euclidean space the equation of the plane bisector of a lattice vector f is ( f, x)"N( f ); so, when F is known, D can be de"ned as * 1 (73) D " x, ∀f3F, " f ) x)"4 N( f ) . * 2
Proposition 5c (VoronomK ). A lattice vector f is a face vector if and only if $f are strictly shorter than the other vectors of their ¸/2¸ coset. Proof of 99if::. Assume that in the same ¸/2¸ coset there are other corona vectors $c, N(c)"N( f ); so l"( f#c)3¸. Then one computes 2( f, l)"N( f )#( f, c)"2N(l); comparing with Eq. (73) this means that f is in the face of center c. With N( f )"N(c), that implies f"c. Proof of 99only if::. Assume that $c are strictly shorter in their ¸/2¸ coset and that the corona vector is not a facet vector: c belongs to the boundary of a face of center f; so (f, c)"N( f ). The middle of the vector c "2f!c is in the same face (it is the symmetric vector of c through the face center); so c is a corona vector in the same ¸/2¸ coset as c. However N(c )"N(c)# 4N( f )!4( f, c)"N(c) which contradicts the hypothesis. This proposition proves that the number of faces is 42(2B!1). A lower bound is 2d; indeed, at least d pairs of parallel hyperplanes are necessary to envelop a bounded domain. Gathering these results and those of Eq. (69) and Eq. (71) This result was "rst obtained by Minkowski (1907) for the packing of a convex body on a lattice. For the VoronomK cells the proof is so much simpler! This inequality was proven "rst by Minkowski (1897).
300
L. Michel / Physics Reports 341 (2001) 265}336
we obtain 2d4"F"42(2B!1)4"C"43B!1 .
(74)
For the dimensions 2, 3, Eq. (74) reads d"2, 44"F"464"C"48,
d"3, 64"F"4144"C"426 .
(75)
From the Proposition 5c, if "F" is maximum, F"C and conversely. If a d-dimensional VoronomK cell has 2d facets, one easily shows that each hyperplane of a parallel pair has to be perpendicular to the hyperplanes of the other pairs. That means that one can take as basis d orthogonal facet vectors; in that basis the quadratic form q(¸) is diagonal. If its diagonal elements are all di!erent, the Bravais group PX &ZB contains all diagonal matrices with elements $1; its Bravais class is usually called * `orthorhombic P-latticea. When the multiplicities of equal elements in the diagonal q(¸) are d , the G Bravais group is the direct product ; O G (Z). As we have seen, in the particular case where q(¸) is G B proportional to the unit matrix, the Bravais group is O (Z) and the VoronomK cell is a d-dimensional B cube. For all the cases in which "F""2d, all k-dimensional faces (04k4d!1) have a symmetry center; it is the middle of a corona vector; so "C""3B!1. We have also proven: Proposition 5d. The Voronon( cell has the symmetry of the Bravais group PX . The cell, its facets and the * smaller faces common with the Voronon( cells of the corona have a symmetry center. 5.2. Delone cells. Primitive lattices In Section 2 we have de"ned Delone cells and in Proposition 2a, we showed that they make a tessellation of the space; that was done in the broader frame of Delone sets with a complete symmetry between the two dual orthogonal tesselations by Delone ("rst treated) and by VoronomK cells. Here we are less general and we start from the results we have obtained on the VoronomK cells. This section is a presentation of a few results of VoronomK (1908) (the logical order is di!erent). A vertex of a VoronomK cell is at the intersection of at least d bisector hyperplanes corresponding to at least d#1 points of ¸. The vertex v is equidistant from these points; in other words: these points are on a sphere R of center v and radius R and this set is the intersection ¸5R . T T T De5nition. The Delone cell D (v) is the convex hull of ¸5R . By de"nition it is a convex polytope * T inscribed in the sphere R of center v. T In the Euclidean space E , a sphere is de"ned by d#1 points in general position (i.e. they are B vertices of a simplex). If more than d#1 points are on a sphere they are not in general position. VoronomK (1908) had de"ned these cells and began to study them for arbitrary lattices. Later, they have been studied more thoroughly by Delone.
L. Michel / Physics Reports 341 (2001) 265}336
301
De5nition. A lattice ¸ is primitive if, and only if, every vertex of its VoronomK tessellation belongs to exactly d#1 cells or, equivalently, if, and only if, every one of its Delone cells is a simplex. Proposition 5e. In the Voronon( tesselation of a primitive lattice, every m-dimensional facet F belongs B to exactly d#1!m adjacent Voronon( cells. Indeed every vertex of this facet is the intersection of d bissector hyperplanes and a m-face is supported by the intersection of d!m hyperplanes; they separate d#1!m adjacent cells. The VoronomK cells belonging to the VoronomK tessellation of a primitive lattice are called primitive Voronon( cells. There is a necessary condition to be satis"ed by primitive VoronomK cells. Proposition 5f. A d-dimensional primitive cell must have 2(2B!1) faces (or, equivalently, F"C). If this condition is not satis"ed there is a corona vector c which is not a facet vector. For instance if c is the center of an m-face (m(d!1) which is the intersection D (c)5D (o); it contains at least * * m#1 vertices. At any one of them, there is a cell whose intersection with D (o) has dimension * m(d!1, so there must be more than d cells meeting D (o) at this vertex v. This proves that the cells * are not primitive since more than d#1 cells meet at v. Beware that the converse is not true but there are no counter examples for d"2, 3; there is one in dimension 4 (which was missed by VoronomK ). The set of m-faces, 04m(d, of a lattice VoronomK tesselation can be decomposed into orbits of the translations. Let us study the intersection of these orbits with a given VoronomK cell D (o). Let * F be one of its m-faces; we have seen in the proof of Proposition 5e that it is the intersection of K d!m facets; let +l ,, 14a4d!m the set of their facet vectors; they give the centers o#l of the ? ? d!m other VoronomK cells which share this m-face with D (o). Each translation !l transforms * ? D (o#l ) into D (o) and therefore F into l #F , another m-face of the VoronomK cell D (o). * ? * K ? K * That proves Proposition 5g. A primitive Voronon( cell contains exactly (d#1!m) m-faces which can be obtained form each other by a translation of ¸. The proof also implies, with Proposition 5f, that for m(d!1 this set of (d#1!m) m-faces is transformed into a di!erent similar set by the symmetry through the origin. We shall call `familya the disjoint union of these two sets; remark that the family is the intersection of the set of m-faces of D (o) with their orbit under the action of the space group P1 , minimal space group of symmetry for * a VoronomK tessellation. We have thus obtained the simple Corollary 5g. In a primitive Voronon( cell the number of m-faces, 04m(d!1, is a multiple of 2 (d#1!m), which is the number of k-faces in the same family of congruent faces (i.e. obtained from each other by Euclidean transformations: here translations and symmetry through a point). One could have said generic, but VoronomK used the word primitive when he introduced this notion for lattices. Moreover, we have already de"ned the generic Euclidean lattices: those with the smallest possible symmetry, i.e. PX "Z (!I ) (the space group is P1 ). The primitive and the generic lattices form two open dense sets of L which do * B B not coincide: as we will see there are primitive lattices whose Bravais group is maximal.
302
L. Michel / Physics Reports 341 (2001) 265}336
From the de"nition of primitive VoronomK cells, d facets meet at every vertex; the corresponding d facet vectors span the space E . They may not form a basis of ¸; they generate only a sublattice. B This was discovered by VoronomK (1909, p. 84) for d"5. We have suggested the following de"nition: De5nition. A primitive lattice L and its Voronon( cell D are called principal if at every vertex of * D the d facet vectors form a basis of ¸. It has been proven that all primitive lattices are principal * for d44. At each vertex v of a primitive principal VoronomK cell the d facet vectors f of the facets meeting at ? v form a basis of ¸; the Delone cell D is a simplex, convex hull of the d#1 points: +o, o#f ,. We T ? remind that the volume of a d-dimensional simplex is given by volD ""det(f )"/d!. The absolute T ? values of the determinant of all bases of ¸ are equal: it is the volume of any fundamental domain of ¸ and we denote it by vol ¸. Let < be the set of vertices of D and "<" is the number of vertices; * Proposition 5g tells us that there are "<"/(d#1) vertices in a fundamental domain. Gathering all results obtained in this paragraph we deduce: for primitive principal D : * "<""(d#1)(vol ¸)/(vol D )"(d#1)! T
(76)
This proves the "rst part of Proposition 5h. In dimension d, a principal primitive Voronon( cell has (d#1)! vertices and (d#1)! d/2 edges. In a primitive lattice, d edges meet at each vertex and each edge has two vertices. This proposition is a very particular case of VoronomK 's remarkable results. Let N (d) be the number of K m-dimensional faces of an arbitrary d-dimensional VoronomK cell; VoronomK (1909, p. 78!83,136) established the upper bound of the N (d)'s. A simple closed form can be given to the K VoronomK expression (Michel 1997a) 04m4d: N (d)"(d#1!m)! SB>\K , K B>
(77)
where the Sl are the Stirling numbers of second kind, e.g. (Abramowitz and Stegun 1964, Graham B et al., 1988). Remark that the values of N (d) for m"0, 1 are those of Proposition 5h and, for K m"d!1, it veri"es Eq. (74). Notice also that N (d)"1; it corresponds to a very natural (and B usual) convention; with it the relation imposed by the Euler}PoincareH characteristic is (!1)KN (d)"1 . K XKXB
(78)
We have given the references for d"2,3 (unique combinatorial type of primitive VoronomK cell). VoronomK (1909) proved that for d"4 there are three types of primitive cells, all principal. For d"5 Baranovskii and Ryshkov (1973), completed by Engel (1986) classi"ed the 222 types of primitive VoronomK cells; 21 of them are non-principal. A famous unsolved problem is to know if every lattice has a basis among its facet vectors.
L. Michel / Physics Reports 341 (2001) 265}336
303
With our knowledge of N (d) for m"0, 1, d!1 we have for primitive cells in dimension d44: K N (2)"N (2)"6, N (3)"24, N (3)"36, N (3)"14 , N (4)"120, N (4)"240, N (4)"150, N (4)"30 . (79)
5.3. The primitive principal VorononK cells of type I VoronomK (1909, p. 137}147) de"nes and studies a combinatorial type of primitive cells; he called it type I. Consider the open set in C (Q ) made of the quadratic forms (" symmetric real matrices) > B satisfying: !q ": j '0, q ! j ": j '0 . GH GH GG GH G H H$G Notice that q is a positive matrix; indeed it de"nes the positive quadratic form 14i, j4d, iOj,
(80)
Q (x )" j x# 2j (x !x ) . (81) H G G G GH G H G G H GH Therefore there exist a basis +b , of E , de"ned up to orthogonal transformations, which satis"es G B (b ) b )"q . This basis generates a lattice ¸. Using proposition 5c on a characterization of facet G H GH vectors and Eq. (81), we "nd that facet vectors of the lattice ¸ de"ned by the quadratic form of Eqs. H (80)}(81) have for coordinates either 1's and 0's or !1's and 0's; explicitly
F(¸ )" $b ,$(b #b ), iOj,$(b #b #b ), G G H G H I H
iOjOkOi,2,$ !b # b , $ b . (82) G H H H H Let us give a more elegant writing of this set of vectors: let N> be the set of the "rst d integers '0 B and X any non-empty subset. Then we de"ne OX-N>, f " b , then F(¸ )"+ f , . (83) B 6 G H 6 GZ6 The proof is straightforward: compute Q(*) for arbitrary integer coordinates of * and transform * C f replacing the even coordinates of * by 0 and the odd ones by 1; * and f are in the same ¸/2¸ coset. One checks that Q(*)'Q( f ). One "nds that "F(¸ )""2(2B!1), the maximal possible value H given in Eq. (74). That is a necessary condition for primitivity; it has been proven by VoronomK who showed that d facets meet at each vertex (see also, Michel, 1997a). Generalizing the work of Selling (1874), written for d"2,3 we introduce some notations and the de"nition of b from the b 's; k,l3Z, 04k,l4d, kOl: G j "0, j "j '0, II IJ JI
b "0 N (b ,b )" j , (b , b )"!j . I I I IJ I J IJ I J
(84)
304
L. Michel / Physics Reports 341 (2001) 265}336
This shows a syntactic symmetry of the domain of the quadratic forms Q 's; it is made by the group H S of permutations of the values of the index k. This permutes the values of the parameters j . B> IJ If these d(d#1)/2 parameters are all equal, one obtains a lattice whose Bravais group contains the permutation group S and its "rst study was made by VoronomK (1909, p. 137) If we put in B> Eq. (81) ∀k,l: j "(d#1)\ then q"I !(d#1)\J . IJ B B
(85)
We have already found this q in Eq. (40); it de"nes the weight lattice of the simple Lie algebra A . B VoronomK showed that the cell of this lattice is primitive principal. He showed that its (d#1)! vertices form one orbit of the Bravais group and that for all positive values of the j parameters, the VoronomK cells have the same combinatorial type. As shown in Coxeter and Moser (1972, end of Section 6.2), the VoronomK cell of AU had already been de"ned in another problem of mathematics as B the Cayley graph of S . B> In a d-dimensional VoronomK cell if a vertex v is common to exactly d facets, we have already noticed that their d facet vectors form a basis of the space. An edge from this vertex is the intersection of d!1 facets, so it is orthogonal to their facet vectors; this shows that the d edges from v are carried by the vectors of the dual basis. Let us assume now that at each vertex of the cell only d facets meet. Since the set of facet vectors F(¸) generates the lattice, the edges of such a cell must be parallel to vectors of the dual lattice. It is well known in the theory of semi-simple Lie algebras that the weight lattices (the irreducible representation of the algebra are labelled by its points) and the root lattice (" the lattice generated by the roots of the algebra) are dual of each other. Moreover we can choose as basis of the dual lattice the fundamental weight whose scalar product with any root is $1 or 0. From these general facts it is easy to prove that the (oriented) edges of D B are the N"d(d#1)/2 positive roots of the Lie algebra A &S;(d#1) and that the combinatorial type of B D B (which is also the type of all the VoronomK cells corresponding to the Q quadratic forms) is that H of a (d, N) zonotope. We have to give its de"nition. De5nition (Zonotope). Given N5d vectors * 3E which span this space and are contained in M B a half-space, a (d, N) zonotope is the convex hull (de"ned up to a translation in the associated Euclidean space) of the points: o#* , X-N , * " * . We use the notation similar to that 6 , 6 MZ6 M of Eq. (83) with the di!erence that N is the set of non-negative integers 4N and X can also be , the empty subset . An equivalent de"nition is: a (d, N) zonotope is the projection of a N-cube on a d-space (when N"d it is a parallelepiped). One calls a zone of a VoronomK cell, a set of parallel edges ("1-face). For a zonotope, the edges of a zone are parallel to one of the * . One proves (see M e.g., GruK nbaum, 1967) that a zonotope and all its m-faces have a symmetry center; indeed they are This d-dimensional representation of S is also the Weyl group of the simple Lie algebra A &S;(d#1). It does B> B not contains !I , so the Bravais group is AM "A ;Z (!I ). B B B B That was not known in the lifetime of VoronomK . He died at 40, just before the appearance of the second part of his memoir. This is the case of primitive cells but also many others. In the VoronomK tesselation one can pass from any lattice point (" center of a VoronomK cell) to any other by going along facet vectors.
L. Michel / Physics Reports 341 (2001) 265}336
305
themselves zonotopes. For D B the d(d#1)/2 generating vectors are the positive roots of the Lie algebra A &S;(d#1). Using this property Michel (1997a) gives, for each type X of m-dimenB sional zonotope which can be combinatorially equivalent to a m-dimensional VoronomK cell, the number N (d, X) of copies contained in D B (these numbers are combinatorial invariants.) K As we show in Eq. (81) the forms Q are expressed in a basis of facet vectors. Let us denote the H vector generating the zonotope by eH since they are in the space of the dual lattice. Then one shows M that the quadratic forms can be written as Q(x)" j (eH, x), j '0 . (86) M M M M We can now answer the question: what happens at the boundary of the domain Q , i.e. when H j becomes 0? Since Q of Eq. (81) is of the form of Eq. (86), we know that the corresponding H generating vector of the zonotope vanishes: this phenomena is called zone contraction: one shrinks simultaneously all edges of a zone and one obtains a new type of zonotope with one less zone. When d'2, several contractions can be made successively; since the "rst contraction destroys the S syntactic geometry, there are often the choice of several inequivalent zone contractions for B> the successive ones to be made. We obtain a family of contractions with a genealogy until we "nd a non-contractible cell, i.e. any contraction would make it collapse into a smaller dimension. Starting from D B , there is a unique end for the family of contractions: the rectangular parallel epiped with d zones. Before going to the dimensions 2,3, let us emphasize that contractions can be made on every contractible VoronomK cell. But we have to distinguish between closed zones (all 2-faces contain 2 or 0 edges of the zone) and open zones (there is at least a 2-face containing only one edge of the zone). A VoronomK cell which has only open zones is non-contractible. In dimension 4 there are two families of contractions: (i) the one of non-zonotopes has two maximal primitive principal cells with 30"2(2!1) faces and 10 closed zones; it has 37 members and the terminal non-contractible one is the regular polytope of symmetry F ; (ii) the one of the 17 zonotopes with two maximal cells, the primitive D B and a non-primitive one with 30 faces and nine zones. These are the only maximal cells missed by VoronomK . There are 84 contraction families in dimension 5 (Engel, 1986). They have not been counted for d"6; Engel (private communication) has studied some of them and he has found contraction families containing only one member! 5.4. VorononK cells for d"2, 3 d"2. Any two variable quadratic form can be written as Q(x)"q x !2q x x #q x . (87) If the coe$cient of the x x term were positive, this form would be obtained by changing the sign of one coordinate. We learned from Selling (1874) that an e$cient way for studying this form is to introduce the parameters j 50: IJ j #j !j , (88) q" !j j #j
306
L. Michel / Physics Reports 341 (2001) 265}336
then Q(x)"j x #j (x !x )#j x . These two equations are the 2-D particular case of the Eqs. (80)}(81) for arbitrary dimension. When the three j are '0, with Proposition 5c (a VoronomK theorem), one can verify by the same method IJ given for any d after Eq. (83), that the basis vectors b (de"ned up to a rotation by Eq. (87)) are facet G vectors as well as b #b and the negative of these three vectors. So the VoronomK cell is a hexagon with a symmetry center. The six edges are orthogonal to the facet vectors and Eq. (88), a particular case of Eq. (86), gives their direction in the dual basis. Corollary 5g tells us that the six vertices of the hexagon form a unique family: so every two-dimensional VoronomK cell is inscribable in a circle. From this property we immediately obtain that when the edges of a pair of facets symmetric through the center shrink to zero, one obtains a rectangle. That is the simplest example of zone contraction; analytically it is obtained of Eq. (88) by making one of the three j coe$cients "0. The sides of the rectangle carry the coordinates axis when j "0. If a second j is 0, then det q"0 which con"rm the obvious fact that a contracted rectangle collapses to a one dimensional segment. Let us apply what we know on the 2-D VoronomK cells to the structure of those in arbitrary dimension. Another important concept for VoronomK cells is that of a belt: consider a (d!2)-face F ; it belongs to a facet. Let F be the symmetric image of F through the center of this facet; it is at the intersection of two facets. Let F be the symmetric image of F through the center of the other facet. Repeating the same operation we obtain a chain F which has to be periodic since the ? set of (d!2)-dimensional faces is "nite; what is its period? De5nition. A belt of a VoronomK cell is the set of (d!2)-faces parallel to a given one. The (d!2)-faces of a belt are orthogonal to a 2-plane. One can show (Venkov 1959) that the orthogonal projection of D on this two-plane is a two-dimensional VoronomK cells. That proves: * Proposition 5i. A belt of a Voronon( cell has six or four (d!2)-faces and therefore six or four facets. In the four facet belts the hyperplanes of adjacent facets are perpendicular. All the belts of a VoronomK cell which have the maximal number of faces ("F""2(2B!1)) have 6 facets. For d"3 the belts and the zones are the same; this will help us for the study of this dimension. d"3. Any three-variable quadratic form can be written in the form of Eq. (80) with (see Eq. (84)) the j 50. It is easy to choose a basis of three vectors b generating the lattice such that their three IJ G scalar products are 40; so the three j are 50. But to have that the 3 j are also 50 one may GH G have to transform this basis. This has been done by Selling (1874) and Delaunay (1932a, b) The Delaunay (" Delone) algorithm for transforming any generating basis of a lattice into a basis (80) is given in the old edition of international tables of crystallography (ITC, 1952, Vol. I, p. 530}535), but has been suppressed in the new one! This implies that all 3D VoronomK cells belong to the family of contractions of the primitive cell corresponding to the quadratic form of Eq. (80). This family It is obvious for a rectangle. For the general cell, I never found this statement and I would be grateful to be given a reference.
L. Michel / Physics Reports 341 (2001) 265}336
307
Fig. 6. The "ve types of VoronomK cells for d"3. P is the primitive cell with minimal symmetry. V, F, H, D, C show the "ve cells with their maximal symmetry. The middle of the facets are marked by #. The middle of corona vectors are marked by a heavy dot 䢇 ; they are vertices or middle of edges common to the cell and its neighbors (which form the corona) in the VoronomK tessellation. or ; marks the other high-symmetry points of the Brillouin cell for lattices with high symmetry (see the next subsection). They are in a direct relation with those of Table 3.
contains "ve types which are shown in Fig. 6; for V, F, D, H, C the cells are shown in the form of their highest symmetry while P is the primitive one in a low symmetry. The # marks the symmetry centers of the facets (" middle of facet vectors); the small 䢇 indicate the middle of the corona vectors. We call these two kinds of points contact points; indeed they are in the intersection of the cell with the neighboring (" member of the corona) cells in the VoronomK tessellation. When they are not vertices, the intersection contains the facets or the edges which contain them as symmetry centers. It is easier to study the primitive VoronomK cell in its highest symmetrical form corresponding to Eqs. (80) and(84) when all j are equal and already found in Eq. (40): q"I !()J . The cell is IJ obtained by truncating the vertices of a cube (or a regular octahedron) by a plane orthogonal to the diagonals and such that some facets are regular hexagons (there are eight of them); what is left of the cube faces are six squares. We shall use also the notation "F"("<") i.e. number of facets followed (in brackets) by the number of vertices for labelling the "ve types of cells. 14(24). V,P. The number of k-faces has been given in Eq. (79): 14, 36, 24. In the case of higher symmetry (Fig. 6, V) its Bravais group is Im3 m. In the orthonormal coordinate system de"ned by the facet vectors of the truncated cube the coordinates of the 14 facet vectors are: eight vectors with (e , e , e ), e"1 (the corresponding facets are eight regular hexagons) and six vectors with G [2e , 0, 0] (when the coordinates are between [ ], apply to them all permutations) whose correG sponding facets are squares. The 24 vertices are trivalent i.e. each one is the intersection of three Beware that, although they are also facet vectors of the primitive cell 14(24), they are not generating the primitive lattice; indeed the lattice determinant is 4 while the basis determinant is 8.
308
L. Michel / Physics Reports 341 (2001) 265}336
Table 5 The "ve types of 3D VoronomK cells "F"
F-edg. 6
4
8 4
6 8 12 6 6
2
"E"
"<"
<-val. 4
14 12 12 8 6
36 28 24 18 12
24 18 14 12 8
2 6
¸/2¸
"C"
3 24 16 8 12 8
2.2.2.2.2.2.2 2.2.2.2.2.2.4 2.2.2.2.2.2.6 2.2.2.2.4.4.4 2.2.2.4.4.4.8
14 16 18 20 26
Zones belts 6 5 4 4 3
z"b 6 6 4 4 1
4
1 3 3
Listing of the columns: "F", "E", "<" give the number of facets, edges, vertices (we recall that "F"!"E"#"<""2); F-edges gives the number of facets (" polygons) with 6,4 edges, <-valence gives the number of vertices at the intersection of 4, 3 edges. ¸/2¸ has 7 non-trivial cosets; the number of shortest vectors in each coset is given (they are corona vectors), the sum of 2's is "F", the total is "C", the number of corona vectors. In 3D-case the zones and the belts are identical; their numbers are given, and the last column gives the numbers of them with 6, 4 elements.
edges (also three facets); their coordinates are: [e , e /2, 0]. It is a zonotope generated by the six vectors [1, e , 0] where [ ] tells us to use only the circular permutations. Any transformation of G this set of six vectors by an element of G¸(d, R) generates a combinatorially equivalent zonotope (Fig. 6, P). It has six belts ("zones) containing four hexagons (with a symmetry center), two parallelograms and six parallel equal edges which form a family (Corollary 5g). In Fig. 6, the symmetry center of a facet ("middle of a facet vector) is marked by the sign #. The information on the "ve types of 3D VoronomK cells are recapitulated in Table 5. 12(18). F (We choose F for Fedorov). In the contraction of any zone of 14(24) the two parallelograms collapse (each one into an edge), the four hexagons are transformed into quadrilaterals, and the six edges become points. So the number of 2, 1, 0-faces of the contracted polytope are 12,28,18; among these vertices two are tetravalent (they come from the collapse of the two zone edges which were between two hexagons) and the 16 others are trivalent. The facet vectors of the two collapsed parallelograms become corona vectors and at least two other corona vectors in the same ¸/2¸ coset must appear in the contraction. Indeed the middle of each of the "rst two corona vectors are the middles of an edge between two hexagons; they belong to a four-element zone ("a four-hexagon belt) so the middle of its four edges are the middle of the four corona vectors (they are marked by a heavy dot in Fig. 6). Moreover, from proposition 5i, a section of this belt by a plane orthogonal to the four edges is a rectangle. For the maximal symmetry (it is tetragonal) this section is a square and the parallelograms are rhombs. The four other zones (" belts) are equivalent; each one contains two hexagons and four parallelograms. The two types of zone will lead to two di!erent contractions. 12(14). D (for rhombohedral dodecagon). It is obtained from the contraction of the four-edge zone (" four-hexagon belt) of 12(18). The four hexagons become four parallelograms; the four edges (and their middle of corona vectors) become tetravalent and middle of corona vectors; that is also the case of the two tetravalent vertices of 12(18). The other eight trivalent vertices are preserved and no facet has disappeared in the contraction, so the number of 2, 1, 0-faces are respectively 12, 24, 14. All facets are parallelograms and there are four zones"belts of six edges and six parallelograms; they are equivalent. Beside the 12 facet vectors, the six four-valent vertices are
L. Michel / Physics Reports 341 (2001) 265}336
309
corona vectors in one ¸/2¸ coset. There are four equivalent zones"belts of six edges, six parallelograms. The maximal symmetry is Fm3 m and the cell is the convex hull of the centers of 14(24) faces (it is easy to see it in Fig. 6 D,V); the centers of the six squares, eight hexagons become the six 4-valence, eight 3-valence vertices, respectively. The 12 facet vectors have four coordinates in the orthonormal basis used for 14(24): [e , e , 0] . Notice that all faces are rhombs with a ratio of diagonal lengths "(2. 8(12). H (" hexagonal prism). This cell is obtained from the contraction of a zone of 12(18) which contains two hexagons and four parallelograms. So four facets disappear and two hexagons are transformed in quadrilaterals; the two other hexagons stay and have no common edges while the parallelograms have one common edge with each of the two hexagons. So combinatorially it is a hexagonal prism: the 2, 1, 0-faces numbers are, respectively, 8, 18, 12. The two hexagons belong to three four-belts. Proposition 5i tells us that in each belt the two pairs of parallel planes are orthogonal; this implies that the six quadrilaterals are rectangles. In Fig. 6 H we put the hexagons horizontally and the rectangles vertically. Obviously the hexagons are VoronomK cells of the 2D horizontal sub-lattice; we showed that they are inscribable in a circle, so all 8(12) cells are inscribable in a sphere. The centers of the facets which have disappeared in the contraction of 12(18) as well as the middle of the corona vectors of the later cell, form 12 corona vectors (four in each of the three ¸/2¸ cosets they belong); their middles are the middles of the horizontal edges. It is easy to visualize from Fig. 6 H that these edges are common to the cells of the corona. The maximal symmetry of this hexagonal prism is obtained when the two bases are regular hexagons; this is the VoronomK cell of the lattices of the hexagonal Bravais class. It is visible in the "gure that the prism collapses in the contraction along the vertical edges (which form one zone). The zone contraction along any of the three other zone"belts contracts the two hexagons into a rectangle (as we have seen it in d"2), so the cells become a rectangle parallelepiped. 6(8). C. We have seen how the hexagonal prism contracts into a rectangle parallelepiped which, in its highest symmetry form is a cube. The numbers of 2, 1, 0-faces are 6, 12, 8. To the three cosets of four corona vectors of the prism 8(12) one has to add a new coset coming from the two disappearing rectangles: it gives one family of eight vectors whose middles are the vertices. The rectangular parallelepiped can also be obtained by the contraction of the rhombohedral dodecagon 12(14) along any of its four equivalent six-zones. To the six corona vectors of 12(14) forming one coset one has to add three cosets of corona vectors: the facet vectors of the three pairs of rhombs which collapse into one edge in the contraction. The middle of these three cosets of corona vectors are the middle of the edges of the three zones of the rectangular parallelepiped each one has four edges in the same orbit of the group of lattice translations. The contact point of 6(8) in the corona are well known and obvious in Fig. 6. This cell has three equivalent zones " belts; since they have four elements, the rectangular parallelepiped collapses in any zone contraction. 5.5. High-symmetry points of the Brillouin cells We have introduced the reciprocal space and the Brillouin zone ("BZ) in Section 4.4. Here we call the VoronomK cell of the reciprocal lattice 2p¸H, the Brillouin cell (indeed it was introduced by Brillouin, 1930). It is a geometrical representation of the Brillouin zone; the torus of BZ is obtained when one identi"es in any pair of facets F with opposite facet vectors $f the points correspond! ing by lattice translations i.e. F U x x#f 3 F . So a point at the intersection of m facets is \ >
310
L. Michel / Physics Reports 341 (2001) 265}336
identi"ed with at least m other points of the Brillouin cell and they all represent the same element of BZ. For instance: Proposition 5j. In a Brillouin cell, the symmetry center (generally denoted by C in the physics literature) represent the 0 of the group BZ; the middles of the corona vectors belonging to a ¸/2¸ non-trivial coset represent an element of order 2 of the group BZ (i.e. 0OkK , 2kK "0). We have shown that there are 2B!1 such points. In Fig. 6 these points are marked by # when they are the center of a facet and by 䢇 otherwise. In this section we want to identify on the Brillouin cell the other `high-symmetrya points of BZ, i.e. those of the zero-dimensional stratum in the actions of the Bravais groups on BZ; those elements are listed in Tables 2 and 3 for d"2, 3. Let us begin with d"2. The "ve Bravais groups are self-dual. Moreover to a Bravais class corresponds a unique type of Brillouin cell; the correspondence (type of Brillouin cell) Q Bravais class (the Bravais group is added in ( )): hexagon for the diclinic (p2), c-orthorhombic (c2mm), hexagonal (p6mm); rectangle for the p-orthorhombic (p2mm), square (p4mm) (the rectangle is a square). As we have seen the hexagon is the primitive cell of the quadratic form Eq. (88) with the three parameters j '0. In the generating basis b ,b we can compute the coordinates of the vertices. IJ They are rational when the j's are rational, so in that case there is an integer m such that for each vertex *, m*3¸. We have already seen that the vertices (for any symmetry) form one family which splits into two orbits for the translations so they represent two points in BZ of order m , m . That is irrelevant; what we want to know is `when do they belong to a stratum of dimension 0 for the action of the Bravais group?a. That occurs for the largest symmetry only, p6mm; then, for the six vertices, 3*3¸. So they correspond to two points of order 3 in BZ; their coordinates in the dual basis are given in Table 2. The rectangular cell has two pairs of facet vectors; they correspond to two points of index 2 in BZ. The third one represents the four vertices; they are the middles of four corona vectors in the same ¸/2¸ coset. d"3. It would be too long to study with the same details the three-dimensional case. Notice that there are no new concepts to introduce from what we have done for d"2. But a new fact is that several types of Brillouin cells may correspond to a Bravais class. For instance, let us consider the open dense domain of generic lattices, i.e. those of minimal symmetry ("P1 ) in L (the topological space made by the set of all 3D lattices); in a dense six-dimensional subdomain (that of j '0), the IJ Brillouin cells are primitive (14(24)), but there is a "ve-dimensional subdomain where they are 12(18) and even a four-dimensional boundary between those two domains where the Brillouin cells are 12(14): see Delaunay, (1932a, b) and Delone et al., (1974), (in Russian, but explained in Michel, 1995). We give the full correspondence between the 14 Bravais classes and the "ve types of VoronomK cells in Table 6. Most of the information * but not all of it * can be found in some classical books, e.g. Zak et al. (1969) and Bradley and Cracknell (1972). They are given in Michel (1995), Eq. 3(22); they are the quotient of two homogeneous polynomials of degree 2.
L. Michel / Physics Reports 341 (2001) 265}336
311
Table 6 The Brillouin cells corresponding to each Brillouin class in 3D. When there are two cells, oblate and prolate qualify the shape of the fundamental domain built on the generating basis. When there are three cells, the domain of 12(14) in C (Q ) > or L , is a boundary between the domain of the two other cells; so its dimension is smaller. P-cubic (Pm3 m), P-tetra (P4/mmm), P-ortho (Pmmm), P-hexa (P6/mmm), C-ortho (Cmmm), P-mono (P2/m), I-cubic (Im3 m) F-cubic (Fm3 m), I-ortho (Immm) I-tetra (I4/mmm) R-rhombo (R3 m) F-ortho (Fmmm), C-mono (C2/m), P-tric(P1 )
6(8) 8(12) 12(14) 14(24) Oblate: 14(24), Prolate: 12(18) Oblate: 14(24), Prolate: 12(14) 14(24), 12(18), 12(14)
With this information, and starting from the highest symmetry lattices, the reader easily identi"es for each Bravais class the BZ points of order '2 in the zero-dimensional strata tabulated in Table 3 with the points marked by a or a ; in Fig. 6. These BZ points have been computed explicitly for building Table 3 with the use of a very natural system of coordinates adapted to the property of BZ to be an Abelian group on a 3D torus. Let us review rapidly these points in Fig. 6. 6(8) There are none of them on the cube or the rectangular parallelepiped. 8(12) Let us "rst consider the two-dimensional sub-lattice ¸ L¸ in the horizontal plane containing the center of the hexagonal prism; it cuts the prism along the hexagon " VoronomK cell of ¸ . If the Bravais class of ¸ is P-hexagonal, that of ¸ is p-hexagonal and as we have seen for d"2, the vertices * of its cell (which are the middles of the vertical edges of the prism marked by ) ? correspond to the elements of order 3 of the Brillouin zones of ¸ ; so they also correspond to the elements of order 3 of the BZ of ¸ (see Table 3). The vertices of the prism have the same horizontal coordinates while their vertical coordinate is p (half of the period 2p); they correspond in the ¸ Brillouin zone to the points of order 6 (the smallest common multiple of 2 and 3) and they are marked by ;. 12(14) For the lattice Im3 m, in the coordinate system of Section 5.4, the eight vertices of valence 3 of this cell can be identi"ed with the middles of the eight hexagonal facets of 14(24); so their coordinates are * "(e , e , e ). The six vertices of valence 4 can be identi"ed with the middles of ? the square facets of 14(24); so their coordinates are * "[e , 0, 0] . The 12 facet vectors generate the M G lattice; their coordinates are f "[e , e , 0] . Since 2* 3¸, the verticies of valence 4 are marked by N M a 䢇: indeed these six vertices represent one of the seven elements of order 2 in BZ while the six others are represented by the middles of opposite facets (we remind that they form one orbit of the Bravais group). Since 4* 3¸, the vertices of valence 3 are marked by a and represent two ? elements of order 4 in BZ; this relation disappears when the lattice symmetry is smaller. 14(24) We have studied in detail the structure of this cell. The middle of the faces of the eight hexagons, six squares form the orbits of 4, three elements of order 2 in BZ (so no points are represented by 䢇). For the symmetry Fm3 m, the 24 vertices satisfy 4* 3¸ and represent the orbit of ? six elements of order 4 in BZ; that is why all vertices are marked by a . For smaller Bravais groups (I4/mmm, Immm) that is no longer true, but a subset of them still represents an orbit of two elements of order 4 in BZ.
312
L. Michel / Physics Reports 341 (2001) 265}336
Remarks. Do not forget that the system of coordinates used in this section (and in ITC) for F and I-cubic system is not a basis generating the lattice that we use generally elsewhere, and in particular in Table 3. But the geometric results (for instance in terms of ¸/2¸ coset in the direct or reciprocal lattice) are independent of the choice of coordinate system. This has also to be true of any physical property of a crystal! The VoronomK cell, which is the Brillouin cell for the reciprocal lattice, has the great quality to be coordinate independent; however, the group structure ;B that carries the Brillouin cell is not directly apparent. Not only several points of the surface of the Brillouin cell describe a unique element kK 3BZ but it is not easy to "nd where is 2kK in the Brillouin cell, except in the special case of middle of facet vectors (marked #) or corona vectors (marked 䢇) for which 2kK is represented by the symmetry center of the cell (the 0 of the group). If kM is of order 4, then 2kK is of order 2: which element among the seven ones existing on BZ? Let us consider for 12(14) the eight vertices of valence 3 * ; from their coordinates given above for the maximal symmetry Im3 m we see that ? 2* !* 3F(¸) and that shows that it is the element of BZ represented by the vertices of valence 4. ? M Similarly, for the maximal symmetry Fm3 m, the 24 vertices of 14(24) represent an orbit of six elements kK of order 4 in BZ. Which centers of faces represent the 2kK ? They must form an orbit ? ? whose number of elements is a divisor of 6; it is the three-element orbit represented by the centers of the square facets and Table 3 tells immediately which square center correspond to 2kK for a given vertex. For 8(12), the hexagonal prism, the 12 vertices (marked by ;) represent an orbit of two elements of order 6 of BZ whose double is represented by the middles (marked by ) of the six vertical edges; those represent two elements of order 3 which constitute with 0 a group &Z . The geometry of the Brillouin cell is interesting; many monocrystals have their shape or the shapes obtained by translating some facets. Simple physical assumptions explain these facts. But for the quantum theory of solids it is the group law of BZ which is fundamental while the geometry of the Brillouin cell is secondary. So we stop here the unnecessary lengthy study for the cases of the last three lines of Table 6, for the "ve Bravais classes whose BZ is represented by two or three types of Brillouin cells.
6. The positions and nature of extrema of invariant functions on the Brillouin zone This section is an application of Section 4.4, mainly the Tables 2,3 and several theorems and methods explained in Chapter I. Physical functions de"ned and measured on the Brillouin zone of The bases generating a lattice are called primitive in ITC. For symmorphic space groups (they are de"ned in Section 7.2) only, the VoronomK cell of their translation lattice has also a natural structure of a group, that of ;B . Brillouin introduced Brillouin cells in papers or books written in French and German. The fundamental paper (Bouckaert, Smoluchowski, Wigner 1936) for the theory of bands introduced the cubic Brillouin cells and studied their strata (the letters they used to label them have been generally kept up to now); however the authors say in conclusion, near the end of the introduction: `Thus a certain topology for the representations must exist and it will be shown that part of this topology is independent of the special B-Z.a. See also, for instance the beginning of their Section IV which inspired the treatment presented here for BZ. Contrary to a popular belief, this paper does not mention Born and von Karman (1912) method which replaces the space group by a "nite quotient G/m¸. This was a remarkable trick when it was invented, but it kills the topological structure of BZ.
L. Michel / Physics Reports 341 (2001) 265}336
313
an isolated crystal are invariant by the space group symmetry. We have explained in Section 4.4 that the e!ective action of the space groups on BZ depends only on its arithmetic class; so we have only 13, 73 cases (for d"2, 3) to study when spin e!ects are irrelevant. We restrict furthermore the scope of this section to time reversal invariant phenomena (e.g. energy of bound electrons, Fermi surface, vibration spectrum, etc.); in Section 4.5 we showed that with this assumption, we have only 7, 24 cases to study. Among them there are the actions on BZ of the 5, 14 Bravais groups PX ; as we * will see, the results are very similar for the 2, 10 other arithmetic classes containing the symmetry !I . Then for di!erentiable functions, Theorem 4b in Chapter I tells us that all these functions B have an extremum at each high symmetry point of BZ, i.e. the points of the strata of 0-dimension in the action of PX on BZ. Such type of arguments were already used in particular cases by Wigner and Seitz (1933) (in direct space), and by Bouckaert et al. (1936, Section IV). Morse theory, explained in Chapter I, Section 6 makes predictions on the nature of these extrema for `Morse functionsa, i.e. the functions whose extrema are isolated. It was "rst used by Van Hove (1953) for this problem, using only the invariance under the lattice translations; he proved that the number of extrema on BZ is 58. In the same paper Van Hove (1953) studied the singularities which are now called by his name; he proved, as an example, that density of vibration states may have a logarithmic singularities for d"2 and singularities in the derivatives of a continuous function for d"3. Morse himself extended his theory to such functions; the topology of their level lines determines the nature of the extrema and tells if they are isolated. Those assumptions are satis"ed for most physical applications of the present chapter. Phillips (1956) and Phillips and Rosenstock (1958) extended the translational invariance to that of the full space group on some cubic examples. But a systematic treatment of this problem had never been done before 1996. To study the consequences of the symmetry under PX, we have also to use Theorem 4c and Corollary 4c from Chapter I for BZ and apply them to the closure of each stratum. For the d"2 the results are given in Section 4, Table 2. We give in Table 7 the results for dimension 3 (this allows me to correct one error * for the I-cubic system * in the previous publication (Michel, 1996)). This table gives for each of the 24 cases the minimum number of extrema for the invariant functions. Most of these extrema belong to critical orbits; when it is not the case we give the conditions which can be established for their location. In Chapter I we have called the simplest Morse functions those which have the minimum number of extrema. To help reading Table 7 we recall that the Hessian H ( f ) is the symmetric 3;3 matrix N ("quadratic form) of the value at the point p of the second derivatives of the function f so its eigenvalues are real. At an extremum, by de"nition of a Morse function, H( f ) has no vanishing eigenvalue ("det HO0) and the number of its negative eigenvalues is called the Morse index of
Singularities can also be added arti"cially. The decomposition of band systems into `distinct bandsa is often done in text books by the following procedure: the "rst band is de"ned by the lowest-energy value at each kK 3BZ; then remove this band from the graph and de"ne each next band in turn by the same procedure. For instance, at each crossing point between the branches of an analytic function on a multi covering of BZ one obtains a maximum with discontinuity in the derivatives. Van Hove already alluded to this procedure and had doubts on its interest. In the Appendix B to Chapter I we sketch an extension of Morse theory to functions with degenerate Hessians at the extrema, which can be very necessary in the case of symmetry under a compact group (not "nite!). That is not the case here; however we apply (in the very easy case of dimension 1) another extension of Morse theory justi"ed in Goresky and MacPhersons (1980).
314
L. Michel / Physics Reports 341 (2001) 265}336
Table 7 Minimum number of extrema and their positions for the functions on the three-dimensional Brillouin zone, invariant by the crystallographic group and time reversal
Column 1 lists the 24 arithmetic classes obtained from Table 4. Columns 2 and 3 give the critical orbits kK "0 and the seven elements 2kK "0 in BZ; they are listed by their number of points. With the same notation, columns 4}6 (depending on the order of kK ) give the number of points of other critical orbits when they exists; [ ] are orbits of maxima or minima. Column 7 gives the minimum number `nba of extrema for any invariant function as a sum of the number of critical and non-critical points. When Morse theory requires that it must have extrema outside the critical orbits, the smallest orbit of those extrema is given between parentheses ( ), + , or +). Columns 8}11 give the orbits of extrema with a given Morse index. The last column gives the corresponding polynomial Q(t).
L. Michel / Physics Reports 341 (2001) 265}336
315
the extremum (it is 0 for a minimum and d (" the dimension of the manifold) for a maximum. At a point p3BZ, we denote as usual PX the stabilizer of PX and R its (real) representation. If f is N a PX-invariant function, its Hessian at p3BZ must satisfy g3PX , H ( f )"R(g)H ( f )R(g)?. If the N N N representation R is irreducible (on the real), the (non-degenerate) H ( f ) has to be a multiple of the N identity operator; this implies that the extremum is either a maximum or a minimum. Orbits which satisfy this condition are given between [ ] in Table 7. In this table we see that for eight Bravais groups (and two others: R3 , Pm3 ) the number of critical points is 8. Every invariant function has extrema at these critical points. There exist functions, the perfect Morse functions, with no other extrema; indeed, the corresponding Q polynomial (introduced in Chapter I, Eq. (132)) vanishes. In the rhombohedral system, these functions have a maximum and a minimum on the orbits of one point. In the P-cubic system, that is true for all invariant functions (in the Brillouin cell the two points correspond to the center of the cube and to the set of eight vertices). For the action of Fmmm there are also only eight critical points; they belong to "ve orbits; but, because there exists an orbit of four points among them, the polynomial Q cannot vanish. So there are no perfect Morse functions and any invariant function must have extrema somewhere. The closures of the one-, two-dimensional strata have, respectively, two and four critical points which are the "xed points of the groups (use Eq. (56) and Table 3); that imposes no condition on the other extrema. The smallest non-critical orbits have two elements (they are on a rotation axis with stabiliser Fmm2); so the functions with the minimal number of extrema have 10 of them. They are the simplest Morse functions; this situation is explained by 8#(2) in the column `nba of Table 7. For four Bravais groups (and three others I4/m, P3 m1, P6/m) the number of critical points are 10, 12, 14. There exist simplest Morse functions with their extrema at these points. Morover for Fm3 m, all invariant functions must have a maximum or a minimum at kK "0. In the case of the (last studied) Bravais group, Im3 m, the minimum number of maxima and minima for the invariant functions is 4; they are located at the center and the quadrivalent vertices of the Brillouin cell (they represent the two "xed points of BZ) and at the trivalent vertices (they represent the critical orbit whose two elements satisfy 4kK "0 and 2kK is the "xed point O0). With the critical orbit of six points, it is impossible to "nd a Morse function which has extrema only on the 10 critical points. The situation is similar to that of Fmmm. We verify that no new conditions for the location of extrema appear from the study of the closure of strata of positive dimension. The largest stabilizers for the non-critical points are those of the conjugacy class of I4mm; the corresponding orbits have six points (they are generic points of the three rotation axes of order 4). This explains the 10 (6) in the column of `nba. So for the I-cubic Bravais class, the simplest Morse functions have 16 extrema. There are "ve arithmetic classes in the hexagonal system which contain !I . For two of them the two two-element orbits in the columns 3kK "0 and 6kK "0 are not critical; indeed the respective stabilizers (obtained from Table 3 and Fig. 3 from Chapter I) P31m"P6 2m5P3 1m and P3" P6 2m5P3 are those of a one-dimensional closed stratum (made of two circles in BZ and represented by the vertical edges of the Brillouin cell). From Corollary 4c of Chapter I, every invariant function must have at least two extrema on each of the two connected components of the
Van Hove and his successors knew it for the P-cubic.
316
L. Michel / Physics Reports 341 (2001) 265}336
stratum. It is possible to have only two extrema, a maximum and a minimum, on each onedimensional component; the two maxima and the two minima form two two-element orbits for the symmetry group. Hence the notations + , for the non-critical orbits, 8#+4, for eight critical extrema (common to all invariant functions) and four con"ned on the two components of a closed stratum. The last case to study is Fm3 . The stabilizer of the orbit of six elements of order 4 is Fmm2"Fm3 5I4 2m (from Table 3 and Fig. 5). Since mm2 is an axial group, it is the stabilizer of a one-dimensional stratum and the orbit is no longer critical. Then the arithmetic class Fm3 has only three critical orbits; Morse theory requires more extrema. We have to study the closure of strata of positive dimension. One of them gives a new condition; it is the stratum with the stabilizer Fmm2. To close this stratum in BZ we must add kK "0 and this closure SM is made of six circles with one common point (kK "0); so SM is not a manifold! However it is obvious that on each circle one must have (at least) another extremum. It is possible to add only one extremum if on the six circles they are on the same orbit Fm3 :Fmm2. This orbit is indicated in the column 4kK "0 by +6). Similarly, in the column `nba there is 8#+6). As a conclusion, we obtain: Theorem 6. The number of extrema for the simplest Morse functions on BZ, invariant by the space group and time reversal, depends only on the Bravais class. It is 16 for I-cubic, 14 for F-cubic, 12 for P-hexagonal, 10 for I-tetragonal, I- and F-orthorhombic and eight for the eight other Bravais classes.
7. Classi5cation of space groups from their non-symmorphic elements 7.1. Action of the Euclidean group on its space The crystallographic space groups are the subgroups of the Euclidean group Eu which contains B a d-dimensional lattice of translations. We "rst introduce notations for the Euclidean group and recall simple properties of its law and its action on the Euclidean space E . B The Euclidean group Eu "RB)O is the semi-direct product of the orthogonal group by the B B translations; moreover one can prove that all subgroups O (Eu are conjugate in Eu . Let us B B B choose an origin o on the Euclidean space E ; then every point x3E can be labelled by the vector B B x which translates o to x. Every element of Eu can be written as the product of "rst, an orthogonal B transformation A and second, of a translation s. We write such an element +s, A,. Its action on the point x3E is B +s, A, ) x"s#Ax .
(89)
As a semi-direct product, the Euclidean group law is +s, A,+t, B,"+s#At, AB,, +s, A,\"+!A\s, A\, .
(90)
To obtain the form of the elements of Eu with the origin o of the coordinate system we conjugate B the elements by the translation o o. Explicitly the conjugation by the translation t gives +t, I,+s, A,+!t, I,"+D t#s, A, with D "I!A .
(91)
L. Michel / Physics Reports 341 (2001) 265}336
317
From Eq. (89) we obtain the condition for the existence of "xed points of +s, A,; let x be such a "xed point +s, A,x"x 0 s"(I!A)x i.e. s3Im D . (92) For instance, if D is invertible, i.e. A leaves no vector "xed ("A has no eigenvalue 1), for every translation s, the Euclidean transformation +s, A, has the unique "xed point x"D\s. This is the case of the connected Euclidean group in the plane, Eu>. A non-trivial A is a rotation and any element of Eu> which is not a translation, is conjugated (by the translations) to a rotation. In the general case D has a kernel Ker D : it is the subspace of vectors w satisfying D w"0. Since A is orthogonal, one proves that E is the direct sum of the two orthogonal spaces Im D and B Ker D : , E "Im D Ker D . (93) B Indeed let s"D t and w3Ker D , i.e. Aw"w. Then (s, w)"(t!At, w)"(t, w)!(At, Aw)"0 since A is orthogonal. If the Euclidean transformation +s, A, has a "xed point o#t, the set of "xed points is of the form o#t#Ker D . We can summarize the obtained results in the Proposition 7a. The Euclidean transformation +s, A, has a xxed point 0 s3Im D , i.e. t, s"D t 0 the component of s in Ker D is trivial. The set of xxed points is t#Ker D . When +s, A, has a "xed point, Eq. (91) shows that by taking this point as origin one transforms +s, A, into +0, A,. It is a trivial remark to say that any non-trivial coset of translations in Eu B contains elements with "xed points (subtract to s its component in Ker D ); that is no longer trivial for a space group because its subgroup of translations, ¸, contains not a continuous set of elements, but only a countable one since it is a discrete subgroup of RB. Hence, unlike the full Euclidean group, a space group may not be a semi-direct product. Let us consider "rst the case of space groups which are semi-direct products. Given a "nite subgroup PX of G¸(d, Z), it acts naturally on a lattice ¸ and we can de"ne the semi-direct product ¸)PX by the law (90). One obtains the same space group by taking another group mPXm\, m3G¸(d, Z), of the same arithmetic class in the m transformed lattice basis. So for d"2,3 there exists, respectively, 13, 73 space groups which are semi-direct products. The crystallographers call them symmorphic. In this section we want to characterize the 17!13"4, 230!73"157 non-symmorphic space groups. They will be characterized by their action on the Euclidean space. We "rst give general features of this action for all space groups. 7.2. Space group stabilizers and their strata (" Wyckow positions) F P"G/¸. Translations have Given a space group G, we recall the group homomorphism GP no "xed points, so a stabilizer G does not contain translations and G &h(G )4P; in plain words V V V the stabilizers are "nite subgroups isomorphic to subgroups of the point group. We recall that two points x and y are in the same stratum when their stabilizers G and G are conjugate in G. Then V W their image by h are conjugated in P. The converse is not true but one can prove that the number of
318
L. Michel / Physics Reports 341 (2001) 265}336
conjugacy classes of "nite subgroups of G mapped by h on a conjugacy class of P is at most 2B and this implies that in the action of G on the Euclidean space the number of strata is "nite. For each space group in dim d"2, 3 ITC give the list of strata and their structure under the heading Wyckow positions. We recall that the set of strata has a natural partial order (induced from the partial order of the conjugacy classes of "nite subgroups of the space group G). Proposition 7b. Every xnite group stabilizes points of space. If it is a maximal xnite subgroup of G, it is a maximal stabilizer of G. Given any "nite subgroup H of the Euclidean group, for any point x of space the barycenter of the orbit H ) x is invariant by H. If H is maximal among the "nite subgroups of G, it is a stabilizer of G. Proposition 7c. If G is a stabilizer of the space group G, then ¸ ) G is an equitranslational subgroup V V of G. Proof. Let us choose x as the origin of coordinates; by Eq. (90), the Euclidean group law of ¸)G V is written as a semi-direct product. Conversely, if G is a semi-direct product, its law can be written with Eq. (90) and P is the stabilizer of the origin of coordinates. That proves: Corollary 7c. The space group G is symmorphic 0 it has at least one stabilizer G &P in its action on V the space. Proposition 7d. ¸ ) H is a maximal equitranslational subgroup of G0H is a maximal stabilizer of G. The proof of N is obvious. Proof of =: admit that H is not a maximal "nite subgroup of G; then it is a strict subgroup of K, a maximal "nite subgroup of G. From Proposition 7b, it is a stabilizer of G and from Proposition 7c, ¸ ) K is an equitranslational subgroup of G; it is strictly larger than ¸ ) H and that is absurd. Proposition 7e. In the action of a space group G on the Euclidean space, the intersection of stabilizers is a stabilizer. Hint: It is su$cient to prove it for any pair of stabilizers. Let G and G be the stabilizers of the V W two points x, y; there is a dense subset of the straight line containing x and y whose points have G 5G as stabilizer. V W Since all stabilizers are "nite, the Palais theorem (Palais, 1961) mentioned in Chapter I Section 2 applies and Theorem 2a (Chapter I) is valid. More precisely: there exists a stratum with trivial symmetry; it is open dense in E . The closure of a stratum is the union of the strata with a higher B symmetry; in particular the strata with maximal symmetry are closed. The number of parameters (among x, y, z) which de"ne a Wycko! position in ITC is the dimension of the strata.
Table 8 gives a statistics on the dimension of the maximal symmetry strata.
L. Michel / Physics Reports 341 (2001) 265}336
319
7.3. Non-symmorphic elements of space groups. Their classixcation We need some criterion for discussing the structure of the non-symmorphic space groups in dimension d"2, 3. It is natural to introduce: De5nition. Let G be a space group and G U g,¸; if no elements of its translation coset g¸"¸g leaves a "x point of space, the coset elements are called non-symmorphic. One can also say that no element of this coset can belong to the stabilizer of a point of space, so the stabilizers of G are isomorphic only to strict subgroups of the point group P. Hence, if a space group G has a non-symmorphic element, from Corollary 7c, it has to be a non-symmorphic space group. As we shall see later the converse is not true! In dimension 3 there exist two nonsymmorphic space groups which have no non-symmorphic elements. Since the linear components A of the elements of a space group G have a "nite order l (it is the smallest positive integer such that AJ"I) it is useful to introduce the operator N (acting on the real vector space E ): B J\ order(A)"l, N "I#A#A#2#AJ\" AH , H AN "N "N A . (94) With D de"ned in Eq. (91) we have the relations N D "0"D N 0 Im D -Ker N , Im N -Ker D . (95) It is easy to prove that the equality holds in the last two relations for these operators on E . Let B *3Ker D , i.e. A*"*; then *"N l(A)\* which proves Ker D -Im N and therefore the last equality. By a similar proof to that for D , Eq. (93) is also true for N ; then we deduce the "rst equality in Eq. (95) from the second one. So we can sharpen Eq. (95) by Ker D "Im N , Im D "Ker N . (96) Beware, that is not true if we restrict the action of these two operators to a lattice ¸. Let +*(A), A, be an element of the space group G written as an element of the Euclidean group for a choice of origin. Then order(A)"l, GU+*(A), A,J"+N *(A), I,0N *(A)3¸ .
(97)
Proposition 7f. The element +*(A), A,3G has a xx point for its action on the Euclidean space if, and only if N *(A)"0. Indeed if +*(A), A, has a "xed point its lth power +*(A), A,J"+N *(A), I, has the same "xed point; this implies N *(A)"0. Conversely, the second equality of (96) shows that *(A)3Ker N implies the condition of Proposition 7a. We can now give a test for determining the non-symmorphic elements of a space group.
320
L. Michel / Physics Reports 341 (2001) 265}336
Corollary 7f. The element +*(A), A,3G is non-symmorphic if, and only if, N *(A),N ¸. Indeed this requires N *(A)O0. Assume the contrary of what is to be proven, i.e. there exists l3¸ such that N *(A)"N l; that is equivalent to say, from Proposition 7f that +*(A)!l, A, has a "xed point and therefore +*(A), A, which is in the same ¸ coset, is not non-symmorphic. As we have shown after Eq. (92), if D is invertible (i.e. A has no eigenvalue "1) Im D is the whole space. From Eq. (96) this is also true for Ker N so Corollary 7c is not satis"ed. We conclude that non-symmorphic elements cannot have a linear part A without eigenvalues "1, so, when d"2, 3, there are no non-symmorphic elements whose orthogonal part A belongs to the following arithmetic elements (we have to add the two trivial arithmetic classes p1, P1 since the corresponding space groups contain only translations): p1, p2, p3, p4, p6, and P1, P1 , P3 , P4 , P6 , R3 , I4 .
(98)
We now show that Corollary 7d eliminates also cm, C2, R3. For
cm"
0 1
1 1 :N " , AK 1 0 1 1
then the vectors N *(cm) are of the form (K) with m integer. But these vectors are in N ¸; indeed AK K AK N ( )"(K). AK K K For C2:
0 1
C2" 1 0
0 ,
0 0 !1
m
m
0
¸5Im N " m , m3Z , ! 0
m
(99)
N 0 " m . ! 0 0 For R3:
0 1 0
m
R3" 0 0 1 , ¸5Im N " m , m3Z , 0 1 0 0 m m
m
N 0 " m . 0 0 m
Hence we can add cm, C2, R3 to the list of the 12 arithmetic elements given in Eq. (98).
(100)
L. Michel / Physics Reports 341 (2001) 265}336
321
We are left to study the 8 arithmetic elements (out of 7#16"23) which can produce nonsymmorphic elements pm, P2, P3, P4, P6, I4, Pm, Cm .
(101)
We now establish the list of non-symmorphic elements they produce. Let us begin by the "rst "ve of them. They have a one-dimensional eigenspace with the eigenvalue 1, i.e.; Im N "Ker D is an axis. It contains a one-dimensional sublattice generated by the lattice vector b (we choose arbitrarily its sign) and N *(A)"kb. For these (p, P)-lattices, this axis of "xed points is orthogonal to Ker N "Im D ; by a choice of origin of coordinates we can bring to zero the component of *(A) in Im D , so it becomes *(A)"l\kb. On the other hand N ¸"+lmb, m3Z,. So we obtain the complete solutions for these "ve cases: A"pm, P2, P3, P4, P6, l" order of A , k *(A)" b, 0(k(l . l
(102)
Explicitly this gives the following 13 non-symmorphic elements in the notation of ITC: pg, Pc, P2 , P3 &P3 , P4 &P4 , P4 , P6 &P6 , P6 &P6 , P6 . (103) With the translations, these 13 non-symmorphic elements generate 13 space groups which are denoted by the same labels as in ITC. The sign & indicates an isomorphism: it is obtained by conjugation with a re#ection through a plane containing the invariant axis. ITC list them as two di!erent orientations which can be distinguished by physical phenomena (e.g. with circularly polarized light); in nature they appear together as domains in crystalline structures or as a result of some phase transitions. The four & of Eq. (103) indicate four of the 11 enantiomorphic pairs [they are listed below in Eq. (109)]. The case of the arithmetic element Pm is similar: (Im N "Ker D )N(Ker N "Im D ) but .K .K .K .K dim Im N "2. Hence ¸5Im N is a two-dimensional lattice. We say that a vector l of this .K .K lattice is visible if no vector m\l, 1(m3Z is a lattice vector; in other words l belongs to the G¸(2, Z) orbit of vectors which can be basis vectors. This non-symmorphic element is denoted by Pc with *(Pc)"l. When the point group contains the non-symmorphic element Pc and has more than two elements, the direction of the glide vector *(Pm) is constrained by the preferred directions of the other point group elements. To distinguish between the di!erent possibilities one of the following symbols are used: Pa, Pb, Pc, Pd, Pn; we refer to ITC for their de"nition. Case Cm:
Cm" 1 0 0 , ¸5Im N 0 0 1
m
m
N 0 " m !K n 2n
m
0 1 0
!K
" m , m, n3Z ,
0
N*(Cc)" 0 .
n
(104)
322
L. Michel / Physics Reports 341 (2001) 265}336
This non-symmorphic element, as well as the space group it generates with the translations, is labelled Cc in ITC. Since its glide re#ection vector is uniquely de"ned, no other label is necessary; however, in ITC, Ca is also used for giving the direction of the plane. Below we list the di!erent notations used when the "rst ("lattice) letter is A, I, F. The last case to be considered is the arithmetic element I4:
I4"
0 1
0
m
0 1 !1 , ¸5Im N " m , m3Z , ' !1 1 0 0 2m
N ¸" 2m , m3Z ' 0
N *(I4 )" . 0
(105)
This non-symmorphic element is denoted by I4 in ITC; its square is symmorphic: it is C2 up to a lattice translation. The square of P4 is the non-symmorphic element P2 . The symbol 4 is completely ambiguous; its meaning is de"ned only when one knows the "rst letter P on one hand, I or F on the other hand. The non-symmorphic space group I4 is a sub-space group with the same translations, of six space groups in the I-tetragonal Bravais class and of F4 32, I4 32 in the cubic crystallographic system. The arithmetic class P2 has one non-symmorphic group P2 . Similarly the non-symmorphic groups of the arithmetic class P222 are P2 22, P2 2 2 and P2 2 2 . The arithmetic class C2 has no non-symmorphic space groups. The groups of the arithmetic class C222, contain three rotations by p around three orthogonal axes; by convention in ITC, the "rst two are of C type and the last one is of the P type. So the only non-symmorphic space group of the arithmetic class C222 is C222 . The rotations by p in the space groups of Bravais classes F, I are of type C2; however, the labels of two space groups of the 24 F, I arithmetic classes contain 2 , i.e. I2 2 2 and I2 3. These two groups are the only non-symmorphic space groups without non symmorphic elements; these labels are a (non-explicitly explained) way for the Hermann}Maughin notation of ITC to signal that these two exceptional groups have no stabilizers G &P for the O points of space, i.e. there is no origin for which all *(A) can be simultaneously removed (note that I2 3 is the automorphism group of its subgroup I2 2 2 when the three fundamental periods of translations become equal). On the other hand F222 is the only space group of its arithmetical class and this is also the case of its automorphism group F23 since it is generated by F222 and the element (R3). A group of the arithmetic class R32 contains only the arithmetic elements P1, R3, C2; so there are no non-symmorphic space groups in the arithmetic class R32. In dimension 2, the only arithmetic element from which one can obtain a non-symmorphic element is pm; from Fig. 1 we see that the only arithmetic classes which contain it are pm, p2mm, p4mm. To summarize, for d"2,3 there are 10, 12 arithmetic classes containing only one space group: the semi-direct product It seems to me that it would have been better to denote it by I4 .
L. Michel / Physics Reports 341 (2001) 265}336
323
("symmorphic one) p1, p2, cm, p4, c2mm, p3, p3m1, p31m, p6, p6mm , P1, P1 , C2, P4 , I4 , R3, R3 , R32, P3 , P6 , F222, F23.
(106)
By a convention similar to that for C222, in the arithmetic class Cmm2, the two re#ections belong to the Cm class while the rotation belongs to the P2 class. So the non-symmorphic space groups of the arithmetic class Cmm2 are Ccc2 and Cmc2 . There is another arithmetic class (of the same geometric class mm2), denoted by Amm2; its rotation is of the C2 class and the two re#ections belong, respectively, to the two di!erent classes Pm, Cm. The three non-symmorphic space groups are Abm2, Ama2, Aba2. The re#ections Cm occur also in the I classes (where they are denoted by Ia, Ib, Ic, Id) and the F class (where they are denoted by Fd, Fc). 7.4. Some statistics on space groups In this section we limit ourselves to d"3. We "rst give a statistics of SpG/AC, the number of space groups per arithmetic class SpG/AC AC SpG
1 12 12
2 32 64
3 6 18
4 15 60
6 3 18
8 2 16
10 1 10
16 2 32
73 230
(107)
The list of arithmetic classes containing only one space group (this symmorphic space group has the same label) is given in Eq. (106). For the 32 arithmetic classes which contain two space groups we list the corresponding 32 non-symmorphic space groups (grouping them by Bravais classes) P2 Pc I4 I4 /a P31c P3 1c
Cc
C2/c
C222 I4 2d
I2 2 2 R3c
Fdd2
Fddd
I4 22 I4 c2 R3 c P3c1 (108) P3 c1 P6 /m P6 c2 P6 2c P2 3 P4 3n Fd3 F4 3d F4 3c I2 3 Ia3 I4 32 I4 3d Ia3 d To "nd the arithmetic class (and the symmorphic space group in it), one suppresses the indices in the labels and replaces the lower case letters a, c, d, n by m. After Eq. (103) we pointed out the existence of enantiomorphic pairs of isomorphic space groups. For the convenience of the reader we give here the list of the 11 enantiomorphic pairs P4 &P4 P4 22&P4 22 P4 2 2&P4 2 2
P3 &P3 P3 12&P3 12 P3 21&P3 21
P6 &P6 P6 &P6 P6 22&P6 22
P6 22&P6 22 P4 32&P4 32
(109)
This geometric class mm2 is the one which has the biggest number of corresponding arithmetic classes: "ve of them. They are denoted by: Pmm2, Cmm2, Amm2, Fmm2, Imm2.
324
L. Michel / Physics Reports 341 (2001) 265}336
Table 8 Number of space groups with given dimension of closed strata 230 dim dim dim dim
104 0 1 2 3
63
5
13
x x x
38
2
x x
x x
4
1
x x
x x x
x
If all non-trivial elements of a space group are either translations or non-symmorphic elements the stabilizers on the Euclidean space are all 1, i.e. there is the unique stratum: the whole space; mathematicians call this action free. This is the case of nine of the space groups of Eq. (103). The "rst of them, pg is the only free acting non-trivial space group for d"2. In dimension 3 there are 13 space groups with free action on the space P1, P2 , Pc, Cc, P2 2 2 , Pca2 , Pna2 , P4 &P4 , P3 &P3 , P6 &P6 . (110) It is interesting to give a statistics of the dimension of the maximal symmetry ("closed) strata of space groups. We reproduce in Table 8 the statistics given in Bacry et al. (1988) for dimension 3. By X-rays, neutron di!raction, etc. one can obtain the reciprocal lattice and in principle the crystal lattice; it has to belong to one of the 14 Bravais classes. The crystal may belong to a smaller symmetry; that is not a generic situation: with a change of temperature, pressure, etc. the lattice Bravais class is likely to go to that of the crystal. One also knows the volume of the fundamental cell. When the chemical composition and the weight density are also known, this determines the number n of the di!erent kinds A of atoms per fundamental cell. We de"ne: G G n "number of atoms A per fundamental cell n "min n . (111) G G K G Let us explain how this knowledge already excludes some space groups. De5nition. The index of symmorphy ("IS) of a space group G is the minimum value of "P"/"G " for V all stabilizers of G. Then we must have the inequality IS4n . K
(112)
Note that IS"1 for a symmorphic space group; so if n "1, i.e. there is a unique atom of a kind K in the fundamental cell, then the space group has to be symmorphic. From Proposition 7d, if we know that G has an equitranslational symmorphic subgroup of index i then IS(G)4i. Let us give some arguments based on the list in Eq. (106) of the arithmetic classes containing only the symmorphic groups. The presence in this list of F23 implies that for the cubic-F groups, IS44 for those of the arithmetic class Fm3 m and IS42 for the others. The presence of R3, R3 , R32 implies
L. Michel / Physics Reports 341 (2001) 265}336
325
that for the other cubic groups IS48 and for the rhombohedral subgroup IS42. The presence of I4 , P4 implies for the cubic groups of the arithmetic classes I4 3m, P4 3m their IS46. In the hexagonal system, the symbols 3 , 3 , 6 , 6 in the label of a space group can disappear only for subgroups of index 3, so IS53. Similarly the symbols 6 , 6 in the label of a space group can disappear only for subgroups of index 6, so IS56. In the tetragonal system, the symbols 4 ,4 in the label of a space group can disappear only for subgroups of index 4, so IS54. So we know here to look in ITC for space groups with IS53. Here is their list: IS"3: P3 , P3 , P3 12&P3 12, P3 21, P3 21, P6 , P6 , P6 22, P6 22. IS"4: P2 2 2 , Pca2 , Pna2 , Pcca, Pccn, Pbcm, Pbca, Pnma, Ibam, Ibca. P4 &P4 , P4 22&P4 22, P4 2 2&P4 2 2, P4 bc, I4 cd, P/ncc, P4 /nbc, P4 /ncm, I4 /ncd, P2 3, I2 3, Pa3 , Ia3 , P4 32&P4 32, I4 32, Fd3 c. IS"6: P6 &P6 , P6 22&P6 22, I4 3d. IS"8: Ia3 d. To summarize, the number of space groups with the di!erent possible values of IS are 1
2
3
4
6 8
73 110 10 31 5 1 For the space group which satis"es Eq. (112) one has also to "nd for each kind of atoms the values of l ""P"/"G " which satisfy n " l without using more than once the strata of dimension V V G V zero. The dynamics of interatomic forces favors generally the high-symmetry strata for the atom positions; it might be a strict symmetry requirement when the n are small. G For instance a little more than half of the chemical elements have the space group symmetries Fm3 m and Im3 m with a small preference for the former which is also the most frequent space group (more than 8%) for all inorganic compounds. As shown in Section 4.4 the strata ("Wycko! positions) of dimension 0 of Fm3 m contain two "xed points (origin and quadrivalent vertices of the VoronomK cell), one orbit of two elements (the trivalent vertices) and one orbit of six elements (the center of faces). Many of the inorganic chemical compounds which crystallize with this symmetry have the stoichiometries XY, XZ , XYZ , XW , XYW , XYZ W . In frequency of occurrence the Fm3 m is followed by the space groups Pnma, Fd3 m, P6 mm, P2 /c, Pm3 m, etc. These six groups represent more than the third of the inorganic compounds. Fd3 m is the space group of the chemical elements C (diamond) and Si, Ge, Se, Sn (gray tin) and the dense packing hexagonal group P6 /mmc is that of C (graphite), As, Sb, Bi, Cr, Ti, etc. (more than 20). P2 /c represents 5% of the inorganic compounds and 30% (the most abundant) of the organic ones (see below). The symmorphic group Pm3 m has four strata of dimension 0; two orbits of one element (center and vertices of the cube) and two orbits of three elements (center of faces and middle of edges). It represents W, O , F and many compounds of the types XY (as NaCl, ionic crystal), XZ , XZ , XYZ and more complicated ones. For organic crystals, which are essentially molecular, the situation is quite di!erent. These molecules can form crystals with larger symmetry point groups than their own symmetry group;
326
L. Michel / Physics Reports 341 (2001) 265}336
indeed there are most often several molecules per fundamental cell and their positions in the cell have the symmetry of the crystal point group (see e.g., Kitaigorodsky, 1973). Half of the organic compounds have the crystal symmetries P2 /c, P2 2 2 and P2 . 8. The unirreps of G and G and their corepresentations with T I 8.1. The unitary irreducible representations of G I In the previous section we have recalled in Eq. (90) the group law of Eu , the Euclidean group, B and in Eq. (91) the e!ect of a change of origin in E , the Euclidean space. Since a space group G is B a discrete subgroup of Eu , we can use for the element of G the notation t3¸ for the translations B and take for each coset of ¸ an element that is traditionally denoted (as in Eq. (97)) +*(A), A, with A an element of the point group P and *(A) a translation often called `imprimitivea to tell that it is not in ¸. The product of two elements of G is s, t3¸, A, B3P , +s#*(A), A,+t#*(B), B,"+s#*(A)#A(t#*(B)), AB, .
(113)
Assuming that the imprimitive translations are known for every point group element, we want *(AB) to appear in the right-hand side; so the group law of G can be written as s, t3¸, A, B3P , +s#*(A), A,+t#*(B), B,"+s#At#*(AB)#z(A, B), AB,
(114)
with the de"nition z(A, B) " : *(A)!*(AB)#A*(B)3¸ ,
(115)
indeed z(A, B) can be at most a translation. We will always make the convention *(I)"0, so z(I, A)"0"z(A, I) .
(116)
The function *(A) depends on the choice of origin in the Euclidean space. From Eq. (91) we know that if we translate the origin o by the translation x"o !o, we make a conjugation in Eu by the B element +x, I,; then * (A), the new imprimitive translation function is related to *(A) by * (A)"*(A)#D x with D "I!A . (117) It is straightforward to verify that z(A, B) is invariant by change of the origin in E . B Corollary 7c tells that for any stabilizer G in the action of G on the Euclidean space, the V semi-direct product ¸)G is a subgroup of G. If a stabilizer G is isomorphic to the point group, V V then the space group is a semi-direct product G"¸)P (crystallographers say G is symmorphic). By taking the origin at such point x the space group law has the same form as that of the Euclidean group in Eq. (90), i.e. all *(A)"0 and therefore all z(A, B)"0. That does not mean that z(A, B) is completely "xed for a given space group G. We leave the discussion of this point to Section 8.3.
L. Michel / Physics Reports 341 (2001) 265}336
327
We use the shorthand unirrep for `unitary irreducible representationa. As we saw in Section 4.4 before Eq. (53), we recall that G (the stabilizer of k in the action of G on BZ) is a space group with I P as point group. Moreover G "G for kK "0 and other points of BZ (listed in Table 3) except for I I the space groups of the Bravais class cubic-F. As we explained in Section 4.4, the subgroup of translations is represented by the onedimensional unirreps labelled by kK 3BZ ¸ Ut C kK (t) " : e k t, kK "class k mod 2p¸H ,
(118)
since for all k32p¸H, i.e. all vectors of the reciprocal lattice, the representation (118) is trivial. It is important to consider the kernel and the image of the unirrep kK of the translation group ¸. We recall the de"nitions: Ker kK " : +t3¸; kK (t)"1,4¸ , Im kK " : kK (¸)&(¸/Ker kK )(; . We recall the nearly obvious proposition:
(119)
Proposition 8a. kK is of xnite order m in BZ 0 the coordinates of k in the reciprocal lattice 2p¸H are rational. Moreover Im kK &Z . K Remember that in Section 4.4 and everywhere else we use the coordinates i of k in the dual G lattice ¸H; so the coordinates of k in the reciprocal lattice are i /2p. G As we saw in Section 4.4, the translations act trivially on BZ, so the space group G acts on BZ through the point group PX; this action (and therefore the decomposition of BZ into strata and orbits) is the same for all space groups of an arithmetic class. We denote by PX the PX-stabilizer of I kK 3BZ. The G-stabilizer G contains all translations (so it is a space group with the same I translations as G) and for two space groups of the same arithmetic class, the two sets of G 's, kK 3BZ I are distinct. Indeed at least some corresponding G are not isomorphic. I Since G is the stabilizer of k, it is easy to verify that Ker kK is an invariant subgroup of G . It is I I natural to consider the corresponding quotient group; this has been done by Herring (1942), so we call this group the Herring group and we denote it by PX(k). To see the structure of this very interesting group we use the theorem of group theory on the simpli"cation of fractions. Let us `simplifya the fraction G /¸ by Ker k which is an invariant subgroup of both numerator and I denominator (see Eq. (119)). We obtain FI PX(k) " : G /Ker k, PX(k)/Im kK "PX , Im kK ¢C(PX(k)) , G P I I I
(120)
the center of PX(k) (use again that G stabilizes k); we recall (see Section 4) that PX is the stabilizer of I I kK in its action on BZ. From the de"nition of PX(k) we verify that the unirreps of G are unirreps of I PX(k). The converse is not true; indeed we have to consider only a subset of PX(k) unirreps. We recall that here the relation ( between groups means `subgroupa; ; , the one-dimensional unitary group, is the group of phases in the complex plane.
328
L. Michel / Physics Reports 341 (2001) 265}336
De5nition. The allowed unirreps of PX(k) are those which represent faithfully the subgroup Im k by the matrices exp(ik ) t)I where n is the dimension of the unirrep of PX(k). This de"nition is tailored for B the following: Proposition 8b. There is a bijective correspondence between the unirreps of G and the allowed I unirreps of PX(k); the corresponding unirreps have the same image. As is well known, in the trivial case of k"0, G "G, PX(0)"PX "PX: the unirreps of the point group P are those of G for k"0. We recall that crystallographers call symmorphic space groups, those G's which are semi-direct products of PX by ¸; i.e. PX is a subgroup of G. That implies that PX is a subgroup of PX(k); since Im kK I is in the center of PX(k), this group is a direct product. To summarize: G"¸ ) PXNPX(k)"Im kK ;PX . I Beware that the converse is not true! We will see counter examples below.
(121)
Proposition 8c. When PX is cyclic, PX(k) is Abelian. Then their allowed unirreps are one-dimensional I and their image is a subgroup of ; . Eq. (121) proves it for symmorphic groups. For the non-symmorphic groups we have studied in the previous section their `non-symmorphic elementsa; for them, the proof is based on the fact that their nth powers, for n5l, are translations. We study in detail the representation of the nonsymmorphic space groups when G"G is generated by ¸ and the arithmetic elements listed in Eq. I (101) (i.e. the list of those which can produce non-symmorphic elements). Let us begin by those of Eq. (102): k A"pm, P2, P3, P4, P6, l"order of A, *(A)" b, 0(k(l . l
(122)
With the di!erent possible values of k we saw that we can form 13 non-symmorphic groups generated by +*(A), A, and the translations pg, Pc, P2 , P3 &P3 , P4 &P4 , P4 , P6 &P6 , P6 &P6 , P6 , (123) when A is a rotation, the index is k and for re#ections k"1. We can take the visible vector b as a basis vector. The points k3BZ such that G "G are those of the form k"ibH. In the I corresponding representation, the translations are represented by t C exp(it ) bHi). So Im k"+exp(ini), n3Z, where n is the component of t on the basis vector b; it is a "nite or in"nite cyclic group depending whether i/p is rational or irrational. From Eqs. (97) and (102) +*(A), A,J is represented by exp(iki); so +*(A), A, is represented by a lth root of this expression. There are l such roots; they are obtained by multiplication of a "xed one by the l roots of 1, i.e. exp(i2po/l), 04o(l. We summarize in the next equation what we established for the l unirreps of the G 's of I That mathematical fact might be abhorred by physicists. It is irrelevant when m (de"ned in the statement of Proposition 8a) is very large; but it is very important when m is small.
L. Michel / Physics Reports 341 (2001) 265}336
329
Eq. (103) labelled by o: 04o(l labels unirreps k"ibH , +t, I, C e GbH t, +*(A), A, C e IG>pM J .
(124)
Because of the division by l in the exponent the equation does not seem to satisfy the periodicity i C i#2p which corresponds to the de"nition of k mod 2p¸H of BZ; indeed i C i#2p, e IG>pM J C e IG>p>pM J"e IG>pM>I J .
(125)
So the set of l unirreps respects the periodicity k mod 2p¸H but when one completes a circle on BZ along the direction bH one obtains a permutation of the l unirreps. If the greatest commun divisor U k, X l of k, l is 1, the permutation is circular and the l unirreps form a unique orbit of the generated permutation group. More generally the number of orbits is l/U k, X l . This phenomenon is called monodromy in mathematics: for the unirreps of space groups, it was "rst observed in the "rst detailed study of them on two examples (Fd3 m and P6 /mmc) by Herring (1937a, pp. 538, 543). To complete this study of non-symmorphic elements, the extension to the last three arithmetic elements of Eq. (101) is straightforward. +*(A), A, is a non-symmorphic element which generates, with the translations, one of the three space groups I4 , Cc, Pc; we have given the corresponding *(A) in Eqs. (104)}(105) as half of the visible vector l which de"nes the rotation axis for I4 or the `glide vectora for Cc and Pc (for the latter case the glide vector can be any visible vector of the re#ection plane). The vectors k"ilH are "xed by these non-symmorphic elements and the corresponding PX(k) has two unirreps which are exchanged by iCi#2p. This monodromy phenomenon is ignored in the tables of space group unirreps and by nearly all solid state physicists. To my knowledge it has never been applied to physical properties up to the very recent paper: (Michel and Zak, 1999). This physical application is explained in Chapter VI. Depending whether the order of kK in BZ is in"nite or of order m, the corresponding Herring groups are isomorphic to Z or to Z . KJ Note that when PX is Abelian but not cyclic, the Herring group PX(k) may be non-Abelian. Let us I study the simple example: P "D (or C ) with kK of order 2. Then Im kK "Z . So PX(k) is I T a eight-element Herring group H such that Z (C(H) and H/Z &Z . It is easy to "nd the four groups which satisfy these conditions; there are four of them and for the non-Abelian group we give the faithful two-dimensional unirreps in terms of Pauli matrices: G: Z , Z ;Z , c "+$I ,$p ,$p ,$ip ,, q "+$I ,$ip ,$ip ,$ip , . (126) T The "rst group is the direct product; the second is Abelian. The last group, q , is called the quaternionic group. The center of the two non-Abelian groups has two elements; they are denoted It corresponds to an action on the set of unirreps, by the homotopy group p of the BZ torus. These two groups are also often mentioned, respectively, as `diamond structurea and `hexagonal close packinga. They are third and fourth rank in the space group frequency of inorganic crystals, each group representing more than 5% of them (see (Mighell et al., 1977)). The list and structure of non-Abelian groups of the order "G"424 are given in the very good book by Coxeter and Moser (1972).
330
L. Michel / Physics Reports 341 (2001) 265}336
by $1. Remark that in these two groups (c , q ), if the product of two elements is not in the T center, these two elements anticommute. Remark also that the square of any element is in the center; the number of elements of square !1 is two for c and six for q . For these two T non-Abelian groups, their allowed representations must represent faithfully the center; so each non-Abelian Herring group has a unique allowed representation which is the one given in Eq. (126). For the "rst Abelian group, Z (Im k) has to be represented faithfully, so there are four allowed representations obtained by making the tensor product of this representation with those of P &Z ; their images are z . For Z ;Z , the allowedness condition requires Z to be represented I faithfully (two such unirreps) and the tensor product with the two unirreps of Z yields four allowed representations of image z . The point kK "0 and the seven other points which satisfy 2kK "0 are the "xed points on BZ for the 40 space groups of the Bravais classes triclinic, P-monoclinic, P-orthorhombic, (see Table 3 and ITC). It is easy to determine the eight corresponding PX(k) for these space groups. The group P2 2 2 illustrates the four examples of Eq. (126). Let us study this space group, its eight G "G and their unirreps. The three generators of the translation lattice ¸ along the rotation I axes of the point group P"D satisfy b ) b "jd . In this basis the three matrices representing G H G GH the non-trivial elements of P are diagonal matrices that we de"ne by their action on the basis R b "b , iOj, G G G
R b "!b . G H H
(127)
Let i, j, k be a circular permutation of 1, 2, 3. The three generators of P2 2 2 are r "+(b #b ), R , , G H G G then r"+b , I,, r r "+* , I,r r , * "b !b !b . G G G H GH H G GH G H I
(128)
In BZ, the eight elements 2kK "0 have coordinates 0 or p in the dual basis (see Table 3). The three elements kK K (p at the coordinate m and 0 for the two other coordinates) are represented by the $ center of the six faces of the BZ-cell (a rectangle parallelepiped); the three elements kK KY (0 at m and # p for the two other coordinates) are represented by the middle of the 12 edges and the element kK , 4 with its three coordinates "p, is represented by the eight vertices. From
* 6 GH " e kK
1 for X"E ,
!1 for X"F, < ,
(129)
we know that the Herring groups P(kK KY) are Abelian and the others are not. To know the nature of # the non-Abelian ones we have to compute their number of elements whose square are !1. From the second equality in Eq. (128) we count number of e k6 bG "!1 is 1, 2, 3 for X"F, E, < .
(130)
The most abundant space groups for the crystals of organic compounds are, according to (Mighell et al., 1977), P2 /c (about 30%), P2 2 2 (more than 12%), P2 (less than 8%). So these three space groups represent half of the organic crystals. Their energy band structure will be studied in Chapter VI.
L. Michel / Physics Reports 341 (2001) 265}336
331
So we can deduce the nature of the eight Herring groups of P2 2 2 : P(kK )&Z , P(kK KY)&Z ;Z , # P(kK K)&c , P(kK )&q . $ T 4
(131)
8.2. The irreducible corepresentations of G[ I As we saw in Section 4.5, time reversal T acts on the Brillouin zone according to T ) kK "!kK mod 2p¸H. So the points 2kK "0 are T invariant. Adding T to G we obtain the cogroup G[ and we can de"ne the stabilizer G[ . Wigner (1932) showed that in quantum mechanics, I the action of T on the Hilbert space of vector states must be represented by an antiunitary operator <(T) which, in absence of spin, satis"es <(T)"I (the identity operator). The elements of G[ , G[ which contain time reversal must be represented by antiunitary operators. The simplest I antiunitary operator acts as complex conjugation on the complex "eld of the Hilbert space; this semilinear operation is denoted by K and K"I. So we can write an antiunitary operator as the product <";K where ;"
(132)
In the absence of spin e!ects, since time reversal T changes the sign of momenta, it is represented on the reciprocal space and on the functions on this space, by the operators: <(T)"(!I)K, T ) f(k)"fK (!k) ,
(133)
where K denotes the complex conjugation. To study the (non-linear) group action on BZ we replace k by kK which represents the class of equivalence: k mod 2p¸H. Hence we have two cases to consider: 2kK "0: Then G is a subgroup of index 2 of G[ . Given an irreducible representation o(G ), to I I I build the corepresentation of G[ we have again to distinguish two cases: I o(G ) is (equivalent to) a real representation; then <(T) commutes with it and the corepresentaI tion has the same dimension; o(G ) is not equivalent to a real representation. Then the direct sum o o of o and its complex I conjugate is equivalent to a real irreducible representation of G ; we are back to the previous case, I but notice that the dimension of the irreducible corepresentation is 2 dim o(G ). I 2kK O0: G and T generate a cogroup which contains both G and G and the cosubgroup G[ is I I \I I not necessarily trivial. This has already been studied by Herring in his thesis, summarized in (Herring, 1937a, b). The simplest case occurs for the space groups P2 or P2 . We assume that the rotation by p is around the axis de"ned by the basis vector b ; note that the two other vectors in a basis generating ¸ have to be orthogonal to b . In the reciprocal space the tip of the vectors k of That is the name usually given in the physics literature; of course it is a group. A state of half-integer angular momentum is an eigenstate of <(T) with eigenvalue !1. In the "rst English edition (Wigner, 1959) of his book on group theory and quantum mechanics, Wigner added a chapter on time reversal. From this time, this topic is dealt with in many text books; so we introduce in this section the minimal content which explains the use of time reversal in Chapter VI. In mathematics, the product of the two operations on a vector space: that of a linear operator and that of an automorphism of the vector space "eld, is called semilinear.
332
L. Michel / Physics Reports 341 (2001) 265}336
coordinates k"(i , i , p) are in the face of the Brillouin cell orthogonal to the dual basis vector bH (since the space groups belong to the P-monoclinic Bravais class and from Fig. 6 and Table 6 we know that it is the horizontal face of the hexagonal prism Brillouin cell). The rotation R (by p around the third axis) of P2, the skew rotation r"+b , R, and T have the same e!ect on these vectors k; they change the sign of i , i"1, 2. So the products RT for P2 and rT for P2 belong to G the respective G[ 's. In the one-dimensional unirreps of the Abelian groups G , the elements R, r are I I represented, respectively, by $1, $i (apply Eq. (124) with l"2, k"1 and i"p). The result is that these co-unirreps are of dimension 1 for P2 and 2 for P2 . Herring (1937a) extended it to the space groups containing P2 : the dimension is doubled when one passes from the representation to the co-representation with T for kK 3BZ represented by the points of the face of the Brillouin cell orthogonal to the skew rotation axis. We have considered some allowed unirreps of Herring groups. One can establish a systematic method for constructing the Herring groups and their allowed unirreps. By a decomposition of the Herring groups similar to what Table 1 from Chapter I is doing for the 32 (geometric) point groups, Michel and Mozrzymas (unpublished) have written the "nite Herring groups in the form P(k)"A(k);P (k) where A(k) is a "nite Abelian group and P (k) is a member of a much smaller list of Herring groups that have been called the skeleton Herring groups. Since the G are themselves space groups, the next section can be applied for "nding their I unirreps. 8.3. The irreducible representations of a space group G We will be very short on this subject since many books deal with it. Several formulas written here are more concise and elegant and few are not in the physics textbooks. The method to build the unirreps of the space group G is by induction from those of subgroups with the same translations. Before doing it, we want to recall some properties of the induction method along lines not found in physics textbooks. To simplify we "rst consider the case of a "nite group G and its sugroup H. The complex valued functions on the group which satisfy
(g g )" (g g ) 0 (g g g\)" (g ) , form a "nite-dimensional Hibert space H with the Hermitian scalar product %
(134)
1u, 2 ""G"\ u(g) (g) . (135) % EZ% Labelling the unirreps of G by a, their characters s? form an orthonormal basis of H . Restricting % % the functions 3H to the subgroup H(G de"nes the linear operator % 0%& (136) H PH . & % We advise the remarkable small book by Serre (1977).
L. Michel / Physics Reports 341 (2001) 265}336
333
By de"nition, its adjoint operator is the induction operator '%& H PH ; & %
Ind% "(Res% )? . & &
(137)
From this de"nition we have immediately the Frobenius reciprocity 1Ind% sM , s? 2 "1sM , Res% s? 2 . & & % % & & % & From the associativity of the linear maps we obtain the theorem of induction by step H(H (H (2(H (H (G K\ K NInd% "Ind% K Ind&KK\ 2Ind& Ind& & & & & &
(138)
(139)
and from the other fundamental properties of linear maps Ind% (sM sN )"Ind% sM Ind% sN . (140) & & & & & & & To construct an induced representation from H to G one chooses representatives of the left cosets G:H G"8 s H . (141) H H Then, given a representation h C D(h) of H, one obtains the induced representation of G as block matrices
D(s\gsl ) if s\gsl 3H , H H D"Ind% D, D l (g)" & H 0, otherwise .
(142)
Obviously that formula can be extended to "nite-dimensional representations of a subgroup H of "nite index in G. That is the case of the space group G and a subgroup H containing the subgroup ¸ of translations. Let us "rst induce from a one-dimensional unirrep kK of the translation group ¸. We choose as representatives of the ¸-cosets, the elements +*(A ), A , de"ned in Eq. (113), where the G G A are the elements of the point group P. Then Eq. (142) yields G (143) D"Ind%kK : D l (+t, I,)"d l e kH t, k "(A?)\k , H H H * H e kH z l if A "AAl , H D l (+*(A), A,)" (144) H 0 otherwise
with the vector z(A, Al ) de"ned in Eq. (115). Remark that the matrix elements of these representations are analytic functions on BZ. These representations of G have dimension "P". When kK is in the open dense stratum of BZ (i.e. G "¸) these representations are irreducible. We notice, since PX "1, that all k are di!erent so any I I H matrix C which commutes with all the diagonal matrices of Eq. (143) must be diagonal. Let us "rst consider the symmorphic group of the arithmetic class PX; then all z(A, Al ) vanish and the matrices of Eq. (144) are permutation matrices (independent of k). Their commutations with the diagonal C permutes the elements of C; so C is an intertwinning matrix, invariant by these permutations, only if it is a multiple of I. The proof can be extended to the non-symmorphic groups.
334
L. Michel / Physics Reports 341 (2001) 265}336
When PX is not trivial, the (unitary) induced representations Ind%kK are reducible. It has been I * proven (Mackey, 1970) that the decomposition into unirreps of all these representations yields the complete set of unirreps of G. For symmorphic groups we have already noticed that the matrices of Eq. (144) are independent of k and they form the regular representation of PX. The decomposition of the induced representation into irreducible components is easy to do. For non-symmorphic groups the reduction into unirreps may not preserve the analyticity of the matrix elements on BZ. Indeed k may be replaced by k/m, m"2,3,4,6 in the exponential, so the analyticity is preserved only on a covering of BZ. That introduces the monodromy of representations that we have introduced earlier (e.g. Eqs. (124)}(125)). Let us consider now the representations of G of the form Ind%I k; their dimension is "P ". From * I I Eq. (143) we see that the matrices representing the translations are e k tI, i.e. multiple of the identity. For the symmorphic G , in the decomposition of this induced representation into I irreducible representations, the multiplicity of the obtained unirreps is equal to their dimension. This extends to the non-symmorphic G . I Eq. (139) shows that the induction can be made by step. Mackey (1970) also proved that by inducing from G to G a unirrep of G , one obtains a unirrep of G and all unirreps of G are obtained I I by induction from all unirreps of all G 's for all orbits 3BZ"PX. So all unirreps of G are I "nite-dimensional; their dimension is a divisor of "P". For non-symmorphic groups, the monodromy groups merge some unirreps into an orbit; the convention to distinguish them by labels is arti"cial and not useful. We "nish this chapter by a small digression about the z(A, B) introduced in Eq. (115). The associativity of the group law applied to Eq. (114) imposes the condition z(A, B)!z(A, BC)#z(AB, C)!Az(B, C)"0 .
(145)
We already noted, at the beginning of this section that a change of origin of coordinates modi"es the *(A)'s but leaves the z(A, B)'s invariant; it is interesting that Eq. (144) depends only on the z(A, B)'s. However there is still some freedom to choose the *(A)'s by adding to it an arbitrary translation c(A)3¸ of the lattice. We denote with a the new vector functions * (A)!*(A)"c(A)3¸ Nz (A, B)!z(A, B)"c(A)!c(AB)#Ac(B) .
(146)
The right-hand expressions of Eq. (115) and of the last equality of Eq. (146) look alike but their meanings are very di!erent: in Eq. (115) the *(A)'s are vectors of the Euclidean space while in Eq. (146) the c(A)'s are lattice vectors. It is convenient and natural to take the *(A) inside the fundamental domain de"ned by the basis vectors generating ¸; then the z(A, B) are uniquely de"ned. Eqs. (115)}(117) and (145)}(146) are basic expressions of the theory of cohomology. This algebraic aspect of crystallography is treated in Ascher and Janner (1965) and Ascher and Janner (1968). The "rst set of lectures on cohomology of groups given in a Physics summer school, is of Michel (1964).
L. Michel / Physics Reports 341 (2001) 265}336
335
References Abramowitz, M., Stegun, I.A., 1964. Handbook of Mathematical Functions. NBS. Since, many corrected editions and reprinting have been published by Dover Publications, NY. Ascher, E., Janner, A., 1965. Algebraic aspects of crystallography. I. Space groups as extensions. Helv. Phys. Acta 38, 551}572. Ascher, E., Janner, A., 1968. Algebraic aspects of crystallography. II. Non primitive translations in space groups. Commun. Math. Phys. 11, 138}167. Bacry, H., Michel, L., Zak, J., 1988. Symmetry and classi"cation of Energy Bands in Crystals, Lecture Notes in Physics, vol. 313. Springer, Berlin, pp. 291}308. Baranovskii, E.P., Ryshkov, S.S., 1973. Primitive "ve-dimensional parallelohedra. Sov. Math. Dokl. 14, 1391}1395. Bieberbach, L., 1910. Uber die Bewegungsgruppen der n-dimensionalen Euklidischen raK ume mit einem endlichen Fundamentalbereich. GoK tinger Nachr. 75}84. Bieberbach, L., 1912. Uber die Bewegungsgruppen der Euklidischen RaK ume Die Gruppen mit einen endlichen Fundamentalbereich. Math. Ann. 72, 400}412. Born, M., von Karman, T., 1912. Uber Schwingungen in Raumgittern. Phys. Z. 13, 297}309. Bouckaert, L.P., Smoluchowski, R., Wigner, E., 1936. Theory of Brillouin zones and symmetry properties in crystals. Phys. Rev. 50, 58}67. Bradley, C.J., Cracknell, A.P., 1972. The Mathematical Theory of Symmetry in Solids.. Clarendon Press, Oxford. Bravais, A., 1850. MeH moire sur les syste( mes formeH s par des points distribueH s reH gulie`rement sur un plan ou dans l'espace. J. Ecole Polytech. 19, 1}128. Brillouin, L., 1930. Les eH lectrons libres dans les meH taux et le ro( le des reH #exions de Bragg. J. Phys. Radium 7, 376}398. Brillouin, L., 1931. Die Quantenstatik und Adwendung auf die Elektronentheorie der Metalle Springer, Berlin. Buerger, M.J., 1956. Elementary Crystallography; An Introduction to the Fundamental Geometrical Features in Crystals. Wiley, New York. Burckhardt, J.J., 1966. Die Bewegungsgruppen der Kristallographie. 2 neubearb. Au# BirkhaK user, Basel. Conway, J.H., Sloane, N.J.A., 1988. Sphere Packings, Lattices and Groups. Springer, Berlin. Coxeter, H., Moser, W., 1972. Generators and Relations for Discrete Groups. Springer, Berlin. Delaunay, B.N., 1932a. Neuere Darstellung der geometrischen Kristallographie. Z. Kristallogr. 84, 109}149. Delaunay, B.N., 1932b. Erratum. Z. Kristallogr. 85, 392. Delone, B.N., Galiulin, R.V., Shtogrin, M.I., 1974. Bravais theory and its generalization to n-dimensional lattices (in Russian). In: Auguste Bravais: Collected Scienti"c Works. Nauka, Leningrad, pp. 538}630. Dirichlet, P.G.L., 1850. Uber die Reduktion der positiven quadratische Formen mit drei unbestimmten gauzen Zahlen. J. Reine Angew. Math. 40, 216}219. Dolbilin, N.P., Lagarias, Senechal, M., 1998. Multiregular points systems. Discrete Comput. Geom. 20, 477}498. Engel, P., 1986. Geometric Crystallography: An Axiomatic Introduction to Crystallography. D. Reidel Publ, Dordrecht. Engel, P., 1998. Investigations on lattices and parallelohedra in rB. Proc. Inst. Math. Acad. Sci. Ukraine 21, 22}60. Fedorov, E.S., 1885. An introduction to the theory of "gures (in Russian). Verh. Russisch-kaiserlichen Mineral. Ges. St Petersbourg 21, 1}279. Frankenheim, M.L., 1826a. Crystallonomische AufsaK tze. ISI Enzyklopadische Zeitung Oken 5, 497}515. Frankenheim, M.L., 1826b. Crystallonomische AufsaK tze. ISI Enzyklopadische Zeitung Oken 6, 542}565. Frankenheim, M.L., 1842. System der Crystalle. Nova Acta Acad. Caesarea Leopoldino-Carolinae Naturae Curiosorum 19, 479}660. Galois, E., 1846. Oeuvres de Galois. J. Liouville, Ser. 1 11. Gauss, C.F., 1805. Disquisitiones Arithmeticae. Goresky, M., MacPhersons, R., 1980. Strati"ed Morse Theory. Springer, Berlin. Graham, R.L., Knuth, D., Patashnik, O., 1988. Concrete Mathematics.. Addison-Wesley, Reading, MA. GruK nbaum, B., 1967. Convex Polytopes.. Interscience, London. Hermite, C., 1850. Second letter to Jacobi on number theory. J. Reine Angew. Math. 40, 279}290. Herring, C., 1937a. E!ects of time-reversal symmetry on energy bands in crystals. Phys. Rev. 52, 361}365. Herring, C., 1937b. Accidental degeneracy in the energy bands in crystals. Phys. Rev. 52, 365}373. Herring, C., 1942. Character tables for two space groups. J. Franklin Inst. 233, 525}543.
336
L. Michel / Physics Reports 341 (2001) 265}336
Hessel, J., 1830. Krystallometrie oder Krystallonomie und Krystallographie. Gehler's Phys. Worterbuch 5, 1023}1360. ITC, 1952. in: Henry, N.F.M., Londsdale, K., (Eds.), International Tables for X-ray Crystallography, Vol. I. Kynoch Press, Birmingham. ITC, 1996. in: Hahn, T. (Eds.), International Tables for Crystallography, Vol. A. Space Group Symmetry. 4th, revised Edition, Kluwer, Dordrecht. Kitaigorodsky, A.I., 1973. Molecular Crystals and Molecules. Academic Press, New York. Lagrange, J.L., 1773. Rech. Arithmetique. Oeuvre III 695}795. Mackey, G.W., 1970. Induced representations of locally compact groups and applications. In: Browder, F. (Ed.), Functional Analysis and Related Fields. Springer, Berlin, pp. 132}166. Michel, L., 1964. Invariance in quantum mechanics and group extensions. In: GuK rsey, F. (Ed.), Group Theoretical Concepts and Method in Elementary Particule Physics. Gordon and Breach, New York, pp. 135}200. Michel, L., 1995. Bravais classes, VoronomK cells, Delone symbols. In: Lulek, S.W.T., Florek, W. (Eds.), Symmetry and Structural Properties of Condensed Matter. World Scienti"c, Singapore, pp. 279}316. Michel, L., 1996. ExtreH ma des fonctions sur la zone de Brillouin, invariantes par le groupe de symeH trie du crystal et le renversement du temps. C. R. Acad. Sci. Paris B 322, 223}230. Michel, L., 1997a. Complete description of the VoronomK cell of the Lie algebra A weight lattice. On the bounds for the L number of d-faces of the n-dimensional VoronomK cells, preprint /P/97/53, IHES. Michel, L., 1997b. Physical implications of crystal symmetry and time reversal. In: Florek, W. (Ed.), Symmetry and Structural Properties of Condensed Matter. World Scienti"c, Singapore, pp. 15}40. Michel, L., Mozrzymas, J., 1989. Les concepts fondamentaux de la cristallographie. C. R. Acad. Sc. Paris 308, 151}158. Michel, L., Zak, J., 1999. Connectivity of energy bands in crystals. Phys. Rev. B 59, 5998}6001. Mighell, A.D., Ondik, H.M., Molino, B.B., 1977. Crystal data space groups tables. J. Phys. Chem. Ref. Data 6, 675}829. Minkowski, H., 1897. Allgemeine LehrsaK tze uK ber konvexe Polyeder. Nachr. Akad. Wiss. GoK ttingen Math.-Phys. Kl., 2. Minkowski, H., 1907. Diophantische Approximationen. Teubner, Leipzig; reprinted by Chelsea, New York, 1957. Moody, R., 1995. Mathematics of the long range order. Proceedings of NATO Conference, Waterloo. Palais, R., 1961. On the existence of a slice for actions of non compact Lie groups. Ann. Math. 73, 295}323. Phillips, J.C., 1956. Critical points and lattice vibration spectra. Phys. Rev. 104, 1263}1277. Phillips, J.C., Rosenstock, H.R., 1958. Topological methods of locating critical points. J. Phys. Chem. Solids 5, 288}292. Schoen#ies, A., 1891. Krystallsysteme und Krystallstructur. Teubner, Leipzig. Schwarzenberger, R.L.E., 1980. N-dimensional Crystallography. Pitman, London. Selling, E., 1874. Ueber die binaK ren und ternaK ren quadratischen Formen. Crelle " J. Reine Angew. Math. 77, 143}229. Senechal, M., 1991. Crystalline Symmetries. Adam Hilger, Bristol. Senechal, M., Galiulin, R.V., 1984. An introduction to the theory of "gures: the geometry of E.S. Fedorov. Topologie Structurale 10, 5}22. Serre, J.P., 1977. Linear Representations of Finite Groups. Springer, New York (translation from French, Hermann, Paris 1967). Shechtman, D., Bleeh, I., Gratias, D., Cahn, J., 1984. Metallic phase with long-range orientational order and no translational symmetry. Phys. Rev. Lett. 53, 1951}1953. Van Hove, L., 1953. The occurence of singularities in the elastic frequency distribution of a crystal. Phys. Rev. 89, 189}1193. Venkov, B.A., 1959. On the projection of parallelohedra. Mat. Sb 49, 207}224. VoronomK , G., 1908. Recherches sur les paralleH loe`dres primitifs. I. ProprieH teH s geH neH rales des paralleH loe`dres. Crelle " J. Reine Angew. Math. 133, 198}287. VoronomK , G., 1909. Recherches sur les paralleH loe`dres primitifs. II. Domaines de formes quadratiques correspondant aux di!eH rents types de paralleH loe`dres primitifs. Crelle " J. Reine Angew. Math. 136, 67}181. Weyl, H., 1952. Symmetry. Princeton Univ. Press, Princeton. Wigner, E., 1932. UG ber die Operation der Zeitumkehr inder Quantenmechanik. Nachr. Ges. Wiss. GoK ttingen Math.Phys. Kl. pp. 546}559. Wigner, E., 1959. Group Theory and its Application to the Quantum Mechanics of Atomic Spectra. Academic Press, New York. Wigner, E., Seitz, F., 1933. On the constitution of metallic sodium. 43 804}810. Zak, J., Casher, A., GluK ck, M., Gur, Y., 1969. The Irreducible Representations of Space Groups. W.A. Benjamin, New York.
Physics Reports 341 (2001) 337}376
Symmetry, invariants, topology. V
The ring of invariant real functions on the Brillouin zone Jai Sam Kim , L. Michel , B.I. ZhilinskimH * Department of Physics, Pohang University of Science and Technology, Pohang 790-784, South Korea Institut des Hautes E! tudes Scientixques, 91440 Bures-sur-Yvette, France Universite& du Littoral, BP 5526, 59379 Dunkerque Ce& dex, France
Contents 1. Introduction 2. Choice of representatives for the 13 and 73 arithmetic classes for d"2, 3 3. Linearization o(PX) of the PX action on BZ; its Molien function 3.1. Case 1. PX is orthogonal; we prove dim o(PX)"2d 3.2. Case 2. PX is hexagonal or I; we prove dim o(PX)"2(d#1) 3.3. Case 3. the eight F-arithmetic classes dim o(PX)"12 4. The module of invariants on BZ for the two-dimensional arithmetic classes 4.1. General remarks 5. The module of invariants on BZ for the three-dimensional P, C, A, R arithmetic classes 5.1. The triclinic, P-monoclinic, P-orthorhombic classes
339 341 344 345 347 351 353 355
355 356
5.2. The C-monoclinic, C-orthorhombic, Ptetragonal classes 5.3. The rhombohedral and P-cubic classes 5.4. The three-dimensional hexagonal system 5.5. Modules of C and A arithmetic groups over BZ of primitive lattices 6. The modules of invariants of the F, I arithmetic classes 6.1. The eight F arithmetic classes 6.2. The eight I arithmetic classes of the orthorhombic and cubic systems 6.3. The eight I arithmetic classes of the tetragonal system 7. Study of the d"2 invariant polynomials on BZ; the orbit spaces 7.1. Invariant functions for 2-D hexagonal classes 8. Conclusion References
357 359 360 361 363 363 364 366 366 371 375 376
Abstract With the coordinates chosen in the previous chapter, we show explicitly how to linearize the action of crystallographic space groups on the Brillouin zone. For two-dimensional crystallography it yields eight four-dimensional representations and "ve six-dimensional representations. For the 73 arithmetic classes in
* Corresponding author. E-mail addresses: [email protected] (J.S. Kim), [email protected] (B.I. ZhilinskimH ). Deceased 30 December 1999. 0370-1573/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 0 0 ) 0 0 0 9 2 - 2
338
J.S. Kim et al. / Physics Reports 341 (2001) 337}376
dimension three, it yields, respectively, 33, 24, 16 linear representations of dimension 6, 8, 12. We give the corresponding Molien functions. For the representations of dimensions four and six, we compute the invariants (up to 96 numerator invariants for the R lattices). We can even extend the results to the 16 hexagonal arithmetic classes. All obtained results are presented in the form of short tables. The comparison with the table of the previous chapter is instructive. Using the possibility to make plots of invariant function for the two-dimensional crystallography we exploit our corresponding results and also study the orbit spaces. 2001 Elsevier Science B.V. All rights reserved. PACS: 61.50.Ah Keywords: Crystallographic groups; Invariant functions
J.S. Kim et al. / Physics Reports 341 (2001) 337}376
339
1. Introduction Many physical properties of periodic crystals of dimension d"2, 3 are described or measured by functions on the Brillouin zone ("BZ) invariant by S, the symmetry group of the crystal ("space group). Physically it is very important to know the conditions imposed by S-invariance on these functions, because these symmetries are excellent physical approximations and their consequences are very strong. The periodic crystals of interest to physicists are of dimension 3 or sometimes 2. The two-dimensional study is simpler, so it has also a pedagogical value. Finally, it is important to know the restrictions that must be satis"ed by the S-invariant functions which are also invariant under the time reversal symmetry T. That is for instance the case of the energy E(k) of the Fermi surface of the electrons or of a one branch energy band (" simple band). The numbers of space groups S are, respectively, 17, 230 for the dimensions d"2, 3. As for the Euclidean group Eu , their B invariant subgroup of translation ¸ acts trivially on the reciprocal space, and the Brillouin zone. So the space group S acts e!ectively through the quotient group S/¸"P which is the "nite point group. As we showed in Chapter I, Section 3, for d"2, 3, the set of crystallographic point groups forms, respectively, 9, 18 isomorphic classes, 10, 32 geometric classes (" conjugacy classes in O(d)) and 13, 73 arithmetic classes (" conjugacy classes in G¸(d, Z)) (in Chapter IV, Section 4). In Chapter I, Section 5.4, Tables 4, 5 give the invariants on our Euclidean space for the geometric class actions. Here for the need of microphysics, we study the action of the arithmetic classes. Indeed, as explained in Chapter I, Section 2, the e!ective actions of the groups P on ¸ are given by an injective homomorphism (i.e. with trivial kernel) PPG¸(3, Z)"Aut ¸. We denote the image by PX. The contragradient action (g C (g\)?) of PX is its action on BZ"¸K , the dual group of ¸ (i.e. the group on its set of unitary irreducible representations). Given an arbitrary function f (k) on BZ, by averaging over the group G we obtain an invariant function: 1 fM (k)" f (g\ ) k) . (1) "G" EZ% This averaging is a projection operator on a vector space of functions on BZ and it has been used in the physics literature; for an impressive review of applications to crystals see e.g. (Cracknell, 1974) (for instance, starting from plane waves one works with `the augmented plane wave methodsa). The sum and the product of invariant functions are invariant functions; so they form a ring. The Schwarz (1975) theorem (mentioned in Chapter I, Section 5) applies to our problem and tells us that, for the global coordinate system we have chosen on BZ, every PX-invariant smooth function is a smooth function of the PX invariant polynomials. So we can limit our study to the ring R.X of invariant polynomials. The aim of this chapter is to compute these rings for the 13#73 arithmetic classes. It is remarkable that for d"2, these rings have only 2, 3 or 4 generators; for d"3 and 62 arithmetic classes, the rings have 3, 4, 5 or
We do know that modulated and aperiodic crystals can be considered as projections of higher-dimensional periodic crystals, but we do not study them here. We gave some more. To be very precise we gave the invariants for the conjugacy classes of the point groups in G¸(3, Q) where Q is the "eld of rational numbers. They are useful for symmetry breaking problems.
340
J.S. Kim et al. / Physics Reports 341 (2001) 337}376
6 generators while 1 (Cm), 3 (C2, R3 , F23), 6 (F222, Fmm2, R32, R3m, I4, I4 ), 1 (R3) have, respectively, at most 7, 9, 10, 15 generators. When time reversal is added the number of generators is always 46, except for R3 . It is also remarkable that, as it is the case for linear representations, these rings have a module structure (see Chapter I, Section 5). These results are new and very useful. To explain the strategy we have used for solving our problem, we apply it to the easy one-dimensional case. The translation group is ¸&Z. There are only two space groups: S "Z and S "Z ) Z (!I) where the two-element point group P"Z "G¸(1, Z) is generated by !I, the symmetry through the origin, so it is the automorphism of Z changing n into !n. In d"1, with time reversal, it is the only group P we have to consider. A unirrep ("unitary irreducible representation) of ¸"Z is of the form Z U n C exp(ink). Since n is an integer, k is a real number de"ned modulo 2p. So the set of unirreps of ¸ forms a group (with the law of addition modulo 2p) which is called the dual group and denoted by ¸K by the mathematicians while the physicists call it BZ (" Brillouin zone). This group ¸K "BZ is isomorphic to ; , the multiplicative group of norm 1 complex numbers. Its topology is that of a circle, S . The action of the point group P on BZ is de"ned by (!I) ) k"!k. Beware that this action, which is the linear representation p on the `reciprocal spacea (here the set of the real numbers k), is not linear on BZ; indeed it has two "xed points k"0 and p. Then we can use two methods for computing the ring of PX invariant polynomials on BZ; since they complement each other, we shall use both of them. (1) For the study of periodic functions of one variable k mod 2p, it is natural to introduce the trigonometric series in c"cos(k) and s"sin(k). The Schwarz (1975) theorem invites us to study "rst the polynomials P[c, s]. Using the algebraic relation cos k#sin k"1, we can transform in each monomial cBsBY the factor (s) BY into (1!c) BY so an arbitrary polynomial in c, s becomes at most linear in s. In other words we have replaced the ring of polynomials P[c, s] by the module of basis (1, s) on the ring P[c] that we denote by P[c]䢇(1, s) with the notation introduced in Chapter I, Section 5, Eq. (77). We computed in Chapter I, Section 5.4.3, Eq. (121) the four-dimensional module of polynomials on the 2-D torus; the eight-dimensional module for the 3-D torus is given in Section 5.1. For the point group P of the other (and larger) one-dimensional space group s is not an invariant, but only a pseudo-invariant. So the P-invariant polynomials form the one-dimensional submodule (" polynomial ring) P[c]. (2) To apply the other method to P, we "rst linearize its action on BZ. To do it, we have to extend the linear representation p to one acting on a two-dimensional orthogonal space < with orthogonal coordinates s, c. With the relations: s"sin k, c"cos k, the equation of BZ in < is that of the unit circle: c#s!1"0. The action of P on BZ is deduced from the linear representation o(P) on < de"ned by !1 0 ,p(!I) c(!I) . (2) o(!I)" 0 1
where c is the trivial representation of Z since cos(!k)"cos k. Notice that o(!I) represents a re#ection in the two-dimensional space; it leaves c invariant and changes the sign of s. As Let e"$1 the elements of the point group Z . The group law of the semi-direct product Z ) Z is (n , e )(n , e )"(n #e n , e e ).
J.S. Kim et al. / Physics Reports 341 (2001) 337}376
341
explained in Chapter I, the ring of its invariant polynomials is the ring of two variable polynomials P[c, s]. The BZ equation de"nes an ideal of this polynomial ring; the quotient is the polynomial ring P[c] (its elements are all polynomial in the variable c"cos k). Those are the PX invariant polynomials on BZ that we were looking for; that result was obvious. We will follow exactly the same method for d"2, 3. As the reader will see, computations become more and more complicated, several new di$culties appear and the results are less and less obvious. Their usefulness is equivalent to that of the one-dimensional result! One has to choose a coordinate system for giving explicitly the invariant polynomials. That is done in Section 2 for the 13#73 PX; we give their generating integral matrices in a basis of vectors generating the translation lattice ¸. This is done in ITC for the P ("primitive lattice). However, for the 16 I and the eight F arithmetic classes, it will appear that simpler results can be obtained with the use of non-primitive cell (as in ITC) by using the orthogonal axes of the corresponding primitive lattice. In Section 3 we build for each PX the orthogonal representation o (whose existence has been proved by the Mostow theorem (Mostow, 1957) which linearizes its action on BZ. This representation is the direct sum: o"p c
(3)
of two representations acting, respectively, on the variable s and c . The dimension of o is 4, 6 for 8, G G 5 two-dimensional arithmetic classes and for dimension three, 6, 8, 12 for the 33 orthogonal arithmetic classes, the sixteen P-hexagonal and sixteen I ones, the eight F ones, respectively. Then we compute the corresponding Molien functions. As we will explain, we need to compute the module of invariant polynomials of the o representations for only a few PX; but we decided not to compute modules with dimension '48. So we do not deal with I or F arithmetic classes in their natural basis. But, as we explain at the end of Section 5, we make the computation in the non-primitive orthogonal coordinates used by the ITC. That is done in Section 6; the results are simple and probably more useful for applications. One needs to make a complete computation for few groups (generally the smallest groups in each family of modules on the same ring). The lengthiest computation is for R3. We obtain the other modules as submodules by using several simple theorems gathered at the end of Section 4. It is also simpler to draw pictures for two-dimensional invariants; we do it in Section 7. In Section 8, to conclude, we emphasize the beauty and usefulness of our results and we suggest some applications.
2. Choice of representatives for the 13 and 73 arithmetic classes for d ⴝ 2, 3 In order to express the invariant polynomials we need to choose a coordinate system or, what is equivalent, to choose matrices l de"ning one of their groups for each of the 13#73 arithmetic classes. This choice will be made by the de"nition of the following matrices (Eqs. (4)}(11)). For some groups, ITC consider several bases. In fact our results are given for all arithmetic classes, in one of the ITC bases.
342
J.S. Kim et al. / Physics Reports 341 (2001) 337}376
For d"2: (p1)"I , (p2)"!I , 0 !1 0 !1 (p3)" "!(p6), (p4)" , 1 !1 1 0
1
(pm)"
0
0 !1
,
(cm)"
0 1 1 0
.
(4)
Notice that (p4)"(cm)(pm). The generator of the group p6 describes the rotation by 5p/3, but it is convenient to introduce the pairs of $ matrices. For d"3, (P1)"I , (P1 )"!I "!(P1), !1 0 0
(P2)"
0 !1 0 "!(Pm) , 0
0 1
0 !1
(C2)" !1 0
0
(5)
0
0 "!(Cm) ,
(6)
0 !1
0 !1 0
0 0 1
(P3)" 1 !1 0 "!(P3 ), (R3)" 1 0 0 "!(R3 ) , 0
0 1
0 1 0
(P6)"(P3 )(Pm)"!(P6 ),
0 !1 0
(P4)" 1 0
(7)
0 1
0 0 "!(P4 ), (I4)" 0 1
0
0 1 !1 "!(I4 ) .
!1 1
(8)
0
The 7#16 matrices we de"ned have been computed in Chapter IV, Section 4.4. Each one represents an arithmetic element (see Chapter IV, Section 4.2, 4.3) and the cyclic groups they generate are the cyclic arithmetic classes (their notation is identical to that of the element). From these matrices and some of their conjugate forms, we can de"ne "nite subgroups of G¸(d, Z), representatives of all non-cyclic arithmetic classes for d"2, 3. It will be convenient to use the following three sets of three re#ections:
1 0
0
0 , (Pm)"m " 0 1 0 0 !1 0 1 0
(Cm)"j " 1 0 0 , 0 0 1
!1 0 0
m "
0 1 0 , 0 0 1
1 0 0
j " 0 0 1 , 0 1 0
1
0 0
m " 0 !1 0 , 0 0 1
0 0 1
j " 0 1 0 , 1 0 0
(9)
(10)
J.S. Kim et al. / Physics Reports 341 (2001) 337}376
0 !1 0
f " !1 1
0 0 , 1 1
1
1
1
f " 0 0 !1 , 0 !1 0
f "
343
0 0 !1 1 1
!1 0
1 .
(11)
0
We check that (R3)"j j "j j "j j , (P4)"m j "j m , (I4)"j f ?"f ?j ,
(12)
(I4)"!f ?, m "!I " f . G G G G
(13)
Notice that the re#ections j , f belong to the arithmetic class Cm; notice also that the j 's generate G G G R3m, the f 's generate Fmmm and the f ?'s generate Immm and (13) shows that (I4) is an I rotation G G by n around axis 3. More generally, with the matrices of (4)}(13), we build a representation group PX for all arithmetic classes in dimension d"2, 3. What we need here is to build for each PX its contragradient representation PX U g C p(g)"g with g " : (g\) ? since we are interested in the PX action on BZ. The contragradient correspondence gPg is a duality on matrices which leaves "xed the orthogonal matrices. It induces a duality on the arithmetic classes; for d"2, 3 the 8, 33 arithmetic classes which are 4O(2, Z)" p4mm,4O(3, Z)"Pm3 m are self-contragradient. That is also the case of 3, 14 other arithmetic classes which are transformed into themselves up to an equivalence. There are 2, 26 non-self-dual arithmetic classes; they form 1, 13 dual pairs: d"2: p3m1 p31m , d"3: F222 I222, Fmm2 Imm2, Fmm Immm ,
(14)
I4 m2 I4 2m, P321 P312, P3m1 P31m , P3 m1 P3 1m, P6 m2 P6 2m .
(15)
F23 I23, Fm3 Im3 , F432 I432, F4 3m I4 3m, Fm3 m Im3 m .
(16)
Let us look at the two-dimensional case in more detail. The relation ( p 3)"(p4)(p3)(p4)\ shows that p3 is self-contragradient (up to a conjugation); but (p4)(cm)(p4)\"!(cm) shows that the arithmetic classes p3m1 and p31m (de"ned in Chapter IV, Section 4.2) are in duality. The contragradient representation of p3m1 is generated by (p 3) and (cm). The conjugation by (p4) shows that the dual class of p3m1 is equivalent to p31m. We give in Table 1 the generators of the contragradient representation p for the chosen PX for the 13#73 arithmetic classes. The corresponding bases coincide with that of ITC for the arithmetic classes whose labels begin by p, P, R, but not for the others. Later, at the end of Section 5 for the C, A classes and in Section 6 for the F, I classes, we will write the invariants in the `primitive basesa.
344
J.S. Kim et al. / Physics Reports 341 (2001) 337}376
Table 1 Generators of the contragradient representation l(PX) of the arithmetic class representatives PX p2: !I c2mm: $(cm) p3m1: (p 3), (cm)
pm: (pm) p4: (p4) p31m: (pI 3),!(cm)
p2mm: $(pm) p4mm: (pm), (cm) p6: !(pI 3)
cm: (cm) p3: (p 3) p6mm: !(pI 3), (cm)
P1 : !I C2: !j3 Pmm2: !m , m Amm2: !j , m ? Fmmm: f G P4: (P4) P4mm: (P4), m I4: (II 4) I4mm: (II 4), f R3: (R3) R3 m: !(R3),$j G P321: (PI 3),!j P3 1m: !(PI 3), m j P622: (PI 6),!j P6/mmm: (PI 6), m , j P4 3m: !m , j ?G G F432: !f ,!j G G Im3 : f , (R3) G
P2: !m Cm: j Pmmm: m G Cmmm: $j , m I222: !f ,!f P4 : !(P4) P4 2m: !(P4),!m I4 : !(II 4) I4 m2: !(II 4), f R3 : !(R3) P3: (PI 3) P3m1: (PI 3), j P6: (PI 6) P6mm: (PI 6), j P23: !m ,(R3) G Pm3 m: m , j G G F4 3m: !f ?, j G G I432: !f ,!j G G
Pm: m C2/m: $j C222: !j ,!m ? F222: !f ,!f ? Imm2: !f , f P4/m: (P4), m P4 m2: !(P4), m I4/m: (II 4), f I4 2m: !(II 4), j R32: (R3),!j G P3 : !(PI 3) P3 m1: !(PI 3), j P6 : !(PI 6) P6 m2: !(PI 6), j Pm3 : m , (R3) G ? F23: !f , (R3) G Fm3 m: f ?, j G G I4 3m: !f , j G G
P2/m: $m P222: !m ,!m Cmm2: j ,!m ? ? Fmm2: !f , f Immm: f G P422: (P4),!m P4/mmm: (P4), m G I422: (II 4),!f I4/mmm: j , f G R3m: (R3), j G P312: (PI 3), m j P31m: (PI 3),!m j P6/m: (PI 6), m P6 2m: !(PI 6),!j P432: !m ,!j G G ? Fm3 : f , (R3) G I23: !f , (R3) G Im3 m: f , j G G
The 12#72 non-trivial arithmetic classes are listed roughly in the order of ITC. The generating matrices are the ones listed in Eqs. (4)}(13) or, when they carry above a %, they are their contragradient which are given in (17).
To use Table 1 we need only the matrices in Eqs. (4)}(13) and the contragradient matrices of the arithmetic elements (p3), (P3), P(6), (I4); they are
(p 3)"
!1 !1
,
1
0
1
1 0
(P 6)" !1 0 0 , 0
0 1
We verify that (I 4)"!f .
!1 !1 0
(P 3)"
(I 4)"
1
0
0 ,
0
0
1
1
1
0
0 !1 .
!1 0
1 0
(17)
3. Linearization q(P z ) of the P z action on BZ; its Molien function In the preceding section we have de"ned the matrices of the l representation of the arithmetic point groups PX; that de"nes a basis in the reciprocal space and on BZ. On it the coordinates are given by the components k mod 2p of the vector k in the reciprocal space EH. We are interested in G B
J.S. Kim et al. / Physics Reports 341 (2001) 337}376
345
the action of PX on functions on BZ, i.e. periodic functions on EH whose lattice period is 2p¸H. This B action is given by PX U g, (g ) f )(k)"f (g \ ) k)"f (g? ) k) .
(18)
Note that this action is a linear action on the vector space of such functions. The necessary and su$cient condition for such a function f on BZ to be invariant under PX is ∀g3PX, f (g ) k)"f (k) .
(19)
The simplest functions of period 2p are the sine and cosine of the coordinates k 's. For a given G k mod 2p¸H we denote their values s "sin k and c "cos k ; they can be considered as 2d H H H H coordinates in a 2d-dimensional space in which BZ is an algebraic manifold of equations: 14j4d, c#s"1 . H H A smooth function on BZ can be written as a Fourier series
(20)
(21) 14i4d f (k )" (a cos(nk )#b sin(nk )) . G GL G GL G G XLZ8 Since the functions cos(nu) and sin(nu) are polynomials in cos u and sin u, the function f is a polynomial series in cos k , sin k . That also suggests the interest of the polynomials themselves. G G To linearize the PX action we have several cases to consider for building the representation o of PX. 3.1. Case 1. PX is orthogonal; we prove dim o(PX)"2d That is the case of 8, 33 arithmetic classes for d"2, 3. They are subgroups of the two maximal arithmetic classes: O(2, Z)"p4m, O(3, Z)"Pm3 m .
(22)
The matrices of O(d, Z) have only one non-zero element per row and per column; this element is $1. When all elements are 1, it is a permutation matrix. So the group O(d, Z) is generated by its diagonal and its permutation matrices; more precisely a matrix of O(d, Z) is the product of one matrix of the Abelian group diag ($1), the group of d;d diagonal matrices of elements $1, and B one matrix of P , the group of d;d permutation matrices. One veri"es that diag ($1) is an B B invariant subgroup of O(d, Z) and O(d, Z)&ZB ) S , "O (Z)""2Bd! . (23) B B For d"2, 3 the groups diag ($1) and P de"ne the arithmetic classes: B B diag ($1)"p2mm, P "cm, diag ($1)"Pmmm, P "R3m . (24) The orthogonal matrices permute up to a sign the three angles k so they transform linearly into G themselves the three functions sin k on one hand and the three functions cos k on the other hand. G G Since the orthogonal matrices are self-contragradient, the three functions sin k transform under G the representation l of PX which is in the present case the represention that we have called p . Since
346
J.S. Kim et al. / Physics Reports 341 (2001) 337}376
cos(!k )"cos k the three functions cos k are permuted without changes of sign. The group G G G O(d, Z) operates on them through its quotient O(3, Z)/diag ($1)"P &S . That yields a linear B B B representation c(PX) obtained by keeping only the absolute value of the elements of the p matrices. The direct sum of these two representations o(PX)"p(PX) c(PX), dim o"2d
(25)
acts on the space of coordinates s , c and leaves invariant the algebraic equations (20) de"ning BZ. G G The restriction of the linear action of o on this manifold is the action of PX on BZ; o is, by de"nition, the linearizing representation. From (24), the kernel of the representation c is d"2, c(G)"G/(G5p2mm), d"3, c(G)"G/(G5Pmmm) .
(26)
Table 2 gives the image of c(PX) in P as another PX. B As we explain in the introduction, we will compute the module of invariants of the representation o(PX). They contain three types of invariants; those of the representation c are polynomials in c , G those of p are polynomials in s and the mixed ones, obtained from the products of two covariants G of the same nature, one from p, the other from c. The next step of the program will be to use the d equations s"1!c of the BZ manifold to eliminate the s. We will "nd that the module G G G structure is preserved and its ring on BZ is the ring of the module of the c representation. These rings are simple since the Im c (second column of Table 2) are all re#ection groups (trivial for p1, P1) except for R3; but this group has the same denominator invariants as R3m. So for d"2, 3 we will have only 2, 3 polynomial rings for all the modules: those of dimension 3 are given in (50). We leave as an exercise to the reader to check that o(R3 ) is equivalent to the regular representation of this six-element cyclic group. From the de"nition given in Chapter I, Eq. (34) we calculate the Molien function of these 8#33 representations o. They are listed, respectively, in Tables 4 and 5. We remark that for the eight groups pX representing the eight orthogonal arithmetic classes in dimension 2, correspond eight groups PX obtained by adding the trivial one-dimensional representation; so for their o representation: PX"pX I No(PX)"o(pX) I .
(27)
Table 2 List of the orthogonal arithmetic classes whose c representations have the same image; it is given in the second column dim
Im c(PX)
PX
d"2
p1 cm
p1 cm
p2 c2mm
pm p4
p2mm p4m
P1 Cm a R3 R3m
P1 C2 P4 R3 R32
P1 Cm P4 R3 R3m
P2 C2/m P4/m P23 R3 m
Pm C222 P422 Pm3 P432
d"3
P2/m Cmm2 P4mm
P222 Amm2 P4 2m
P4 3m
Pm3 m
Pmm2 Cmmm P4 m2
Pmmm P4/mmm
J.S. Kim et al. / Physics Reports 341 (2001) 337}376
347
These PX have the same symbol in ITC except that the "rst letter p is replaced by P. Their Molien functions are de"ned by M X "M X (1!t)\ . N .
(28)
3.2. Case 2. PX is hexagonal or I; we prove dim o(PX)"2(d#1) This case splits into three subcases: the two-dimensional hexagonal system, and, for dimension 3, the hexagonal system and the sixteen I arithmetic classes. It contains 5#16#16"37 arithmetic classes. Case 2a: the "ve two-dimensional hexagonal classes. The simplest arithmetic class belonging to this case is p3. Its p representation is generated by (p 3) given explicitly in (17). It transforms the coordinates of the vector k"k bH#k bH according to k !1 !1 k !k !k k k " , p 3 " p 3 " . (29) k 1 0 k k k !k !k Let us de"ne k by the relation k #k #k ,0 mod 2p . (30) This equation is preserved by the action of p 3 since this matrix acts on the k , a"0, 1, 2 by ? a circular permutation: k , k , k Pk , k , k . That is exactly the action of the matrix (R3) of Eq. (7). This orthogonal matrix is self-contragradient and must have the same action on the direct space. Let us verify it. In the direct basis i"1, 2, (b , b )"(3d !1)/2, we have G H GH (31) k"k bH#k bH"((2k #k )b #(k #2k )b ) . We can introduce in the direct space a third vector de"ned by
b #b #b "0 , (32) these three vectors form a regular 3-branch star since the two vectors b , b have the same length and form an angle of 2p/3. Then with (29), k can be written in a unique way a"0, 1, 2, k" k b . (33) ? ? ? This concludes our veri"cation; it shows that the linearization of the actions of the arithmetic classes p3 and R3 on BZ have the same o representation. The third line of Table 1 shows that, besides $(p 3), the other generators of the "ve groups representing the two-dimensional hexagonal arithmetic classes are $(cm) which are orthogonal matrices. One veri"es the identity of the p representations and, therefore, of the six-dimensional o linear representations. This identity of the o representations can be extended to the four other natural pairs between the groups of the 2-D hexagonal Bravais class and the groups of the
And 2mm is replaced by mm2; we do not know why this exception had been made by ITC.
348
J.S. Kim et al. / Physics Reports 341 (2001) 337}376
rhombohedral Bravais class: o(p3)"o(R3), o(p3m1)"o(R3m), o(p31m)"o(R32) , o(p6)"o(R3 ), o(p6mm)"o(R3 m) .
(34)
Let us show it for p3m1 and p31m. The actions of $(cm) on the coordinates k are ? (cm) ) k "k , (cm) ) k "k , !(cm) ) k "!k , !(cm) ) k "!k . (35) The Molien functions depend only on o and so the equalities given in (34) extend to the Molien functions; they are given in Table 4. Case 2b: the three-dimensional hexagonal arithmetic classes. There are 16 of them. Through the faithful contragradient representations of the two- and three-dimensional hexagonal arithmetic classes given in Table 1, one sees several relations subgroup ( group between the two- and three-dimensional classes. The "rst one is obtained for the three-dimensional group acting trivially on the third component k mod 2p of the BZ: h(H: p3(P3, p3m1(P3m1, p31m(P31m , p6(P6, p6mm(P6mm .
(36)
So the eight-dimensional linear representation o(H) is de"ned by o(H)"o(h) I , (37) where I acts on the coordinates s , c . From these groups we can generate all other hexagonal arithmetic classes except for three of them and compute their o representations. That is done in Table 3. It is easy to compute the Molien functions of the groups of the third and fourth lines from that of the "rst or second line groups (Tables 4 and 5) 1 1 , M F "M . M "M & F (1!t)(1!t) & F (1!t)
(38)
Table 3 Construction of the representations o(PX) for the two and three-dimensional hexagonal systems, from the o(PX) of the rhombohedral system. The arithmetic classes P321, P312, P622 do not appear. In the last column the I , !p acts on s , c . The identity of the o representaions of the "rst two lines was established in (34) l.s.
Group quintuplet
R h H H F H \
R3 p3 P3 P6 P3
R3m p3m1 P3m1 P6 m2 P3 m1
Generators R32 p31m P31m P6 2m P3 1m
R3 p6 P6 P6/m P6/m
R3 m p6mm P6mm P6/mmm P6/mmm
h 1 1H, m 2 1H,!I 2
o representation o(R) o(h)"o(R) o(h) I (o(h) I )6(o(h) !p ) o(H)6(!p(H) c(H))
J.S. Kim et al. / Physics Reports 341 (2001) 337}376
349
Table 4 Molien functions of the o representations of the two-dimensional arithmetic groups AC
Generators
D(t)
N(t)
p1 p2 pm p2mm cm c2mm p4 p4mm p3 p3m1 p31m p6 p6mm
I !I (pm) $(pm) (cm) $(cm) (p4) (pm), (cm) (pI 3) (pI 3), (cm) (pI 3),!(cm) !(pI 3) !(pI 3), (cm)
(1!t) (1!t)(1!t) (1!t)(1!t) (1!t)(1!t) (1!t)(1!t) (1!t)(1!t) (1!t)(1!t)(1!t) (1!t)(1!t)(1!t) (1!t)(1!t)(1!t) (1!t)(1!t)(1!t) (1!t)(1!t)(1!t) (1!t)(1!t)(1!t)(1!t) (1!t)(1!t)(1!t)(1!t)
1 1#t 1 1 1#t 1#t 1#2t#t 1#t 1#2t#6t#2t#t 1#t#2t#t#t 1#t#3t#5t#t#t 1#5t#6t#2t#3t#4t#2t#t 1#2t#3t#t#t#2t#t#t
Their denominator and numerator are D(t) and N(t). The Molien functions of the 3D rhombohedral classes are those of the 2D hexagonal class with the correspondence: o(p3) o(R3), o(p3m1) o(R3m), o(p31m) o(R32), o(p6) o(R3 ), o(p6mm) o(R3 m).
The new invariants are s , c in the "rst case and s , c in the second case. It will be also very easy to obtain for these 13 three-dimensional hexagonal arithmetic classes, the module of invariants on BZ from the correspondence of Table 3 with the two-dimensional hexagonal classes. The arithmetic classes P321, P312, P622 must be treated separately (see Section 5). Case 2c. The 16 I-arithmetic classes. With the components k of the vector k we introduce G k #k #k #k "0 mod 2p . (39) Then we obtain the tranformations of k by the matrices generating the contragradient representations of the I-tetragonal arithmetic classes:
k k k k (40) f ) k "! k , f ) k "! k , k k k k k k k k f ) k "! k , (I 4) ) k "! k . (41) k k k k They induce permutations up to a sign, of the 4 k . In the classical cycle notation for the ? permutation they can be written with the change of sign: f P!(14)(23), f P!(13)(24), f P!(12)(34), (I 4)P!(4231) .
(42)
350
J.S. Kim et al. / Physics Reports 341 (2001) 337}376
Table 5 Molien functions of the o representations of the three-dimensional arithmetic groups of P, C, A lattices AC P1 P1
Generators
I !I P2 !m Pm m P2/m $m C2 !j Cm j C2/m $j P222 !m ,!m Pmm2 !m , m Pmmm m G C222 !j ,!m Cmm2 j ,!m Amm2 !j , m Cmmm $j , m P4 (P4) P4 !(P4) P4/m (P4), m P422 (P4),!m P4mm (P4), m P4 2m !(P4),!m P4 m2 !(P4), m P4/mmm (P4), m G P3 (PI 3) P3 !(PI 3) P312 (PI 3), m j P321 (PI 3),!j P3m1 (PI 3), j P3 m1 !(PI 3), j P31m (PI 3),!m j P3 1m !(PI 3), m j P6 (PI 6) P6 !(PI 6) P6/m (PI 6),m P622 (PI 6),!j P6mm (PI 6), j P6 m2 !(PI 6), j P6 2m !(PI 6),!j P6/mmm (PI 6), m , j
D(t)
N(t)
(1!t) (1!t)(1!t)
1 1#3t
(1!t)(1!t) (1!t)(1!t) (1!t)(1!t)
1#t 1 1#t
(1!t)(1!t) (1!t)(1!t) (1!t)(1!t)
1#3t 1#t 1#t#2t
(1!t)(1!t) (1!t)(1!t) (1!t)(1!t)
1#t 1 1
(1!t)(1!t) (1!t)(1!t) (1!t)(1!t) (1!t)(1!t)
1#t 1#t 1#t 1#t
(1!t)(1!t)(1!t) (1!t)(1!t)(1!t) (1!t)(1!t)(1!t) (1!t)(1!t)(1!t) (1!t)(1!t)(1!t) (1!t)(1!t)(1!t) (1!t)(1!t)(1!t) (1!t)(1!t)(1!t)
1#2t#t 1#t#4t#t#t 1#2t#t 1#t#t#t 1#t 1#2t#t 1#t#2t 1#t
(1!t)(1!t)(1!t) (1!t)(1!t)(1!t)(1!t) (1!t)(1!t)(1!t) (1!t)(1!t)(1!t) (1!t)(1!t)(1!t) (1!t)(1!t)(1!t)(1!t) (1!t)(1!t)(1!t) (1!t)(1!t)(1!t)(1!t) (1!t)(1!t)(1!t)(1!t) (1!t)(1!t)(1!t) (1!t)(1!t)(1!t)(1!t) (1!t)(1!t)(1!t)(1!t) (1!t)(1!t)(1!t)(1!t) (1!t)(1!t)(1!t) (1!t)(1!t)(1!t) (1!t)(1!t)(1!t)(1!t)
1#2t#6t#2t#t 1#t#7t#10t#5t#5t#10t#7t#t#t 1#t#3t#5t#t#t 1#2t#4t#10t#4t#2t#t 1#t#2t#t#t 1#t#3t#5t#2t#2t#5t#3t#t#t 1#t#3t#5t#t#t 1#3t#5t#3t#2t#5t#4t#t 1#5t#6t#2t#3t#4t#2t#t 1#2t#6t#2t#t 1#5t#6t#2t#3t#4t#2t#t 1#2t#6t#4t#2t#4t#3t#2t 1#2t#3t#t#t#2t#t#t 1#t#2t#t#t 1#t#3t#5t#t#t 1#2t#3t#t#t#2t#t#t
J.S. Kim et al. / Physics Reports 341 (2001) 337}376
351
Table 5 Continued AC
Generators
D(t)
N(t)
P23 Pm3 P432 P4 3m Pm3 m
(R3),!m G (R3), m G !m ,!j G G !m , j G G m ,j G G
(1!t)(1!t)(1!t)(1!t) (1!t)(1!t)(1!t)(1!t)(1!t) (1!t)(1!t)(1!t)(1!t) (1!t)(1!t)(1!t)(1!t) (1!t)(1!t)(1!t)(1!t)(1!t)
1#3t#2t#2t#3t#t 1#3t#2t#2t#3t#t 1#t#t#3t 1#t#t#t#t#t 1#t#t#t#t#t
Their denominator and numerator are D(t) and N(t). The Molien functions of the 3D rhombohedral Bravais system are those of the 2D hexagonal system; the correspondence which gives these functions are given in the caption of Table 4.
The matrices in (40), (41) and the three j of (10) generate the contragradient representations of the G I-cubic arithmetic classes. The matrices j generate the permutation group S which is a subgroup G of S ;Z of permutations up to a sign and this group is isomorphic to O . So the p representation F of Im3 m is the natural four-dimensional representation of S ;Z (I ), and the c representation is obtained by forgetting the factor !I . We have solved the linearization of the action of Im3 m on BZ and also for the 15 other I-groups PX; indeed their o representation is obtained by restriction to these subgroups. It is straightforward to compute the corresponding Molien functions. They are given in the "rst of Table 6. 3.3. Case 3: the eight F-arithmetic classes dim o(PX)"12 The contragradient representations of the PX groups of the eight F arithmetic classes are generated by the matrices $f ? and j . The matrices f ? transform the coordinates k of a vector of G G G G the reciprocal space either into themselves or into the di!erence of two coordinates. To linearize the action on BZ we are led to introduce the following expressions: k "k !k , k "k !k , k "k !k , so k #k #k "0 . (43) ? The action of the f on the k , i"1, 2, 3 induces an orthogonal action on the k , a"1, 2, 3, 4, 5, 6; G G ? indeed the corresponding orthogonal symmetric matrices p( f ?) are: G 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 !1 0 0
0
0
0 0 !1 0
0
0
0 0
0
1
0
0
0 1
0
0
0 !1 0 0
0
0
0
0
1 0
0
0 0
0 0
0
0 0 !1
1 0
0
0 0
0
0
0 0
0
0 1
0
0
0 0 !1 0 0
0
,
Similarly, the p representation of the j is G p( j )"j !j . G G G
0
0 1
,
0
0 0
0
1 0
0
0 1
0
0 0
!1 0 0
0
0 0
0
1 0
0
0 0
0
0 0
0
0 1
.
(44) (45)
352
J.S. Kim et al. / Physics Reports 341 (2001) 337}376
Table 6 Molien functions of the o representations of the three-dimensional arithmetic groups of I and F lattices AC
Generators
D(t)
N(t)
I222 Imm2 Immm
!f ,!f !f , f f G (I4I ) !(I4I ) (I4I ), f (I4I ),!f (I4I ), f !(I4I ), f !(I4I ), j j ,f G !f , (R3) G
(1!t)(1!t)(1!t) Same as I222 (1!t)(1!t)
1#4t#10t#11t#12t#11t#10t#4t#t Same as I222 1#7t#7t#t
(1!t)(1!t)(1!t) Same as I4 (1!t)(1!t)(1!t) (1!t)(1!t)(1!t) (1!t)(1!t)(1!t) Same as I422 Same as I4mm (1!t)(1!t)(1!t)
1#3t#12t#12t#8t#12t#12t#3t#t Same as I4 1#9t#14t#8t#8t#14t#9t#t 1#t#8t#11t#11t#11t#11t#8t#t#t 1#2t#6t#5t#4t#5t#6t#2t#t Same as I422 Same as I4mm 1#5t#6t#4t#4t#6t#5t#t
(1!t)(1!t)(1!t) (1!t) (1!t)(1!t)(1!t) (1!t)(1!t) (1!t)(1!t)(1!t) (1!t) (1!t)(1!t)(1!t) (1!t) (1!t)(1!t)(1!t)
1#2t#7t#4t#7t#2t#t
I4 I4 I4/m I422 I4mm I4 m2 I4 2m I4/mmm I23 Im3
f , (R3) G
I432
!f ,!j G G
I4 3m
!f , j G G
Im3 m
f ,j G G
(1!t)(1!t) ?
?
1#2t#5t#8t#12t#14t#12t#14t#12t#8t #5t#2t#t 1#t#6t#7t#9t#9t#7t#6t#t#t 1#t#2t#4t#2t#4t#2t#4t#2t#t#t 1#2t#4t#4t#4t#6t#6t#6t#4t#4t #4t#2t#t
F222 Fmm2 Fmmm
!f ,!f !f ?, f ? ? f G
(1!t)(1!t) (1!t)(1!t) (1!t)(1!t)
1#9t#27t#27t#27t#27t#9t#t 1#6t#9t#9t#6t#t 1#3t#13t#15t#15t#13t#3t#t
F23
!f ?, (R3) G
(1!t)(1!t)(1!t)
Fm3
f ?, (R3) G
(1!t)(1!t)(1!t) (1!t)(1!t)
F432
!f ?,!j G G
(1!t)(1!t)(1!t) (1!t)
F4 3m Fm3 m
!f ?, j G G f ?, j G G
Same as F432 (1!t)(1!t)(1!t)
1#21t#58t#70t#104t#178t#178t#104t #70t#58t#21t#t 1#13t#33t#58t#127t#221t#332t#501t#641t #726t#803t#803t#726t#641t#501t #332t#221t#127t#58t#33t#13t#t 1#t#10t#32t#67t#135t#220t#335t#447t #494t#480t#422t#329t#231t#144t #66t#29t#11t#2t Same as F432 1#7t#18t#35t#75t#137t#233t#364t#484t #613t#730t#770t#759t#714t#614t #489t#358t#226t#142t#85t#36t#15t #6t#t
(1!t)(1!t)
Their denominator and numerator are D(t) and N(t).
J.S. Kim et al. / Physics Reports 341 (2001) 337}376
353
To obtain the matrices of the c representations one replaces the elements !1 of the p matrices of (44), (45) by 1. We have thus obtained the o"p c linearizing representation; it has dimension 12. The corresponding Molien functions are given in the last part of Table 6.
4. The module of invariants on BZ for the two-dimensional arithmetic classes We have established the linear four-dimensional o representations of the eight orthogonal two-dimensional arithmetic classes in Section 3.1 and given the corresponding Molien functions in Table 4. It is easy to write the module of o(PX) invariants. Then, using the algebraic equations (20) of the BZ we can eliminate the s variables from the module ring (after it has been made into a form G containing only s's). We "nd that the "nal result is still a module of invariants. G Let us illustrate the general method by the simplest case: p2"Z (!I ). Since o(!I )c "c , G G o(!I )s "!s , as suggested by the Molien function in Table 4, the corresponding module is G G RN"P[c , c , s , s ]䢇(1, s s ). Using the BZ equations: s"1!c we obtain for the module of G G invariants on the BZ: RN"P[c , c ]䢇(1, s s ). That is the generalization of the one-dimensional case studied at the end of the introduction; because of the existence of a non-trivial numerator invariant in the module corresponding to o(p2), the module on the BZ, RN has dimension 2. The trivial case p1"1 is slightly more complicated! Corresponding to the trivial o representation R"P[c , c , s , s ] (obviously, all polynomials). But, to use the BZ equations we have to transform the polynomial ring R into a four-dimensional module over a sub polynomial ring, repeating twice the transformation explained in Chapter I, Section 5: RN" P[c , c , s , s ]䢇(1, s )(1, s ). We obtain on the BZ: RN"P[c , c ]䢇(1, s )(1, s ) . (46) That is Eq. (121) from Chapter I, Section 5.5.3, obtained from a di!erent point of view, of the module of polynomial on a 2-D torus. The two other cases of the "rst line of Table 2, the c representations are trivial so the ring of the modules on BZ are the same: P[c , c ]; so we have to obtain a submodule of R.. For pm we want to treat together the two groups pm , i"1, 2 G depending on whether the symmetry axis is the axis 1 or 2. Then s is invariant. The group p2mm is G generated by the two groups pm and pm so its ring of invariants is the intersection of the two rings. It is P[c , c ], i.e. a polynomial ring; it is the direct generalization of the one-dimensional case. Notice that p2mm is a group generated by re#ection. The four arithmetic classes of the second line of Table 2 have the same c representation; its image is cm, a re#ection group. The ring of its invariant is P[c #c , c c ]. On this module we obtain for P1 (i.e. for all polynomials on BZ): (47) RN"P[c #c , c c ]䢇(1, s )(1, s ) . We want again to compute the invariants for both of the two-element re#ection groups Z (cm); we denote them by cm (cm is the cm of Table 1). With e"$1 we make both computations together: ! > RAK"P[c #c , c c , s #es , (s !es )]䢇(1, (c !c )(s !es )) "P[c #c , c c , s #s , s s ]䢇(1, s #es )(1, (c !c )(s !es )) "P[c #c , c c , s #s , s s ]䢇(1, s #es )(1, s s )(1, (c !c )(s !s )) .
354
J.S. Kim et al. / Physics Reports 341 (2001) 337}376
Table 7 Modules of invariant polynomials on the Brillouin zone for the 13 arithmetic classes in dimension 2 Class
h
h
u
u
B T B
s #s s !s s s (c !c )s s
s s s s
\>
\>
\\
>\
\\
T B
p3 p3m1 p31m p6 p6mm
c c c c c c c c c c c c c c c c (c c !s s ) c c (c c !s s ) c c (c c !s s ) c c (c c !s s ) c c (c c !s s )
s s s s G
cm > cm \ c2mm p4 p4mm
c c c c c #c c #c c #c c #c c #c c #c #c c !s s c #c #c c !s s c #c #c c !s s c #c #c c !s s c #c #c c !s s
s
B
p1 p2 pm G p2mm
B
u
dim s s
c s #c s !c s #c s
>\" \> \\
4 2 2 1 4 4 2 2 1 4 2 2 2 1
Time reversal restricts to the seven cases indicated in the 1st column by T, or B when it is a Bravais group. The dimension of the modules is given in the last column. As rings, all the modules of the table have from 2 to 4 generators.
\>"s #s !(c s #c s ),
\\"s !s #c s !c s #2(c !c )(c s #c s ), ( \>)"1!2h #h !4h . ( \\)"2#4h #h #20h !2h #20h h !h #4h h !4h .
Using the two equations s"1!c of the BZ, we obtain G G (48) P[c #c , c c ]䢇(1, s #es , s s , (c !c )(s !es )) . This is indeed a submodule of (47) and could have been obtained directly. In Table 7 we give a di!erent expression for the last numerator invariant by using the relation: (c !c )(s !es )" (c #c )(s #es )!2(ec s #c s ). Since c2mm is generated by the groups cm , its module of polynomial invariants is the ! intersection of the module of these two groups. Let us compute the module of invariants of o(p4) as a submodule of (47). The matrix (p4) transforms s into s and s into !s . So s , s form the basis of a two-dimensional, irreducible on the real, representation of the group p4 and s s is a pseudoinvariant. As we saw, from Table 2, the c representation of (p4) is the matrix (cm) which exchanges c , c ; hence c !c is a pseudo invariant and the module of p4 is (49) RN"P[c #c , c c ]䢇(1, (c !c )s s ) . The group p4mm is generated by its two subgroups c2mm and p4; hence the module of invariant polynomials of p4mm is the intersection of the modules of its two generating subgroups. Thus we have "nished the second part of Table 7. The results for the "ve hexagonal arithmetic classes are obtained from the modules of the "ve rhombohedral classes and given in Table 7. They are modules of invariants, but with
J.S. Kim et al. / Physics Reports 341 (2001) 337}376
355
inhomogenous polynomials. The four generators of RN have been obtained through brute force computations. Then it is easy to "nd the submodules corresponding to the four other hexagonal groups. The invariants of p3m1 must be invariant by its subgroup cm ; that choose \>. > Similarly the invariants of p31m must be invariant by its subgroup cm ; that choose } }. The } } \ group p2 leaves invariant c and changes the sign of s . So it changes the sign of \> and and G G leaves invariant their product >\; hence the basis of the module of p6, submodule of that of p3. The group p6mm is generated by the four other hexagonal groups; its module is the intersection of their modules. In Table 7 the module structure for each group is veri"ed by forming all equations (Chapter I, Section 5, Eq. (78)) for the u polynomials. The veri"cation is easy for the orthogonal classes; it is ? given in the caption of Table 7 for the hexagonal ones. 4.1. General remarks We want to gather di!erent remarks we made for the construction of the Table 7; indeed they are very general and allow us to build a strategy for making the tables of the 3-D arithmetic classes. (i) Study together the PX whose modules have the same ring P[h (c ), h (c ), h (c )]; e.g. for the 33 G G G orthogonal arithmetic classes, that gives three families which are given in Table 2. The ring depends only on the image of the representation c and for the last two lines of Table 2, R3 and R3m have the same denominator invariants (cf. Chapter I, Section 5, Table 4). Then we have to compute the module of the o representations of the smallest groups in each family as well as those not generated by their subgroups in the family and take their quotient by the ideal de"ned by the equations of BZ. It is remarkable that in our case the module structure passes to the quotient (notice that the ideal is contained in the ring of the module of o), but we do know from the "rst method that the ring of invariant of any PX is a submodule of R.. (ii) The module of invariants of a PX is a submodule of the modules of its subgroups. More precisely, when PX is generated by some subgroups its module is the intersection of the modules of the generating subgroups. The intersection is very easily made inside a family with a common ring for the modules. (iii) When a small P X is an invariant subgroup of PX in the same family and the quotient PX/P X is Abelian (e.g. P X is a subgroup of index 2 of PX) we can generally choose the basis of its module of P X-invariants on BZ such that the basis polynomials are either invariants or pseudo-invariants (i.e. change of sign) for the larger group PX. Then the module of PX is the submodule of that of P X whose basis is made of the P X basis polynomials invariant by PX. (iv) The weak condition that the invariants of the groups containing PX as a subgroup must be invariant by PX is a good check for the consistency of the tables and can even help to build them!
5. The module of invariants on BZ for the three-dimensional P, C, A, R arithmetic classes The results of this section are summarized in the four Tables 8}10, 12. We begin "rst with the 33 orthogonal classes. Table 2 shows that there are only 4 di!erent c representations. The groups P1
356
J.S. Kim et al. / Physics Reports 341 (2001) 337}376
Table 8 Bases of the modules of invariant polynomials on the three-dimensional BZ for the eight arithmetic classes P triclinic, monoclinic, orthorhombic
B
B
B
Class
s
s
s
s s
s s
s s
s s s
d
P1 P1 P2 Pm P2/m P222 Pmm2 Pmmm
x
x
x
x x
x x
x
x
x
x x x x x
8 4 4 4 2 2 2 1
x
x
x x
The common ring of these modules is P[c , c , c ]. The module dimensions are 1, 2, 4, 8 (last column). Their bases contain 1 and the polynomials marked by the x's. Time reversal restricts to the three cases indicated by a B (for Bravais group) in the 1st column. Four of these groups are invariant subgroups of Pm3 m, the symmetry group of the cube. In this group, for each of the four other classes, there are three groups, labelled by the preferred axis i"1, 2, 3, forming one conjugacy class. With i, j, k as a permutation of 1, 2, 3, here are their modules: R(Pm )"P[c , c , c ]䢇(1, s , s , s s ), G H I H I R(P2 )"P[c , c , c ]䢇(1, s , s s , s s s ), G G H I R(P2/m )"P[c , c , c ]䢇(1, s s ), R(Pmm2 )"P[c , c , c ]䢇(1, s ). H I G G G As rings the modules of the table have from 3 to 6 generators.
(trivial), Cm, R3m are re#ection groups. Their rings of invariants are, respectively, R. "P[c , c , c ], R!K"P[c #c , c c , c ] , R0K"P[c #c #c , c #c #c , c c c ] . (50) The group R3 is a unimodular subgroup of the re#ection group R3m, so its module of invariants has the same ring. The three rings of (50) are also the ring of the module of invariants over BZ. 5.1. The triclinic, P-monoclinic, P-orthorhombic classes So the "rst polynomial ring of (50) (all polynomials in c ) is the ring of the modules of the eight G arithmetic classes of line 3 ("rst line for d"3) of Table 2. We give the eight corresponding modules in Table 8. We have already explained in the previous section the construction of the module for the trivial group: R."P[c , c , c ]䢇(1, s )(1, s )(1, s ); this is the eight-dimensional module of the polynomials on a 3-D torus. The seven other modules of Table 8 are submodules of R.. Since the p representations of the seven other groups of Table 8 are diagonal, the eight elements of R. are either invariants or pseudoinvariants for each of these groups. Hence we can apply the strategy of Section 4.1(iii): The three elements s , s , s of the basis of the module R. generate the four following elements in Table 8; to construct the other modules one needs only the action of PX on the s : Pm changes the sign of s , P2 that of s and s , P1 changes the signs of the three s . Hence the G G next three lines of Table 8 are obtained by keeping only the invariants of these three groups. P2/m is generated by all previous groups in Table 8; so its module is the intersection of the four previous ones: its basis is (1, s s ). The next two groups, P222, Pmm2, contain P2 and do not
J.S. Kim et al. / Physics Reports 341 (2001) 337}376
357
Table 9 Bases of the modules of invariant polynomials on the three-dimensional BZ for the 15 arithmetic classes whose label begins with C, A, P4: monoclinic C, orthorhombic C, tetragonal P arith. cl.
B
B
T
B
C2 Cm C2/m C222 Cmm2 Amm2 Cmmm P4 P4 P4/m P422 P4mm P4 2m P4 m2 P4/mmm
s
s >
x
x
s \
s >
s
x
x x x
x x x x x x x
x x
s
c s \
c s \ >
x
x
x
c s \ \
c s \ \
x
x x x
c s \
x
c s \
d
x
8 8 4 4 4 4 2
x
x x
x x
x
x x x
x
x x x x
4 4 2 2 2 2 2 1
These modules are over the polynomial ring P[c , c , c ],P[c #c , c c , c ]. The PX groups of the Table have > 2, 4, 8, 16 elements and the dimension of the corresponding modules are, respectively, 8, 4, 2, 1. Time reversal restricts to the arithmetic classes indicated in the "rst column by T or by B for the Bravais groups of lattices. In order to make easier the veri"cation that the module of an arithmetic class G is a submodule of the modules of the smaller classes ((G), we add to the table the eight-dimensional module of C2 , represented by the matrix m j and Cm represented by !m j . R!Y"P[c #c , c c , c ]䢇(1, s , c s , c s , s , s , c s , c s ), > \ \ \ \ \ > \ R!KY"P[c #c , c c , c ]䢇(1, s , s , c s , s , s , c s , s ). \ \ > \ \ > As rings, the number of generators of these 15 modules is 3 (for P4/mmm), 4, 5 (for C222, Cmm2, P4), 6 (for C2/m, Amm2, P4 ), 7 (for Cm), 9 (for C2). We use the following shorthands: a for c or s : a "a $a , a "a a , a "a (a $a ), a "a a a . G G G ! !
contain P2/m; so each one keeps one of the two invariants of P2 not invariants of P2/m. By its de"nition, Pmm2 leaves s invariant. Finally, since Pmmm contains all seven previous groups, its module has dimension one, i.e. it is the polynomial ring in the c 's. This is also obvious from the fact G that p(Pmmm) is a re#ection group containing three re#ections changing the sign of each s separately. These results are gathered in Table 8. G 5.2. The C-monoclinic, C-orthorhombic, P-tetragonal classes The 15 orthogonal arithmetic classes of the fourth and "fth lines of Table 2 have the same c representation, that of Cm; the ring of their modules of invariants is the second of (50). On this ring we have for the module of polynomials on BZ: (51) R."P[c #c , c c , c ]䢇(1, c !c )(1, s )(1, s )(1, s ) . The two smallest groups in this list are C and C . Table 1 gives their p representations: Z (!j ) K and Z ( j ), respectively. The eigen polynomials of the c and p representations are linear
358
J.S. Kim et al. / Physics Reports 341 (2001) 337}376
Table 10 Bases of the modules of invariant polynomials on the three-dimensional BZ for the 10 arithmetic classes of the rhombohedral and the P-cubic Bravais classes arith. cl. s
T
B T
B
R3 R3 R32 R3m R3 m P23 Pm3 P432 P4 3m Pm3 m
s.s
x x x x x x x
s.s.s
cs
c.s.s
cs
c.s.s
f
(s)(f)
(s.s)(f)
(s.s.s)(f)
c[s] c[s] cs[s] cs[s] d
x
x
x
x
x x
x
x
x
x
x
x x x x x
x x
x
x x x x x
x
x
x
x x
x x
x
x x
x x
x x
16 8 8 8 4 4 2 2 2 1
These modules are over the ring P[c, c, c.c.c],P[c #c #c , c #c #c , c c c ]. The groups PX of 3, 6, 12, 24, 48 elements have modules of dimensions 16, 8, 4, 2, 1, respectively. The arithmetic classes satisfying time reversal are indicated in the "rst column by T or, when it is a Bravais class of lattices, by B. The invariant polynomials of R3 m, R3 (respectively, those of R3m, R32 not invariant by R3 m) are of even (odd) degree in the variables s ; the invariant polynomials of R3 m, R3m (respectively, those of R3 , R32 not invariant by R3 m) are even G (odd) under permutations of the variable indices. As rings, the numbers of generators of these modules are 3, 4, 5, 6, 8 (for R3 ), 9 (for R3m), 10 (for R32), 14 (for R3). For R3 notice that three invariants are a product of two other invariants and that s.s"((s)#c!3). Notations. We use a short code for labelling the invariant polynomials. Let i, j, k a circular permutation of 1,2,3. is the sum over circular permutations. a , b are expressions with index i, e.g. c , c, s , c s , cs ; we introduce the G G G G G G G G G shorthands: a" a , ab" a b , a.a" a a , a.a.a"a a a , a.b" (a b #b a ), a.b.b" a b b "b.b.a, G G G G G G H G H G G G H I a[b]" a (b !b )"!b[a]. If the obtained polynomial is multiplied by an overall integer factor O1, drop it. For G H I product or power of these invariants, we write each factor between ( ). Example for the code: f":c[c]" c (c!c). G H I
combination of c 's and s 's. To write them, it is convenient to use the following shorthands: G G a for c or s : a "a $a , a "a a , a "a a , a "a a . (52) G G G ! ! ! Then (51) can be written as (53) R."P[c , c , c ]䢇(1, c )(1, s , s , s )(1, s ) . \ > \ > The eigen polynomials of the common c representation are c , c , as invariants and c as > \ pseudoinvariants. For p(Cm), change c by s in the previous statement. For p(C2) exchange the words invariants and pseudoinvariants from p(Cm). This leads to the module of these two groups; their basis are, respectively, N "1, s , s , s , s , s , c s , c s , !K > > \ \ \ \ N "1, s , s , s , c s , c s , c s , c s . (54) ! \ > \ \ > \ \ \ These results form the "rst two lines of Table 9. The third line of the table is their intersection since C2/m is generated by C2 and Cm. The three next groups of Table 9 are generated by the following
J.S. Kim et al. / Physics Reports 341 (2001) 337}376
359
pairs of groups (see Table 1): C222"1C2, P22, Cmm2"1Cm, P22, Amm2"1C2, Pm2 .
(55)
As we show in (51), one transforms the modules of Table 8, into modules with the ring of Table 9 by multiplying their module basis by (1, c ); that yields for P2 and Pm: \ R."P[c , c , c ]䢇(1, c )(1, s , s , s ) , > \ R.K"P[c , c , c ]䢇(1, c )(1, s , s , s ) . (56) > \ Thus, from (55), we obtain the modules for C222, Cmm2, Amm2 by taking the intersections of the modules in (56) with those of the "rst two lines of Table 9. The group Cmmm contains the "rst six groups of Table 9; they generate it. So the Cmmm module is the intersection of the six modules of the lines above it. We have noted earlier that the groups Cmm2, P4, P4mm are, respectively, the direct sums c2mm I, p4 I, p4mm I. So the basis polynomials of their respective modules are s for all of them, and for Cmm2, P4, respectively, s , c s from Table 7 and their products with s . The \ cyclic group P4 is generated by the matrix (P4 )\"p4 !I. So, as for P4, c s is one of its basis \ invariant polynomial while s is a pseudoinvariant; since c and s are also pseudoinvariant, the \ two other invariants are s c and s s . \ From Fig. 2 of Chapter I, Section 3 we verify that the next "ve groups of Table 9 are each generated by two subgroups of the same table: P4/m"1P4, P4 2, P422"1P4, C2222, P4mm"1P4, Cmm22 , P4 2m"1P4 , Cmm22, P4 m2"1P4 , C2222 .
(57)
So the module of invariants of these "ve groups is the intersection of the modules of invariants of the two subgroups. The group P4/mmm contains the 14 other groups of Table 9 as subgroups; they generate it. So its module of invariants is of dimension 1. 5.3. The rhombohedral and P-cubic classes The module R0 of invariant polynomials on BZ, has dimension 16. This module is reproduced in the "rst line of Table 10. R3 "R3;Z (!I ) where !I is the symmetry through the origin, changes the sign of s and G leaves invariant c in its o representation. So the module of R3 is the submodule of R3 whose basis G elements ("numerator invariants) are of even degree in s . G The group R3m is the group of permutation of coordinates of both, the direct and the reciprocal space; R3 is the subgroup of even permutations. So the module of R3m is the submodule of R3 whose basis elements are also invariant by odd permutations (i.e they do not contain expressions of the type a[b]). Similarly the module of R32 is the submodule of R3 whose basis elements are invariant under the product of an odd permutation and a change of sign of s . The module of R3 m is the intersection of G the four previous modules.
360
J.S. Kim et al. / Physics Reports 341 (2001) 337}376
Table 2 shows that the P-cubic groups belong to the same family. The "ve cubic-P groups are generated by their common invariant subgroup P222 and their respective rhombohedral subgroup (see for instance Chapter 1, Section 3, Fig. 2). The module of invariants of P222 is given in Table 8: R."P[c , c , c ]䢇(1, s.s.s). As a direct application of Chapter I, Section 5, Theorem Chevalley 2 the ring of polynomials in the three variables c is a dimension "P3m""6 module G on the polynomial ring P[c, c, c.c.c] of the family we study, so R. is a dimension 12 module on the family polynomial ring. Its intersection with R0 is P[c, c, c.c.c]䢇(1, c[c])(1, s.s.s)"R., the 6th line of Table 10. The intersection of this module with those of R3 , R32, R3m, R3 m can be read directly from the "rst six lines of the table and yields its last four lines. 5.4. The three-dimensional hexagonal system This system has 16 arithmetic classes. Between the maximal one: "P6/mmm""24 and the minimal one "P3""3 there are two sets of seven classes whose groups have 12 and six elements, respectively. Fig. 4 of Chapter I gives the incidence relations between these two sets of seven groups; for the convenience of the reader we give these data in an easier form in Table 11. We know from Table 3 that the c representation of these 16 groups is either R3 I or R3m I; that leads to the same ring for all the modules of Table 12: P[h , h , c ], where the h are the two G invariants of the polynomial ring of the modules of the 2-D hexagonal groups. Table 3 also tells us how to proceed for constructing the module bases for the 3-D hexagonal family. To pass from the line h to the line H of Table 3 we add the invariant s and its products with the
's. This gives the invariants of the module basis of P3. This group is an invariant subgroup of the 15 other groups of Table 12. Their modules are strict submodules of the eight-dimensional module of P3. The rest of the line H of Table 3 gives the module of P3m1, P31m, P6, P6mm. To pass to the line H , since m changes the sign of s independently, keep only the invariants. This gives F G the module of P6 , P6 m2, P6 2m, P6/m, P6/mmm. To pass from line H to H , one has to keep only the invariants quadratic in the s : s , s , \ G
. That gives the modules of P3 , P3 m1, P3 1m and again of P6/m, P6/mmm. Table 11 Partial order of 14 hexagonal arithmetic classes for d"3 P3 P622 P3 1m P3 m1 P6/m P6 m2 P6 2m P6mm
x x x
P312
P321
P6
x x
x
x
P31m
P3m1
P6
x x
x x
x
x x x
x x
x x x
x
For a given six-element group, each column gives the three 12-element group containing it. For a given 12-element group, each line gives the three 6-element groups that it contains. This table has to be symmetric through its "rst diagonal.
J.S. Kim et al. / Physics Reports 341 (2001) 337}376
361
Table 12 Module of polynomial invariants of the three-dimensional hexagonal arithmetic classes
T
T T T
B
arithm. cl.
s
s
s
s
d
P3 P3 P312 P321 P6 P31m P3m1 P6
x
x
x x
x
x x
x x x
x
8 4 4 4 4 4 4 4
P622 P3 1m P3 m1 P6/m P6 m2 P6 2m P6mm
x x
x x
x x x
x
x x x
x x x
x x
x x x x x
x x x
P6/mmm
2 2 2 2 2 2 2 1
The ring of the module is: P[h , h , h ] with h "c #c #c c !s s , h "c c (c c !s s ), h "c . The 's polynomials are those of Table 7:
"s #s !(c s #c s ), "(c !c )(s #s )!(c #c !2c c )(s !s ). The arithmetic classes with groups of 3, 6, 12, 24 elements have modules of invariants of dimension 8, 4, 2, 1, respectively. Time reversal is satis"ed by the "ve classes indicated in the "rst column by T or by B for the Bravais class of the hexagonal lattice. As rings, all the modules of the table have from 3 to 6 generators.
As we noted at the end of Section 3.3, case 2b, using this systematic method, we have solved the problem for 13 out of the 16 hexagonal classes. Since the seven invariants must distinguish the seven classes of 12-element group, s has to be the numerator invariant of P622. The numerator invariant of any 12-element group PX is also a numerator invariant of its three 6-element subgroups: they are given in Table 11. With this remark we obtain the modules of P312 and P321; thus we have completed Table 12. 5.5. Modules of C and A arithmetic groups over BZ of primitive lattices For all groups PX we have studied up to now, we have used one of the coordinate systems used in ITC except for the groups of C-lattices. For the orthorhombic, tetragonal and cubic system, the 4, 2, 3 Bravais classes of lattices have three orthogonal symmetry axes; it is very tempting to use them as coordinate axes and it has some advantages (and some drawbacks). We shall conclude this For some groups, these tables propose several sytems of coordinates (e.g. for the rhombohedral groups).
362
J.S. Kim et al. / Physics Reports 341 (2001) 337}376
section by computing the modules of invariants of the 7 C and A Bravais classes in a system of coordinates used by ITC. Beware that we are studying the invariants of a group PX not on its Brillouin zone but on the BZ of another group whose BZ of PX is a submanifold but not a subgroup!. Let us "rst show the method for a two-dimensional lattice. Let us consider a lattice p2mm. In an orthogonal system of coordinates the lattice points have integer coordinates: i"1,2, k 3Z. As we G explained in Chapter IV, Section 3.3, we obtain the orthorhombic c2mm, sublattice of index 2 of p2mm, by imposing the supplementary condition on the integral coordinates of vectors: k #k 32Z, i.e. the sum of the vector coordinates is even. By duality the lattice p2mm becomes a sublattice of index 2 of c2mm; i.e., in the reciprocal space, the c2mm lattice has a `centringa, i.e. a point at the center of the fundamental rectangle 04k , k (2p of p2mm. The c2mm and p2mm lattices have the same axes of symmetry; since the mid-point between two lattice points is a symmetry center of the lattice, c2mm has moreover four symmetry centers in a fundamental rectangle. They form an orbit of the point group of p2mm since one orbit point has coordinates k "k "p/2. By the symmetry through this point, k is transformed into p!k so c are changed G G G into !c . With this new symmetry, the module of invariants of c2mm on BZ becomes a submodule G of the module of p2mm-invariants which is given in Table 7: RNKK"PNKK[c , c ],PNKK[c , c ]䢇(1, c )(1, c ) .
(58)
So (59) on BZ(p2mm), RAKK"P[c , c ]䢇(1, c c ) . Similarly, from the module of invariants of pm given in Table 7 and the fact that pm changes the sign of s : (60) on BZ(pm), RAK"P[c , c ]䢇(1, c c , c s , c s ) . We can easily pass to the three-dimensional C-Bravais classes. Nothing is changed for the orthogonal third axis; indeed the centring of the three-dimensional C-lattices is in the plane of the coordinates 1, 2. Crystallographers call it a one face centring. We want to consider this situation in the reciprocal space; we showed how to obtain it in dimension two to form a c2mm sublattice of p2mm. Similarly in the reciprocal space one obtains the reciprocal lattices of the Bravais groups C2/m, Cmmm from those of P2/m, Pmmm by adding a new period of coordinates w(p, p, 0). That imposes the transformations: c !c , c !c , s !s , s !s , c c , s s . On the Brillouin zone of Pmmm we have the equivalence of modules: R.KKK"P[c , c , c ],P[c , c , c ]䢇(1, c )(1, c ) . The module of Cmmm on this BZ is obtained by adding conditions (61):
(61)
(62)
(63) on BZ(Pmmm): R!KKK"P[c , c , c ]䢇(1, c c ) . For the Bravais group P2/m one has to choose a preferred axis: the rotation axis by p which is also, from the convention m,2 the direction normal to the re#ection plane. For the P-lattices the
J.S. Kim et al. / Physics Reports 341 (2001) 337}376
363
consensus chooses the axis 3. For the C-lattices, ITC pp. 114}121 present many possible choices. Using the module presentations given in the caption of Table 8, it is easy to make the computations corresponding to any of these choices. As an illustration we choose the "rst (and most used later) choice given by ITC, `unique axis b, cell choice onea; it corresponds to the de"nition of the matrices m , j . The preferred axis (that of the rotation by p) is 2: R. K "P[c , c , c ]䢇(1, s s ) (64) ,P[c , c , c ]䢇(1, c )(1, c )(1, s s ) . It is then straightforward to compute the module of the subgroups of C2/m. For the other three larger groups, we need to know two of their generating subgroups (see Table 1): C222"1C2, P2 2, Cmm2"1Cm, P2 2, Amm2"1C2, Pm 2 .
(65)
6. The modules of invariants of the F, I arithmetic classes 6.1. The eight F arithmetic classes In a Molien function the value N(1) (of the numerator for t"1) gives the dimension of the module of invariants. In Tables 4 and 5, for the o representations, the highest value of N(1) is, respectively, 24 (for p6) and 48 (for P3 ); in Table 6, the highest value is 6912 (for four of the "ve F-cubic groups). So we cannot make the similar computations done up to now for "nding the module of invariants in a generating basis. As we did in the previous case (Table 13), we will use the orthogonal bases used in ITC. For the Pmmm (" primitive) lattices, there is a basis with three orthogonal vectors: (b , b )"j d . In this basis the P-lattice vectors have arbitrary integer coordinates k while the G H G GH G Table 13 Bases of the modules of invariant polynomials on the three-dimensional BZ of primitive P-lattices for the seven arithmetic classes whose label begins with C, A: monoclinic C, orthorhombic C
B
B
arith. cl.
c
s
c s
C2 Cm C2/m C222 Cmm2 Amm2 Cmmm
x x x x x x x
x x x
x
c s
c s
c s
x
x
x
c s
c s
c s
c s
d
x x x
x
x
x
x
x
8 8 4 4 4 4 2
x
x x
c s
x x
These modules are over the ring P[c , c , c ]. The PX groups of the table have 2, 4, 8 elements and the dimension of the corresponding modules are respectively 8, 4, 2. Time reversal restricts to the arithmetic classes indicated in the "rst column by B for the Bravais groups of lattices. As rings, all the modules of the table have from 3 to 10 generators. Notations: a"c, s, a "a a , a "a a a . GH G H
364
J.S. Kim et al. / Physics Reports 341 (2001) 337}376
Table 14 Modules of the F-orthorhombic arithmetic classes. Bases of the modules of invariant polynomials on the threedimensional BZ of Pmmm 3 F arithmetic classes of the orthorhombic system
B
arith. cl.
c
c
c
c s G
c s G
c s G
c s G
F222 Fmm2 G Fmmm
x x x
x x x
x x x
x
x
x
x
c s
c s
c s
c s
d
x
x
x
x
8 8 4
The modules are over the ring P[c , c , c ]. The PX groups of the table have 4, 8 elements and the dimension of the corresponding modules are respectively 8, 4. Time reversal restricts to the arithmetic classes indicated in the "rst column by B for the Bravais group of lattices. As rings, all the modules of the table have from 3 to 9 generators. Notations: a"c, s, a "a a , a "a a a . GH G H
coordinates of the vectors of the F-lattices must satisfy: k 32Z (i.e. their sum is even). That gives G G the F-lattices as sublattice of index 2 of the corresponding primitive lattice. By duality, up to the scaling of the basis vectors: bH"jG nvb , the primitive lattice is transformed into itself and becomes G G G a sublattice of FH"PH6w#PH, with w"(, , ); that is exactly the `body centringa used in ITC for the I-lattices. We had already explained this F I duality. We have also explained how to pass from the corresponding primitive lattice to the reciprocal lattice of an F lattice: one has to introduce a new period for the invariant functions 2pw"(p, p, p). That condition changes the signs of c and s , so the invariant polynomials are homogeneous of even degrees in these six variables. G G That leads us to use for the module of invariants of Pmmm the equivalent form (see Table 8): (66) R.KKK"P[c , c , c ],P[c , c , c ]䢇(1, c )(1, c )(1, c ) . Using the last three lines of Table 8 we write with the same polynomial ring the modules of P222, Pmm2; then we keep only the even degree polynomials to obtain Table 14. The same method applies for the F-cubic arithmetic classes. First we have to choose for the polynomial ring of Pm3 m a polynomial ring of even degree polynomials (see Table 10): R.K K"P[c #c #c , c #c #c , c c c ] "P[c #c #c , c c #c c #c c , c c c ]䢇(1, c #c #c )(1, c c c ) . (67) With this presentation we write, from Table 10, the modules of the other P-cubic arithmetic classes. In each module, keeping only the even degree polynomials, we obtain the modules of the F-cubic classes (Table 15). 6.2. The eight I arithmetic classes of the orthorhombic and cubic systems The dual of these lattices are F-lattices. In ITC they are presented as P-lattices with three face centrings added: (, , 0), (, 0, ), (0, , ). So to pass to the reciprocal lattices we have to add the three In German `Inner zenntrita.
J.S. Kim et al. / Physics Reports 341 (2001) 337}376
365
Table 15 Modules of the F-cubic arithmetic classes. Bases of the modules of invariant polynomials on the three-dimensional BZ of Pm3 m for the 5 F arithmetic classes of the cubic system
T
B
Class
c.c
c.s
c[c]
c .c[c]
s .c[c]
c s
c s .c[c]
d
F23 Fm3 F432 F4 3m Fm3 m
x x x x x
x
x x
x x
x
x
x
8 4 4 4 2
x x
x x
These modules are over the ring P[c #c #c , c c #c c #c c , c c c ]. The groups PX of 12, 24, 48 elements have modules of dimensions 8,4,2, respectively. The arithmetic classes satisfying time reversal are indicated, in the "rst column, by T or by B when it is a Bravais class of lattices. As rings, all the modules of the table have from 3 to 9 generators. Notations. Let i, j, k be a circular permutation of 1, 2, 3. We use a short code for labelling the invariant polynomials (listed here by increasing degree): c" c , c "c c c , s "s s s , c[c]" c (c!c), c[c]" c(c !c ). G G G G H I G G H I
periods (p, p, 0), (p, 0, p), (0, p, p). That corresponds for the c , s to the simultaneous changes of sign G G for each of the three possible pairs of indices. Remark that each square c, s is invariant for any of G H the three transformations; that is also the case for the products a a a when each a, a , a is either c or s. For the orthorhombic system, the presentations of the module of Pmmm invariants given in Eq. (66) is exactly what we need; the only basis element (O1) which is invariant for Immm is c c c . So (68) R'KKK"P[c , c , c ]䢇(1, c c c ) . From Table 8 we obtain immediately: R'"P[c , c , c ]䢇(1, c c c )(1, s s s ) , R'KKG "P[c , c , c ]䢇(1, c c c )(1, c s ) . G G The presentation of the module of Pm3 m invariants we need is
(69)
(70) R.K K,P[ c, c c c , c]䢇(1, c )(1, c c #c c #c c ) . G G G No basis element of this module is invariant under the three changes of pairs of signs. So the ring of Im3 m invariants is the polynomial ring in (70); remark that it is formally that of the re#ection group ¹ in Table 4 of invariants of Chapter I. The basic element of degree 3 in (70) can be transformed, B modulo the polynomial ring, into ( c )(c c #c c #c #c )!c c c . The product of this G expression with the invariant of Pm3 in Table 10 [ " c (c!c)] is the sixth degree invariant G G H I (c !c )(c !c )(c !c ) which coincides with the of ¹ in Table 4 of Chapter I where x, y, z F should be replaced by c , c , c . We verify that no other similar product satis"es the three changes of pairs of signs. This completes the construction of the modules of the cubic I arithmetic classes: R'"P[ c, c c c , c]䢇(1, s s s )(1, (c !c )(c !c )(c !c )) . G G R'K "P[ c, c c c , c]䢇(1, (c !c )(c !c )(c !c )) . G G
(71) (72)
366
J.S. Kim et al. / Physics Reports 341 (2001) 337}376
R'"P[ c, c c c , c]䢇(1, s s s (c !c )(c !c )(c !c )) . G G R' K"P[ c, c c c , c]䢇(1, s s s ) . G G R'K K"P[ c, c c c , c] . G G
(73) (74) (75)
6.3. The eight I arithmetic classes of the tetragonal system We recall that these classes are self-dual except for the following pair of classes I4 m2 I4 2m which are exchanged. In ITC they are presented as the primitive tetragonal ones with the centring w"(, , ). We can use this presentation directly on the reciprocal lattice (replacing w by 2pw). That seems to ful"ll the needs of the user; but one should be aware that it means that in the direct space, the coordinates used are not (up to a scale) the ones of ITC. With the centring 2pw the I invariant polynomials are homogeneous in c , s and of even degree, G G a convenient presentation of the P4/mmm module is (see Table 9) R. KKK"P[c #c , c c , c ] (76) "P[c #c , c c , c ]䢇(1, c #c , c , (c #c )c ) . Then R' KKK is the two-dimensional module of even degree polynomials. From the bottom half of Table 9, it is straightforward to build the next Table 16. 7. Study of the d ⴝ 2 invariant polynomials on BZ; the orbit spaces In Section 4 we established the structure of the module of the invariant polynomials for each of the 13 arithmetic classes; the results are given in Table 7. We noticed that, as rings, these modules Table 16 Bases of the modules of invariant polynomials on the three-dimensional BZ of P4/mmm for the eight arithmetic classes I-tetragonal
T
B
arith. cl.
c >
c s
c s >
I4 I4 I4/m I422 I4mm I4 2m I4 m2 I4/mmm
x x x x x x x x
x
x
x
c s \
c s >\
c s >\
c s \
c s >
c s
x
x
x x x
x x x
x
x
x x x
x
x
c s \
c s d >\
x
x
x
x
8 8 4 4 4 4 4 2
These modules are over the ring P[c #c , c c , c ]. The PX groups of the table have 4, 8, 16 elements and the dimension of the corresponding modules are, respectively, 8, 4, 2. Time reversal restricts to the arithmetic classes indicated in the "rst column by T or by B for the Bravais group of lattices. As rings, all the modules of the table have from 3 to 9 generators. Notations: c "c $c , c "(c $c )c , c "(c !c ), c "(c !c )c , s "s s , s "s s s . ! ! >\ >\
J.S. Kim et al. / Physics Reports 341 (2001) 337}376
367
have from 2 to 4 generators. Here, from the explicit knowledge of these generators, we build some of the orbit spaces BZ"PX; they are orbifolds whose singularities are related to the critical orbits. We recall in Table 17 (Michel, 1996), the critical orbits for the actions of the PX's on the Brillouin zone (we add also T invariance) and the positions and nature of the extrema for the simplest Morse functions (i.e. those with the minimal number of extrema); Chapter IV, Section 6 explains the building of this table. One can `showa an invariant function by drawing its level lines on BZ. It is interesting to do it for the generators of some rings of invariants. Fig. 1 shows the level lines of the three functions h "cos(k )#cos(k ), h "cos(k )cos(k ), u"sin(k )sin(k ) which generate the ring RAKK. To abbreviate the notation we will write cos(k ),c and sin(k ),s . Thus h "c #c , h "c c , G G G G and u"s s . Table 17 shows that c2mm has three critical orbits on BZ: O, R, AB. The 6-side Brillouin cell has four sides of same length; their middle represents the 2 points A, B on the same orbits. Morse perfect functions have only 4 extrema on the 2-torus: one maximum ("M), one minimum ("m) and two saddle points ("sp). For the perfect Morse functions invariant by c2mm on BZ, since A, B belong to the same orbit, they must be saddle points; that is the case of h . The functions h and u in Fig. 1 have eight extrema on BZ: 2M, 2m, 4sp; notice that u has saddle points not only at A, B but also at O, R while for h , the points O, R are maxima and the points A, B form an orbit of minima. Table 17 Extrema common to all functions on the two dimensional Brillouin zone, invariant by the crystallographic group and time reversal cr. syst.
BZ
sg
arithm. class
k"0
2k"0
Diclinic Orthorhombic
6 4 6 4 4
2 5 2 1 2
p2 p2mm c2mm p4 p4mm
p1 pm cm
O O O [O] [O]
R, A, B R, A, B R, AB [R], AB [R], AB
6 6
2 3
p6 p6mm
p3 p3m1 p31m
[O] [O]
RAB RAB
Square Hexagonal
3k"0
[CC ] [CC ]
Nb
0, 2
1
2, 0
Q(t)
4 4 4 4 4
1 1 1 [1] [1]
1, 1 1, 1 2 2 2
1 1 1 [1] [1]
0 0 0 0 0
6 6
[2] [2]
3 3
[1] [1]
1, t 1, t
Column 1 gives the crystallographic system; each contains one Bravais class except the orthorhombic one which contains two: pmm and cmm. Column 2 indicates the number of sides of the Brillouin cell. Column 3 gives the number of corresponding space groups. Columns 4, 5 list, respectively, the arithmetic class containing !I, (so T"time reversal is implied) and those which yield that arithmetic classes when !I is added to them. Columns 6, 7, 8 list the critical orbits for the arithmetic classes of column 4. When the Brillouin cell has six sides, the three points satisfying 2k"0 are R, A, B, the middle of the sides. We choose R to be a "xed point for c2mm and to correspond to the pair of shrinking symmetric edges (for pm or c2mm) when the Brillouin cell is transformed into a 4-side one (rectangle). Then R is represented by the four vertices and is invariant by the full point group. In the hexagonal system, the two points C, C satisfy 3k"0 and represent the six vertices of the Brillouin cell. The points of the orbits between [ ] have to be maxima or minima because the stabilizer acts as a two-dimensional representation irreducible on the real. Column 9 gives the minimal number of extrema. Columns 10}12 give, for the simplest Morse functions, the orbits of extrema with a given Morse index. Column 13 gives the corresponding polynomial Q(t) (de"ned in Chapter I, Eq. (132)).
368
J.S. Kim et al. / Physics Reports 341 (2001) 337}376
Fig. 1. Level lines for denominator and numerator invariants for c2mm class: upper left } h "c #c ; upper right } h "c c , center } u"s s .
We recall that the most general polynomial in c , s on BZ, invariant by c2mm is of the form G G f"p(h , h )#q(h , h )s s with h "c #c , h "c c , (77) where p, q are arbitrary polynomials in two variables. It is not very di$cult to discuss such a function when the degrees of p, q are low. From the relations u"(h !h #1)(h #h #1)50 , (c !c )"h !4h 50, h #150 , (78) we can build the orbit space BZ"c2mm of the action of c2mm on BZ in the space of basic (denominator) invariants h . The coordinates of the critical orbits in this space can be obtained G from the (k , k ) coordinates of the corresponding points in BZ: O"(0, 0), R"(p, p), A"(p, 0), B"(0, p). So coordinates (h , h ): O"(2, 1), R"(!2, 1), A"B"(0,!1) . (79) The orbit space BZ"c2mm shown in Fig. 2 is built from the three inequalities of (78); the "rst one gives the two straight lines R!(AB) and (AB)!O, the second one gives the parabola tangent to the two straight lines at R and O, the third one limits the domain to that between the parabola and
J.S. Kim et al. / Physics Reports 341 (2001) 337}376
369
Fig. 2. Orbifold for c2mm class.
its two tangents at R, O. The two basic invariants h and h do not specify completely the orbit. If the auxiliary invariant uO0, there are two orbits with the same values of h and h and opposite values for u. So we represent in Fig. 2 the space of orbits as a two-piece puzzle with corresponding points on the boundary having u"0 glued together: i.e. glue together the two parts u'0 and (0 along the common boundary R!(AB)!O. The two boundaries R!O stay distinct since they correspond to opposite signs of uO0. A schematic representation of the orbit space for c2mm is a disk with the critical orbit (AB) in the interior and the two critical orbits R and O on the boundary. The geometrical form of the orbifold depends on the choice of integrity basis polynomials. This choice is ambiguous as we remarked several times. For example, we can use instead of h the combination ah #h with an arbitrary parameter aO0. The choice a"!4 gives in particular the h "(c !c ) as a basic polynomial. In coordinates (h , h ) the space of orbits would have the geometrical form with R!O boundary being straight line and both R!(AB) and (AB)!O boundaries being parabola. More generally one should remark that the geometry of space of orbits changes with changing the basic polynomials but the type of singularity at vertices remains the same: the two critical orbits O, R are represented by cusp points and the critical orbit AB by the crossing of two lines at an angle O0. Using the same Fig. 2 we can explain the orbit space for the p4 arithmetic class. The two basic invariants for p4 are the same as for c2mm but the numerator invariant u is di!erent. Its square is N a product of the three factors: u "(h !h #1)(h #h #1)(h !4h )50 , (80) N so u "0 on the whole boundary O!(AB)!R!O. This means that to represent the space of N orbits for p4 we should glue together the two sub-orbifolds u'0 and (0 through the whole boundary. The result is the topological S sphere with three marked points corresponding to critical orbits. The space of orbits for p4mm class is "nally just one part of the c2mm or p4 orbifold. We can ful"ll on this orbifold the same topological analysis of the level sets of a simple function written in terms of h and h as that realized in Chapter I for the O group (see Section 5.4 of Chapter I). F To see the location of stationary points for an arbitrary functions f (h , h ) it is su$cient to consider the parallels f (h , h )"constant on the orbifold. In particular, for the h function the topology of the level h "const changes only when this level passes through the critical orbits, i.e. for h "!2, 0, 2.
370
J.S. Kim et al. / Physics Reports 341 (2001) 337}376
For the h function, the level lines h "const which pass through the critical orbits have the isolated values h "!1, 1. Besides that, there is another exceptional level line h "0' which is tangent to the boundary R!O at h "0, h "0. This means that for the h function two orbits (0, 0) on both parts (u'0 and u(0) of the c2mm orbifold are orbits of stationary points (four saddle points), but they are not critical points. In fact h and h functions are invariant with respect to p4mm class and for these four points form one orbit for the p4mm class. On the reciprocal space, the patterns of level lines of h and u are identical, with alternate (vertical or horizontal) bands of maxima and minima. Indeed, by a translation (p/2, p/2) on BZ the translated function hI coincides with u: hI "cos(k !p/2)cos(k !p/2)"sin(k )sin(k )"u . (81) Moreover, by a dilation of one period one makes the two periods equal and one obtains the h functions of p4mm. G As indicated by Table 17, for the symmetry p4mm or p4 the points O (center of the square BZ cell) and R (the four vertices) must be maximum or minimum of any invariant Morse function. From Fig. 1 after scaling transformation which makes the lattice quadratic one can show that it is true for the perfect Morse functions h (see Table 7). At the same time the auxiliary invariant function u is G N not a Morse function since its Hessian vanishes identically at O and R. This function shown in Fig. 3 has two non-degenerate saddle points on the orbit AB. It has also one p4-orbit of 4 maxima and one of four minima; these extrema are on a circle of BZ centered at O and the M's and m's are alternate: indeed u is p4 invariant and not p4mm invariant. The complete list of stationary points N of the u function "nally includes four maxima, four minima, two non-degenerate saddle points N and two degenerate saddle points. Slight perturbation of u by a Morse-type function removes N two degenerate saddle points producing the non-degenerate maxima (or minima) at O and R and four saddle points near O and near R. An interesting relation between functions h and h can be seen from Figs. 1. This relation has a very simple form for the p4mm class. To go from one function to another one needs to make a rotation by p/4 and to scale with factor (2. A corresponding transformation of the k , k variables can be written with the matrix
1 !1 1
1
,
(82)
Fig. 3. Level lines for the numerator invariant polynomial u . N
J.S. Kim et al. / Physics Reports 341 (2001) 337}376
371
whose determinant is 2. The corresponding relation between h and h has the form h (k , k )"h (k #k , k !k ) . (83) The factor 1/2 is unimportant because invariant polynomials are de"ned up to a scalar factor. The correspondence in Eq. (83) is the re#ection of the identity 2cos(A)cos(B)"cos(A#B)#cos(A!B) .
(84)
7.1. Invariant functions for 2-D hexagonal classes Let us analyze "rst the invariant functions hNKK"c #c #c c !s s and hNKK" c c (c c !s s ) (where the superscripts relate to the arithmetic class p6mm in Table 7), which are the basic invariant polynomials for all hexagonal arithmetic classes p3, p6, p31m, p3m1 and p6mm. These two invariant functions are represented in Fig. 4. There is a simple relation between these two functions: (85) h (k , k )"(1#h (2k , 2k )) . This relation follows from the explicit form of these functions taking into account the trigonometric identity 4cos(A)cos(B)cos(C)"cos(A#B#C)#cos(A#B!C) #cos(B#C!A)#cos(C#A!B) ,
(86)
with C"A#B. It is easy to see that function hNKK is a Morse-type function with a minimal (compatible with symmetry) number of stationary points. All stationary points of h are on critical orbits of p6mm class. h possesses one maximum, two minima, and three saddle points. (Naturally, if we change the sign of the function, the minima and maxima are interchanged.) If we consider the hexagonal cell, one critical orbit (orbit O in the notation of Table 17) lies in the center (one-point orbit); another critical orbit (CC ) corresponds to vertices of the hexagon. It is a two-point orbit; due to k , k periodicity three vertices correspond to the same point on the torus whereas one point of this orbit can be transformed into another by a geometrical operation (rotation by p/3). At last, the third
Fig. 4. hNKK (left) and hNKK (right) invariant functions for p6mm (or p3, p6, p31m, p3m1) class.
372
J.S. Kim et al. / Physics Reports 341 (2001) 337}376
critical orbit (RAB) corresponds to the middle of the edges. Opposite points on the BZ should be identi"ed (they correspond to the same point on the torus) and thus we have a three-point orbit. The same system of stationary points can be easily seen on k , k rhombohedral representation of the torus. Three stationary points are inside the rhomb. One point O (center of rhomb) forms itself the orbit, two others (CC ) are related through a geometric operation and form one two-point orbit. Four vertices of the rhomb correspond to one point R of the torus. Four middle points of the edges correspond in fact to two points (A, B) of the torus. Vertices and middle-edges are related by geometrical operations and form one three-point orbit (RAB) on the torus. We just remind that the minimal number of stationary points of a Morse-type function on torus (without symmetry) is four (one minimum, one maximum and two saddle points). Number of stationary points lying on critical orbits in the presence of p6mm symmetry is six. Thus h is a Morse-type function with the minimal possible number of stationary points compatible with p6mm symmetry. The function h is also a Morse-type function. But the number of its stationary points is much larger than the minimal number required by critical orbits. Besides the critical orbits it has three other orbits of stationary points. Each of these three orbits includes six points related by geometrical symmetry operations. The total number of stationary points is 24 (eight minima, four maxima, and 12 saddle points). In fact, the symmetry of h function is higher than the symmetry of h due to the translational symmetry on a half of period. The re#ection of this fact is the equivalence between stationary points forming di!erent orbits (for example, one-point orbit of the center and three-point orbit of the middle edges become equivalent by half of the translation of the lattice). To understand better the correspondence between an arbitrary invariant function and its system of stationary points we turn again to the representation of level lines for di!erent functions directly on the space of orbits drawn in terms of invariant polynomials. The space of orbits for p6mm is shown in Fig. 5. To "nd the admissible values of h and h we take into account restrictions imposed on h G by natural inequalities 50 and 50. As soon as "s #s !(c s #c s ) and
"s !s #c s !c s #2(c !c )(c s #c s ) are invariants of p3m1 and p31m, respec tively, and are pseudoinvariants of p6mm (invariants of subgroup of index two) and their squares are polynomials in h and h . Thus the equations for the boundary of the orbifold of p6mm can be written as (RAB)!E!O: h "(h !1), !14h 43 , (CC )!D!F!O:
(87)
h "h #h #!(2h #3) , !4h 43 , (CC )!(RAB):
(88)
(89) h "h #h ##(2h #3) , !4h 4!1 . In Fig. 5 the boundary (RAB)!E!O corresponds to "0 whereas the boundary (RAB)!(CC ) and (CC )!D!F!O corresponds to "0. The geometrical form of the orbifold is rather complicated but as soon as it is known we can use simple geometrical construction to "nd the system of stationary points of functions just by
J.S. Kim et al. / Physics Reports 341 (2001) 337}376
373
Fig. 5. Orbifold for p6mm class in hNKK, hNKK variables.
analyzing the constant levels of the functions on the orbifold. Let us take the h function. h "const levels are simple vertical lines in Fig. 5. Exceptional sections of the orbifold correspond to orbit O (maximum of h ), to orbit (RAB) (saddle point) and to orbit (CC ) (minimum). All these three orbits are critical and they are clearly seen in Fig. 4. Let us now take the h function. Its levels correspond to horizontal lines in Fig. 5. Section h "1 is exceptional. It corresponds to two di!erent orbits: O and (RAB). Both these orbits are maxima. Next, exceptional section corresponds to h "0. We have again two independent by symmetry orbits D and E. These two orbits are saddles. At last the section h "! is again an exceptional section with two independent orbits CC and F. All these stationary points are clearly seen on the contour plot of h function shown in Eucleadean space in Fig. 4. We can analyze equally for example, what will be the system of stationary points for a linear combination h #tan(a)h with aO0. The level set of this function is represented by a set of parallel lines forming with axis h an angle a. This function has generically no additional symmetry and on each critical level there is typically only one orbit of critical points. Positions of stationary points can be graphically found as points where constant level functions are tangent to the border of the orbifold. An interesting example is given by the function h !h . Levels h !h "const are tangent to orbifold at points O and (CC ). Remark that two boundaries of the orbifold are tangent at these points. Consequently, the orbits O and (CC ) are degenerate points of the function. Orbit O is a degenerate maximum and orbit (CC ) is a degenerate saddle. Fig. 6 con"rms this system of stationary points. The apparent number of stationary points on the BZ hexagonal (or rhombohedral) cell for this function is six. These points cannot be non-degenerate because they do not satisfy the Morse inequalities. In fact there are one maximum in the center O, two degenerate saddles (CC ) (vertices of hexagon), and three non-degenerate minima (RAB) at the middle of edges of the hexagon. Invariant functions for such subgroups as p6, p31m, p3m1, and p3 can be represented as well on the same hexagonal Brillouin cell but the symmetry of the function is naturally lower. Auxiliary invariant polynomials , , and are plotted in Fig. 7. First of all remark that all three functions , , and are of non-Morse type. Each has a degenerate stationary point at the center O of the hexagonal cell. The point O is a critical orbit with the symmetry at least p3. This means that this point should be stable (maximum or minimum) for a Morse-type function.
374
J.S. Kim et al. / Physics Reports 341 (2001) 337}376
Fig. 6. hNKK!hNKK invariant function for p6mm (or for any another hexagonal) class.
Fig. 7. Graphical representation for auxiliary invariant polynomials for p3 class: upper left } \> invariant function for p3m1 class; upper right } } \ invariant function for p31m class; center } >\ invariant function for p6 class.
The number of stationary points on the torus for is three (one degenerate saddle in the center of hexagonal cell, one maximum and one minimum on the vertices of the hexagon). This number is less than the number four required by Morse theory for a minimal number of non-degenerate stationary points of the function on a torus. The minimal number of stationary points of a smooth function on a manifold is given by Lusternik}Schnirelman category of the manifold (Lusternik and Schnirelmann, 1930; Lusternik and Schnirelmann, 1934; Fomenko and Fuks, 1989). The category
J.S. Kim et al. / Physics Reports 341 (2001) 337}376
375
for d-torus is equal to d#1. Thus the minimal possible number of stationary points on the 2-torus is three and the function supplies the example of a function with the minimal possible number of stationary points on the torus. Function is related to through a linear transformation of variables. This transformation is slightly more complicated than simple scaling between h and h . Namely, we can remark that if we turn the image of function by p/6 (or equivalently by p/2"p/6#p/3 ) and scale by a factor of 1/(3 we will have . The exact transformation reads
} }(k , k )" \>(2k #k , k !k ) .
(90)
Again the factor is unimportant whereas the determinant of the transformation in k , k variables which is equal to 3 is important. This determinant characterizes the increase of the area of the cell which triples under this transformation. Consequently, the number of stationary points for on the torus is three times the number of stationary points for . We see on BZ cell three degenerate saddles O, (CC ) (one in the center and two in the vertices of hexagon) and three maxima and three minima inside the hexagon. An auxiliary invariant function is the product . It has three degenerate saddles O, (CC ), three non-degenerate saddles (RAB), and six minima and six maxima at two generic orbits. To conclude our examples of the orbifold representation we remark that the space of orbits of index two subgroups of p6mm (namely of p6, p31m, p3m1) can be represented as two identical copies of p6mm orbifold glued together through the identi"cation of boundary points corresponding to zero value of auxiliary (numerator) invariant of the subgroup. Thus to get the p6 orbifold one must glue two copies of p6mm orbifold along the whole boundary. The function which is a numerator invariant of p6 vanishes along the whole boundary. For p3m1 the identi"cation should be done along O!(RAB) and for p31m along O!(CC )!(RAB). Finally the p3 orbifold in hNKK, hNKK variables can be represented as a four-body decomposition with certain identi"cation of boundaries.
8. Conclusion We have not only given a minimal set of generators for the ring of invariant polynomials on the Brillouin zone for each arithmetic class, but also we have given the (richer structure of the) free modules on a polynomial ring P: any invariant polynomial is a linear combination of the polynomials of the module basis with coe$cients in P. The generators of P and the module basis are homogenuous polynomials except for the 2D hexagonal system. This mathematical structure has a meaning independent of the coordinate system; the choice of coordinates is necessary for writing explicitly the invariants. For each arithmetic class we have chosen one of the choices made by the international tables of crystallography (ITC, 1996). We recall that this choice of coordinates is not "xed. It is a family of bases depending on some arbitrary parameters (among the values of the elements (b , b )) of the Gram matrix), their number depending on the crystallographic system: G H 6 for triclinic, 4 for monoclinic, 3, 2 or 1 dilations of axes for the other systems; these variations of parameters preserve not only the symmetry, but also the integral matrices of the representation of the point group. The computation of invariants is based on these matrices.
376
J.S. Kim et al. / Physics Reports 341 (2001) 337}376
The ring of invariants is evident in a few cases and in many other cases it is easy to construct one by one invariant polynomials. It is amazing that the complete set of generators for the 73 arithmetic classes can be written in a few short tables. For each problem in solid state physics, physicists who have to solve it for studying a given material, know how to introduce the crystal symmetry; so their results satisfy all requirements of the symmetry. These methods (e.g. LAPW"linearized augmented plane wave) work well and are presently introduced in computer codes; with them specialists can make the symmetry preserving computations they need. Of course these computations always use an approximation method and it could be interesting to express the results in a development in the invariant polynomials. These computations can help to think more about physics laws. We o!er these free modules of invariant polynomials for a di!erent approach of thinking about the implications of crystal symmetry. For instance it is easy to form from the tables, the orbit spaces BZ " PX. This has been done in Section 7, for 2D, on some examples. In the "rst three chapters we have shown many examples of the use of the orbit spaces. There has been some trend, in several domain of physics, to use them more; it may also be done in solid state physics. We hope that the tool we give in this chapter can "nd many applications by their users.
References Cracknell, A., 1974. Group theory in solid-state physics is not yet dead alias some recent developments in the use of group theory in solid-state physics. Adv. Phys. 23, 673}866. Fomenko, A.T., Fuks, D.B., 1989. Homotopic Topology. Nauka, Moscow. ITC, 1996. In: Hahn, T. (Ed.), International Tables for Crystallography. Vol. A. Space Group Symmetry. 4th, revised ed. Kluwer, Dordrecht. Lusternik, L.A., Schnirelmann, L.G., 1930. Topological Methods in Variational Problems. Moscow State University press, Moscow. Lusternik, L., Schnirelmann, L., 1934. Methodes topologiques dans problemes variationnels, ActualiteH s scient. et indust., Paris. Mostow, G., 1957. Equivariant embedding in Euclidean space. Ann. Math. 65, 432}446. Schwarz, G., 1975. Smooth functions invariant under the action of a compact Lie group, Topology 14, 63}68.
Or equivalently, in trigonometric series, since there are well-known algebraic transformations of cos nx and sin nx into nth degree polynomials in cos x, sin x.
Physics Reports 341 (2001) 377}395
Symmetry, invariants, topology. VI
Elementary energy bands in crystals are connected L. Michel , J. Zak * Institut des Hautes E! tudes Scientixques, 91440 Bures-sur-Yvette, France Department of Physics, Technion, Israel Institute of Technology, 32000 Haifa, Israel Contents 1. Introduction 2. The history of energy band symmetry 3. Elementary band representations. Elementary bands 4. The representations of the G 's in an elementary I band representation 5. Contacts between the branches of an elementary band
378 379 381 384 386
6. Some examples of the band structure of a space group 6.1. The non-simple elementary bands of Cmm2 6.2. The elementary bands of the three most frequent space groups of organic compounds 7. Conclusion References
389 389
390 393 394
Abstract A short review is given of elementary band representations of space groups and correspondingly a de"nition is presented of elementary energy bands in solids. Using these notions one can conclude that all the branches of an elementary energy band are connected. We present in this chapter a topologically unavoidable degeneracy in the band structure of solids. 2001 Elsevier Science B.V. All rights reserved. PACS: 71.28.#d; 02.20.Fh; 02.40.Pc Keywords: Narrow band systems; In"nite groups; General topology
* Corresponding author. E-mail address: [email protected] (J. Zak). Deceased 30 December 1999. 0370-1573/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 0 0 ) 0 0 0 9 3 - 4
378
L. Michel, J. Zak / Physics Reports 341 (2001) 377}395
1. Introduction In crystals the energy spectrum of bounded electrons "lls continuous intervals (called energy bands), separated by gaps. Bloch (1928) explained this structure by the new quantum mechanics from the ¸ ("translation lattice) symmetry. With Fermi statistics the electron states could be "lled up to some energy level. These basic facts explained many physical properties of crystals (e.g. the conduction of electricity and heat by metals) which had not been properly understood with the `olda quantum mechanics. For instance, Peierls (1929) studying the Hall e!ect, showed that a change of sign occurs when one considers a nearly "lled band instead of an nearly empty one. In many experiments the electron energy is observed as a function E(k) over the Brillouin zone ("BZ). This function is often multivalued, but each branch corresponds to a continuous function over BZ. Connected branches have to belong to the same energy band. The converse is not necessarily true, but energy branches cannot meet if they belong to distinct bands of the energy spectrum. In this paper, from the decomposition of their representations, we show that energy bands can be decomposed into elementary bands (i.e. bands which cannot be decomposed any further) whose symmetry can be de"ned by quantum numbers. The aim of this paper is to de"ne the quantum numbers of elementary bands and show that each quantum number "xes a number of branches; then to explain the essentially new result: the set of branches of an elementary energy band is connected. This can be a new de"nition of elementary bands. A mathematical proof of this theorem has to be very technical and is out of the scope of this journal. Here we explain on examples the several new concepts which are necessary for making the proof. The "rst part of this chapter deals with the symmetry of bands, reviewing rapidly the history of the problem in Section 2 up to the de"nition of elementary bands ("EB) and their classi"cation which are the subject of Section 3. The de"nition of elementary bands representations ("EBR) and of their quantum numbers was "rst given in Zak (1980); their classi"cation for all space groups G was completed in Bacry et al. (1988b). Since electron energy bands are stationary states, they are T invariant. Including this invariance, we give here the revised de"nition of EB and sketch the slight modi"cation that it yields for the EB classi"cation. In Section 4 we review the method for obtaining the decomposition of an EB representation into unitary irreducible representations of the G 's; these groups were introduced by Bouckaert et al. (1936) and are the groups of local I symmetry on BZ. In Section 5 we "rst review the known symmetry tools which predict contacts between branches; they are not su$cient for proving the connectivity of multi-branch elementary bands because these tools are local on BZ. Then we present the topologically global concepts necessary for the proof. In Section 6, their use is shown and explained on several examples of space groups: we study the structure of their elementary bands. As already pointed out by Herring (1937a, bottom of p. 364) the symmetry arguments used for the study of energy bands apply also to the vibration spectrum u (k) with its 3n branches (labeled J by l), where n is the number of atoms in a fundamental domain of translations. The 3n branches are connected for some families of crystals with small n values. Although that was well known experimentally (e.g. for diamond-like crystals), but a proof from symmetry arguments for some of these families, is rather recent (Michel et al., 1995). We will deal with this subject in another paper. For the notations and the concepts not explained in this chapter, the reader is implicitly referred to Chapter IV. To simplify the presentation we neglect spin e!ects.
L. Michel, J. Zak / Physics Reports 341 (2001) 377}395
379
2. The history of energy band symmetry The "rst and most quoted paper on the subject is by Bouckaert et al. (1936). In the introduction the authors explain that for a given k the energy spectrum is discrete and that the symmetry group is G ; so we can use the same symmetry arguments as those used for the states of atoms (eventually, I in an external "eld) and molecules. We have to know the representations of G on the Hilbert space I H spanned by the discrete set of vector states. Since, the Hamiltonian H commutes with the I action of G on H , the states belonging to a G irreducible representation of dimension n'1, I I I have the same energy E ; so we have at the point k of BZ a contact between n branches (these I contacts cannot occur on the BZ dense stratum but only on strata with higher symmetry. On the other hand, the set of eigenfunctions is in"nite and it is shown in the paper (Bouckaert et al., 1936) that, along a branch, E is a continuous function of k. We quote some sentences of their I introduction. `Thus a certain topology for the representations must exist and it will be shown that part of this topology is independent of the special BZ.a In their introduction the last sentence is: `The investigation of the `topologya of representations will be essentially the subject of this paper, from the mathematical point of view.a Indeed, in Bouckaert et al. (1936) the authors prove Corollary 4a of Chapter I, which reads here: There exists a neighborhood V(k )LBZ of a given k such that for any k3V(k ) the groups G 's are, up to a conjugation, subgroups of G (that we also denote by 4G ) and the authors I I I established the `compatibility conditionsa: k3V(k )LBZNs I -Res%II s I , (1) % % % where Res means `restriction of the group representation to the subgroupa and - indicates that s I is contained in the direct sum of components appearing in the decomposition into unirreps of % the representation to which it applies. The next step is Herring's thesis published in two papers: (Herring, 1937a,b). In the "rst paper, Herring shows that T ("time reversal) predicts contacts between some pairs of branches belonging to complex conjugate unirreps of G . These contacts may occur at the points kK of BZ I satisfying 2kK "0 or on a whole circle of BZ corresponding to a rotation axis in direct space, or on a whole face of the Brillouin cell. In the second paper, Herring studies the `accidental contactsa; they are not due to symmetry, they can occur between branches belonging to the same band of enegy spectrum, but they might belong to distinct elementary bands. These accidental contacts can occur on any kK of BZ and they move when the potential in the SchroK dinger equation changes (either by varying temperature, pressure, or changing an element into another one in the same column of the Mendeleev's table, without changing the crystal symmetry, i.e. same G and same EB quantum numbers); eventually, by moving these accidental contacts one can remove them. That is certainly the case when these contacts are due to the disappearance of an energy gap. In this
As is clear in Chapter IV, Section 8, the images of the G representations are either "nite or their closure is I a one-dimensional compact Lie group; so Corollary 4a of Chapter I applies. In Chapter IV, Eq. (51) we denote by kK the points of BZ: they are the class of vectors k in the reciprocal space, modulo the reciprocal lattice.
380
L. Michel, J. Zak / Physics Reports 341 (2001) 377}395
chapter, we will show the existence of another type of contact between the branches of an elementary band which was unknown up to now: These contacts can be moved but not removed because they are a consequence of symmetry and topology. From the thirties, and up to now, most studies of electron energy bands have been carried out by using their atomic spectroscopic labeling. Such a labeling re#ects the chemical composition of the crystal but it does not take in account its crystalline symmetry which is given by a space group G. Burneika and Levinson (1961) and des Cloizeaux (1963) have developed a local orbital approach for electronic energy bands which is based on the space group symmetry G. This is as follows: let t (x!a) be wave functions, fast decreasing at in"nity (with m labeling the local symmetry in the K independent particle approximation), which is localized around the site a and with spin e!ects neglected. The wave functions t (g\(x!a)), for all g3G span a Hilbert space H of functions K which also contains the state vectors of delocalized electrons in a band. G acts linearly on H; by de"nition of induced group representations, the linear representation of G on H is Ind%? cM? where % % cM? is the representation of the stabilizer G of the site a. Des Cloizeaux (1963, p. 561) gave an ? % obvious de"nition for an energy band as having a connected graph. He also explicitly discussed the four branch bands of valence and conduction electrons for the diamond structure as based on the local space group approach. In the 1960s the situation was clearer for the vibration spectrum u (kK ) of a crystal. The space of J eigenstates corresponding to this spectrum of eigenvalues carries the representation obtained by the `Frobenius methoda as a direct extension of the work of Wigner (1930) on the vibrations of the methane molecule; e.g. the excellent text book (Streitwolf, 1967). Kovalev (1975) has given a detailed description of the local space groups symmetry approach for phonons which, in more modern terminology, is based on induced representations: (2) Ind%U <(G ), <(G ) is the vector representation of G U U U U % and the direct sum is over all Wycko! positions occupied by atoms; e.g. Michel and Mozrzymas (1982) which also give for each kK 3BZ the explicit reduction in unirreps of G . I For electronic states, Zak (1980, 1982a,b) has introduced the concept of band representations of space groups based on the kq-representation. The concept of band representations was later extended by Evarestov and Smirnov (1984, 1986) who have also written a book (Evarestov and Smirnov, 1993) on site symmetry in crystals for di!erent excitations in solids. The concept of a band representation of a space group G is a unitary representation of G induced from any unitary representation of a stabilizer G where q is any point of a fundamental translation domain (i.e. not O necessarily an atom position). By de"nition, the stabilizers of points belonging to the same stratum ("Wycko! position) are conjugate in G so they induce equivalent band representations. We will generally denote the stabilizer up to a conjugation by indicating only its Wycko! position G , U using for w the label a, b, c,2, k given by ITC (International Tables of Crystallography (ITC, 1996)) and we denote a band representation by the couple (w, sMU ) where the second symbol is the % inducing representation. Using the step induction theorem (Chapter IV, Eq. (139)) we see that
Essentially, the continuity of the energy function on each branch. That might not be good approximation for molecular crystals.
L. Michel, J. Zak / Physics Reports 341 (2001) 377}395
381
several di!erent couples of symbols may de"ne the same band representation ("BR); for example, if G (G , then U UY BR"(w, sMU ),(w , sMY ) when sMY "Ind%UYU sMU . (3) % % % %UY %UY 3. Elementary band representations. Elementary bands Zak (1980) de"ned the elementary band representations as those which cannot be decomposed as a direct sum of other band representations. He gave the following conditions on w and the representation sMU for inducing elementary band representations: % (i) the unitary representation sMU must be irreducible (evident since Ind commutes with (see % Chapter IV, Eq. (140)); (ii) w is a maximal symmetry stratum (we know from Theorem 4a, Chapter I that these strata are the closed ones). Let w be a non-maximal (symmetry) Wycko! position such that w (w and let p be an irreducible representation of G . By step induction we know that sN UY and UY % sMU "Ind%UUY sN UY induce the same band representation; to be an elementary one requires that % % % sMU is irreducible. That justi"es condition (ii). % Condition (ii) can also be formulated as: the inducing subgroups are the maximal xnite subgroups of G. Indeed, let H be such a maximal "nite subgroup of G and Hx the H-orbit of x. The barycenter b""H"\ hx of this orbit is invariant by H and since H is a maximal "nite subgroup, H is the FZ& stabilizer G . It just happens that the necessary condition (ii) is not su$cient for de"ning @ elementary band representations. The "rst counter example was found by Evarestov and Smirnov (1984): the band representation of R3 c labeled by the Wycko! position a (de"ned in ITC), and induced from the two-dimensional irreducible representation of G "R32, is not elementary. ? The complete classi"cation of elementary band representations for all space groups G is made in Bacry et al. (1988b). This paper proves that among the more than 3000 band representations satifying conditions (i) and (ii) 40 of them are not elementary. That is due to the following situation: let G and G be maximal "nite subgroups of the space OY O group G and G "G 5G . From Proposition 7e of Chapter IV we know that G is also O O OY O a stabilizer. Similarly to Eq. (3) we have three symbols for de"ning the same BR, with w, w , w the Wycko! positions of the points q, q , q , respectively: BR"(w, sN),(w , sMY ),(w , sMO ), with sMY "Ind%OYO sNO , sMO "Ind%OO sNO . (4) % % % % % O %OY % %OY "Ind%OYO sNO is irreducible. Since the irreducible representations of point Let us assume that sMY % % %OY groups are of dimensions 1}3, we must have dim sNO "1. Bacry et al. (1988b) have shown that these % conditions imply that G 3+D , D , D , D ,, dim sMY "2 . OY B %OY
(5)
He called them irreducible band representations and commented (on p. 1026) `The irreducible band representations serve as elementary building bricks in the symmetry de"nition of bands in solidsa.
382
L. Michel, J. Zak / Physics Reports 341 (2001) 377}395
Table 1 List of the 40 band representations induced from two-dimensional unirreps of maximal isotropy groups and not elementary; they belong to 25 space groups. After the nM (in bold face for symmorphic space groups) and the symbol of the space group, the columns of this table give the Wycko! position, the maximal isotropy group G . All these BR are U induced from the two-dimensional unirrep of G . Note that D has two such unirreps U 124
"P4/mcc
131
"P4 /mmc
132
"P4 /mcm
139 140
"I4/mmm "I4/mcm
163 165 167
"P3 1c "P3 c1 "R3 c
a c e f b d d a b a a a
D D D B D B D B D B D B D D B D D D
188
"P6 c2
190 192
"P6 2c "P6/mcc
193 207
"P6 /mcm "P432
208
"P4 32
210
"F4 32
a c e a a a c d c d b c c d
D D D D D D D D D D D D D D
211
"I432
215
"P4 3m
217 222 223
"I4 3m "Pn3n "Pm3n
224 226 228 229 230
"Pn3m "Fm3c "Fd3c "Im3m "Ia3d
b c c d b b c d e d c b d b
D D D B D B D B D D B D B D D B D B D D B D
If sMO "Ind%OO sNO is reducible, then the BR of Eq. (4) is non-elementary although its second symbol % % % in Eq. (3) satis"es conditions (i) and (ii). We reproduce in Table 1 the list of the 40 exceptions given in Table 7 of Bacry et al. (1988b). If sM O "Ind%OO sN O is also irreducible, the authors (Bacry et al., 1988b) show that G &G ; this % % % OY O yields their Table 6 that we summarize by Proposition 2a. There are 32 pairs of equivalent band representations induced from two-dimensional unirreps of non conjugate isomorphic maximal isotropy groups belonging to geometric classes D , D , D , D . These pairs belong to 22 space groups. B ι
De5nition. Two isomorphic subgroups H P H of the group G are quasi-conjugate if each pair of their corresponding elements are conjugate in G; or in mathematical terms ∀h3H , g3G, n(h)"ghg\. When g is independent of h the two subgroups are conjugate. Using Frobenius reciprocity one shows immediately that for a "nite group G and any representation o of two quasi-conjugate subgroups H , i"1, 2 the two induced representations Ind% sM and G & & Ind% sM are equivalent. As we said in Chapter IV, we assume the validity of Frobenius reciprocity & & for the band representations. Even so, a subtlety appears: intertwinning unitary operators transforming one representation into the other cannot be continuous on BZ Bacry et al. (1988a) so such equivalent band representations can still be distinguished by a physically observable topological invariant. It had been already de"ned in a larger setting by Zak (1989a,b, 1991) who called it the Berry phase of the Bloch functions for the entire energy band. Its application to band representations is given in Michel and Zak (1999). We reproduce here, as Table 2, Table 4 from (Bacry et al., 1988b) with a modi"ed caption. There exist four more pairs of equivalent but distinguishable EBR; the isotropy groups are not even quasi-conjugate but that is true of their index 2 subgroup ¹; this equivalence occurs only for
L. Michel, J. Zak / Physics Reports 341 (2001) 377}395
383
Table 2 The 17 pairs of maximal isotropy groups which are quasi-conjugate but not conjugate; they belong to 14 space groups. They yield 58 pairs of equivalent elementary band representations which can be distinguished by an observable topological invariant and 5 pairs of completely equivalent band representations. These are induced by the twodimensional unirrep of the stabilizer D ; four of them, marked by * are elementary BR, while the one marked by L is not elementary (see Table 1). After the nM and the symbol of the space group, the columns of this table give the isotropy group and the pair of Wycko! positions 22"F222 68"Ccca 70"Fddd 94"P4 2 2 98"I4 22
D D D D D D
a,b c,d a,b a,b a,b a,b
118"P4 n2 163"P3 1c 182"P6 22 196"F23 203"Fd3
D D D ¹ ¹ ¹
c,d c,Hd c,Hd a,b c,d a,b
210"F4 32 212"P4 32 213"P4 32 214"I4 32
¹ D D D D
a,b c,Ld a,Hb a,Hb c,d
the two-dimensional representations of the isotropy group [see Table 5 from Bacry et al. (1988b)]. We list them here by giving the symbols of the space group, the Wycko! position and the stabilizer: 209"F432, ab, O; 216"F4 3m, ab, cd, ¹ ; 227"Fd3 m, ab, ¹ . (6) B B Finally, Bacry et al. (1988b) determine two sets of equivalent EBR at the same Wycko! position, for inequivalent inducing unirreps: (a) If N (G ), the normalizer in G of G , is strictly larger than G , it acts non trivially on the set of % U U U G unirreps and those in the same N (G ) orbit induce equivalent G representations. That U % U occurs only if G leaves invariant an axis (more precisely G "C , C , C , C ) and it yields 23 U U T pairs of equivalent EBR in 15 space groups (Table 2 of the paper), (b) Let H(G, H strict subgroup of G; then every conjugacy class of H is contained in a conjugacy class of G. If a conjugacy class of G contains several conjugacy classes of H, then it is possible that di!erent unirreps of H induce the same representation of G. That is the case for space groups G when the subgroup G belongs to the geometrical classes: C , D , D , ¹. That yields U T F 34 pairs of equivalent EBR at the same Wycko! position; they belong to 25 space groups (Bacry et al., 1988b, Table 3). To summarize the classi"cation of Bacry et al. (1988b): outside the list of 40 BR which are induced from unirreps of maximal G but which are not elementary, they list 95 pairs of EBR which have U distinct symbols but are fully equivalent and 62 pairs of EBR which are mathematically equivalent but physically distinguishable. An electron energy band whose band representations is elementary, is simply called elementary energy band. Since electron energy bands are stationary states, they are T ("time reversal) invariant. To take into account this physical symmetry, we have to modify the quantum numbers of EB: De5nition. An elementary band is de"ned by a maximal symmetry Wycko! position and a representation of its stabilizer sMU irreducible on the real except for the 40 exceptions listed in Table 1 (these % exceptions are unchanged because the two-dimensional irreducible representations of the four di!erent stabilizers of this table are all real).
384
L. Michel, J. Zak / Physics Reports 341 (2001) 377}395
From this de"nition and by taking into account all equivalence we have established, we can classify all inequivalent elementary bands of a space group. That has been done for the 230 space groups. Let us mention here that the 13 space groups acting freely on the Euclidean space (they are listed in Chapter IV, Eq. (110)) have only one type of elementary bands.
4. The representations of the Gk :s in an elementary band representation The fundamental papers of Bouckaert et al. (1936) (which introduced G ) and Herring (1937a,b), I show that the representations of G play an essential role in the study of the symmetry of energy I bands. As soon as the quantum numbers of elementary bands were de"ned (Zak, 1980), Zak (1982b) computed their decomposition into G 's unirreps for all elementary bands of Fd3 m (diamond I structure). The method of computation was known for the "nite groups; the computation was done with the Born}von Karman boundary conditions. For the sake of topology on BZ, we want to solve the same problem for the space groups. We do not know a mathematical text book or paper useful for this problem. When we made the classi"cation of elementary bands, we assumed the validity of Frobenius reciprocity for the band representations. Now we feel strongly that the assumption is true. With this assumption one needs only to use the side of Frobenius reciprocity which deals with "nite-dimensional representations: they are the unirreps sI ? of the space group % G and its restrictions. We have only to study the restriction Res%U sI ?; its components on the di!erent unirreps of % % G are U Res%U sI ?" mI ? sMU , M U M % % % (7) mI ? "1Res%U sI ? " sMU 2 U , integer50 . % % U M % % Since G is the symmetry group of the problem at each point kK 3BZ we have to introduce it in I Eq. (7). Let us "rst begin by the simple, but important case: G "G. We recall the de"nition of the I Herring group PX(k) and of h given in Chapter IV, Eq. (120): I FI PX(k)"G /Ker k, G &h (G )4PX(k) , (8) G"G P I U I U I the isomorphism is due to the absence of pure translations in G . Then Eq. (7) can be written for U any allowed unirrep (a) of PX(k): G"G , Res.IXkU s?X k " mI ? sMI U . M U M F % F % . I
(9)
Mackey, 1970 extended the theory of representation induction to locally compact groups. The space groups can be considered as (degenerate) examples; but Mackey's theory and all papers using only measure theory are useless: they give only the number b of branches! We need to take into account the analyticity properties which are essential to this problem of physics. We plan to prove it.
L. Michel, J. Zak / Physics Reports 341 (2001) 377}395
385
In the particular case k"0, h ,h already de"ned in Chapter IV, Eq. (52): h(G)"P and P(0)"P; we introduce the notation h(G )"P . Then, U U (10) kK "0: Res.U s?" m? sMU 0 Ind.U sMU " m? s? . M UM . . . ? UM . . . The dimension of this induced representation gives the number b(w, o) of branches of the elementary band (Bacry et al., 1988b, Eq. (21)): "P" b(w, o)"number of branches" dim sMU . % "G " U
(11)
For the other kK of the BZ, we have to replace in Eq. (7) the unirrep sI ? of the space group G by the % expression: sI ?"Ind%I s?I where s?I is an allowed unirrep of G (see Chapter IV, Section 8.3). % I % % % Then Eq. (7) is transformed into mI ? "1Res%U Ind%I s?I "sMU 2 U . U M % % % % %
(12)
To "nd the value of these coe$cients we use a relation due to Mackey which tells how to commute Res and Ind (see e.g. Serre, 1977, Sections 7.4 and 7.5). First, we have to restrict the representation s?I to a subgroup K(G such that we can induce % I this representation from a conjugate subgroup sKs\(G to G . This is possible by choosing U U successively s in the di!erent double cosets G :G:G . With the de"nition K "G 5s\G s, U I Q I U starting from Eq. (12), Mackey's formula yields mI ? " 1Ind%UQ \ Res%IQ s?I "sMU 2 U Q) Q ) % % % U M QZ %U % %I
" 1Res%IQ s?I "Res%UQ \ sMU 2 Q ) % Q) Q % ) Q " "K "\ s?I (g)sMU (sgs\) . (13) Q % % Q EZ)Q When either G or ¸ ) G are invariant subgroups of G, then G ) G is a subgroup of G so the I U U I double coset decomposition can be replaced by a simple coset decomposition. That is the case when P "1 so G "¸, i.e. kK belongs to the generic stratum of BZ; then the number of cosets is I I "P"/"G " and Eq. (13) gives again the number b(w, o) of branches in Eq. (11) since "K ""1 and U Q sMU (1)"dim sMU . We will also use the special case when G , G generates the space group (then % I U % there is no summation over s) G 5 G "K: U I G"1G , G 2: mI ? "1Res%I s?I "Res%U sMU 2 ""K"\ s?I (g)sMU (g) ) % ) % ) U I U M % % EZ) and in the more particular case K"1: G"1G , G 2, G 5G "1: mI ? "(dim sMU )(dim s?I ) . U I U I U M % %
(14)
(15)
386
L. Michel, J. Zak / Physics Reports 341 (2001) 377}395
5. Contacts between the branches of an elementary band In this section, we will present new types of contacts which occur between the branches of elementary bands; they are based on global topological arguments. We "rst review the already known predictions of contacts between EB ("elementary bands) branches. Contacts correspond to a degeneracy in the energy spectrum: two or more states have the same energy. As was "rst explained in Bouckaert et al. (1936) this occurs at kK when the G representation I contains irreducible ones of dimension '1. Here is a simple example that we shall need later. Assume that G &P; as proved in Corollary 7c of Chapter IV, that requires that the space group U G is symmorphic. Then the number of branches is b"dim sMU . This representation is also that of % G "P at the BZ point kK "0; since, by de"nition of EB, sMU is irreducible, the b branches of this I % EBR meet at kK "0. With Eq. (10) we can list the EBR with all b branches meeting at kK "0 when P (P is a strict U subgroup of P: they are those for which the b-dimensional representation Ind.U of P"G is . irreducible so its dimension 43. From Eq. (11) that requires b"["P"/"P "]dim sMU "2 or 3; it can U % occur only for "P"/"P ""2 or 3 and dim sMU "1. Hence, this can occur only when P is an Abelian U U % subgroup of P which must be non-Abelian and have a unirrep of dimension "P"/"P ". Note that if U P contains !I (the inversion symmetry in 3D-space), then P must also contain it. Because the U Herring groups PX(kK ) of symmorphic space groups are direct products Im kK ;P [see Chapter IV, I Eq. (121)], for these groups the highest dimension of unirreps of their G 's is 3 and that is the I maximum degenracy which can occur in their energy bands. The situation is di!erent for non-symmorphic space groups; we have shown in Chapter IV, Eq. (126) the existence of Herring groups of Abelian point group with allowed unirreps (which are unirreps of the corresponding G ) of dimension 2. There are in the cubic system G 's with allowed I I unirreps of dimension 6. As we already stated, the contact between pairs of band branches due to time reversal have been "rst studied by Herring (1937a). He used (Wigner, 1932) where it is shown that in quantum mechanics, T has to be represented by a anti-unitary operator which satis"es, in the absence of states with half integral angular momenta, T"I. As we showed in Chapter IV, the action of time reversal on BZ changes the sign of kK ; so the 8 points of BZ which satisfy 2kK "0 are T invariant. The enlarged symmetry group G[ which leaves kK invariant, is usually called the co-group of kK ; for I these 8 BZ points, it is the direct product G ;Z (¹) while for the other points of BZ, the co-group I G[ is a subgroup of a larger group containing both G and G (see Chapter IV, Section 8.2). The I I \I co-representation of a co-group is made with unitary, antiunitary operators depending whether the group element does not or does contain time reversal. As we showed in Chapter IV, Section 8.2, for the 8 points satisfying 2kK "0 the restriction to G of I the irreducible co-representation of G[ is an irreducible representation of G on the real. It is the I I initial unirrep o(G ) if o is (equivalent to) a real representation. Otherwise, it is o o, the direct sum I of the complex representation o and of its complex conjugate. So the dimension of the co-unirrep of G[ is 2 dim o. That is independent from the fact that the complex representations o and o are or are I This point of BZ is traditionally called C in the solid state literature. Our policy is always to use the notation containing the maximal information; so here we prefer 0, since it is the 0 element of the Abelian group BZ.
L. Michel, J. Zak / Physics Reports 341 (2001) 377}395
387
not equivalent. When they are equivalent Frobenius and Schur (1906a,b) showed that the complex representation must have even dimension; so the dimension of the corresponding irreducible co-representation is divisible by 4. For the other kK (which do not satisfy 2kK "0) Herring has studied in his thesis the contacts imposed by T (in the summary of his paper Herring (1937a) gives only a partial list). These contacts must occur when the co-group G[ has co-unirreps of dimension '1. For example, Herring I (Herring, 1937a) proves the existence of a two-fold degeneracy on the face of a Brillouin cell perpendicular to the axis of the non-symmorphic element P2 (see Chapter IV, Section 8.2). Remark that all types of contacts we have reviewed predict the contacts between branches at well de"ned kK of BZ, either at a "x point, or on an axis (a loop in BZ), or on a whole face (generally a two-dimensional torus on BZ). Indeed, these predictions use only local properties. Let us now consider global arguments. The values of a real function, e.g. energy, on BZ de"ne a 3-dimensional submanifold of a four-dimensional manifold of coordinates k , k , k , E. Two energy branches de"ne two such submanifolds: if they intersect, the generic dimension of such an intersection is 2; but in non-generic cases (often favored by symmetry conditions), it can be a curve or an isolated point. The main aim of the second part of Herring's thesis (Herring, 1937b) was to study the accidental degeneracies: when one modi"es the potential (in the SchroK dinger equation) two energy bands separated by an energy gap may begin to intersect. Typically, these intersections move over the BZ when the potential is changed, without changing the space group symmetry (e.g. di!erent pressure, temperature, chemical compounds). Herring (1937b) proves interesting theorems about the accidental degeneracies. The monodromy of unirreps on BZ, discovered in Herring (1942), and explained in Chapter IV, Section 8.2 is typically a global topological phenomenon. The equivalence classes of the unirreps of the G 's are labeled by an orbit PX ) kK and a discrete index a distinguishing the di!erent unirreps of I G . If the stratum of kK in BZ is not simply connected, it may happen that when one comes back to I kK after going along a non-contractible loop in this stratum, the di!erent a representations are permuted. For instance, we have shown in Chapter IV, Section 8.1 that for the non-symmorphic groups P2 , P3 &P3 , P4 &P4 , P6 &P6 , on the circle in BZ corresponding to the rotation axis there are l ("the order of the rotation) inequivalent unirreps of G "G [Chapter IV, I Eq. (124)]; we have labeled these (one-dimensional) unirreps by o mod l and shown in Eq. (125) of Chapter IV that after a turn on the cycle the representation labeled by o is changed into that labeled by (o#1) mod l. This corresponds to a circular permutation of the representations. These groups (which have only one stratum on the Euclidean space) have a unique type of elementary bands. The l inequivalent one-dimensional unirreps of G "G give the symmetry of the I l branches of the elementary band. Let us start from a kK for which the energy of the l branches are distinct and let us follow the continuous energy function E(k) along the circle of the closed stratum;
Do not forget that the disappearance of an energy gap means only the appearance of an overlap in the energy spectrum and it does not nessarily imply the intersection of the graphs of two energy functions on BZ. Since the induction from G to G of a unirrep of G yields a unirrep of G, the pairs kK , a also label the unirreps G. I I
388
L. Michel, J. Zak / Physics Reports 341 (2001) 377}395
we follow the branch of symmetry o but after a turn we do not "nd the same energy value since the corresponding unirrep is labeled by o#1. By repeating this process, we shall go successively on each branch. That proves that the graph of the l branches is connected. This proof has been recently published in Michel and Zak (1999), where we have drawn several typical graphs of energy for these elementary bands. When the order of the l energy levels at kK "0 is such that the number of contacts is minimum, each contact is only between two branches and it occurs at a kK of BZ satisfying 2kK "0. If one moves the energy level at k (or at any other kK ) and makes two levels cross, new contact appears: they are accidental. In the quoted paper we drew a graph of the P4 elementary band with two accidental contacts over the stratum and we showed how to remove them by changing the order of the energy levels. To prove the connectivity of the energy graph of an elementary band, it is enough to prove it over a stratum. So our proof seems to apply to all space groups which contain a non-symmorphic element. We also proved that the graph of energy is connected for all elementary bands of the groups I2 2 2 and I2 3, the two non-symmorphic groups which have no non-symmorphic elements (that is explained in Chapter IV, Section 7). So our conjecture seems true for the 157 non-symmorphic groups. What can we say about the 73 symmorphic groups? The conjecture is obviously true for 27 of them; their maximal stabilizers G are all isomorphic to P, their point group. For the 19 groups U with Abelian P, Eq. (11) shows that all their elementary bands have only one branch ("simple bands). For the 8 groups with non Abelian G &P, the number of branches of an EB is U the dimension d"1, 2, 3 of the inducing unirrep of G . As we have seen in the beginning of this U section, this representation is also that of G for kK "0 and the d branches meet at this point. So we I are left to study 46 symmorphic groups at 97 Wycko! positions with G isomorphic to a subgroup U of P. We have found a new phenomenon which will probably solve the problem. It applies the `compatibilitya condition at di!erent point of a well chosen closed loop. On this loop, there are singular points; each open segment between two singular points is in one (non-generic) stratum in the action of PX on BZ. At each singular point, the G is stricly larger than those of the two I neighbor segments. So we apply the compatibility condition on the two sides of each singular point. Starting from a point with the representation of a branch, after a complete turn on the loop, we come back with a di!erent branch representation (we will illustrate this phenomenon with the four 2-branch elementary bands of C2mm in the next section). So the two branches must have crossed somewhere on the loop. In the chosen loop, the symmetry does not impose contacts at the singular points or any other point of the loop, so we cannot localize the contact; indeed it will move when the potential is changed. But the situation is completly di!erent from the accidental degeneracy. The symmetry, with the continuity of the energy function and the local compatibility conditions (these two properties were proven in (Bouckaert et al., 1936)), impose on the branches of an elementary band the existence of a contact which can be moved but not removed.
The proof for I2 2 2 is very simple. This group has six 2-branch elementary bands, induced from 3 di!erent Wycko! positions. It is su$cient to "nd in a table of characters of the G 's unirreps, that there is a point of BZ for which I all unirreps have dimension 2. For I2 3 the proof is given in Michel and Zak (1999, p. 6000).
L. Michel, J. Zak / Physics Reports 341 (2001) 377}395
389
6. Some examples of the band structure of a space group 6.1. The non-simple elementary bands of Cmm2 The case of the symmorphic group Cmm2 seems hopeless. The Herring groups are of the form Im kK ;Z at the 7 points of BZ of order 2, since Im kK "Z , their allowed unirreps have z (i.e. $1) as image. Morover, as Table 3 from Chapter IV shows, the BZ of Cmm2 has no other critical points than eight 2kK "0. We assume that the two-fold rotation axis is vertical. Since the point group acts trivially on this axis the proof is identical with that of c2mm, the 2D space group of the sub-lattice in a horizontal plane; we will save some writing of indices by writing the proof for c2mm. We choose a generating basis of the lattice with two vectors of the same norm. The list of the matrices of the four elements of the point group P and of the characters of the four one-dimensional unirreps of this Abelian group are
0 1 , cm "!cm , I ,!I , cm" 1 0
(16)
o(I )"1, o(!I )"m, o(cm)"g, o(!cm)"mg, m"1"g , (17) i.e. the four unirreps o are labeled by a pair of signs m, g. The only Wycko! position generating multi-branch elementary bands is labeled c in ITC
c" , 0
0
1 , G "Z (c), c"+l,!I ,, l" . (18) A 0 We label the two elementary bands (c, f) where f(c)"f"$1; each one has two branches. In BZ with the dual basis, we will use the two points O, R whose coordinates mod 2p are O: (0, 0), R: (p, p) and two closed curves M: (k, k) and M : (k,!k) which correspond to the two symmetry axes of the lattice and of the reciprocal lattice. These curves meet at O and R. The corresponding P 's I are P "P "P"c2mm, P "Z (cm) and P "Z (!cm). For the G groups, G "P, 0 + +Y I G "Z ;P; in the allowed representations of G this Z (image of the translation group ¸) is 0 0 represented faithfully so the allowed unirreps of G can also be labeled by m, g. We denote by p, p 0 the unirreps of P , P and label them by the value $1 of p"p(cm), p "p (!cm). + +Y By the restriction of the unirreps of G for A, B we obtain that the two branches of the elementary I band (c, f) belong to the unirreps c "
at O: (m"f, g) at R: (m"!f, g) .
(19)
On the axes M and M , we are exactly on the case G G "G, G 5G "1, treated in Eq. (15), so O I O I m+ N"1, i.e. the two unirreps p of P and p of P are the symmetry of the two branches over A D + + these lines of BZ for each elementary band (c, f). But p and p must satisfy the compatibility
Moreover, we can check that the closed curve OMRM O cannot be shrunken to a point by continuous deformation.
390
L. Michel, J. Zak / Physics Reports 341 (2001) 377}395
Table 3 Connection of the two-branch elementary bands (c,$1) of cmm2. Each branch starts from one state at the point O of BZ and after a full turn on the closed path (O, M, R, M , O) it reaches the other state at O; so there must be a point of the cycle at which the two bands cross. A second turn completes the full cycle band (c, f)
O mg
M p
R mg
M p
O mg
M p
R mg
M p
O mg
(c, 1)
## #!
# !
!# !!
! #
#! ##
! #
!! !#
# !
## #!
(c,!1)
!# !!
# !
## #!
# !
!! !#
! #
#! ##
! #
!# !!
conditions in Eq. (1); we use Eq. (17) for the notations and incorporate the results of Eq. (19): at O: p"g, p "mg"fg at R: p"g, p "mg"!fg .
(20)
The change of sign for p is the crux of the proof; indeed in Table 3 we follow each branch for each elementary band (c,$1) by giving the value of their quantum numbers. Table 3 shows that for each of the two elementary bands (c,$1), each branch starts from one state at the point O of BZ and after a full turn on the closed path (O, M, R, M , O) it reaches the other state at O; so there must be a point of the cycle at which the two bands cross. That proves the connection of the two branches of the energy graph. A second turn completes the full cycle. The stability of a crossing under perturbation in quantum theory has been studied by von Neumann and Wigner (1929); the instability is due to the possibility of an o!-diagonal term between the two bound states; as is well known (e.g. in atomic physics) the two branches avoid each other when this term appears. In the present case the two states have opposite parity, so the crossings are stable as long as parity violating interactions can be neglected. 6.2. The elementary bands of the three most frequent space groups of organic compounds As we noted in Chapter IV, Section 8.1, footnote 64, these three groups are, in order of decreasing frequency, P2 /c, P2 2 2 , P2 ; they represent about half of the organic compounds listed in Mighell et al. (1977). The group P2 : It has been dealt with in Michel and Zak (1999). We just recall the results. There is only one type of elementary band: it has two branches which have the same energy on the hexagonal faces of the hexagonal prism of the Brillouin cell; it is a two-dimensional torus in BZ. The group P2 /c: Exceptionally, 8 pages are devoted to this group in ITC and "ve di!erent coordinate systems are used; we choose the one of p. 180. The rotation axis carries the basis vector b (vertical) and it is orthogonal to the (horizontal) glide plane which contains the glide vector b ; the vector b is chosen among the other visible vectors of this plane (e.g. a shortest one) such that (b , b (0). In this coordinate system, the rotation matrix R is the diagonal (!1,!1, 1), the matrix of the plane re#ection is S"!R. The representative of the three non-trivial cosets of ¸ can be chosen as: u"+0,!I,, r"+(b #b ), R,, s"+(b #b ), S, .
(21)
L. Michel, J. Zak / Physics Reports 341 (2001) 377}395
391
The multiplication table of these elements is (in shorthand): r
s
u
r s u
b b u b> s
b u b b> r
s r I
(22)
where I"+0, I,, b "+b , I,, b! "+!(b $b ), I,. This table yields the commutation relations: G G rs"b\ sr, ur"b> ru, us"b> su .
(23)
ITC gives four Wycko! positions with maximal symmetry G &Z (!I); they are labeled with U a, b, c, d. Therefore, there are eight inequivalent elementary bands (a,$1), (b,$1), (c,$1), (d,$1). The Brillouin cell is the hexagonal prism of Fig. 6 of Chapter IV. In BZ, the center O of the cell represents (0, 0, 0) and the center O of the horizontal faces: (0, 0, p); the centers F , F , F of the vertical faces represent: (p, 0, 0), (p, p, 0), (0, p, 0), the corresponding middles of the horizontal edges E , E , E represent: (p, 0, p), (p, p, p), (0, p, p). The multiplication table in Eq. (22) shows that r and s are square roots of the translation b , b ; this gives us their one dimensional representation outside the 2kK "0 points. The axes invariant by r are the vertical symmetry axes and the vertical axes de"ned by the three segments F E ; their G has only one-dimensional allowed unirreps in G G I which r is represented by exp(ik /2). The situation is similar for the G of the horizontal planes I containing O and O ; the s is represented by exp(ik /2). With these informations we can make a thorough study of the symmetry and topology of the eight elementary bands. Here we will only study the contacts between their two branches. In Chapter IV, Eq. (126) we calculated the Herring groups for the seven points of order 2 in BZ when the point group P&Z . There are four di!erent groups: Z , Z ;Z , c , and q the T quaternionic group. From Eq. (23), the diagonal part of the matrix in Eq. (22) and the notations de"ned for the seven elements of order 2 we "nd for the nature of the Herring groups: Z at F , Z ;Z at E , E , c at F , F , O , E . T
(24)
This proves that for the eight elementary bands, their two branches have six contacts imposed by symmetry at the same points of BZ. They occur at F , F , O , E , E , E . The last two contacts are imposed by time reversal, but it is in fact a particular case of the general result of Herring (1937a) for the non-symmorphic element rotation P2 : time reversal imposes the degeneracy of the two branches on the two (identi"ed in BZ) hexagonal faces of the Brillouin cell. The group P2 2 2 : This group is one of the 13 space groups with only one stratum (stabilizer G "1) on the Euclidean space. So it has only one type of elementary bands: they have four U branches. The geometry of this group and its Herring groups at the seven BZ points of order 2 have been studied in Chapter IV, end of Section 8.1. For the convenience of the reader we write here a summary. The three generators of the translation lattice ¸ along the rotation axes of the point group P"D satisfy b ) b "jd . In this basis the three matrices representing the non-trivial element G H G GH
392
L. Michel, J. Zak / Physics Reports 341 (2001) 377}395
of P are diagonal matrices that we de"ne by their action on the basis R b "b , iOj, R b "!b . G G G G H H Let i, j, k be a circular permutation of 1, 2, 3. The three generators of P2 2 2 are: r "+(b #b ), R , , H G G G then,
(25)
r"+b , I,, r r "+* , I,r r , * "b !b !b . (26) G G G H GH H G GH G H I The point group P has 8 "xed points on BZ (see Table 3 from Chapter IV), they are the eight elements satisfying 2kK "0. They have coordinates 0 or p in the dual basis. The three elements kK K (p at the coordinate m and 0 for the two other coordinates) are represented by the center of the $ six faces of the Brillouin cell (a rectangle parallelepiped); the three elements kK KY (0 at m and p for # the two other coordinates) are represented by the middles of the 12 edges and the element kK "(p, p, p) is represented by the eight vertices. We obtain the Herring groups of these seven 4 elements of order 2 in BZ by looking at the representation, at these seven points of BZ, of the commutation relations of the r and of their squares (a translation) given in Eq. (26). We obtain for G the seven Herring groups: P2 2 2 : P(kK KY)&Z ;Z , P(kK K)&c , P(kK )&q . (27) # $ T 4 As we have seen in Chapter IV (see Eq. (126) and the following text), c and q have only one T allowed unirrep each; they are two-dimensional and are given in Eq. (126) of Chapter IV. The Abelian group Z ;Z has four inequivalent allowed unirreps; they are one-dimensional and their image is z . Since for these three cases G "1, G "G Eq. (15) tells us the representations of the U I three types of Herring groups: the direct sum of the four di!erent unirreps for Z ;Z and the direct sum of two equivalent two-dimensional unirreps for the two non-Abelian groups. By adding time reversal invariance, nothing is changed for the two real representations of c and the two (equivalent) complex representations of q form a unique co-representation. Hence, T The four branches meet at the point kK "(p, p, p) represented by the eight vertices. 4 On the edges, time reversal transforms the four one-dimensional unirreps of image z into two two-dimensional equivalent co-unirreps; the unitary part of each one is the real irreducible representation of image c . This is compatible with the Herring (1937a) theorem of the two-fold degeneracy on the face orthogonal to an axis of a P2 screw rotation: from this and from our study we can conclude that on the whole surface of the Brillouin cell the four branches form two pairs of two-fold degeneracies (i.e.; there are two distinct doubly degenerate energy values) except at the vertices where the degeneracy is four-fold. On the open segments ]OF [ of the rotation axes ("coordinate axes), the Herring group is G generated by the translations along this axis and r ; we have seen that r is the translation +b , I, so G G G it is represented by $exp(ik /2). Their direct sum of 4 of these representations with two # and G two ! signs gives the symmetry of the four-branch band on each axis. At F (a center of face), by G time reversal, two contacts appear, each one with a # and a ! branch; their representations form, with the two other r's, two unirreps equivalent to the direct sum of two two-dimensional faithful representations of the Herring group P(kK G )&c . T $
L. Michel, J. Zak / Physics Reports 341 (2001) 377}395
393
At kK "0, represented by the center of the Brillouin cell, the Herring group is isomorphic to the point group D . Eqs. (10) and (15) tell us that its representation is the regular one: in each unirrep each r is represented by g "$1 with g g g "1. There are four distinct unirreps and the regular G G representation is their sum; so its characters satisfy s(r )"0. We denote the four distinct energy G levels by the set of values of the g . Let us assume that there is no contact on each semi-open axis G [OF [; then for the two upper and the two lower energy levels at O, each pair (labeled by i) of G g must have opposite signs. This is not possible!; indeed the product of the g 's which label the G G unirrep of a level is 1; and the product (!g )"!1: it is not a label of a unirrep. So contacts G G must appear. One is enough as is shown in the sequence of unirreps for the levels ordered by increasing energy: ### #} } }#} } }# There might be accidental contacts but there is no necessity of contacts on [OF [ and [OF [; one contact is su$cient on [OF [, the crossing of the branch # of the second level with the band ! of the third level. This is a beautiful (and more sophisticated) example of a contact which can be moved but not removed because it is required by symmetry and global topology.
7. Conclusion We recall in the introduction the successful paper of Bloch (1928) explaining the energy band structure by the new quantum mechanics. It was natural to ask the implications of crystal symmetry and time reversal (understood since (Wigner, 1932)) on the symmetry and topology of bands. The "rst given answers in Bouckaert et al. (1936) and Herring's thesis are fundamental. They prove the continuity of the (multivalued) energy function, they introduce G as the local I symmetry group on BZ and Herring's thesis uses the co-representations of G[ . The two works I complement each other to predict where to look for contacts between the branches of the energy function. We read in the introduction of the BSW paper: `The investigation of the topology of representations will be essentially the subject of this paper.a Indeed the compatibility conditions in Eq. (1) are proven in this paper. It was unclear at this period how to classify the bands by quantum numbers; we have tried to sketch the history of this subject in Sections 2 and 3 and quote some relevant papers. The discontinuity of the function kK C G on the Brillouin zone (except for the 13 I space groups for which G is constant and equal to ¸) discouraged global topology considerations. I Now we are in a new era of this history. One can decompose bands into elementary bands with clear quantum numbers (the exceptions make it a little more tricky). For the topology of elementary bands we had made a conjecture which could become an equivalent de"nition: The set of branches of an elementary energy band is connected. The symmetry and local topology tools of 1936 are su$cent for some space groups (for example, I2 2 2 and I2 3). It is obvious that for the other space groups tools from global topology were The thesis was quoted in a footnote of the paper (Bouckaert et al., 1936). The paper emphasizes the word by putting it between ` a.
394
L. Michel, J. Zak / Physics Reports 341 (2001) 377}395
needed. One was available, the monodromy of the G -unirreps on some cycles on BZ discovered by I Herring (1942). With it (and the two examples quoted) we think that we have transformed our conjecture into a physics truth for the 157 non-symmorphic space groups (Michel and Zak, 1999). In this paper, we presented a new (global topology on BZ) tool: a well chosen loop with discontinuity of the G on BZ and monodromy of the multivalued function G -unirreps. With it we I I discovered (see the Cmm2 example) a new type of contact: It can be moved but not removed. With this tool we think to prove our conjecture for the 46 symmorphic space groups (see end of Section 5) which require a proof (three of them resist us). We found this new type of contact with a well-chosen loop on a non symmorphic group (e.g. Pna2 ). The example P2 2 2 is fascinating because the domain of the new type of contact is not a loop but a trihedra with open ends. One can predict a new era of the history of energy bands when one gives the international number of a space group to his/her personal computer and he/she will receive the list of inequivalent elementary bands and the di!erent types of imposed contacts between branches with their location: "xed points or within a loop, a trihedra, etc. 2 Indeed, we could have made a complete table this year. We have preferred to explain to the reader how the new or the old global topological tools work for getting the answers. We wanted to follow the advice of the Chinese proverb: If you see a man suwering from hunger, it is good to give him a xsh from the nearby lake; it is even better to teach him to xsh.
References Bacry, H., Michel, L., Zak, J., 1988a. Symmetry and analyticity of energy bands in solids. Phys. Rev. Lett. 61, 1005}1008. Bacry, H., Michel, L., Zak, J., 1988b. Symmetry and Classi"cation of Energy Bands in Crystals, Lecture Notes in Physics, vol. 313, Springer, Berlin, pp. 291}308. Bloch, F., 1928. Uber die Quantenmechanik der Elektronen in Kristallgittern. Z. Phys. 52, 555}600. Bouckaert, L.P., Smoluchowski, R., Wigner, E., 1936. Theory of Brillouin zones and symmetry properties in crystals. Phys. Rev 50, 58}67. Burneika, I.P., Levinson, Y.B., 1961. Use of global and local symmetry for the calculation of band structure in the tight binding approximation. Tr. Akad. Nauk Lit. SSR B 4, 21}34. des Cloizeaux, J., 1963. Orthogonal orbitals and generalized Wannier functions. Phys. Rev. 129, 554}566. Evarestov, R.A., Smirnov, V.P., 1984. Application of the band representation of space groups in the theory of electronic states of crystalline solids. II. Band representations for some space groups. Phys. Stat. Sol. (B) 122, 559}567. Evarestov, R.A., Smirnov, V.P., 1986. Application of band representations of space groups in the theory of phase transitions and point defects in crystals. Phys. Stat. Sol. (B) 136, 409}415. Evarestov, R.A., Smirnov, V.P., 1993. Site Symmetry in Crystals. Springer Series in Solid State Sciences, Vol. 108. Springer, Berlin. Frobenius, F.G., Schur, I., 1906a. UG ber die aK quivalenz der Gruppen linearer Substitutionen. Sit. Ber. Preuss. Akad. Wiss. fur Berlin, pp. 209}217. Frobenius, F.G., Schur, I., 1906b. UG ber die reelen Darstellungen der endlichen Gruppen. Sit. Ber. Preuss. Akad. Wiss. fur Berlin, pp. 186}208. Herring, C., 1937a. E!ects of time-reversal symmetry on energy bands in crystals. Phys. Rev. 52, 361}365.
We have explained the holes which exist for a possible proof from the mathematical physicist's point of view.
L. Michel, J. Zak / Physics Reports 341 (2001) 377}395
395
Herring, C., 1937b. Accidental degeneracy in the energy bands in crystals. Phys. Rev. 52, 365}373. Herring, C., 1942. Character tables for two space groups. J. Franklin Inst. 233, 525}543. ITC, 1996. In: Hahn, T. International Tables for Crystallography. Vol. A. Space Group Symmetry, 4th Revised Edition. Kluwer, Dordrecht. Kovalev, O.V., 1975. Induced representations and corepresentations of space groups. Sov. Phys. Solid State 17, 1106}1114. Mackey, G.W., 1970. Induced representations of locally compact groups and applications. In: Browder, F. (Ed.), Functional Analysis and Related Fields. Springer, Berlin, pp. 132}166. Michel, L., Mozrzymas, J., 1982. Representation de vibration d'un cristal construite par induction de representations. C. R. Acad. Sci. Paris 295, 435}437. Michel, L., Walker, M.B., Zak, J., 1995. Continuity of phonon spectra for diamond and hexagonal close packed structures. Phys. Rev. Lett. 74, 4871}4874. Michel, L., Zak, J., 1999. Connectivity of energy bands in crystals. Phys. Rev. B 59, 5998}6001. Mighell, A.D., Ondik, H.M., Molino, B.B., 1977. Crystal data space groups tables. J. Phys. Chem. Ref. Data 6, 675}829. Peierls, R.E., 1929. Zur Theory des Hall-E!ekts, Phys. Z. 30, 273. Serre, J.P., 1977. Linear Representations of Finite Groups. Springer (Trans. from French, Hermann, Paris, 1967), New York. Streitwolf, H.-W., 1967. Gruppentheorie in der FestkoK rperphysik. (English translation: Group Theory in Solid-state Physics), Geest and Portig, Leipzig. von Neumann, J., Wigner, E., 1929. UG ber das Verhalten von Eigenwerten bei adiabatischen Prozessen. Phys. Z. 30, 467}470. Wigner, E., 1930. UG ber die eslastischen Eigenschwingungen symmetrischer Systeme. Nachr. Ges. Wiss. GoK ttingen Math.-Phys. Kl., 133}146. Wigner, E., 1932. UG ber die Operation der Zeitumkehr in der Quantenmechanik. Nachr. Ges. Wiss. GoK ttingen Math.Phys. Kl., 546}559. Zak, J., 1980. Symmetry speci"cation of bands in solids. Phys. Rev. Lett. 45, 1025}1028. Zak, J., 1982a. Continuity chords of bands in solids. The diamond structure. Phys. Rev. B 25, 1344}1357. Zak, J., 1982b. Band representation of space groups. Phys. Rev. B 26, 3010}3023. Zak, J., 1989a. Berry's phase for energy bands in solids. Phys. Rev. Lett. 62, 2747}2750. Zak, J., 1989b. Berry's geometrical phase for non-cyclic Hamiltonians. Europhys. Lett. 9, 615}620. Zak, J., 1991. Berry's Phase and Wycko! Positions for Energy Bands in Solids, Lecture Notes in Physics, Vol. 382. Springer, Berlin, pp. 581}587.
