Theory of Molecular Rydberg States (Cambridge Molecular Science)

THEORY OF MOLECULAR RYDBERG STATES

Molecular Rydberg states have many unusual properties, lending themselves to a diverse range of experimental applications. This book is designed to unravel the mysteries of molecular Rydberg states that lie beyond the scope of accepted spectroscopic theories. It is the first single-authored text to focus on the application of multi-channel quantum defect theory (MQDT) and ab initio theory to this special class of molecular systems, introducing readers to novel theoretical techniques. The scattering techniques of MQDT are examined, along with a unified description of bound states and fragmentation dynamics. Connections with established spectroscopic theory are also described. The book concludes with an account of the spherical tensor and density matrix theories required for the interpretation of multiphoton experiments. While the main text focuses on physical principles and experimental applications, appendices are used to handle advanced mathematical detail. This is a valuable resource for researchers in chemical, atomic and molecular physics. mark child is a distinguished theoretical chemist and internationally recognised authority on quantum defect theory and its applications to molecular systems. He was Coulson Professor of Theoretical Chemistry at the University of Oxford, a post from which he recently retired.

‘This excellent book is ripe with insights and practical methods, and it will be indispensable for anyone interested in learning the theory of molecular Rydberg states.’ Christopher H. Greene, Professor of Physics, Fellow of JILA, University of Colorado ‘In this book, Mark Child presents the crucial ideas, techniques, and surprises of Rydberg states with authority, clarity, and glorious attention to interconnections. The physically sensible approximations upon which MQDT is based are explained and illustrated, both pictorially and rigorously: why MQDT works so well to describe many things that the Born–Oppenheimer approach cannot; the power of MQDT to deal in a unified manner with all aspects of molecular structure and dynamics; and the extraordinary compactness of infinite-member channels rather than individual electronic states as fundamental building blocks . . . Theorists and experimentalists ignore MQDT at their peril.’ Robert Field, Haslam and Dewey Professor of Chemistry, Department of Chemistry, Massachusetts Institute of Technology ‘Mark Child’s monograph is not only comprehensive and authoritative but also delightfully readable, teaching clearly the subtleties of how to describe molecules in which one electron has been excited to near the ionization limit. Moreover, a unifying description is developed based on the ideas of electron-molecule scattering (multichannel quantum defect theory), a description that shows a seamless transition between bound Rydberg states and the unbound ionization continuum. I enthusiastically recommend this book to anyone desiring to understand the nature and dynamics of molecular Rydberg states. It is the starting place for all serious students of this topic.’ Richard N. Zare, Chair, Marguerite Blake Wilbur Professor in Natural Science, Department of Chemistry, Stanford University ‘This book fills a gap, and will be of great value to graduate students and researchers interested in highly excited molecular systems.’ Christian Jungen, Directeur de Recherche au CNRS, Universit´e de Paris-Sud

Cambridge Molecular Science As we move further into the twenty-first century, chemistry is positioning itself as the central science. Its subject matter, atoms and the bonds between them, is now central to so many of the life sciences on the one hand, as biological chemistry brings the subject to the atomic level, and to condensed matter and molecular physics on the other. Developments in quantum chemistry and in statistical mechanics have also created a fruitful overlap with mathematics and theoretical physics. Consequently, boundaries between chemistry and other traditional sciences are fading and the term Molecular Science now describes this vibrant area of research. Molecular science has made giant strides in recent years. Bolstered both by instrumental and theoretical developments, it covers the temporal scale down to femtoseconds, a timescale sufficient to define atomic dynamics with precision, and the spatial scale down to a small fraction of an Angstrom. This has led to a very sophisticated level of understanding of the properties of small molecule systems, but there has also been a remarkable series of developments in more complex systems. These include: protein engineering; surfaces and interfaces; polymers; colloids; and biophysical chemistry. This series provides a vehicle for the publication of advanced textbooks and monographs introducing and reviewing these exciting developments. Series editors Professor Richard Saykally University of California, Berkeley Professor Ahmed Zewail California Institute of Technology Professor David King University of Cambridge

THEORY OF MOLECULAR RYDBERG STATES M. S. CHILD Physical and Theoretical Chemistry Laboratory and University College University of Oxford

cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, S˜ao Paulo, Delhi, Tokyo, Mexico City Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521769952 C M. S. Child 2011 

This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2011 Printed in the United Kingdom at the University Press, Cambridge A catalogue record for this publication is available from the British Library Library of Congress Cataloging in Publication data Child, M. S. Theory of molecular Rydberg states / M.S. Child. p. cm. – (Cambridge molecular science) Includes index. ISBN 978-0-521-76995-2 (hardback) 1. Rydberg states. 2. Energy levels (Quantum mechanics) 3. Ionization. QC454.A8C45 2011 539.7 – dc22 2011012793

I. Title.

ISBN 978-0-521-76995-2 Hardback

Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

Contents

Preface

page xi

1 Molecular Rydberg states 1.1 The nature of Rydberg states 1.2 Organization of the text

1 1 7

2 The quantum defect picture 2.1 Introduction 2.2 Coulomb wavefunctions 2.3 Single-channel quantization 2.4 Coupled channels

17 17 18 25 27

3 Ab-initio quantum defects 3.1 Traditional quantum chemistry 3.2 Constrained ab-initio wavefunctions 3.3 The R-matrix matching procedure 3.4 The Wigner–Eisbud R-matrix 3.5 Variational R-matrix theory 3.6 Rydberg–valence interactions 3.7 The influence of positive ion dipoles

45 45 52 54 58 62 74 78

4 Frame transformations and channel interactions 4.1 Physical assumptions 4.2 Rotational channel interactions 4.3 Vibrational channel interactions 4.4 Vibronic channel interactions

90 91 97 105 114

5 Competitive fragmentation 5.1 Perturbation model for diatomic species 5.2 Diatomic predissociation

125 126 129 vii

viii

Contents

5.3 Dissociative recombination and related phenomena 5.4 R-matrix formulation 5.5 Vibronically induced dissociative recombination of H+ 3

130 140 148

6 Photo-excitation 6.1 Introduction 6.2 n-photon discrete absorption 6.3 Spherical tensor representation 6.4 Spatial selectivity 6.5 Resonant two-photon excitation 6.6 Multiphoton band structure 6.7 Angular momentum decoupling in high Rydberg states 6.8 ZEKE intensities

157 157 158 163 167 170 171 177 183

7 Photo-ionization 7.1 Boundary conditions and cross-sections 7.2 The photo-ionization matrix element 7.3 Integrated cross-section 7.4 Differential cross-section 7.5 Fixed molecule angular distribution 7.6 Resonant two-photon ionization 7.7 Orientation and alignment 7.8 Spin polarization

191 192 195 199 202 215 219 226 232

8 Manipulating Rydberg states 8.1 Rydberg wavepackets 8.2 The Stark effect

239 239 255

Appendix A MQDT normalization A.1 Open channels A.2 Closed channels

273 273 275

Appendix B Alternative MQDT representations B.1 Standard representation B.2 Sine–cos representation B.3 Mixed representation

278 278 279 280

Appendix C Rotational frame transformations C.1 Hund’s cases for diatomic molecules C.2 Parity considerations C.3 Basis functions C.4 Diatomic frame transformations C.5 Asymmetric tops

282 282 284 285 286 290

Contents

ix

Appendix D Optical transition and photo-ionization amplitudes D.1 Discrete absorption amplitudes D.2 Photo-ionization amplitudes D.3 Dipole radial matrix elements and Cooper minima

295 295 297 303

Appendix E Generalized MQDT representation

307

Appendix F Notation F.1 Angular momenta F.2 Reduced matrix elements F.3 Other special brackets

310 310 310 312

Index

314

Preface

My initial aim was to introduce the powerful but relatively under-used techniques of multichannel quantum defect theory (MQDT) to graduate students in atomic and molecular physics. The methods are particularly attractive in two ways. They provide an elegant, computationally tractable approach to the treatment of molecular Rydberg states, which invalidate the normal molecular assumption that the electronic motion is overwhelmingly rapid compared with other degrees of freedom. In addition the theory offers a unified description of the discrete molecular states below an ionization limit and those above in the ionization continuum. At the same time the novelty of the MQDT method makes it essential to point to the links with the familiar techniques of ‘normal’ molecular physics. While writing, I realized that workers in two other fields would benefit from a more general treatment of molecular Rydberg states. In the first place there is a huge literature on electronic structure theory or ‘quantum chemistry’, which can, however, handle only the very lowest Rydberg states, owing to the very long range of the excited orbitals. A chapter has been written to show how the familiar quantum chemical techniques can be adapted to handle arbitrary members of the infinite Rydberg series. Secondly, to meet the demands of modern experiments, the chapters involving interaction with radiation take account of developments in the theoretical description of coherent multiphoton excitation and resonant multiphoton ionization. The book is primarily written for graduate students in atomic and molecular physics. Many will have seen and admired the striking success of the theory. My hope is that the book will enable them to apply it for themselves. To avoid undue mathematical complexity, the main text focuses on physical principles and illustrative applications, leaving technical details to be covered in the appendices. Knowledge of scattering theory will be helpful but not essential for the early chapters, while the later ones require familiarity with angular momentum algebra.

xi

xii

Preface

I myself was introduced to the mysteries of quantum defect theory at a lecture by Ugo Fano. I later had the benefit of working with Christian Jungen and Chris Greene and, of course, learned much from my own collaborators, notably Rick Gilbert, Colin Batchelor, Miyabi Hiyama and Adam Kirrander. I am particularly indebted to Christian Jungen for constant advice and encouragement during the preparation of the book. It is also a pleasure to acknowledge the hospitality of the Chemistry Department of ETH in Zurich and Laboratoire Aim´e Cotton at Orsay. Discussions with Mireille Aymar and Frederik Merkt were particularly helpful. I also owe debts to Oxford colleagues, John Brown, John Eland, Brian Howard, David Manolopoulos and Tim Softley for expertise and advice. In addition thanks are due to John Freeman and Yuan-Pin Chang for assistance in the preparation of various diagrams. Financial support by the Leverhulme Trust during the early stages of writing is gratefully acknowledged. Acknowledgement is also due to the following for permission to reproduce copyright material: the American Institute of Physics, the American Physical Society, the Institute of Physics, Springer Verlag, Taylor and Francis and the Royal Society of Chemistry. I am also grateful to Michelle Carey and Claire Poole at Cambridge University Press for help, encouragement and advice in preparing the text. M. S. CHILD Oxford September 2010

1 Molecular Rydberg states

1.1 The nature of Rydberg states The nature of atomic Rydberg states is well described by Gallagher, though with less emphasis on theory [1]. Those of molecules are severely complicated by the additional nuclear degrees of freedom, in a way that gives them quite different properties from those treated in most spectroscopic texts [2, 3, 4, 5]. The essential difference is that established spectroscopic theory is rooted in the Born– Oppenheimer approximation, whereby the frequencies of the electronic motion are assumed to be so high compared with the vibrational and rotational ones that the nuclear motions may be treated as moving under an adiabatic electronic energy (or potential energy) surface. In addition the vibrational frequency usually far exceeds that of the rotations, which means that every vibrational state has an approximate rotational constant. Such considerations provide the basis for a highly successful systematic theory. Modern ab-initio methods allow the calculation of very reliable potential energy surfaces and there are a variety of efficient methods for diagonalizing the resulting Hamiltonian matrix within a functional or numerical basis. Electronically non-adiabatic interactions between a small number of electronic states can also be handled by this matrix diagonalization approach, even including fragmentation processes, if complex absorbing potentials are added to the molecular Hamiltonian. The difficulty in applying such techniques to highly excited molecular electronic states is that the Rydberg spectrum of every molecule includes 100 electronic states with principal quantum number n = 10, separated from the n = 11 manifold by only 100 cm−1 , which is small compared with most vibrational spacings and comparable to rotational spacings for small hydride species. Figure 1.1 shows a simplified level scheme for a species with a positive ion rotational constant, B = 30 cm−1 , and a vibrational interval, ω = 2322 cm−1 , appropriate to H+ 2 . The levels

1

2

Molecular Rydberg states

(1,0) 0.02

10

0.01

(0,4) (0,2)

(0,0)

0

7 10

0.01

10

10

7 0.02

7

7 0

5

5 10

Figure 1.1 A schematic molecular Rydberg system, showing series terminating on different (v, J ) energy levels of the positive ion. The isolated solid lines mark the series limits. The wavy dotted lines are the attached continua. Small labels are the principal quantum numbers, n. Energies are measured in atomic units from the (0, 0) ionization limit.

are calculated by the Rydberg formula Env+ J + = Ev++ N + −

Ry , (n − μ)2

(1.1)

with the quantum defect μ = 0.169 appropriate to the np 1 u+ system of H2 . Each series, whose continuum is marked by a wavy dotted line, is labelled by the vibrational rotational quantum numbers (v + , N + ). The small symbols, marking individual levels are the principal quantum numbers, n. One sees for example that the n = 10 level of the (0, 2) series lies above the n = 11 level of the (0, 0) series, which means that the electronic energy splitting is smaller than the N + = 0 → 2 rotational interval. In addition, the perturbations arising from the resulting nonadiabatic coupling in the discrete spectrum go over to auto-ionization as soon as discrete members of a higher series lie in the continuum of a lower one. The situation with regard to the 1 g+ series of H2 is further complicated by interaction with ion-pair valence states, which give rise to the famous double minimum pairs (E,F), (G,H), etc. It is therefore evident that Rydberg systems require a theory that can handle the presence of bound and slowly fragmenting states in a unified way. The method of choice is ‘multichannel quantum defect theory’ or MQDT, which visualizes

1.1 The nature of Rydberg states

3

the dynamics as a scattering process in which collisions between the loosely bound electron and a short-range positive ion core lead to phase changes and energy transfer probabilities that determine the physical observables [6, 7, 8, 9]. An important difference from normal scattering theory is that the electron is scattered into an attractive Coulomb field, rather than a field-free region. Thus for example a low-lying bound electron may be scattered to and fro between neighbouring bound states, leading to spectral perturbations. Another possibility is that the collisions may excite the positive ion into a dissociative electronic state, leading to molecular predissociation. The dynamics are further complicated for higher-lying bound states by the possibility of scattering into the ionization continuum. Fortunately there is a major simplification, in that the energy range of most interest in molecular physics is small compared with the total energy of the system. Consequently the parameters, loosely termed ‘quantum defects’, that characterize the fundamental collision process, may often be treated as almost independent of energy over the physically interesting range. The dynamical complications arise principally from the boundary conditions of the coupled system. The quantum defects may be treated at one level as phenomenological parameters, to rationalize the observations. As a simple example, the Rydberg formula for the energies of alkali atoms En = I −

Ry , (n − μ )2

(1.2)

with μ2 > μp > μd , etc., is well known to allow fairly accurate predictions of entire series, from observation of a single member – even without allowing for the weak energy dependence of the quantum defect parameters, μ . Similarly, as a slightly more complicated example, analysis of the perturbations between bound levels below the first ionization limit in Fig. 1.1 may be used to predict the rates of autoionization above the limit. Another benefit of the weak energy dependence concerns the relevance of ab-initio electronic structure theory. Standard techniques [10] can now yield highly accurate low-lying potential energy functions for small molecules, including Rydberg states up to n  4, but the extension to higher members of the series is prohibitive because the very diffuse outer parts of the orbitals are poorly approximated by Gaussian functions. By contrast, MQDT techniques work with exact Coulomb wavefunctions in the outer region, and the necessary interactions with the core may be extracted as quantum defect functions, from the molecular analogue of (1.2). In the simplest case of a diatomic molecule Vnλ (R) = V + (R) −

Ry , [n − μλ (R)]2

(1.3)

4

Molecular Rydberg states

n = 15

f

n = 10

d

s p

n=5 Figure 1.2 Semiclassical p orbits with  = 1.5 and n = 5, 10 and 15, compared with a core of radius 10a0 . The inset shows that the two latter orbits are indistinguishable in the core region, with a small but discernible difference from the inner n = 5 orbit. The dotted curves in the inset follow the s, d and f orbits for n = 10.

where Vnλ (R) is the ab-initio potential function for n = 3 or 4 and V + (R) is the corresponding curve for the positive ion. An alternative, more powerful, ab-initio approach, particularly in the presence of potential surface crossings, is to solve the ab-initio equations within a relatively small box around the ionic core and to join the resulting Rydberg orbitals to combinations of the exact Coulomb functions in the outer region by what are called R-matrix methods [11, 12, 13]. The reader may wonder at this stage why it is legitimate to employ the output from ab-initio calculations, which are performed within the fixed nucleus ‘Born– Oppenheimer’ approximation, despite the stated aim in the second paragraph to handle situations in which the electronic energy separations are small even compared with rotational ones. The answer again lies in the fact that the relevant perturbations to the electronic motion occur at short range, where the electron velocity is high. Figure 1.2 shows the equivalent classical orbits of a hydrogenic p electron, with n = 5, 10 and 15. The total orbit times are roughly equal to n3 times the atomic time unit, τ = 2.417 × 10−17 s, but an application of classical angleaction theory1 shows that the fractional time within a core of radius 10 atomic units decreases as n−3 . Hence the transit time, and even the shape of the track in the inset 1

Appendix E.5 of Child [14] shows that r = n2 a0 (1 + ε cos u),

1.1 The nature of Rydberg states

5

in Fig. 1.2, are almost independent of n. Finally, as discussed in the footnote, the transit time itself varies from 1.2 × 10−16 s for an s orbit to 1.6 × 10−16 s for an f orbit, which is short compared with any molecular vibrational or rotational period. The Born–Oppenheimer approximation is therefore fully justified over the period during which the electron is perturbed by the core. This classical argument has two other significant consequences. In the first place the n−3 dependence of the transit time for any bound orbit translates into an n−3/2 dependence of the wavefunction amplitude within the core, which gives rise to a variety of Rydberg scaling laws. Secondly, the abruptness of the transit allows the use of a powerful ‘frame transformation’ technique [6]. The uncoupled states |i in the outer region apply to the combination of a positive ion in a well-defined quantum state and an independent incoming or outgoing electron with defined angular momentum . The interaction region on the other hand supports Born–Oppenheimer states, |α, in which the electron occupies a molecule fixed orbital with fixed instantaneous bond lengths. Moreover, an electron starting from an asymptotic state |i typically switches to a Born–Oppenheimer core state |α, so rapidly that the transition amplitude may be approximated as the simple projection i |α. The influence of the positive ion core is then taken into account via the quantum defects, μα , which cause phase changes between the incoming and outgoing wavefunctions. As a result the scattering from incident state |i to final asymptotic state |j  may be characterized by matrices of the exponential or trigonometric forms, with elements of the form   Sij = i |α e2iπμα α |j  or Kij = i |α tan π μα α|j . (1.4) α

α

where ε is the eccentricity

 ε=

n2 − ( + 1/2)2 n

and u is an auxiliary variable, in terms of which the classical angle αn is given by αn = u + ε sin u. It is readily verified that the closest approach at r = an = n2 a0 (1 − ε), which is approximately independent of n for n2  ( + 1/2)2 , is reached at u = αn = π . Since αn is designed to vary linearly with time, the fractional time to cross a spherical core of radius rc is given by [αn (rc ) − π ] /π . One may assume a quadratic expansion for cos u about u = π, which is valid for c (u − π )2 = 2  1, n ε where c is the scaled displacement 2(rc − an )/a0 . The corresponding cubic expansion for sin u rearranges to  √  c 3( + 1/2)2 + c τcore αn (rc ) − π = , = τorbit π 6π n3 which demonstrates the n−3 dependence. A computation based on following the evolution of αn and r as functions of the auxiliary variable u readily shows that the fractional angle action range within a core of radius 10 a0 varies between 4.9n−3 for an s orbit to 6.7n−3 for an f orbit, which translates into a range of transit times from 1.2 × 10−16 s to 1.6 × 10−16 s.

6

Molecular Rydberg states

Π A* + B

A+B Σ+

Figure 1.3 Schematic potential curves for a hypothetical AB molecule. Two continua are shown by wavy lines. One starts from the A + B asymptote of the repulsive  curve. The other has an attached ladder of vibrational levels terminating on the A∗ + B dissociation limit of the bound  + curve.

We shall see in later chapters that the choice of an S- or a K-matrix depends on whether the scattering is formulated in terms of incoming and outgoing waves, or in terms of sine-like and cosine-like standing waves. It is also helpful to introduce the concept of a ‘channel’, which plays an important role in any scattering formulation. Seen in a familiar framework, consider the potential curves of a hypothetical diatomic molecule in Fig. 1.3, each of which may be labelled by the electronic symmetry, by the states of its separated atom fragments, and by the total angular momentum. Each ‘channel’ then consists of the entire family of bound and continuum states, supported by the relevant potential curve, or ‘channel potential’. Since the diagram is actually drawn for J = 0, the bound levels of the  + potential belong to the rotationless vibrational states. The corresponding channel potentials for J = 0 include an additional centrifugal term, J (J + 1)¯h2 /2μR 2 , and the bound levels are rotational–vibrational ones. As another useful terminology, a channel is said to be ‘open’ (to fragmentation) at energies above the dissociation limit, ‘closed’ at energies between Vmin and the dissociation limit, and ‘strongly closed’ or ‘forbidden’ for E < Vmin . Thus the  + channel is closed and the  channel strongly closed below the A + B asymptote, while the  channel opens at energies above this dissociation limit. Finally both channels are open above the A∗ + B dissociation limit. One also speaks of the bound levels of the  + channel ‘lying in the  continuum’ at energies between the A + B and

1.2 Organization of the text

7

A∗ + B limits. Symmetry forbids any interaction between the two J = 0 channels, but added angular momentum can lead to ‘heterogeneous predissociation’ with a selection rule J = ±1, induced by Coriolis coupling [15]. The channels in Fig. 1.3 are termed molecular dissociation channels, labelled by different electronic states of the fragment atoms. In the Rydberg context, the ionization channels are equally important. The relevant channel potentials are then centrifugally corrected Coulomb potentials, each supporting infinite series of electronic states; and the asymptotes belong to different electronic, vibrational and rotational energies of the positive ion. Figure 1.1 illustrates electronic energy ladders (or Rydberg series) terminating on the |v + , N +  = |0, 0, |0, 2, |0, 4 and |1, 0 states of H+ 2 . There are in fact many such ladders with different angular momenta. One of the first molecular applications of multichannel quantum defect theory [6] demonstrated the important connection between spectral perturbations below the |0, 0 limit and auto-ionization in the interval between the |0, 0 and |0, 2 limits, arising from rotationally induced interactions between the np series that are designated as npσ 1 u+ and npπ 1 u at low energies, where the Born– Oppenheimer approximation applies; and as np0 and np2, according to the relevant value of N + , at energies close to ionization. To see how the interaction arises, it is interesting to refer to the structure of the scattering matrices in (1.4). The labels |i = |N + J  designate the positive ion rotational state, the electronic angular momentum  and the total angular momentum J ,2 while the Born–Oppenheimer labels are |α = | J , where is the electronic angular momentum that gives the σ or π character. It follows from (1.4) that the scattering matrices will contain no offdiagonal terms unless the quantum defects μα differ from one Born–Oppenheimer channel to another. In other words, by reference to (1.3) the energies of the npσ 1 u+ and npπ 1 u electronic states must differ. The beauty of the MQDT formulation is that the n dependence of such energy separations is determined by the almost energy-independent quantum defects, μpσ and μpπ . 1.2 Organization of the text Four of the following chapters cover aspects of multichannel quantum defect theory, with an emphasis in the early chapters on connections with traditional spectroscopic theory. Two subsequent chapters focus on photo-excitation and photo-ionization in the context of modern n-photon experiments. The final chapter concerns the manipulation of Rydberg states. Readers should note that much of the technical detail is placed in the appendices. In particular, those unfamiliar with molecular terminology will find the notation explained in Appendix F. In addition details 2

We ignore vibrational complications for simplicity.

8

Molecular Rydberg states

of the construction of parity-adapted basis states for different angular momentum schemes are included in Appendix C. The quantum defect picture Chapter 2 outlines the essential elements of multichannel quantum defect theory (MQDT) [7, 8, 9]. It starts from the assumption that the scattering problem between the outer Rydberg electron and the positive ion core has been solved – by methods that will be outlined in later chapters. The relevant quantum defects, μα , and scattering K-matrices, that monitor the strength of the Rydberg–core interaction are assumed to be known. The chapter is therefore concerned with the behaviour of this outer electron, moving in a pure Coulomb field, and with how this behaviour is modified by the K-matrix parameters and the boundary conditions as r → ∞. The flexibility to handle arbitrary Rydberg systems, from atoms to polyatomic molecules is introduced by showing how the supposedly known K-matrix elements determine combinations of Coulomb basis functions in each channel, which are simply independent solutions of the radial Schr¨odinger equation, at the chosen energy and orbital angular momentum. An analysis of the boundary conditions as r → ∞ leads to the characteristic structure of the MQDT working equations, which depend on matrices with dimensions equal to the number of interacting channels, which is orders of magnitude smaller than the number of basis states required for a conventional diagonalization. The nature of typical solutions is illustrated by demonstrating close similarities between the bound state structure immediately below the ionization limit and the auto-ionizing resonances immediately above. Connections with the matrix formulation of traditional spectroscopic theory are also discussed. Ab-initio methods Chapter 3 concerns the ab-initio determination of the quantum defects and scattering K-matrices, which may be derived as functions of the nuclear coordinates. A short introductory section uses the case of H2 to show how analogues of (1.3) and (1.4) may be used to extract diagonal and off-diagonal quantum defects from families of possibly strongly interacting high level ab-initio surfaces. Factors affecting the bond length dependence of the quantum defect functions, μ (R) are also discussed. Subsequent sections concern the formulation and implementation of what are termed R-matrix techniques, which allow the direct determination of the nuclearcoordinate-dependent K(Q) matrix at arbitrary energies up to and above the ionization limit [11, 12, 13]. The argument is that the outer parts of all Rydberg wavefunctions behave as known Coulomb functions, outside the range of the Rydberg–core interactions. Thus the ab-initio effort may be restricted to a finite

1.2 Organization of the text

9

volume. The resulting inner wavefunction is then joined to the outer one by appropriate log-derivative boundary conditions at the core boundary, at which a strongly energy-dependent matrix R(E) ensures continuity between the inner and outer wavefunctions in the form   (1.5) ψE (a) = R(E) aψE (a) + bψE (a) , where the components of the column vector ψE (a) are the amplitudes of the Rydberg wavefunction in different channels at the core radius r = a, while b is a parameter appropriate to the nature of the internal basis functions. Continuity of the log-derivative at the boundary normally leads to the very weakly energydependent K-matrix, because the energy dependence of the R-matrix is cancelled by the log-derivatives of the outer basis functions. Two types of R-matrix theory are discussed, according to whether the inner ab-initio equations are solved for electronic eigenvalues in a suitable continuum basis, perhaps with fixed log-derivatives in the early Wigner–Eisbud approach [16], or for eigenvalues of the log-derivative, b(E), at a fixed energy in the eigenchannel method [13]. The convergence properties of the two methods are discussed, including the introduction of so called Buttle corrections to compensate for incompleteness in the Wigner method [17]. Illustrative applications of the two methods are described. The treatment of species with large ion core dipoles is tackled in the final section. Frame transformations and channel interactions The strength of the theory is greatly enhanced by the use of frame transformations (Chapter 4), which are justified on one hand by the brevity of the ‘core transit time’, and more explicitly by demonstrating the insensitivity of the radial wavefunction to energy changes comparable to those associated with vibrations and rotations of the ion core. Both these lines of argument justify the conclusion that the transition amplitude between coupled and uncoupled representations may be accurately approximated by the simple internal overlap i |α. Subsequent sections of the chapter deal with the machinery required to perform rotational, vibrational and vibronic frame transformations, each of which is illustrated by appropriate physical applications. Rotational frame transformation theory is complicated by the variety of angular momentum coupling schemes from one molecule to another [5, 18]. The treatment in the main text is therefore limited to the simplest case, in which spin is ignored, leaving more complicated situations to be handled in Appendix C. Applications are chosen to show how MQDT theory handles the familiar topics of -doubling and -uncoupling [2, 5, 19]. It is also shown how decreasing electronic energy spacings

10

Molecular Rydberg states

as the energy increases are reflected in ‘stroboscopic’ fringes in the spectrum as periods of the electronic motion tune through integer multiples of the fixed rotational period. The third section covers vibrational channel interactions, for which the vibrational wavefunctions, R|v + , of the positive ion play the part of frame transformation elements between the fixed nucleus Born–Oppenheimer and the uncoupled positive ion states. The forms of the resulting K-matrix elements v + | tan π μλ (r)|v +  are well displayed by the np series of H2 , in view of the strong nuclear coordinate dependence of the pσ quantum defect. Applications to the discrete level structure and vibrational auto-ionization are given. The final section of Chapter 4 starts with a brief discussion of vibronic curvecrossing interactions. A longer account of Jahn–Teller induced coupling throughout a Rydberg series is then described. Particular emphasis is given to the scaling of the coupling strength according to changes in the principal quantum number. The mechanism of Jahn–Teller-induced auto-ionization is also discussed as a prelude to the theory of dissociative recombination in the following chapter. Predissociation and dissociative recombination Chapter 5 extends the theory to include both ionization and dissociation channels as illustrated by the processes AB+ + e  AB∗  A + B. Thus the central species AB∗ , which designates a Rydberg state, may auto-ionize to left to produce a positive ion and an electron, or predissociate to the right into neutral fragments. Alternatively a collision between the electron and positive ion may yield neutral fragments in a process known as ‘dissociative recombination’ [20]. The reverse possibility, whereby the neutrals collide to produce ions is ‘collisional ionization’. Two treatments of these composite processes are described. The first employs a perturbation model, which is particularly appropriate to curve-crossing situations. It rests on combining the vibrational and rotational channel interaction theory of the previous chapter with a ‘generalized MQDT’ treatment of the dissociation dynamics, details of which are given in Appendix E. The two strands are linked by the perturbative construction of a global K-matrix with both ionization and dissociation channels. The same formal construction allows the treatment of dissociative recombination at higher energies, although the theory is more easily expressed in terms of a composite S-matrix than the corresponding Kmatrix. The factors responsible for the magnitude and energy dependence of the cross-section are illustrated by reference to the dissociative recombination of H+ 2 and NO+ .

1.2 Organization of the text

11

The second approach combines the MQDT channel coupling techniques of Chapter 4 with a non-perturbative R-matrix treatment of the dissociation channel, analogous to the ‘eigenchannel’ or ‘variational’ ab-initio R-matrix treatment of the electrons in Chapter 3. Two types of R-matrix boundary condition are applied. A large set of ‘closed’ solutions, which vanish at the boundary, is used to converge the MQDT part of the calculation, while a small set of open solutions, close to the energy of interest, provides the dissociation flux. Applications to the competitive auto-ionization and predissociation of H2 are described. Finally, the theory is extended to the Jahn–Teller-induced dissociative recombination of H+ 3 , which plays an important role in dense interstellar clouds [20]. A significant refinement involves the use of complex R-matrix boundary conditions in a major computational study, which played a large part in reconciling differences between theory and various types of experiment [21]. It is also shown how these results can be beautifully rationalized by reference to the spectroscopy of H3 [22]. Photo-excitation Since our knowledge of Rydberg states frequently comes from optical excitation by more than one photon, it is useful to start Chapter 6 with an introduction to the perturbation theory of multiphoton spectroscopy, along lines initiated by G¨oppertMayer [23]. The relevant transition operators generalize from the three-component dipole operator for a single photon to the nine-component polarizability tensor for two photons, 27-component hyper-polarizability tensor for three photons, etc. A systematic spherical tensor treatment of such n-photon interactions is given, including tables to indicate how the weighting of the different spherical tensors depends on the relative polarizations of the light sources. As a prelude to the discussion of photo-ionization in Chapter 7, it is also shown how even discrete absorption leads to ‘orientation and alignment’ of molecular angular momentum with respect to the polarization axis of the light beam. The convenience of the spherical tensor approach is illustrated in relation to the rotational branch structure associated with n-photon transitions. Tables of the symmetry selection rules for up to three photons, along the lines initiated by Dixon et al. are also included [24]. A spherical tensor formulation is particularly convenient in allowing one to take over the normal angular momentum machinery [18]. Two further sections relate to the transition intensities to very high-lying Rydberg states. The first describes a systematic change in the character of the spectrum as the excited electron uncouples from the nuclear framework – a change that anticipates the photo-ionization selection rules given in Chapter 7. The second concerns the nature of the ‘quasi-continuum’ of bound states just below the ionization limit, which is accessed by pulsed field zero-kinetic energy (ZEKE-PFI) spectroscopy. ‘Intensity anomalies’ in such spectra, which arise from the extreme polarizability of

12

Molecular Rydberg states

Rydberg states with n ∼ 200, are shown to be related to auto-ionization resonances above the limit. Photo-ionization The treatment of photo-ionization in Chapter 7 starts with a discussion of the boundary conditions in a simplified model with a single open ionization channel – leaving details of the general case to Appendix D. It is also shown how the spherical tensor treatment of photo-excitation goes over to photo-ionization. The creation of two ionized fragments naturally complicates the angular momentum algebra. Much of the detail concerning different spin-coupling cases is again deferred to the appendix – along lines initiated by Buckingham et al. [25]. The main text largely follows the formal development given by Fano and Dill [26, 27]. One surprise is that it proves convenient for the purpose of averaging over a bulk sample to work in terms of the angular momentum, Jt , transferred from the parent molecule to the ion, rather than in terms of the total angular momentum. The former is found to centre attention on the electronic angular momentum components 

of the orbital from which the electron is excited, while the latter focuses on the angular momentum  of the outgoing electron. It proves illuminating to compare the photo-ionization selection rules from these two perspectives. Attention then turns to the photo-electron angular distribution, which is typically determined by interference between two or more outgoing orbital angular momentum waves. One sees that the broad properties of such distributions may be characterized by a limited number of ‘asymmetry parameters’, governed by the orders of the relevant spherical tensor operators – regardless of the values of the photo-electron angular momenta. The extent to which the optical transition amplitudes and phase terms can be inferred from such experiments is also discussed. The normal angular distribution is observed for a bulk sample, but a recent class of experiments allows the observation of ‘fixed molecule’ angular distributions, which allow detection of the electron in coincidence with the ionized fragment of a rapidly dissociating species. The theory of such experiments, which was worked out by Dill many years ago [28], is shown to provide direct evidence on the shape of the orbital from which the electron is excited [29]. The chapter concludes with sections in which the spherical tensor formulation of the theory is conveniently replaced by a density matrix treatment [18, 30]. The first example is resonant two-photon ionization of NO, for which the photo-electron angular distributions obtained with different laser polarization axes and different polarizations can yield direct information on the transition amplitudes and phases associated with the contributing partial waves of the outgoing electron [31]. The two final sections concern the ‘orientation and alignment’ of the positive ion angular

1.2 Organization of the text

13

momentum vector J+ and spin-polarization of the photo-electron. The first, which was briefly touched on in Chapter 6, is now handled more formally, along lines established by Greene and Zare [32]. The discussion of the second follows Raseev and Cherepkov [33]. Manipulating Rydberg states The final chapter describes two examples of the way in which Rydberg states may be manipulated for experimental purposes. Each requires the inclusion of significant theoretical background material to bring out important aspects of the Rydberg dynamics. For example, the section on Rydberg wavepackets includes an analysis of the recurrences and revivals in the wavepacket auto-correlation function, as developed by Averbukh and Perelman [34]. The resulting information on the phase variation within the packet is then exploited to show how the outcome of a photo-fragmentation experiment can be controlled by a suitable sequence of picosecond pulses [35]. It is also shown how the MQDT theory of photo-ionization in the previous chapter can be modified to illustrate the time dependence [36]. The second section concerns the Stark effect, including the field-ionization of the high Rydberg states, which lies at the heart of high resolution ZEKEPFI spectroscopy [1, 37]. It is prefaced by a relatively extended account of the O(4) super-symmetry responsible not only for well-known separation of the Schr¨odinger equation in parabolic coordinates, but also for a frame transformation between angular momentum states |nm and parabolic states |n1 n2 m [38, 39] The treatment of hydrogenic and non-hydrogenic Stark effects follows standard lines, but the final section, which lays the foundations for a generalized MQDT treatment of arbitrary highly excited species, will be new to all but specialist readers. Appendices The text is organized to allow the reader to follow the argument without too much technical detail – leaving the latter to be handled in the appendices. The first of these concerns the normalization of MQDT states, which is complicated by employing internal forms for the Coulomb basis functions in Chapter 2, which pass smoothly from open to closed channels. Moreover the normalization is required to apply to the fully coupled states, not only those in a single channel. Cases involving N o open channels (regardless of the number of closed ones) have N o solutions, each of which has N o open-channel coefficients. The normalization is therefore expressed in terms of an N o × N o matrix T , which is required to be orthogonal, T T T = I , under the energy normalization conditions appropriate to one quantum state per

14

Molecular Rydberg states

unit energy. Details of the manipulations required to ensure normalization of the purely bound states to unity are also given. Appendix B presents various different formulations of the working MQDT equations, to assist the reader in following the literature. The text itself follows what we term the ‘standard’ or K-matrix representation, in which the states are defined in terms of the coefficients Zi of the uncoupled states |i, and in which the openchannel coefficients satisfy a simple eigenchannel equation. Awkward divergences of tan π μ can however be avoided by employing a ‘sin–cos’ formulation, at the cost of introducing a singular generalized eigenvalue equation. In addition, the early literature is expressed in what we term a ‘mixed’ form, in which the rows and columns of the working equations are defined with respect to uncoupled and coupled states |i and |α, respectively. Appendix C contains details of the angular momentum algebra required to determine rotational electronic frame transformations between coupled and uncoupled states, |α and |i, respectively. It includes an introduction to Hund’s coupling schemes for diatomic molecules, with expressions for the relevant parity-adapted wavefunctions. Specific expressions are given for the important frame transformations from case (a) to case (e) and from case (b) to case (d). In addition a detailed symmetry discussion for asymmetric tops is followed by the specific form for the case (b) to case (d) frame transformation. Appendix D covers details of the angular momentum theory for both photoexcitation and photo-ionization, on the assumption that defined |

λ

 electronic angular momentum components of the initial molecular orbital can be identified. Familiar expressions for the transition amplitudes between bound states are included for the sake of completeness [5, 18]. The account of photo-ionization starts with an MQDT-based exposition of the boundary conditions for multiple open ionization channels – showing how the optical transition amplitudes may be combined with scattering matrix elements arising from inter-channel coupling in the photo-excited state. Explicit expressions are also given for the photo-ionization matrix elements appropriate from transitions between various angular momentum coupling schemes in the neutral molecule and the positive ion, along lines pioneered by Buckingham et al. [25]. To aid the incorporation of dissociation as well as ionization channels, Appendix E outlines a generalized MQDT description of the states supported by arbitrary potentials. A convenient but ambiguous phase-amplitude representation for the generalized basis functions is available via equations given by Milne [40, 41]. It is shown how the ambiguity may be resolved by comparison with a high-order Jeffreys–Wentzel–Kramers–Brillouin (JWKB) wavefunction [42]. Finally, for the sake of easy reference, the last appendix contains a guide to the notation employed.

References

15

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35]

T. Gallagher, Rydberg Atoms (Cambridge University Press, 1994). G. Herzberg, Spectra of Diatomic Molecules, 2nd edn (Van Nostrand, 1950). G. Herzberg, Infrared and Raman Spectra (Van Nostrand, 1945). G. Herzberg, Electronic Spectra of Polyatomic Molecules (Van Nostrand, 1967). P. R. Bunker and P. Jensen, Molecular Symmetry and Spectroscopy (NRC Research Press, 1998). U. Fano, Phys. Rev. 2, 353 (1970). M. J. Seaton, Rep. Prog. Phys. 46, 167–257 (1983). C. H. Greene and C. Jungen, Adv. At. Mol. Phys. 21, 51 (1985). C. Jungen, Molecular Applications of Quantum Defect Theory (IOP Publishing, 1996). A. Szabo and N. S. Ostlund, Modern Quantum Chemistry (Dover Publishing, 1996). P. G. Burke and K. A. Berrington, Atomic and Molecular Processes: An R-Matrix Approach (IOP, 1993). P. G. Burke, R-Matrix Theory of Atomic Collisions: Application to Atomic, Molecular and Optical Processes (Springer, 2011). M. Aymar, C. H. Greene and E. Luc-Koenig, Rev. Mod. Phys. 68, 1015 (1996). M. S. Child, Semiclassical Mechanics with Molecular Applications (Oxford University Press, 1991). H. Lefebvre-Brion and R. W. Field, Perturbations in the Spectra of Diatomic Molecules (Academic Press, 1986). E. P. Eisbud and L. Eisbud, Phys. Rev. 72, 29 (1947). P. J. A. Buttle, Phys. Rev. 160, 719 (1967). R. N. Zare, Angular Momentum (Wiley Interscience, 1988). H. Lefebvre-Brion and R. W. Field, The Spectra and Dynamics of Diatomic Molecules (Academic Press, 2005). M. Larsson and A. E. Orel, Dissociative Recombination of Molecular Ions (Cambridge University Press, 2008). V. Kokoouline and C. H. Greene, Phys. Rev. A 68, 012703 (2003). C. Jungen and S. T. Pratt, Phys. Rev. Lett. 102, 023201 (2009). M. G¨oppert-Mayer, Ann. Physik. 9, 273 (1931). R. N. Dixon, J. M. Bayley and M. N. R. Ashfold, Chem. Phys. 84, 21 (1984). A. D. Buckingham, B. J. Orr and J. M. Sichel, Phil. Trans. Roy. Soc. London, A 268, 147 (1970). U. Fano and D. Dill, Phys. Rev. A 6, 185 (1972). D. Dill and U. Fano, Phys. Rev. Lett. 29, 1203 (1972). D. Dill, J. Chem. Phys. 65, 1130 (1976). R. R. Lucchese, A. Lafosse, J. C. Brenot et al., Phys. Rev. A 65, 020702 (2002). K. Blum, Density Matrix Theory and Applications (Plenum Press, 1981). D. J. Leahy, K. L. Reid, H. Park and R. N. Zare, J. Chem. Phys. 97, 4948 (1992). C. H. Greene and R. N. Zare, Phys. Rev. A 25, 2031 (1982). G. Raseev and N. A. Cherepkov, Phys. Rev. A 49, 3948 (1990). I. S. Averbukh and N. F. Perelman, Phys. Lett. A 139, 449 (1989). H. H. Fielding, Ann. Rev. Phys. Chem. 56, 91 (2005).

16 [36] [37] [38] [39] [40] [41] [42]

Molecular Rydberg states F. Texier and C. Jungen, Phys. Rev. A 59, 412 (1999). K. M¨uller-Dethlefs and E. W. Schlag, Ann. Rev. Phys. Chem. 42, 109 (1991). D. Park, Z. Physik 159, 155 (1960). U. Fano, Phys. Rev. A 24, 619 (1981). W. E. Milne, Phys. Rev. 35, 863 (1930). W. E. Milne, Am. Math. Mon. 40, 863 (1933). C. Jungen and F. Texier, J. Phys. B 33, 2495 (2000).

2 The quantum defect picture

2.1 Introduction Multichannel quantum defect theory uses scattering methods to provide a uniform treatment of spectroscopic and fragmentation phenomena. It rests on the idea that the exchange and correlation interactions between an outer Rydberg electron and the positive ion core act over a relatively short range, so that the detached electron moves in a purely Coulomb field at larger distances. One therefore thinks, even in the bound state context, of the scattering effect of the non-Coulomb core on the Coulomb wavefunctions. Put in explicit terms this means that the outer parts of the Rydberg orbitals are solutions of the Coulomb equation, the phases of which are determined by matching to the inner wavefunction at the core boundary. There can also be more complicated situations, in which the non-Coulomb interactions lead to energy transfer from the core, which ‘auto-ionizes’ the detached electron from a bound to a continuum state. The general solutions are normalized to allow a uniform description at energies above and below the ionization limit. Further ramifications, which are deferred to a later chapter, allow the inclusion of simultaneous ionization and dissociation. Reviews that emphasize molecular aspects of the theory are given by Greene and Jungen [1] and Ross [2]. There is also a collection of seminal papers, edited by Jungen [3]. This exposition starts with a description of the properties of Coulomb wavefunctions at arbitrary energies, using definitions that provide a uniform description of both bound and continuum states. The second important step is to eliminate all exponentially divergent parts of the physical wavefunction as r → ∞, thereby leading to a related set of quantization conditions for bound, continuum and resonant states. In addition, care must be taken to ensure a consistent set of normalization conditions for the three types of state. Illustrative examples are used to bring out different threads of the argument.

17

18

The quantum defect picture

2.2 Coulomb wavefunctions The properties of the bound-state solutions of the hydrogen atom may be found in any quantum mechanics text [4, 5, 6, 7]. However, the nature of the solutions at arbitrary energies are less readily available. The defining equation   h¯ 2 2 e2 − ∇ +I −  = E, (2.1) 2m 4π 0 r where m = me M/(me + M) separates, by the substitution 1 (2.2) (r, θ, φ) = ψ (r)Ym (θ, φ) r into radial and angular equations. The angular functions, Ym (θ, φ), are the familiar spherical harmonics [8]. Since the Coulomb waves will be exclusively employed in the external uncoupled region, in which the radial motion is governed by the reduced mass, it is most convenient to follow Seaton in employing ‘natural’ units, such that the radial factors satisfy the scaled equation [9]1  2  d 2 ( + 1) ψ = 0, ++ − (2.3) dr 2 r r2 where  = (E − I )/Ry

(2.4)

Ry = (m/me )R∞ , in which I denotes the ionization energy. Of course the individual quantities E, I and Ry can be expressed in any convenient units. To illustrate some important ideas, Fig. 2.1 depicts the functional forms of pairs of independent solutions of (2.3) at positive and negative energies. The solid oscillatory traces follow the so-called regular solutions, f (, r), which are defined to vanish at the origin, while the second solution of each pair g (, r), which diverges as r → 0, is taken to be ‘out of phase’ with f (, r) in the sense illustrated 1

The corresponding form in atomic units is less convenient because the reduced mass, m, appears as a separate parameter in the equations   1 d2 1 ( + 1) +  + − ψ = 0, 2 2 2m dr r 2r where  = (E − I )/Eh Eh = 2R∞ . However the approximation m  1 is valid for heavy molecular species, in which case  = /2. The reader should also take note of consequent changes in the wavefunction normalization in Appendix A.

2.2 Coulomb wavefunctions

19

10

500

5

0

5

10

0

50

100

150

200

250

r/a Figure 2.1 Forms of the basis functions f (, r) (solid lines) and g (, r) (dashed lines) for  = 1 and  = (E − I )/Ry = ± (1/10.5)2 . The dotted lines indicate the common envelopes for f (, r) and g (, r) and the heavy solid line is the outer branch of the channel potential, U (r) = 2/r − ( + 1)/r 2 . The left hand scale is both a reduced energy scale, mutiplied by 500, and the scale of f (, r) and g (, r), measured from the appropriate horizontal axes. Notice the close similarity between the positive and negative envelope functions close to r = 0. One should also note that f (, r) diverges at infinity, while g (, r) happens to decrease exponentially to zero at the half odd integer value, ν = 10.5.

in Fig. 2.1.2 The scale of the diagram is too small to show the divergence of g (, r) at the origin, but three essential features are well displayed. (i) The upper pair of open channel solutions, with positive energy, oscillate within a common envelope towards the asymptotic limit r → ∞, where [9]   f (, r) ∼ (π k)−1/2 sin kr + k −1 ln(2kr) − π/2 + η (E) (2.5)   g (, r) ∼ − (π k)−1/2 cos kr + k −1 ln(2kr) − π/2 + η (E) , 1/2  and the Coulomb phase is given by in which k = (E − I ) /Ry η (E) = arg ( + 1 − i/k), 2

(2.6)

The reader should note that the early Coulomb literature is complicated by rival nomenclature. The symbols f (, r) and g (, r) follow the notation of Fano [10] and Greene et al. [11]. By contrast, Seaton uses s (, r) and -c (, r) for the same functions, which are scaled versions of Seaton’s own f and h functions [9]. In turn, h is a linear combination of Seaton’s f and g, which are defined to be analytic across the ionization limit,  = 0. The precise combinations are listed in Table 2.1.

20

The quantum defect picture

where (z) is the gamma function [12]. The chosen asymptotic amplitude in (2.5) ensures that (see Appendix A)  ∞ f (, r)f ( , r)dr = δ( −  ), (2.7) 0

which is consistent with a continuum density of one quantum state per ‘natural’ energy unit. Finally f (, r) and g (, r) satisfy the Wronskian relation W [f , g ] = f (, r)g (, r) − g (, r)f (, r) = π −1 ,

(2.8)

which is valid for all r values.3 (ii) The lower pair of functions in Fig. 2.1, with a negative reduced energy, also oscillate within a common envelope over the classically accessible range, where they may be represented in the phase-amplitude forms f (, r) = π −1/2 α (, r) sin φ (, r) g (, r) = −π −1/2 α (, r) cos φ (, r),

(2.9)

subject to the ancillary condition φ (, r) = [α (, r)]−2 , which again ensures that W [f , g ] = π −1 . This use of a common Wronskian condition to fix the wavefunction amplitudes at both positive and negative energies, which is sometimes loosely termed energy normalization of the bound states, allows a systematic uniform treatment of bound, resonant and continuum states – at the cost of the later introduction of a special normalization condition for the true bound states. Details are discussed in Section 2.4 and Appendix A. Note also the close similarity between the detailed short-range forms of the positive and negative energy solutions, with a given angular momentum – which may be traced in physical terms to the fact that the kinetic energy at small r, which fixes the local de Broglie wavelength, is large compared with the energy range of normal spectroscopic interest. The practical consequence is that short-range non-Coulomb interactions, even in species other than the H atom, are found, to a first approximation, only to cause an energy-independent change in the phase of the outer part of the regular wavefunction, which is reflected, as we shall see later, in an almost energy-independent ‘quantum defect’. (iii) The third important observation concerns the asymptotic forms of the negative energy solutions of (2.3). Both typically diverge as r → ∞, although the special energy choice,  = −1/(10.5)2 , corresponding to the half-integer effective quantum number ν = 10.5, ensures that the function g (, r), which diverges at r = 0, accidentally decays exponentially to zero as r → ∞. 3

The implicit √ dependence of (2.7) on the energy scale requires that f (, r) and g (, r) should be multiplied by factors of 2 if ‘atomic’ rather than ‘natural’ units are employed, because the natural values of  are twice as large as the atomic ones. Consequently W [f , g ] = 2/π in the atomic unit system.

2.2 Coulomb wavefunctions

21

Routines for the evaluation of these basis functions at arbitrary energies,  = (E − I )/Ry , are given by Seaton [13]. 2.2.1 Closed-channel solutions: E < I The important properties of the open-channel solutions, at energies above the ionization limit, are given by (2.5)–(2.7). However the negative energy case,  = (E − I ) /Ry = −1/ν 2 , requires more discussion. Following Sections 2.6–2.7 of Seaton [9] and bearing in mind the difference in notation (see footnote 2), the present f (, r) and g (, r) may be expressed in terms of functions with the following asymptotically increasing and decreasing forms [9] −ν ν r  r 2r 2r ξ (, ; r) ∼ exp exp − , (2.10) ; θ (, ; r) ∼ ν ν ν ν in terms of which

1/2   sin π (ν − )  (ν − ) cos π (ν − ) +1 A(ν, ) f =ν ξ− θ 2 π  (ν +  + 1)

1/2   cos π (ν − )  (ν − ) sin π(ν − ) +1 A(ν, ) ξ+ θ , g = −ν 2 π  (ν +  + 1) (2.11) where [9]





22 2  (ν +  + 1) 12 1 − 2 ··· 1 − 2 . A (ν, ) = 2+1 = 1− 2 ν  (ν − ) ν ν ν

(2.12)

The properties of the (z) functions [12] for ν >  + 1 allow the following standard form for (2.11): f (, r) = sin β (E)F (+) (, , r) − cos β (E)F (−) (, , r) g (, r) = −cos β (E)F (+) (, , r) − sin β (E)F (−) (, , r), where β (E) = [ν(E) − ] π =



 Ry / (I − E) −  π.

(2.13)

(2.14)

Finally the re-scaled asymptotically increasing and decreasing functions are given by  −1 F (+) (, , r) = (2ν)1/2 π K ξ (, , r),

   3/2 K θ(, , r), F (−) (, , r) = ν 3 /2  −1/2 K = ν 2  (ν +  + 1)  (ν − ) . (2.15)

22

The quantum defect picture

In addition it may be confirmed that the pairs (f, g) and (F (+) , F (−) ) both satisfy the required Wronskian relation   (2.16) W [f, g] = W F (−) , F (+) = π −1 . Equation (2.14) merits special discussion, as one of the central equations of the theory. The argument of β (E) is written as the unscaled physical energy, for convenient reference in later chapters. The related function ν (E) will be identified later as the quantum number function, which changes by unity between adjacent eigenvalues. Moreover the energy derivative π ν3 dν dβ =π = dE dE 2Ry

(2.17)

is related to the eigenvalue density, which has important consequences for the scaling behaviour of Rydberg systems. The following additional identity:   W f, F (−) , tan β (E) = −  (2.18) W g, F (−) is used in later sections to show how changes in the definition of the base pair (f, g) give rise to changes in the accumulated phase function β (E) . Equation (2.13) shows that both the regular and irregular functions, f (, r) and g (, r), typically diverge exponentially as r → ∞. However the coefficients of the two divergent terms vanish for integer and half-odd integer values of ν −  respectively, as illustrated in the latter case in Fig. 2.1. Thus in the case of the hydrogen atom, which has no core, the proper wavefunctions, which vanish at r = 0, contain only an f (, r) component. The condition β(E) = (n − )π , which eliminates the F (+) component of f (, r), therefore leads, by rearrangement of (2.14) to the well-known hydrogenic Rydberg formula En = I −

Ry . n2

(2.19)

More generally, however, the outer part of a typical atomic wavefunction is a particular linear combination of f (, r) and g (, r), dependent on the influence of the non-Coulombic core. The bound states then occur at energies at which the coefficients of the two divergent terms cancel. 2.2.2 Strongly closed channels As written, (2.19) carries no restriction on the principal quantum number n, although the careful reader will have noted the restriction ν >  + 1 on the validity

2.2 Coulomb wavefunctions

23

of (2.13), which eliminates false 1p, 2d, etc., eigenvalues. Nevertheless the determination of ab-initio quantum defects, using the methods in Chapter 3, requires knowledge of suitable basis functions in the strongly closed channels, which are defined as those with energies below the minimum of the channel potential, i.e.  < −1/( + 1)2 . General readers may ignore this specialist topic, without loss of continuity. √ The earliest approach, by Ham, recognized that the factor A (ν, ) in (2.11) causes f (, r) to vanish identically at the series of ‘false’ 1p, 2d, etc., hydrogenic eigenvalues at which ν −  = 0, −1, −2, etc. [14]. At the same time g (, r) diverges, because the Wronskian retains the fixed value W [f , g ] = π −1 . Ham overcame this difficulty by rescaling f (, r) and g (, r) by factors of [A (ν, )]−1/2 and [A (ν, )]1/2 , respectively, so that by the use of (2.18)   1 W f , F (−) tan π(ν − ) (Ham) =  (E) = − tan β . (2.20) (−) A (ν, ) W g , F A (ν, ) The zeros of tan π(ν − ) at ν −  = 0, −1, etc., are cancelled by those of A (ν, ) , leaving 0 < β Ham () < π over the strongly closed channel region. However, as seen in Fig. 2.2, the resulting phase functions tan β (Ham) (E) tend to oscillate between the hydrogenic energy levels n = I − Ry /n2 . More recently, Guerout et al. have found a way to construct alternative base pairs, f˜ (, r) and g˜  (, r) appropriate to any desired form for the accumulated phase function [15]. They are related to the functions f(Seat) (, r) and g(Seat) (, r) above by the equation  

 −1/2 0 A () f(Seat) (, r) f˜ (, r) = , (2.21) 1/2 1/2 g˜  (, r) g(Seat) (, r) A () tan λ () A () which is the most general form consistent with the regularity of f (, r) and f˜ (, r), the irregularity of g (, r) and g˜  (, r), and the Wronskian relation W [f , g ] = W [f˜ , g˜  ] = π −1 . The appropriate forms for A () and tan λ () for the three base pairs discussed by Seaton are listed in Table 2.1, in which the function G (ν, ) is given by [9] G (ν, ) =

A (ν, ) [ψ(ν +  + 1) + ψ(ν − ) − 2 ln ν], 2π

where ψ (z) is the digamma function [12]. It follows that   W f˜, F (−) tan β(Seat) () ˜   A () tan β () = − , = ˜ F (−) W g, 1 − tan λ () tan β(Seat) ()

(2.22)

(2.23)

24

The quantum defect picture

Table 2.1 Transformation variables A and tan λ and accumulated phase functions tan β appropriate to different ˜ as given by (2.21). base pairs ((f˜), (g)), Seaton (f, g) (f, h) (s, −c)

Present  (Seat)



f (Seat) , g  (Ham) (Ham)  ,g f  (Fano) (Fano)  ,g f

A

tan λ

tan β

1

0

1

G (ν, ) G (ν, )

tan π (ν−) A(ν,)+G(ν,) tan π(ν−) tan π (ν−) A(ν,)

1 A(ν,)

tan π (ν − )

which rearranges to 1

1 () A () tan β˜ ()   1 A (ν, ) − = G (ν, ) + . tan π (ν − ) A () tan β˜ ()

tan λ () =

tan β(Seat)

−

(2.24)

Consequently the form of tan λ () may be chosen to generate any desired form for the product A () tan β˜ (). The flexibility in having two independent functions A () and β˜ () is helpful in controlling the energy dependence of the basis functions. Guerout et al. obtain smooth energy variations in both β˜ () and the base pair ˜ [f (, r), g˜  (, r)] at a particular radius r, by choosing β˜ () to increase linearly with ν over two linear segments, joined by a curved transition region close to the position of the lowest eigenvalue, ν   + 1 [15]. The first segment covers the strongly closed region, 0 < ν  , over which β˜ /π = b (ν/) and A = a (ν/)−2 ,

(2.25)

with a  b  0.2. The behaviour at ν = 0, corresponding to  → −∞ is irrelevant in practice, but the choice that β˜ should vanish is physically reasonable on the grounds that there should be no accumulated phase in this strongly closed limit, while the corresponding choice for A ensures that the final term in (2.24) vanishes as ν → 0 [15]. The second range covers the normally closed region  + 1 − δ  ν < ∞, over which the choice β () = π (ν − ) ,

tan λ = G (ν, ) and A =

1 A (ν, )

(2.26)

recovers the identity tan λ = G (ν, ) appropriate to the ‘Fano’ line of Table 2.1. Finally quadratic spline functions for both β˜ and A−1  are adopted over the transitional region,  < ν <  + 1 − δ .

2.3 Single-channel quantization Ham

25

Modified

2

2

1

1

β 1/π (a) 0

1

2

(b) 3

0

2

2

1

1

1

2

3

β 3/π (c) 0

2

3

ν

4

(d) 5

0

2

3

ν

4

5

Figure 2.2 Accumulated phase functions as given by the ‘Ham’ and ‘modified’ prescriptions of (2.20) and (2.22)–(2.24), respectively. Panels (a) and (b) apply for  = 1, and (c) and (d) for  = 3. The upper parts of the dashed lines in (a) and (c) vary as β = (ν − )π and the lower parts follow β = π/2.

Figure 2.2 illustrates a comparison between the oscillatory forms for β(Ham) () for  = 1 and 3, and the much smoother modified functions β˜ (). The upper parts of the dashed lines in the Ham panels are the linear forms, β(Fano) () = (ν − ) π, whose extrapolation into the strongly closed region predicts the forbidden eigenvalues. The lower parts follow β = π/2. 2.3 Single-channel quantization The case of bound state quantization, in a single channel, brings out some important physical principles, which apply equally in multichannel contexts. The first step is to recognize that the presence of an attractive non-Coulomb core will typically reduce the de Broglie wavelength of the regular wavefunction in the inner region, thereby causing a phase change in the outer one, compared with the corresponding hydrogenic wavefunction. The resulting form in the outer Coulomb field region is expressed as the phase-shifted linear combination ψ (, r) = cos π μ f (, r) − sin π μ g (, r).

(2.27)

It follows from (2.13) that ψ (, r) ∼ sin [β () + π μ ()] F(+) (, r) − cos [β () + π μ ()] F(−) (, r). (2.28)

26

The quantum defect picture

Elimination of the exponentially divergent term therefore leads, via (2.14), to the quantization condition

  Ry / (I − E) −  + μ () π = [n − ] π, β () + π μ () = (2.29) which converts to the famous Rydberg formula En = I −

Ry . [n − μ ()]2

(2.30)

As written, (2.30) simply points to the influence of the phase shift μ () on the resulting energy levels. For example an attractive core decreases the de Broglie wavelength in the inner region, thereby increasing the phase and lowering the bound state energy levels, compared with those of the H atom. The quantum defects therefore typically vary according to the penetration of the relevant orbital. Thus μs > μp > μd > μf  0.

(2.31)

The practical value of (2.30) is, however, greatly increased by the fact that the energy dependence of the quantum defect is often extremely small, to the extent that a weak linear energy dependence of the form μα () = μ0α + μ1α /ν 2 ,

ν 2 = Ry /(I − E)

(2.32)

suffices, even for the most accurate work.4 Figure 2.3 illustrates results for the K atom. This dependence can sometimes be further reduced by working in the Ham representation, in which the so-called ‘η defects’, as distinct from ‘μ defects’, are determined by5 tan β Ham (E) + tan π η = 0,

(2.33)

where β Ham (E) is given by (2.20). A final point concerns the normalization of the associated wavefunctions, whose amplitudes are fixed in Fig. 2.1 to conform with those of the energy normalized continuum functions. Mathematical details are given in Appendix A, where it is shown that

 ∞ dν ν3 1 dβ 2 = = ψn (r)dr = , (2.34) π d d 2Ry 0 4

5

This slow variation is, however, restricted to normal closed channels. It is necessarily replaced by a variation such that π μ () ∝ β () to avoid false 1p, 2d, etc., solutions of (2.29) at energies such that ν <  + 1 [11]. ˜ with a modified phase function β˜ () (see Appendix B) the quantization Similarly in any representation (f˜, g) condition is expressed in terms of ‘mu tilde’ defects, tan β˜ () + tan π μ˜ = 0. This form is, however, relevant only in the strongly closed region, because the definitions adopted by Guerout et al. ensure that β˜ () = β () and μ˜ = μ for  > −1/( + 1)2 [15].

2.4 Coupled channels

27

1.5

s

μ or η

1

p 0.5

d 0

0

0.1

0.2

1/ν

0.3

2

Figure 2.3 An Edlen plot of the μ quantum defects (solid line) and η quantum defects (dashed line) for the 2 S1/2 , 2 P1/2 and 2 D5/2 levels of the K atom. Points indicate the atomic eigenvalues.

in which ν() is the number function, which takes values ν = n − μ at the eigenvalues. The final equality means that the internal amplitudes of unit normalized bound states scale as ν −3/2 , which carries important consequences for the scaling properties of Rydberg states. 2.4 Coupled channels The coupled N channel case is more complicated, because one must now eliminate the exponentially increasing wavefunction components in all closed channels. Each channel |i is specified by an appropriate fragment wavefunction i , which includes labels for the positive ion core and for the Rydberg angular momentum . There are now 2N radial factors fi (, r) and gi (, r) in the outer region, which must be matched to N regular inner solutions at the core boundary, leaving N independent outer functions, which are conveniently specified in terms of the scattering Kmatrix6 as linear combinations of the form (j )

ψi (, r) = fi (, r) δij − gi (, r) Kij , 6

(2.35)

Details of the connection at the boundary between between inner and outer regions are discussed in Chapter 3. In the conceptually simplest situation the column vector ψ (rc ) of inner radial factors may be related to its derivative ψ (rc ) in the R-matrix form ψ = Rψ .

28

The quantum defect picture

in which the superscript j = 1, 2, . . . N specifies a column of the total outer wavefunction matrix, with amplitudes in each of the i = 1, 2, . . . N channels. To ensure linear independence, the N distinct solution vectors, ψ (j ) , are each defined to have one regular component, with unit amplitude in channel i = j and irregular components in all N channels, with amplitudes Kij , which reflect the influence of the core. Notice that a rescaling of (2.28), by a factor sec π μ would yield K = tan π μ in the single channel case. Equally, as discussed in Appendix B, the elements δij and Kij in (2.35) might be replaced by Cij and Sij , respectively, with the matrices S and C related to K in the form K = C−1 S. The next step is to define a partial wavefunction  (j ) of the full system by (j ) combining the radial terms ψi with the channel functions (or target states), i , which include all other degrees of freedom. Thus     (j ) r (j ) = (2.36) i ψi = i fi δij − gi Kij . i

i

The asymptotic forms in (2.13) show that any particular solution  (j ) typically diverges exponentially in the closed channels, and the aim is to eliminate this divergence by summing over the superscript j to create the total wavefunction     r = r (j ) Zj = i fi δij − gi Kij Zj . (2.37) j

ij

The boundary conditions on the coefficients Zj differ according to whether the channel |i belongs to the open set {P } or the closed set {Q}. The most convenient The matching condition at the boundary rc therefore requires that the same equation must apply for the outer functions; thus f − gK = R(f −g K), which rearranges to

 −1   K = g − Rg f − Rf .

Hence knowledge of the N × N inner log derivative matrix and of the N basis function pairs fi and gi [13] is sufficient to determine the N × N K-matrix. The connection between K and the scattering S-matrix may be obtained by comparing the matrix form  = [fI − gK] A, where A is a normalizing matrix, with the S-matrix representation    = ψ − I − ψ + S B, in which ψ ± = −g±if It follows by simple substitution that S = (I−iK)−1 (I+iK).

2.4 Coupled channels

29

requirement for the open-channel wavefunctions is that their radial components ψi (r) should satisfy the asymptotic boundary conditions ψi (r) ∼ [fi (r) cos π τ − gi (r) sin π τ ] Ti ,

(2.38)

with the same ‘eigenphase’ τ in each channel.7 In other words, the coefficients Zj in (2.37) must be chosen to eliminate off-diagonal terms in (2.37). The corresponding condition for the closed channels is that exponentially increasing components given by (2.13) should cancel. In other words, the total wavefunction must behave as  (o)  i r ∼ [fi cos π τ − gi sin π τ ] δij Ti i⊂P

+



j

(c) i

i⊂Q

  Fi(+) sin βi δij + cos βi Kij

j

(−) 

+ Fi



cos βi δij − sin βi Kij



Zj ,

(2.39)

where the sums over j include all channels. As derived in Appendix A, the normal ization requirement is that i Ti2 = 1. Comparison between the coefficients of the open channel fi and gi components in (2.37) and (2.39) shows that Zi(o) = cos π τ Ti and   Kijoo Zjo + Kijoc Zjc = sin π τ Ti = tan π τ Zio ; i ⊂ P . (2.40) j ⊂P

j ⊂Q

The requirement that coefficients of Fi(+) should vanish for the closed channels leads to a second set of equations ⎡ ⎤   Kijco Zjo + Kijcc Zjc ⎦ = 0; i ⊂ Q. (2.41) sin βi Zic + cos βi ⎣ j ⊂P

j ⊂Q

Taken together, (2.40) and (2.41) may be expressed in the block matrix form

o

oo Z K oc K − tan π τ (E)I oo = 0, (2.42) co cc K K + tan β(E) Zc in which tan π τ (E) is a scalar and tan β(E) is a diagonal matrix. The notation is intended to emphasize the relatively strong energy dependence of τ (E) and β(E) compared with that of the K-matrix elements. 7

The eigenchannel treatment of the coupled equations is particularly convenient for spectroscopic applications, but may also be readily transformed to allow photo-ionization from a particular target channel or scattering from a selected incident channel into all other channels. Details relevant to photo-ionization are given in Appendix D.2.1.

30

The quantum defect picture

Equation (2.42), which is the main working equation of MQDT theory, may also be expressed in the alternative forms given in Appendix B. Three important physical situations, which differ according to the presence or absence of open or closed channels, are discussed below. Here we emphasize that the dynamics are governed first by the K-matrix, which transfers amplitudes between different channels in response to the influence of the core. Secondly, the accumulated phase functions βi (E) take into account the closed-channel boundary conditions at infinite separation. Moreover, each cycle of tan βi () corresponds to an increase in the principal quantum number, so that the size of any computation is independent of the energy. Finally, as discussed in detail next, this single set of equations allows a uniform treatment of continuum states, purely bound states and bound–continuum resonances. 2.4.1 Open-channel case If all channels are open, (2.42) takes the form of an eigenvalue equation, with N eigenvalues tan π τρ (), ρ = 1, 2, . . . N, and N eigenvectors, with components Ziρ , subject to the normalization condition o = cos π τρ (E)Tiρ , Ziρ

(2.43)

where T is a unitary matrix, T T † = T † T = I . By virtue of this normalization each eigensolution corresponds to one quantum state per unit energy, whose amplitude components Tiρ are unchanged by the scattering. In addition the energy dependence of the eigenphases τρ () reflects only the weak energy dependence of the K-matrix. 2.4.2 Resonant case In the more interesting resonance case, we assume a total of N channels, of which N o are open and N c are closed. The energy dependence of the elements βi (E) is given by (2.14), while that of τρ (E) is dictated by the solution of (2.42). We anticipate, by analogy with the open-channel situation that there will be N o solutions, labelled by an index ρ, the forms of which are conveniently obtained by using the second block row of (2.42) to express the vector Zρc in terms of Zρo . Thus  −1 co o Zρc = − K cc + tan β(E) K Zρ . (2.44) It follows by substitution in the first block row that    −1 co K − tan π τρ (E) I oo Zρo = 0. K oo − K oc K cc + tan β(E)

(2.45)

In other words tan π τρ () is an eigenvalue of the ‘physical’ K-matrix  −1 co K phys = K oo − K oc K cc + tan β(E) K ,

(2.46)

2.4 Coupled channels

31

which varies strongly with energy in the vicinity of ‘resonances’ close to zeros of the determinant det [K cc + tan β(E)], which will be shown to correspond to the bound states of the closed-channel spectrum. The resulting singularities of the inverse matrix [K cc + tan β(E)]−1 are found to cause jumps by π in particular eigenphases, π τρ (), although the resonances  are normally detected by looking for jumps in the eigenphase sum, ρ π τρ (E). o determine the branching ratios The open-channel amplitudes, Tiρ = sec π τρ Ziρ between different open fragmentation channels, while the closed-channel coefficients, which are related to the open ones by (2.44), reflect the extent to which a particular eigenchannel is coupled to the pseudo-bound state responsible for the resonance.8 Illustrative example: the np auto-ionization series of H2 As an illustrative example, we consider the interacting np series of H2 , with J = 1, which are seen in Fig. 2.4 to terminate on the (v + , N + ) = (0, 0) and (v + , N + ) = (0, 2) rotational levels of H+ 2 – a system which Fano employed as the first introduction of MQDT techniques to molecular physics [10]. Our immediate 8

An interesting normalization condition on the closed-channel components Z c may be derived from the matrix identity d −1 dA −1 A = −A−1 A , d d by assuming that the K-matrix elements have negligible energy dependence across a resonance. After pre T multiplying (2.45) by Zρo , setting A = K cc + tan β(E) and applying the unitarity condition, TρT Tρ =   T sec2 π τρ Zρo Zρo = 1, one finds that  o T oc  cc  −1 −1 co o dτρ Zρ K K + tan β(E) sec2 ββ K cc + tan β(E) K Zρ = π . dE This leads, after using (2.44) to substitute for Z c , to the probability density  2   T dτρ c Pρ (E) = Zρc π −1 sec2 ββ Zρc = π −1 sec2 βi βi Zρi = , d i⊂Q

which means that



En +/2

En −/2

 Pρ (E) dE =

1/2 −1/2

dτρ = 1,

where  denotes the resonance width. The physical consequence of an abrupt jump by unity in τρ (E) is therefore that Pρ (E) traces out a sharp peak with unit integrated area. Alternatively  En +/2   c 2 π −1 sec2 βi βi Zρi (E) dE = 1. En −/2 i⊂Q

These two equations express the connection between the normalization of bound and continuum states. As shown in Appendix A, the Wronskian normalization of the Coulomb basis functions in Section 2.1 ensures continuum normalization to one quantum state per unit energy. Resonant states, which also lie in the continuum, are seen to have their closed-channel amplitudes concentrated into a narrow range such that the integrated probability density corresponds to a single quantum state. It will be seen that the normalization of a discrete bound state again involves a sum of the terms π −1 sec2 βi βi Zi2 analogous to those in the final equation.

32

The quantum defect picture 700

(E − 124000) / cm

−1

600

500

H2+ +

+

+

+

+

+

H2

(v , N = 0, 2)

np2

40

(v , N = 0, 1) 30 np0

(v , N = 0, 0) 400

40 300

20 30 J=1

200

J=1

Figure 2.4 Level diagram for the interacting np(J = 1) series of H2 , which terminate on the (v + , N + ) = (0, 0) and (0, 2) vibrational–rotational levels of H+ 2.

interest is in the auto-ionizing resonances at energies between the N + = 0 and N + = 2 limits. The K-matrix is conveniently expressed in the frame transformation form (see Section 4.2.1), with elements [10]

 cos θ − sin θ T Kij = Uiα tan π μα Uαj , U = , (2.47) sin θ cos θ α=σ,π

where the transformation matrix U is obtained by an angular momentum projection of the case (d) states |i = |NN +  onto the case (b) ones, |α = |N . The specific form appropriate to a p series is given by (4.22) in Chapter 4. The quantum defects, μα are extracted from the spectral perturbations to the bound states. These and other parameters of the model are given in Table 2.2. The solutions are readily written down because K phys in (2.46) reduces to a single term. Figure 2.5 illustrates two important features of the resulting auto-ionizing series. The upper panel shows that the phase function τ (E) incurs jumps of unity, which are well parameterized by the Breit–Wigner form, [K oc ]2 K cc + tan β (E)  ,  tan π τ0 + 2(En − E)

tan π τ = K oo −

(2.48)

2.4 Coupled channels

33

Table 2.2 Parameters of the interacting np Rydberg series of H2 , which terminate on the (v + , N + ) = (0, 0) and (0, 2) levels of H2 + [16]. μσ = 0.203 μπ = −0.082 I1 = 124417.42 cm−1 I2 = 124591.5 cm−1 −1 Ry = 109707.42 √ cm cos θ = 1/3

0.4

τ (E)

0.2 0 0.2 0.4 450

460

470

480

(E − 124 000) / cm−1

8

28

n2

29

30

32

31

6 4

−4

10 P2(E) / cm

10

2 0

450

460

470

480

(E − 124 000) cm−1

Figure 2.5 Auto-ionizing resonances arising from J = 0 → 1 excitation to the interacting p series of H2 . The upper panel shows a characteristic sequence of unit jumps in the phase function τ (E). The lower panel shows a related sequence of Lorentzian peaks in the closed-channel probability density P2 (E).

around the energies En where K cc + tan β(E) = 0. One may also verify, by combining a linear expansion around these roots with the identity in (2.17), that  2 4Ry (K oc cos βn )2   2 K oc cos βn βn −1/2 = . π νn3

(2.49)

Secondly, the lower panel shows that the probability density in the closed channel follows a sequence of the Lorentzian peaks, P2 (E) =

 dτ 1  .  dE 2π (E − En )2 +  2 /4

(2.50)

34

The quantum defect picture

The resonance widths,  ∼ 1 cm−1 , are of the same magnitude as those observed experimentally [16]. Notice, however, that any observed auto-ionization profiles may be distorted from the simple Lorentzian forms in Fig. 2.5 by interference between the optical transition amplitudes in the two channels. The origins of the resulting asymmetric Beutler–Fano line-shapes [17, 18] are discussed next. It is interesting to examine the factors that influence the resonance width in (2.49). On inserting the explicit forms for K oc one finds that =

Ry sin2 2θ sin2 π(μα − μβ ) cos2 β(En ) . π νn3 cos2 π μa cos2 π μb

(2.51)

where tan β(En ) + K cc = 0 at the resonance. Consequently the line-width increases with the frame transformation mixing angle, θ , over the range 0 < |θ| < π/4 and with the difference in quantum defects, with a maximum at |μ| = 0.5. The mixing angle reflects the difference between the forms of the coupled and uncoupled basis states |α and |i, respectively, while the quantum defect difference is a measure of the anisotropy of the positive ion core. Beutler–Fano line-shapes As first observed by Beutler and later discussed by Fano, the shapes of the resonance peaks in the bound state probability in Fig. 2.5 may differ profoundly from those of the absorption intensity I (E) = |Z1 (E)D1 + Z2 (E)D2 |2 ,

(2.52)

when the interfering transition amplitudes, D1 and D2 to the open and closed channels, respectively, are taken into account [17, 18] . We take as an illustration the N = 1 → 2 excitation of the np series of H3 , in Fig. 2.6, as modelled and observed by Bordas et al. [19]. The auto-ionization arises between the open N + = 1 and closed N + = 3 channels, for which the two channel  version of (2.42) again applies, with the K-matrix given by (2.47) with cos θ = 2/5. This spectrum is preferred over that of H2 because the latter is strongly perturbed by an interloping feature in an excited vibrational series [20]. To understand the origin of the asymmetric line-shapes, the solutions of (2.43)– (2.44) and (2.49) may be expressed in the form Z1 (E) = cos π τ (E) Z2 (E) = [sin π τ (E) − K11 cos π τ (E)] /K12 2 / [K22 + tan π ν (E)]  K11 + tan π τ (E) = K11 − K12

(2.53)  . 2 (En − E)

2.4 Coupled channels

Photo-ionization intensity

4

20

35 30

40 50

3

ν3

N=2 N=0

2

1

0 12 850

12 940 Energy

13 030 (cm–1)

13 120

above

3s2A′1(N

13 210

13 300

= 1)

Figure 2.6 Calculated (upper trace) and experimental (lower trace) Beutler– Fano profiles, in the np Rydberg series of H3 , observed by double resonance from the N = 1 level of the 3s2 A 1 state. Taken from Bordas et al. [19] with permission.

Written in another way, this means that δ + K11  , 2 1 + K11 1 + ε2 (2.54) where ε is a scaled displacement from the resonance centre En , given by sin π τ (E) = 

K11 δ − 1  , 2 1 + K11 1 + δ2

cos π τ (E) = 

  2 )/  (E − En ) − K11 . ε = 2(1 + K11

(2.55)

It readily follows that I (E) = I0 f (ε, q), where f (ε, q) is the celebrated Fano line-shape function [11] (q + ε)2 , f (ε, q) =  1 + ε2

(2.56)

with the asymmetry parameter given in this context by q = K11 −

2 )D2 (1 + K11 , K12 D1

which takes the value q  −0.1 for the model illustrated in Fig. 2.6.

(2.57)

0 −10 1

−5

0

5

10

q=5

0.5 0 −10

−5

0

5

10

1 q=2 0.5 0 −10

−5

0

ε

5

10

f (ε,q) / (1+q 2)

0.5

f (ε,q) / (1+q 2)

f (ε,q) / (1+q2) f (ε,q) / (1+q2)

1 q = 10

f (ε,q) / (1 + q 2)

The quantum defect picture f (ε,q) / (1+q2)

36

1 q=1 0.5 0 −10

−5

0

5

10

0

5

10

0

5

10

1 0.5

q = 1/2 0 −10

−5

1 0.5

q = 1/10 0 −10

−5

ε

Figure 2.7 Scaled Beutler–Fano line-shape profiles for selected asymmetry parameters, q. Solid and dashed lines are used for positive and negative values of q, respectively.

The striking influence of this latter parameter is illustrated in Fig. 2.7, in which each plot has been scaled by the maximum amplitude fmax (q) = (1 + q 2 ). The profile is seen to change from a simple symmetric absorption-like peak for |q|  1 to a symmetric ‘window resonance’ for |q|  1, via a sequence of increasingly asymmetric line-shapes as |q| → 1 from either limit, subject to the requirement that f (ε, q) must touch zero at ε = −q. In addition, the identity f (ε, −q) = f (−ε, q) is confirmed by the reflection symmetry between profiles for positive and negative q values. The physical origin of a particular line-shape is most easily understood for systems with K11 = 0, because q then depends only on the ratio of absorption amplitudes between the open and closed channels, bearing in mind, according to (2.53), that the continuum channel coefficient Z1 falls to zero at the resonance centre τ (E) = (n + 1/2)π, while |Z2 | reaches a peak. Thus a large value of |q| in (2.57) implies very weak continuum absorption, with the result that the peak in Fig. 2.7 for q = 10 follows the peak in Z22 . By contrast, at the opposite extreme, q  1, discrete absorption may be neglected and the ‘window resonance’ follows the dip in the continuum probability |Z1 |2 . Intermediate situations allow the possibility of interference between the continuum and discrete absorption amplitudes, which have their most striking effect for q  1. The situation is more complicated when K11 = 0, except that |q| again becomes very large as D1 → 0. For example, window resonances can then arise from cancellation of the terms on the right of (2.57), rather than from the vanishing of D2 .

2.4 Coupled channels

37

2.4.3 Closed-channel case In the absence of open channels, (2.42) reduces to [K + tan β(E)] Z = 0,

(2.58)

which can be satisfied only at discrete energies such that det [K + tan β(E)] = 0.

(2.59)

The corresponding condition for normalization to unity, given by (A.29) in Appendix A, is that the normalized components Xi in each channel are related to the Zi derived from (2.58) by9  1/2 Zi sec βi βi Xi =  , (2.60)

2 2 i sec βi βi Zi where βi = (dβi /d) . The factors (dβi /d) clearly come from the single-channel equation (2.34), while the terms sec βi diminish the amplitude Zi in accordance with the energy separation between the single-channel bound level, given by βi (ni ) = ni π, and the true eigenvalue given by (2.59). For example Zi = 0 at energies at which βi (ni ) = (ni + 1/2)π. Successive cycles of the tangent terms in (2.58) correspond to successive principal quantum numbers, which means that the roots include all the eigenvalues of the coupled N channel system. We know, however, that the lowest groups of roots, for n = 2 or 3 say, are usually well handled by traditional spectroscopic techniques. It is therefore illuminating to relate (2.58)–(2.60) to a conventional matrix diagonalization in situations where the energies, Ein , at which tan βi (Ein ) + Kii = 0 fall into localized groups. It is readily verified by use of the expansion, tan βi () + Kii  sec2 βi βi (E − Ein )

(2.61)

that (2.58) reduces to the schematic form ⎞⎛ ⎞ ⎛ 2 K12 Z1 K13 sec β1 β1 (E − E1n ) ⎠ ⎝Z2 ⎠ = 0. ⎝ K23 K21 sec2 β2 β2 (E − E2m )  K31 K32 sec2 β3 β3 E − E3p Z3 (2.62) 9

An additional term such that

1  2 dβi 2 1  dKij sec βi Zi Zi + Zj = 1 π d π d i

ij

may be included to allow for any energy dependence of the K-matrix elements.

38

The quantum defect picture

Simple rearrangement yields the local matrix eigenvalue equation ⎞⎛ ⎞ ⎛ H12 H13 X1 E1n − E ⎠ ⎝ ⎝ H21 E2m − E H23 X2 ⎠ = 0, H31 H32 E3p − E X3

(2.63)

where  −1/2 Kij = Hj i Hij = − cos βi cos βj βi βj =−

2Ry cos βi cos βj Kij π (νi νj )3/2

1/2

Xi = sec βi βi

Zi .

(2.64)

The second line follows from (2.34); the normalized eigenvector components in the third are seen to conform exactly with (2.60). Coupled and uncoupled channel limits It is interesting follow the changes in the solutions of (2.58) as the principal quantum number increases. It is assumed that the coupled and uncoupled basis states, |i and |α respectively, are related by a frame transformation matrix with elements Uiα = i |α, in terms of which (2.58) may be written   (2.65) [K + tan β(E)] Z = U tan π μU T + tan β(E) Z = 0. The two limits occur according to whether energy separations arising from differences in the quantum defects are large or small compared with differences between the ionization limits Ii , which are reflected in differences between the accumulated phase terms  Ry βi (E) = π . (2.66) Ii − E The coupled or case (b) limit, in which Ii − Ij  Ii − E, allows the approximation ¯ tan β(E)  tan β(E)I,

(2.67)

Z = U cos π μA

(2.68)

in which case the substitution

casts (2.65) into the uncoupled form   ¯ U tan π μ + tan β(E) cos π μA = 0.

(2.69)

2.4 Coupled channels

39

The resulting eigenvalues are given by the Rydberg formula Ry , (2.70) Enα = I¯α − [n − μα ]2  T where I¯α is the average ionization energy appropriate to β¯α (E) = i Uαi βi (E)Uiα . Each solution has a single non-zero coefficient Aα , and the factors cos π μα in the explicit expansion of (2.68) mean that the normalization condition A.29 corresponds exactly with (2.34), because   T Zi2 sec2 βi βi  π −1 sec2 β¯ β¯ Aα cos π μα Uαi Uiβ cos π μβ Aβ π −1 i

iαβ

= π −1 sec2 β¯ β¯ cos2 π μα A2α = π −1 β¯ A2α = 1.

(2.71)

The uncoupled case (d) limit arises when the electronic origins in (2.70) become small compared with the separations between the ionization limits Ii appropriate to different states of the positive ion. Series of the form Ry , (2.72) Eni = Ii − [n − μ¯ i ]2 typically converge on the separate ionization limits, each with a suitably averaged quantum defect  T tan π μ¯ i = Uiα tan π μα Uαi , (2.73) α

and a single non-zero coefficient Zi . However, the relatively dense series converging on lower limit in, for example, Fig. 2.4 are subject to perturbations by interloping members of higher series. The connection between such perturbations and the resonant behaviour above the lowest ionization limit is described in the following section. A related stroboscopic effect in the level structure of species with very small positive ion rotational constants is described in Section 4.2.4. Perturbations between interpenetrating series It is illuminating to see how the auto-ionization spectrum in Fig. 2.5 relates to perturbations between the two np Rydberg series of H2 series in Fig. 2.4, which were analyzed by Herzberg and Jungen [16]. One sees from the latter plot that perturbations by the n2 = 19, 20 and 21 levels of the more widely spaced upper, np2, series, are centred around the levels n1 = 30, 33 and 37 of the lower, np0, series; and similar perturbations can arise up to the last level, n2 = 24, below the ionization limit. Equation (2.59) reduces in this case to the two-channel form

K12 K11 + tan π ν1 () = 0, (2.74) det K21 K22 + tan π ν2 ()

40

The quantum defect picture

 where νi () = Ry / (Ii − E). As emphasized above, this 2 × 2 matrix form covers the whole sequence of perturbations because each cycle of the tangent functions introduces a new principal quantum number. Herzberg and Jungen illustrate this cyclic behaviour by plotting deviations from the Balmer formula for the lower series, as a function of energy, but Lu and Fano adopt a more elegant procedure, by recognizing that each observed energy level, En may be ascribed two effective quantum numbers ν1 (n ) and ν2 (n ), according to whether it is referenced to the upper or lower ionization limit [21]. In addition (2.74) may be expressed in the form 2 = 0, F (ν1 , ν2 ) = [tan π ν1 + K11 ] [K11 + tan π ν2 ] − K12

(2.75)

where the function F (ν1 , ν2 ) repeats with period νi = ±1 over the (ν1 , ν2 ) plane.10 Moreover, the discrete roots such that En = I1 −

Ry Ry = I2 − 2 , 2 ν1 ν2

(2.76)

which lie, for the present example, in cells with ν1 = 28−40 and ν2 = 19−21, may be mapped onto a single unit cell, by the construction 1 (2.77) arctan(tan π νi ), π because tan π ν is independent of the integer part of ν. Consequently, the perturbed eigenvalues may be displayed in the form of the Lu–Fano plot shown in Fig. 2.8, in which the points are taken from the experimental data. The curves are the ‘Fano function’ given by (2.75) using the parameters in Table 2.2. The scatter arises mainly from perturbations between the present v + = 0 Rydberg series and the series terminating on higher vibrational energy levels of 11 Finally, the dashed lines in Fig. 2.8 show a construction for the quantum H+ 2. defects μσ and μπ , which is based on the special property that intersections between the Fano function and the line ν¯ 1 = ν¯ 2 = ν¯ are given by   det U t tan π μU + tan π ν¯ I = det [tan π μ + tan π ν¯ I ] (2.78) ν¯ i =

= [tan π μσ + tan π ν¯ ] [tan π μπ + tan π ν¯ ] = 0. Thus ν¯ = −μσ or −μπ , as illustrated in the diagram. The values μσ = 0.203 and μπ = −0.082 obtained by this analysis of the high n perturbations may be compared with the corresponding values of 0.197 and −0.080 derived from published Te values for the 2pσ 1 u+ and 2pπ 1 u states [22]. 10 11

To the extent that the energy dependence of the K-matrix may be ignored. The paper by Herzberg and Jungen includes an an analysis of such vibrational perturbations, with an appropriate Lu–Fano plot for v + = 1 and 2 [16].

2.4 Coupled channels

41

0.4

ν2

0.2

−μπ

0

0.2

−μσ

0.4 0.4

0.2

0

0.2

0.4

ν1

Figure 2.8 A Lu–Fano [21] plot of the perturbations between the interacting np0 and np2 series of H2 [16]. Points are the experimental data and the lines are the Fano curves determined by (2.74), using the parameters in Table 2.2.

As an alternative presentation, Fig. 2.9 plots the solutions of (2.58) and (2.74) in a form analogous to that for the auto-ionizing resonances in Fig. 2.5. To avoid the scatter, the points representing the eigenvalues were derived from the parameters in Table 2.2. The upper panel shows that the negative of the reduced effective quantum number, −¯ν1 , in the lower series experiences a sequence of jumps analogous to those of τ (E) in Fig. 2.5, except that the observations are restricted to discrete points. The underlying curves are derived from the Lu–Fano plot by using the second form of (2.76) to replace the ν2 axis in Fig. 2.8 with an energy axis. Similarly, the lower panel shows that the probabilities |X2 |2 belonging to the upper series clump into peaks analogous to those in the lower panel of Fig. 2.5, such that the sum (rather than an integral) over a clump now approximates to unity. Spin-decoupling in s configurations As a third example we consider the transition between spin-coupled Hund’s case (a) 1 1 and 3 1 states of an s configuration and the uncoupled case (e) series that converge on the 2 1/2 and 2 3/2 limits of the positive ion. The states in question are connected by the frame transformation





1 !

1 1 |1 |α 1 −1 = 3 ! =√ |2 |β 1 2 1 1 !

s1/2 |1 |sα+ β ! = , (2.79) = |sβ+ α |2 s3/2

42

The quantum defect picture

− ν1

0.4 0.2 0 0.2 0.4 300

250

350

(E − 124 000) /

0.5

n2 = 18

19

20

0.4

X 22

cm−1

21

0.3

22

0.2 0.1 0 250

300

350

(E − 124 000) / cm−1

Figure 2.9 Upper panel: perturbations between the interacting np series of H2 plotted as reduced effective quantum numbers against energy. The discrete eigenvalues are shown as points on a Fano function to guide the eye. Lower panel: upper series probabilities |X2 |2 at the various eigenvalues.

where (α, β) in the final term denote spin orientations, |+  implies = +1, and the matrix in the first line corresponds to U T in the notation of (2.47). The quantization equation   U tan π μU T + tan β (E) Z = 0 (2.80) therefore requires that   det tan π μ + U T tan β (E)

1 (t1 − t2 ) tan π μα + 12 (t1 + t2 ) 2 = 0, = det 1 (t − t2 ) tan π μβ + 12 (t1 + t2 ) 2 1

(2.81) (2.82)

where ti = tan βi (E). We assume an ‘inverted’ level structure, appropriate to HI for example, with the 1/2 limit higher than the 3/2 one. Thus " Ry β1,2 (E) = π . (2.83) ¯ I ±−E In addition the electrostatic contributions to the 1,3  energies are denoted as Ry

Eα,β = I¯ − 

n − μα,β

2 .

(2.84)

References

43

Expansions for t1 + t2 about the energies Eα or Eβ , with a similar expansion for t1 − t2 about their mean, leads to

  − 2

E − Eα = 0, (2.85) det sec ββ − E − Eβ

after assuming a common value of sec2 ββ for sufficiently low n values. Written in another way, with  = −A/2, (2.85) means that the  = 1 energy levels of configuration ns correspond to the eigenvalues of the familiar spin–orbit coupling Hamiltonian [23]

1  −A/2 E    . (2.86) H = −A/2 E 3  It is also trivial to show that     E 3 0 = E 3  − A/2,

E

3

   0 = E 3  + A/2,

(2.87)

because |3 0  = |sβπ1 β and |3 0  = |sβπ1 β converge on the 2 1/2 and 2 3/2 limits respectively. The importance of this example lies first in showing the connection with conventional spectroscopic theory. Secondly, contributions to the spin–orbit constant, A, arising from the positive ion are seen to be introduced into the MQDT quantization equation, via the ionization limits. Additional contributions from Rydberg orbitals with  = 0 would be introduced as off-diagonal quantum defects in (2.80), the effects of which would fall off as ν −3 . References [1] C. H. Greene and C. Jungen, Adv. At. Mol. Phys. 21, 51 (1985). [2] S. C. Ross. In Half Collision and Resonance Phenemena, ed. M. Garc´ıa-Sucre, G. Raseev and S. C. Ross (AIP Conf. Proc. No 225, 1991). [3] C. Jungen, ed., Molecular Applications of Quantum Defect Theory (IOP Publishing, 1996). [4] A. Messiah, Quantum Mechanics (North Holland, 1961). [5] L. I. Schiff, Quantum Mechanics (McGraw-Hill, 1968). [6] L. D. Landau and E. M. Lifshitz, Quantum Mechanics, Nonrelativistic Theory (Pergamon Press, 1965). [7] C. Cohen-Tannoudji, B. Diu and F. Laloe, Quantum Mechanics (John Wiley and Sons, 1977). [8] R. N. Zare, Angular Momentum (Wiley Interscience, 1988). [9] M. J. Seaton, Rep. Prog. Phys. 46, 167–257 (1983). [10] U. Fano, Phys. Rev. 2, 353 (1970). [11] C. H. Greene, A. R. P. Rau and U. Fano, Phys. Rev. A 26, 2441 (1982). [12] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1965).

44

The quantum defect picture

[13] M. J. Seaton, Comp. Phys. Commun. 25, 87 (1982). [14] F. S. Ham, Solid State Phys. 1, 127–92 (1955). [15] R. Guerout, C. Jungen, H. Oueslati, S. C. Ross and M. Telmini, Phys. Rev. A 79, 042717 (2009). [16] G. Herzberg and C. Jungen, J. Mol. Spec. 41, 425 (1972). [17] H. Beutler, Z. Physik 93, 177 (1935). [18] U. Fano, Phys. Rev. 124, 1866 (1961). [19] M. C. Bordas, L. J. Lembo and H. Helm, Phys. Rev. A 44, 1817 (1991). [20] C. Jungen and D. Dill, J. Chem. Phys. 73, 3388 (1980). [21] K. T. Lu and U. Fano, Phys. Rev. A 2, 81 (1970). [22] K. P. Huber and G. Herzberg, Constants of Diatomic Molecules (van Nostrand, 1979). [23] H. Lefebvre-Brion and R. W. Field, Spectra and Dynamics of Diatomic Molecules (Elsevier, 2004).

3 Ab-initio quantum defects

The previous chapter laid out the principles of multichannel quantum defect theory, showing in particular how knowledge of the quantum defects or scattering K-matrices are built into a unified theory of Rydberg spectroscopy and ionization dynamics. This chapter deals with the ab-initio determination of these quantum defects. We know from the discussion in Chapter 1 that they arise from interactions between the positive ion core and the Rydberg electron, which were seen to occur on a timescale far shorter than that of the molecular vibrations and rotations. It is therefore natural to compute them within the fixed nucleus Born–Oppenheimer approximation. Useful information on the lowest members of a given series may be obtained by traditional Hartree–Fock and configuration interaction techniques [1]. Carefully designed diffuse Rydberg orbitals are, however, required [2]. The resulting information is normally limited to the potential energy surfaces for principal quantum numbers n  4, from which it may be difficult to extract the desired forms of the quantum defects, as functions of the nuclear coordinates, particularly for polyatomic molecules. An alternative is to recognize that the distant outer parts of the Rydberg wavefunction may be expressed as Coulomb functions. Thus the ab-initio effort may be restricted to a finite volume, chosen to be large enough to allow a proper treatment of all Rydberg–core interactions [3, 4, 5, 6]. The inner and outer wavefunctions are then joined at the core boundary by a so-called R-matrix, from which the scattering K-matrix may be obtained directly, without reference to information on any potential energy surfaces. Details of these various approaches are outlined in this chapter. There is also a final section on species with large positive ion dipole moments, which require special methods. 3.1 Traditional quantum chemistry The conceptually simplest approach is to perform a high level ab-initio computation with a basis set constructed to include the diffuse orbitals. It is rarely 45

46

Ab-initio quantum defects 0.5

n=4 n=3 +

HH

0.6

−

E / au

n=2 1

Σg+

0.7

1 1

Σu+

Πu

1

0.8

0

2

4

6

8

10

12

Πg 14

R / a0

Figure 3.1 Ab-initio potential curves [7, 8, 9] for the 1 g+ (solid lines) and 1 u+ (dashed lines) states of H2 . The labels n = 1, 2, 3 and H+ H− apply to the asymptotes of the indicated curves.

possible to obtain converged results for principal quantum number n > 4, but the coordinate-dependent quantum defect functions, μα (Q), may be derived by comparison between high-level potential energy surfaces for the parent molecule and the positive ion. Thus the Rydberg formula generalizes to Vnα (Q) = V + (Q) −

Ry , [n − μα (Q)]2

(3.1)

where the subscript α includes all relevant angular momentum and symmetry labels, and the coordinate Q defines the nuclear geometry. Significant complications to this simple construction are illustrated by the potential curves for the lowest 1 g+ and 1 u+ states of H2 [7, 8, 9], which are shown in Fig. 3.1. One sees a sequence of nested bound-state curves, which belong to the Rydberg series (1sσg nσg ) 1 g+ and (1sσg nσu ) 1 u+ , both of which converge on 1 + the bound 2 g+ curves of H+ 2 . In addition the g series is perturbed by strongly avoided crossings with an ion-pair curve, marked H+ H− , which give rise to the famous double minimum states (E,F) and (G,H), etc. [10]. Seen in Rydberg terms, this ion pair curve correlates with the doubly occupied anti-bonding state, (2pσu )2 1 + g , which is the lowest member of the (2pσu npσu ) 1 g+ series, that converges on 1 + the anti-bonding 2 u+ state of H+ 2 . Figure 3.1 also shows a second, u , ion-pair state, which correlates with the singlet component of the (1sσg 2pσu ) configuration, whose spatial wavefunction retains a pure ion-pair character, within an LCAO

3.1 Traditional quantum chemistry

approximation;   1sσg (1)2pσu (2) + 2pσu (1)1sσg (2) = [1sa(1)1sa(2) − 1sb(1)1sb(2)] .

47

(3.2)

The resulting bound B 1 u+ state is therefore described by Mulliken as a ‘semiRydberg’ state, with a rather larger bond length than the norm [11]. A further complication is that certain of the higher curves in Fig. 3.1 show maxima, owing to interaction between bound and less strongly repulsive curves, which are attributed by Mulliken [11] to the low-lying members of the Rydberg series terminating on the repulsive 2 u+ state of H+ 2 . Another pronounced feature is the close similarity between the long-range behaviour of the two ion-pair curves, including their avoided crossing with curves that correlate with the H(1s)+H(2s, 2p) dissociation limit, which are marked n = 2 in the diagram. As a result the two lowest curves of both symmetries correlate with the the n = 2 limit, which will be seen to have consequences for the energy and bond-length dependences of the quantum defects. Finally, it should be noted that the high-lying curves between the n = 3 and n = 4 limits experience similar avoided curve-crossings with their ion-pair counterparts at R  35 a0 [8, 12]. Single channel quantum defects for the 1 u+ series Despite the above complications, it is instructive to examine the forms of the quantum defect functions, μλ (R), derived from the 1 u+ potential curves in Fig. 3.1 and the corresponding 1 u curves, by inverting (3.1) in the form  Ry μ (R) = n − , (3.3) V + (R) − Vn (R) with the help of the known ab-initio potentials Vn (R) and V + (R) [13, 14]. The higher npσ , nfσ and 4fπ functions in Fig. 3.2 have been truncated to eliminate distortions at longer bond lengths, arising from interaction with the 2sσ and 3pπ Rydberg curves converging on the repulsive 2 u+ potential of H2 O+ .1 The various μλ (R) functions show the expected approximate energy (or n) independence, except that the 2pσ curve deviates strongly downwards for R > 2.5, as a consequence of the ‘semi-Rydberg’ character of the B 1 u+ state, whose potential curve passes from the (1s2p) united atom limit to the n = 2 dissociation limit, via the outer curve-crossing. Thus the principal quantum number returns to its united atom value as R → ∞. The quantum defect μ2pσ (R) is therefore close to zero in both united and separated atom limits. However, as noted by Mulliken and later by 1

Such interactions are taken into account by the coupled-channel method, which is used below to handle the 1  + series. g

48

Ab-initio quantum defects 1.5

5pσ

μ(R)

1

3pσ

4pσ

0.5

2pσ (4,5)fσ 0

4fπ

(2,3)pπ 0.5

1

2

3

4

5

6

7

8

9

10

R / a0 Figure 3.2 Quantum defect functions for low members of the npσ (solid lines) and nfσ (dashed lines) Rydberg series of H2 that terminate on the 2 g+ potential curve of H+ 2 . Dotted lines indicate the npπ and nfπ defect functions.

Jungen and Atabek, the higher npσ curves are constrained by the non-crossing rule to correlate with the n − 1 dissociation limit [11, 15]. Hence the quantum defect must increase by one unit to compensate. There is no such complication in the npπ case because there is no ion-pair state with 1 u symmetry; and the 2pπ and 3pπ quantum defect functions are indistinguishable, on the scale of Fig. 3.2. The forms of these curves have various implications for the Rydberg spectrum. The difference between the equilibrium quantum defects μpσ (Re ) and μpπ (Re ) at Re ≈ 2 a0 is responsible for the perturbations in Figs 2.8 and 2.9. The strong nuclear coordinate dependence causes vibrational channel interactions, which are discussed in Section 4.3. Finally, the distortion to the 1 u+ potential energy curves due to the jump in the μpσ (R) leads to competitive ionization and dissociation, which is treated in Section 5.4. Note also that the above behaviour, whereby an anti-bonding molecular orbital takes on a Rydberg character at short bond lengths, is sufficiently common to have been granted the title Rydbergization [16]. It is particularly well displayed by the recent ab-initio R-matrix calculations on NO [17], which are discussed in Section 3.3. Again one finds unit jumps, as R increases, in the pσ quantum defect functions, but in no other μ (R) function, because the highest anti-bonding

3.1 Traditional quantum chemistry

49

orbital, denoted σ ∗ , Rydbergizes at short range to the 2pσ orbital, while the (π 4 σ ∗ ) A 2  + state correlates with ground state atomic fragments [18]. Coupled channel K-matrix elements It is impossible to obtain isolated channel quantum defect functions for the 1 g+ series of H2 except at very short bond lengths, owing to the very strong shortrange interactions with the repulsive branch of the ion-pair state. In addition the maximum in the third 1 g+ potential curve points to interactions with a second repulsive state. Fortunately, the molecular orbital descriptions of the two repulsive states, (2pσu )2 and (2pσu 3pσu ), in the binding region show that they may be treated as the first two members of the npσu series terminating on the repulsive 2 u+ state of H+ 2 . Similarly, the bound Rydberg states belong to the nsσg and ndσg series terminating on the bound 2 g+ state. Ross and Jungen have shown how the channel interactions responsible for the avoided crossings in Fig. 3.1 may be taken into account by inverting the MQDT quantization equation [19]2 det [K(R) + tan π ν (E, R)] = 0,

(3.4)

where ν (E, R) is a diagonal matrix with elements  Ry , να (E, R) = + Vα (R) − E(R)

(3.5)

in which the eigenvalues E(R) are the interacting adiabatic potential energy curves and Vα+ (R) is either the 2 g+ or the 2 u+ potential of H+ 2 [14]. Ross and Jungen actually prefer the Ham variant of (3.4), namely (see (2.28))   tan π ν (E, R) = 0, (3.6) det K(R) + A(ν, ) with the K(R) matrix elements expressed in terms of η(R) defect functions in the form Kij (R) = tan π ηij (R).

(3.7)

The problem in hand involves three channels, which are labelled as s: 2

(1σg )nsσg

p:

(1σu )npσu

d:

(1σg )ndσg ,

There is a notorious difficulty in such inversion procedures owing to the occurrence of ‘false roots’ of (3.4) with ν <  + 1. As discussed in Section 2.1.2, such problems are partly met by employing the Ham variant in (3.6). A cruder but more effective approach is to set ν = 1/2 over energy ranges where ν <  + 1/2 [20, 21].

50

Ab-initio quantum defects

E (a.u.) –0.30 Diabatic potential curves –0.40 1σu

3p 2p

–0.50

H+2

1σg

4d,s 3d,s

–0.60

2s Hazl et al.

–0.70

Guberman –0.80

1

2

3

4

5

6

7

R (a.u.) Figure 3.3 Diabatic potential curves defined by the diagonal K-matrix elements in eqn (3.6). Taken from Ross and Jungen [19] with permission.

in which (1σg ) and (1σu ) denote the 2 g+ or the 2 u+ states of H+ 2 , respectively. The system of diabatic potential curves obtained by setting the off-diagonal K-matrix elements to zero in (3.6) is shown in Fig. 3.3. The diagram, which was taken from Ross and Jungen [19], includes curves from earlier work by Hazi et al. [22] and Guberman [23]. The diagonal quantum defects that were used to determine this diagram were obtained by a least-squares fitting procedure, which was constrained to ensure smooth functional forms. The limiting values of ηss (R) and ηdd (R) were required to ensure the correct He atomic energies at R = 0. There is, however, no similar constraint on ηpp (0) because the (1σu )2 configuration correlates with the doubly excited (2s)2 1 S state of He, which lies in the auto-ionization continuum. It was found convenient to employ constant interaction terms ηsp and ηdp over the range R  5 a0 and to continue them to zero for R  7 a0 . Given these forms for the main interactions, there was found to be sufficient information to determine a weak linear energy dependence for ηss (R) and a small off-diagonal s–d interaction term ηsd (R) over the interval 1.4 < R < 4.0 a0 . The resulting η (R) functions, which are shown in Fig. 3.4, reproduce the three lowest 1 g+ potential curves in Fig. 3.1 to within 8 cm−1 . As a more severe test, an MQDT calculation of the vibronic eigenvalues of the double minimum EF, GK ¯ states is comparable in accuracy with the best state-by-state calculations and HH by traditional spectroscopic methods [24]. This model will also be employed in Chapter 5 to illustrate the MQDT treatment of competitive auto-ionization and

3.1 Traditional quantum chemistry

51

0.6 0.4

ηdd

ηdp

0.2

ηsp

0.0

ω

ηsd –0.2

ηss

–0.4

2p

ηpp

–0.6

3p

–0.8 0

1

2

3

4

5

6

7

8

R (a.u.) Figure 3.4 Forms of the derived η (R) functions over the range 0 < R < 8 a0 , with the energy-dependent term, ηss (R), evaluated at the ionization limit. The ηpp (R) function and the points, from an earlier study by Guberman [23], have been shifted down by one unit to avoid congestion. Taken from Ross and Jungen [19], with permission.

predissociation and the related phenomenon of dissociative recombination; ∗ e + H+ 2  (H2 )  H + H,

in which (H2 )∗ denotes a Rydberg state. Polyatomic molecules Traditional ab-initio techniques can, of course, be extended to the Rydberg states of polyatomic molecules, by inclusion of carefully designed diffuse Rydberg orbitals [2]. Particular attention has been given to the water molecule, in view of extensive experimental information [25]. Figure 3.5 shows that the Rydberg system consists of an intricate pattern of interacting ‘bent’ and ‘linear’ states, which converge on ˜ 2 B1 states of H2 O+ [26]. Moreover, this ˜ 2 A1 and A the Renner–Teller coupled X latter coupling is known to cause predissociation throughout the Rydberg spectrum [25]. It is therefore highly desirable to obtain the ab-initio quantum defect surfaces to model the various competing fragmentation pathways (see Chapter 5). Unfortunately, there are serious problems in extending the above inversion procedures to polyatomic systems in view of the higher dimensionality of the nuclear configuration space [27]. The long-term aim is to employ one of the following ab-initio R-matrix procedures, that compute the K-matrix directly, without recourse to the

52

Ab-initio quantum defects A

A

B

B

Figure 3.5 Ab-initio bending potentials for the water molecule. Taken from Hirst and Child [26] with permission of Taylor and Francis.

potential energy curves. However, at the time of writing results are not yet available for species with the complexity of H2 O. 3.2 Constrained ab-initio wavefunctions Ab-initio R-matrix methods are designed to perform high-level electronic structure calculations for the inner parts of the Ne + 1 electron wavefunction, within a radius, r = a, which is large enough to allow a proper treatment of the Coulomb, exchange and electron correlation interactions between the Ne electron core and the excited Rydberg electron [3, 4, 5, 6]. Outer parts of the appropriate Coulomb functions are then joined to single-electron inner ‘continuum components’ by matching radial logarithmic derivatives at a suitable core radius. The first advantage is that the eigenvalues of the constrained inner problem readily span energies up to and well beyond the ionization limit, thereby including the energy ranges that are encountered in atomic and molecular physics. Secondly, the matching procedure allows a direct calculation of the scattering K-matrix, as a function of nuclear geometry, without the need to extract this information from families of interacting potential energy functions.

3.2 Constrained ab-initio wavefunctions

The total Ne + 1 electron wavefunction is expressed in the form $ #  1 akα α (ξ ,ˆr )ukα (r) + bv χv (ξ ,r) ,  (ξ ,r) = A r kα v

53

(3.8)

where ξ includes the coordinates of the inner electrons and the angular variables, rˆ , of the Rydberg electron, and r is the Rydberg radial variable. The ‘target functions’ α (ξ ,ˆr ) are properly symmetrized combinations of positive ion electronic states and body-fixed angular momentum projections of the Rydberg angular momentum, while the radial components, ukα (r), are families of ‘continuum functions’ in the radial variable of the excited electron, centred on the molecular centre of mass, whose forms vary from one variant of R-matrix theory to another. The suffix k counts the number of nodes in the radial continuum function ukα (r). Finally, the functions χv (ξ ,r) are compact N + 1 electron terms designed to represent the valence states. The continuum orbitals employed in diatomic applications are often obtained as numerical solutions of an appropriate isotropic model, because it is feasible to evaluate the electronic structure integrals numerically. Expansions in terms of Gaussian functions [28, 29] are however preferred for polyatomic molecules, to facilitate the integral evaluation. The details of computer codes designed to handle these problems have been published by Morgan et al. [30]. As an illustrative example, Rabad´an and Tennyson employed first four and later twelve target functions α in (3.8), which were taken as configuration interaction (CI) expansions involving the N electrons of the positive ion [31, 32]. The orbitals were obtained from a self-consistent field (SCF) calculation in a basis of (10σ, 6π, 1δ, 1φ) slower-type orbitals (STOs) centred on each nucleus [33]. A total of 16 target molecular orbitals (9σ, 5π, 1δ, 1φ) were determined by SCF calculations on several of the target states. The CI wavefunctions of NO+ were constructed with a six-electron frozen core, with the remaining eight electrons allocated to four active space (CAS) orbitals. The numbers of configuration state functions (CSFs) and resulting vertical excitation energies for the four target states, at the bond length R = 2.175 a0 , are given in Table 3.1. Two types of inner valence configuration, χv , were constructed as N + 1 electron determinants built on the target orbitals. In one set the final electron was placed in a target virtual orbital in order to allow for high  effects near the nucleus, while the second set was designed to permit interaction with the continuum orbitals. The forms of the continuum orbitals, ukα , themselves differ according to the method employed to construct the R-matrix, details of which are given in later sections. In brief, the Wigner approach employs basis functions with known log-derivatives at the boundary, in terms of which the N + 1 electronic Hamiltonian is diagonalized in the basis {, χ} to obtain a set of

54

Ab-initio quantum defects

Table 3.1 Experimental and calculated vertical excitation energies (in eV) for the first four states of NO+ at R = 2.175 a0 . State X 1+ a 3+ b 3 w 3

CSFs

Eexp [34]

Ecalc [31]

76 70 126 67

0.00 6.47 6.92 7.56

0.00 6.73 7.00 7.97

ab-initio eigenvalues, En , and Rydberg orbital boundary amplitudes  akα,n ukα (a), ψnα (a) =

(3.9)

k

(a). Alternatively, in the variational, or eigentogether with their derivatives ψnα channel, approach the N + 1 electronic Hamiltonian diagonalization at a fixed energy, E, serves to determine a set of eigen-log-derivatives, which are common to all channels, plus sets of Rydberg orbital boundary amplitudes.

3.3 The R-matrix matching procedure Before exploring different methods, we concentrate on general aspects of the matching procedure at the R-matrix boundary, r = a. The notation is simplified by treating the boundary amplitudes ψnα (a) as elements of a column vector, ψ n (a). The vector solutions, ψ n (r), in regions inside but close to the boundary may be taken as close-coupled eigenfunctions of a one-electron radial Hamiltonian with elements3  2 2 d h ¯ (0) ˆ αβ (r) = − ˆ ,r)β dξ δαβ + α H(ξ H 2m dr 2 =−

h¯ 2 d2 δαβ + Vαβ (r), 2m dr 2

(3.10)

ˆ , r) is the residual Ne + 1 electronic Hamiltonian, after taking out in which H(ξ ˆ (0) (r) must be the kinetic energy of the Rydberg electron. The resulting operator, H augmented by a diagonal Bloch operator [35]   2 h ¯ d ˆ = L δ(r − a) a + b I, (3.11) 2ma dr 3

The term h¯ 2 /2m is included for generality. It would be replaced by unity or 1/2 when using the natural or atomic unit systems, respectively.

3.3 The R-matrix matching procedure

55

when imposing the boundary condition aψ n (a) = −bψ n (a)

(3.12)

ˆ then has the necessary symmetry to ensure at r = a. It is easy to see that H(0) + L real eigenvalues and orthogonal eigenvectors, because the identity

   a d2 d b φq (r)dr φp (r) − 2 + δ(r − a) Tpq = + dr dr a 0  a b φp (r)φq (r)dr + φp (a)φq (a), (3.13) = a 0 shows that Tpq = Tqp for any pair of regular functions φp (r) and φq (r), regardless of whether or not they satisfy (3.12). Sections 3.4 and 3.5 show how (3.10)–(3.11) may be combined with the ab-initio outputs En and ψ n (a) to construct an R-matrix, R (E), at any chosen energy, such that ψ E (a) = aR(E)ψ E (a),

(3.14)

where the factor a ensures that the resulting matrix is dimensionless. The matching procedure is completed by expressing the outer wavefunctions in terms of the boundary values of the Coulomb basis functions, fα (, a) and gα (, a), which were introduced in Section 2.2. Note that the energy  in this context is measured in Rydberg units, with respect to the vertical ionization energy appropriate to the channel in question,  = (E − Vα+ )/Ry . The requirement is simply that the outer wavefunction, given by (2.35), namely ψ E = f − gK,

(3.15)

  f − gK = aR f − g K ,

(3.16)

&−1 % & % f − aRf . K = g − aRg

(3.17)

should also satisfy (3.14). Thus

which rearranges to

Provided the core radius is chosen to be sufficiently large compared with the range of Rydberg–core interactions, the a dependence of the residues of R, in (3.28) below, will be found to be matched to that of the basis functions fα (, a) and gα (, a), and their derivatives, fα (, a) and gα (, a), in such a way that the K-matrix is strictly independent of a. Quantum defects for the various series may be obtained by diagonalizing the blocks associated with different target states.

56

Ab-initio quantum defects

0.2

sσ

Quantum defect

0.1

fσ

gσ

0.0

−0.1

pσ

dσ

−0.2

−0.3

−0.4 0.00

0.05

0.10

0.15

1/

0.20

0.25

ν2

Figure 3.6 An Edlen plot of the σ quantum defects for the series converging on the ground state X 1  + of NO+ . Filled and open points are experimental values [36] and those reported by Rabad´an and Tennyson [31], respectively. The continuous lines are the MQDT results. Taken from Hiyama and Child [17], with permission.

The corresponding eigenvectors determine the angular momentum composition of the Rydberg series, while the residual interactions between the 2 blocks  are respon+ 1 + sible for configuration interaction, between for example the  g nsσ, ndσ g 2 +  series in Fig. 3.1 and the partially ionic u 2pσ 1 g+ state. It is usually found that the energy dependence of the R-matrix combines with that of the basis functions to leave only weakly energy-dependent quantum defects. Exceptions, which can arise from perturbations due to very weak poles, associated with compact valence states, are discussed in Section 3.6. Illustrative application to NO Illustrative results from an R-matrix/MQDT study of NO [17] are shown in Figs 3.6 and 3.7. Those in Fig. 3.6 show the energy variation of the resulting quantum defects of the σ Rydberg series terminating on the ground, 1  + , target state of NO+ at the bond length R = 2.175 a0 . The MQDT values, which are shown as continuous lines, are obtained at arbitrary quite widely spaced energies, from which the electronic eigenvalues may be obtained by an easy iteration for solutions of the

3.3 The R-matrix matching procedure 0.4

1 +

Σ

μ

0.2

pσ

0.4

sσ

0.2

0.0

dσ

−0.4 2.0

1.5

μ

0.2

2.5

3.0

pσ

3 +

Σ

sσ

0.0

Π

−0.2

dπ

−0.4

pπ 2.0

1.5

0.4 0.2

3

2.5

3.0

Δ

0.0

dσ

−0.2 −0.4 1.5

3

0.0

−0.2

0.4

57

dδ

−0.2 −0.4

2.0

2.5

3.0

1.5

R / a0

2.0

2.5

3.0

R / a0

Figure 3.7 Bond length variations in the quantum defects for series with total symmetry 2  + , converging on the target states indicated in the upper left hand  corner of each panel. The data are given for ν = Ry /(E + − E) = 3, where E + = V + (R) for the target state in question. Taken from Hiyama and Child [17], with permission.

equation ν = n − μ(ν).

(3.18)

The open circles are eigenvalues obtained by Rabad´an and Tennyson, who used a shooting method to match the R-matrix log-derivatives on the boundary to properly decaying functions in all 24 channels at infinity [31]. They are seen to lie on or very close to the MQDT-determined lines. However, the solid points, representing experimental values [36], are systematically larger than the calculated ones, a discrepancy that is reduced but not eliminated by increasing the number of target states from four to twelve [32, 37]. The largest discrepancy implies an error of order 2000 cm−1 for the energy of the 3sσ A2  state – an error that decreases as ν −3 for higher members of the series. The quantum defects are seen to vary approximately linearly with energy,  = ν −2 , over the physically observable region, apart from a sharp jump in the sσ quantum defect as the dσ channel changes from strongly closed, ν <  + 1,

58

Ab-initio quantum defects

to normal, which is a well-known consequence of core polarization [38].4 The composition of the corresponding orbitals, obtained by diagonalization of the 1  + block of the K-matrix show a sudden change from almost pure s character at n = 2 to almost 50% sσ and 50% dσ character for n  3. By contrast there is minimal -mixing for the p orbitals, partly because NO+ appears as approximately homonuclear at short bond lengths [18],5 and partly because the f orbitals that might mix with p by core polarization have minimal penetration into the core. The reported results also show remarkable strong mixing between the close lying fπ/gπ and fδ/gδ orbitals [17]. Further information on the quantum defects for states with 2  + symmetry is shown in Fig. 3.7 by plotting their variation with bond length at energies corresponding to ν = 3 [17]. The label in the corner of each panel indicates the symmetry of the target state. The most interesting feature is the insensitivity of all but one of the μ (R) functions to changes of R, while the pσ defect function increases by almost one unit as R increases from 1.7 to 3.0 a0 – a pattern that is remarkably similar for all target states [17]. The explanation lies in Rydbergization [16] of the highest anti-bonding, pσ ∗ , orbital to form the 3pσ Rydberg orbital, which leads to a jump in the pσ quantum defect comparable to that seen for the higher μnpσ (R) functions of H2 in Fig. 3.2. Minor irregularities in many of the quantum defect plots and the large fluctuation in the pπ function in the 3  panel indicate valence state perturbations, which are discussed in Section 3.6. 3.4 The Wigner–Eisbud R-matrix 3.4.1 R-matrix derivation As mentioned, variants of R-matrix theory differ according to the chosen behaviour of the continuum basis functions. The choice in the simplest Wigner–Eisbud formulation, is that radial components ukα (r) in (3.9) should satisfy the same logderivative boundary condition [39] aukα (a) = −bu kα (a),

(3.19)

in all channels, usually with b = 0.6 To obtain the proper joining condition at the boundary, the inner eigenfunctions, ψ n (r) satisfy the matrix equation   (0) (3.20) H (r) + L − En I ψ n = 0, 4 5 6

Properly adapted Coulomb basis functions are required to handle this transition (see Section 2.2.1). Rabad´an and Tennyson [31] report strong pd mixing in orbitals with n ∼ 10, by propagating the R-matrix under the influence of the small NO+ dipole from a = 15 a0 to a = 250 a0 . In the previously cited NO application [31] they were taken taken as the numerical wavefunctions of a spherical Coulomb field, with angular momenta  = 0 − 5 and energy up to 5 au.

3.4 The Wigner–Eisbud R-matrix

59

where   d h¯ 2 a + b Iδ(r − a), L= 2ma dr while the outer solutions ψ E (r), at arbitrary energies, satisfy   (0) H (r) − EI ψ E = 0,

(3.21)

(3.22)

without the Bloch operator, because there is no prescribed condition at r = a. The symmetry implied by (3.13), between components of ψ E (r) and ψ n (r), may be used to determine the components of the expansion  ' ! (3.23) ψ n (r) ψ Tn ψ E . ψ E (r) = n

! ' Notice that since  n and  E are column vectors, the notation ψ TE A ψ n implies a sum over the channel indices α and β;  !  a ' T ψEα (r)Aαβ ψnβ (r)dr. (3.24) ψE A ψn = αβ

0

Thus (3.20) and (3.22) combine to yield ! ' ! ' 0 = ψ TE H(0) (r) + Ln − En I ψ n = ψ Tn H(0) (r) + Ln − En I ψ E ' ! ' ! = (E − En ) ψ Tn ψ E + ψ Tn Ln ψ E (3.25) ' !   h¯ 2 T = (E − En ) ψ Tn ψ E + ψ n (a) aψ E (a) + bψ E (a) , 2ma which rearranges to   ! ' T h¯ 2 a ψ Tn (a)ψ E (a) − ψ T n (a)ψ E (a) ψn ψE = , (En − E) 2ma

(3.26)

because ψ Tn (a)b = −aψ T n (a). It follows, after substituting in (3.23) and collecting the terms in ψ E (a) and ψ E (a) that   (3.27) ψ E (a) = R(E) aψ E (a) + bψ E (a) , where R(E) =

h¯ 2  ψ n (a)ψ Tn (a) , 2ma n (En − E)

(3.28)

60

Ab-initio quantum defects

in which the rows and columns of the square matrix ψ n (a)ψ Tn (a) belong to the channels α, β, etc. Written in terms of components, the elements of the R-matrix are therefore given by Rαβ (E) =

h¯ 2  ψnα (a)ψnβ (a) . (En − E) 2ma n

(3.29)

Equation (3.27) clearly reduces to (3.14) if the radial components ukα (r) in (3.19) are defined with b = 0 in all channels. The advantages and disadvantages of working in such a basis are discussed next.

3.4.2 Buttle correction The restriction to a common surface log-derivative means that the Wigner–Eisbud method is normally employed in situations where the electronic structure integrals can be evaluated numerically [31, 32]. There is, however, the serious defect that any Wigner–Eisbud basis, no matter how large, is incomplete on the boundary because it contains only a single log-derivative. The correction proposed by Buttle [40] may be understood in the common case with ψ n (a) = 0. We note first that many of the pole terms in the connection equation # $ N h¯ 2  ψ n (a)ψ Tn (a) ψ E (a) = (3.30) ψ E (a) 2ma n=1 (En − E) are required simply to cancel the disappearance ψ E (a) at the eigenvalues, E = En , leaving finite values of ψ E (a) at E = En . 2 (a), may be used to Secondly, the magnitudes of the diagonal residues, ψnα organize most of the poles into orderly progressions. Figure 3.8 shows, as an example, the energies of the pσ poles, belonging to the ground (X1  + ) state of NO+ , in a 24-channel study of the 2  + states of NO [17]. Notice that only two of the levels lie in the bound electronic energy region below the ionization limit. One also sees only ten well-converged (1  + )pσ poles, compared with the complete set assumed by (3.23). On the other hand, despite their limited number, the spacings between the higher ab-initio eigenvalues, En , are seen to correspond with the dashed eigenvalues of the reference particle-in-a-box system. This correspondence provides the clue to understanding the Buttle correction. The argument is most conveniently illustrated for s-wave plane wave components, but it is readily extended to Coulomb waves and to  = 0. If E0 denotes the bottom of the box, a reference set of particle-in-a-box eigensolutions, subject to

3.4 The Wigner–Eisbud R-matrix

Ab initio

Box poles

p-wave poles

E / au

2

61

0

Figure 3.8 The energies of poles assigned to pσ channels of the lowest NO+ target state, obtained by an ab-initio R-matrix/MQDT study of NO [17] at the bond length R = 2.178 a0 . The dashed levels are those for the p waves of a particle in a spherical box. Energies are measured from the lowest vertical ionization limit.

the boundary condition ψn (a) = b = 0, are taken as " ψn0 (r)

=

2 sin kn r, ψn0 (a) = a

En0 = E0 +

"

2 a

(n + 1/2)2 π 2h¯ 2 , 2ma 2

(3.31)

in which the superscripted quantities En(0) , etc., apply to the reference system. Thus the reference R-matrix is given by 2 ∞  0 2  ψ (a) h ¯ n R (0) (E) = . 2ma n=0 En0 − E

(3.32)

Written in another way, with E = E0 + k 2h¯ 2 /2m, and ψE0 (r) = A sin kr, R (0) (E) conforms to the identity [41] ∞ 2  1 ψE0 (a) tan ka = 2 = ;

0 2 ka a n=0 kn − k 2 aψE (a)

kn =

(n + 1/2)π , a

(3.33)

where k 2 = 2m(E − E0 )/¯h2 . Hence, if the spacing between the two progressions coincide for n > N , a judicious choice of E0 allows completion of the diagonal

62

Ab-initio quantum defects

terms in (3.29) in the form 2 N  h¯ 2  ψn0 (a) + R Butt , R(E) = 2ma n=0 En − E

(3.34)

where R

Butt

−1

= (ka)

N 1 2  tan ka − 2 . 2 a n=0 kn − k 2

(3.35)

The same idea may be extended to higher partial waves by replacing the sine functions with spherical Bessel functions [42] ψ (kr) = krj (kr),

(3.36)

with the eigenvalues kn2 for b = 0 given by aψ (ka) = −bψ (kn a).

(3.37)

The resulting Buttle correction then becomes 1  [ψ (kn a)]2 ψ (ka) − , aψ (ka) a n=0 kn2 − k 2 N

Butt = R

(3.38)

where k 2 = 2m(E − E 0 )/¯h2 again makes allowance for the energy off-set E 0 . Alternatively, ψ (kr) may be taken as Coulomb functions. 3.5 Variational R-matrix theory 3.5.1 Standard method Variational R-matrix theory eliminates the need for a Buttle correction by solving directly for the energy dependence of the boundary log-derivative, in a basis which is assumed to be complete, both internally and on the boundary r = a. To simplify the argument, the main discussion is restricted to the single channel case along the lines suggested by Jackson [43] and Manolopoulos and Wyatt [44]. The multichannel generalization is outlined at the end of the section. The single channel wavefunction ψ(r) satisfies the Schr¨odinger equation   Hˆ − E ψ(r) = 0, (3.39) and the log-derivative, Y (r), which is defined by ψ (r) = Y (r)ψ(r),

(3.40)

3.5 Variational R-matrix theory

63

is related to the R-matrix in the dimensionless form R(a) = a −1 Y −1 (a).

(3.41)

Moreover ψ(r) must vanish at r = 0 and the homogeneity of (3.39) and (3.40) allows a normalization to unity on the boundary, ψ(a) = 1, except on an energy set of measure zero at which ψ(a) = 0. In other words, the variations of ψ(r) can be restricted to the interior region. To proceed further, (3.39) and (3.40) are combined in the form δ(r − a)Y (r)ψ(r) =

 2m  ˜ H − EI ψ(r), 2 h¯

(3.42)

in which the modified Hamiltonian

  h¯ 2 d d h¯ 2 d2 ˜ + V (r) H =H+ − δ(r − a) δ(r − a) = − 2m dr 2m dr 2 dr

(3.43)

is Hermitian on the interval 0  r  a by virtue of the surface term. An approx˜ imation Y˜ (a) to the exact value Y (a) is now defined by substituting ψ(r) = ˜ ψ(r) + δψ(r) in (3.42), multiplying by ψ(r), integrating over the interval [0, a], and remembering that ψ(a) = 1:   2m a ˜  ˜ ˜ ψ(r) H − E ψ(r)dr. (3.44) Y˜ (a) = 2 h¯ 0 We know, however, that the true solution ψ(r) satisfies (3.39), that the operator H˜ − E is Hermitian, that ψ(a) = 1 and that δψ(a) = 0. Hence, after evaluating the surviving surface derivative term,  a    2m

˜ ˜ δψ(r) H − E δψ(r)dr Y (a) = ψ (a) + 2 h¯ 0 = Y (a) + O(δψ 2 ).

(3.45)

In other words Y (a) is an extremum of Y˜ (a), which is the desired variation principle. ˜ Suppose now that ψ(r) is expanded in a basis, ˜ ψ(r) =

N 

uj (r)cj = uT (r)c.

(3.46)

j =1

The extremal values of Y˜ (a), subject to the constraint ψ(a) = uT (a)c = 1, are given by the stationary points of the functional   F (c, λ) = cT Kc − λ uT (a)c − 1 ,

(3.47)

(3.48)

64

Ab-initio quantum defects

where the matrix K has elements7 )  a( 2m

ui (r)uj (r) + ui (r) 2 [V (r) − E] uj (r) dr Kij = Kj i = h¯ 0  2m  = 2 H˜ ij − ESij . h¯ The required extremum of F (c, λ) is given by

(3.49)

∂F ∂F = 2Kc − λu(a) = 0 and = uT (a)c − 1 = 0, (3.50) ∂c ∂λ which rearrange to the following unique expressions for the optimal values of λ and c 1 λ = , T 2 u(a) K−1 u(a)

and c =

K−1 u(a) , u(a)T K−1 u(a)

(3.51)

provided that det K = 0. Consequently aR(a) =

1 1 = T = u(a)T K−1 u(a) Y (a) c Kc

  h¯ 2 ˜ − E −1 u(a). (3.52) u(a)T H 2m It remains to express the inverse K-matrix in terms of the solutions of the generalized N × N matrix eigenvalue equation appropriate to the basis expansion (3.46);   ˜ − En S cn = 0, H (3.53) =

˜ n = En and cTn Scn = 1. Thus so that cTn Hc h¯ 2  cn cTn 2m n=1 (En − E) N

K−1 =

(3.54)

and R(a) =

N N h¯ 2  ψn (a)ψn (a) h¯ 2  (u(a)T cn )(cTn u(a)) = , (En − E) 2ma n=1 2ma n=1 (En − E)

(3.55)

which is the required single channel variational R-matrix expression, at energies arbitrarily close to but not equal to the eigenvalues En , where det(K) = 0. The derivation also breaks down on the energy set of measure zero, where ψ(a) = 0 and hence R(a) = ψ(a)/ψ (a) = 0. 7

K in this context, which is sometimes denoted as  [5], should not be confused with the MQDT K-matrix.

3.5 Variational R-matrix theory

65

In view of the formal similarity between (3.55) and the Wigner–Eisbud expression (3.28) it is important to emphasize the distinction between the two methods. The essential difference is that the incompleteness of the Wigner basis means that the sum in the latter equation is inevitably seriously inaccurate over finite energy ranges around all the roots of ψ(a) = 0 (assuming a basis with b = 0). This weakness can be eliminated by a Buttle correction, but it never arises for an adequately converged variational calculation (see below), because the variational basis contains functions with a variety of boundary log-derivatives. Moreover the variational principle is designed to optimize the accuracy of R(a), within the given basis, over the whole energy region, apart from on the sets of measure zero that were discussed above. Some comments on the practical application of the two methods are in order. The first question concerns convergence of the solutions of (3.53), with regard to the number of basis functions. Accurate representations of R(a), by either method, are restricted to the energy range spanned by the converged eigenvalues En , whose wavefunctions must have vanishing surface derivatives ψn (a) = 0 in order that the poles in (3.55) should replicate the behaviour of ψn (a)[ψn (a)]−1 . Differences between the two methods arise over the treatment of the higher ‘unconverged’ solutions. They are simply discarded in the Wigner–Eisbud method, because the Buttle correction relies on merging the finite set of converged pole terms with those of the inverse log-derivative of a suitable reference function ψn0 (r). By contrast all N terms in (3.55) must be retained in the variational method, because the higher ‘unconverged’ solutions contain the information from the surface derivatives of the basis functions ui (r), which are essential for controlling the proper behaviour of R(a) at energies between the eigenvalues. An illustrative example is given in Section 3.5.3. This section concludes with a brief outline of the generalization to M channels. Each of the channel functions ψ α (r) is expanded in its own set of, say, N basis functions uαi (r), which must again contain a variety of surface values and surface derivatives.8 The obvious analogue of (3.29) takes the form NM h¯ 2  ψnα (a)ψnβ (a) Rαβ (a) = , 2ma n=1 (En − E)

(3.56)

in which En are the eigenvalues of the full NM × NM extension of (3.53) and ψnα (a) = α |ψ(a) =

N 

(α) u(α) i (a)cin

(3.57)

i=1 8

The number N may vary from one channel to another, but a common number is assumed, to simplify the exposition.

66

Ab-initio quantum defects

is the surface projection of the nth eigenfunction in channel |α. Again it is essential to include all NM terms in the R-matrix sum. Truncated Gaussian basis This variational method is ideally suited for the ab-initio R-matrix treatment of polyatomic molecules for which the electronic structure integrals are optimally evaluated in a truncated Gaussian continuum basis centred on the molecular centre of mass. Sets of suitable exponents are chosen by optimizing fits to the Bessel or Coulomb basis sets over a range 0 < r < r0 , which encloses the intended R-matrix radius [28, 29, 45]. The required variety in the boundary values and derivatives is obtained by truncating the basis elements at a common radius r = a, which means in practice (a) including the surface term in (3.43), in order to symmetrize the kinetic energy matrix elements; and (b) subtracting the tail contributions to the long-range one electron integrals, which can be evaluated analytically [46]. Problems of over-completeness are eliminated by diagonalizing the overlap matrix and omitting combinations with eigenvalues less than a prescribed value of order 10−7 Computer codes to implement this scheme are given by Morgan et al. [30].

3.5.2 Eigenchannel method The eigenchannel variational approach is similar to the ‘standard’ method in employing a basis with inhomogeneous boundary conditions. It differs however in working with a set of energy dependent eigenfunctions and eigenvalues of the b term of the Bloch operator, instead of the log-derivative matrix Y(a). The multiplicity of eigenvalues bt (E) provides sufficient flexibility to ensure completeness on the boundary. The theory was first set up in an iterative form by Fano and Lee [47], but this account follows a close-coupled exposition of a non-iterative formulation from Greene [48] and Le Rouzo and Raseev [49]. Applications to the spectroscopy of complex atoms have been reviewed by Aymar et al. [5]. As indicated above the difference from the standard approach is that one treats the equation   (0)   h¯ 2 Iδ(r − a) aψ (r) + b(E)Iψ(r) , H − EI ψ(r) = − 2ma

(3.58)

as an eigenvalue problem for b(E), such that solutions satisfy the same boundary log-derivative equation ψ (a) = −[b(E)/a]ψ(a)

(3.59)

3.5 Variational R-matrix theory 0.6

67 (a)

U nl o

0.3 0.0

–0.3

–0.6 0.6 (b) U nl o

0.3 0.0 –0.3

–0.6 0.0

4.0

8.0 12.0 r (au)

16.0

20.0 r0

Figure 3.9 s-wave continuum orbitals of Ba+ (a) in the closed channels and (b) in the open channels, as used in an eigenchannel R-matrix calculation with a = r0 = 20 a0 . Taken from Aymar et al. [5] with permission.

in all channels. Hence the term ‘eigenchannel’ method. The wavefunctions in each channel |α are expanded in an basis set  (α) ui (r)ci(α) , (3.60) ψ (α) (r) = i

such as the truncated Gaussian basis with inhomogeneous boundary conditions. Alternatively, in cases with sufficiently high symmetry that the electronic structure integrals can be evaluated numerically, it proves convenient to employ numerically determined continuum orbitals of the forms in Fig. 3.9 [5]. The large set of ‘closed’ functions in panel (a) serves to converge the ab-initio calculation, while the much smaller set of ‘open’ functions in panel (b) provides the necessary flexibility on the boundary. The second set must be relatively small (one–three per channel), to avoid problems of over-completeness, and it is desirable that their mean energies are close to the fixed energy E of interest.

68

Ab-initio quantum defects

The allowed eigenvalues bt (E) in (3.58) arising from the expansion (3.59) are determined by the generalized eigenvalue equation Kc = −a −1 cb, where [5]9 2m Kij = 2 h¯

ij =

 0

a

(3.61)

uTi (r)(H − EI)uj (r)dr − uTi (a)u j (a) =

uTi (a)uj (a).

 2m  ˜ Hij − ESij 2 h¯ (3.62)

The matrices K and  are symmetric, in the former case by virtue of (3.13), and αβ β  is block-diagonal in the channel index, ij = uαi (a)uj δαβ . With the solution vectors labelled ct , the normalization may be chosen such that cTs ct = δst ,

(3.63)

which also ensures the orthonormality of the surface amplitude vectors in (3.60), namely  TEs (a) Et (a) = δst .

(3.64)

The connection with the R-matrix is that each eigenvalue bt (E) has associated channel amplitudes  (α) ψt(α) (a) = ui (a)cit(α) (3.65) i

and, in view of (3.59), Rαβ (a) = −

 ψt(α) (a)ψt(β) (a) bt (E)

t

.

(3.66)

The relationship with the standard variational representation (3.56) is discussed by Robicheaux [50]. 9

When written in terms of the N + 1 electronic wavefunctions, n of (3.8), the K matrix elements (which are written by Aymar et al. as ij = −Kij ) appear as a combination of volume and surface integrals     1 ∂ rψ j 2m i (H (0) − E)j dV − a Kij = Kj i = 2 i d ∂r h¯ V  r  2m  ESij − Hij − Lij , h¯ 2 where S is the overlap matrix and H and L are the matrices of the Hamiltonian and the Bloch operators. The matrix  consists of surface overlap integrals [5].  i j d.

ij = =



3.5 Variational R-matrix theory

69

The nature of the solutions of (3.61) requires some discussion because the number of non-trivial solutions is fixed by the rank of the matrix , which is typically small. For example, in the case of a single channel,  takes the form of a rank-one outer product  = u(a)uT (a),

(3.67)

with a single non-zero eigenvalue, λ = uT (a)u(a). Since  is block diagonal, the same argument applies in each channel. Hence there are M non-zero eigenvalues for an M channel system, and (3.61) has rank M. It follows that diagonalization of the  matrix provides an orthogonal transformation ˜ = XT KX, K

˜ = XT X, 

c˜ = XT c

such that (3.61) goes over to oo

o

˜ oo ˜ oc ˜ K c˜ K (b/a) =− ˜ co K ˜ cc c˜ c K 0

0 0



(3.68)

c˜ o c˜ c

,

(3.69)

which may be folded, by the substitution cc = −(Kcc )−1 Kco c0 , to the reduced form

 cc −1 co  o ˜ oo c˜ o , ˜ ˜ oc K ˜ ˜ oo − K c˜ = −(b/a) (3.70) K K oo

˜ is diagonal, with elements λα . where  The method of solution is similar to that for the standard variational approach ˜ cc )−1 in in that the analogue of (3.54) provides a spectral representation for (K terms of the energy eigenvalues of the ‘closed block’ transformed Hamiltonian matrix, which will typically have dimension (N − 1) × M. The resulting M × M generalized eigenvalue equation (3.70) is then solved for the allowed values of bt (E) at any given energy. The procedure is computationally easier but algebraically more awkward in a basis of the form in Fig. 3.9, because the ij matrix elements vanish unless both ui (r) and uj (r) belong to the ‘open’ set in panel (b). There is therefore an automatic initial separation into Kcc or Koo matrices according to whether they couple basis functions within sets (a) or (b), respectively. In other words the tilde symbols can be dropped from (3.70). The orthogonality of the basis functions within the large closed set also brings the computational advantage that the spectral representation for Koo is based on the solutions of an ordinary rather than the generalized eigenvalue problem in (3.53) At this stage (3.61) has been reduced to the form Kco = −(b/a)co ,

(3.71)

70

where

Ab-initio quantum defects

 −1 co  K . K = Koo − Koc Kcc

(3.72)

The dimensions of this system are fixed by the total number of closed basis functions (of order between 2M and 3M). Finally (3.71) can be reduced to the K analogue of (3.70) by concentrating the elements of  into an M × M diagonal block, and folding the resulting equation. Another slightly awkward feature is that one must ensure that at least one open basis function in each channel is chosen such that its diagonal Koo matrix element is close to zero for the energy range in question. Otherwise the R-matrix will lack the essential poles required to represent zeros of the surface derivatives. Basis sets of this latter type are used in Section 5.4 to handle the competition between ionization and molecular dissociation. The present ab-initio coupling to the core is replaced by an MQDT treatment of the ionization channels and the dissociation channels are handled by using analogues of (3.61) and (3.62). The structure of the combined equations makes it convenient to combine the results from separate MQDT computations based on sets that are either ‘open’ or ‘closed’ on the dissociation boundary. In other words, using R for the dissociation coordinate, the ui (R) are treated as eigenfunctions of H in (3.62) and the coupling arises from the overlap between the two basis sets.

3.5.3 Convergence tests The following tests are performed for an analytically soluble plane wave model, simply to illustrate the numerical convergence of the variational and eigenchannel approaches. Variation method As a test of the variation method, the plane wave basis set  un (r) = 2/d sin k˜n r, k˜n = (n − 1/2)π/d

(3.73)

with d = 10 a0 was truncated at r = a = 9 a0 . The variational eigenvalues En , surface amplitudes ψn (a), and surface derivatives ψn (a) in Table 3.2 were obtained by solving (3.53) in a basis restricted to n = 1 − 15. The comparative reference √ values were derived from the exact plane wavefunctions, ψn (r) = 2/a sin kn r with kn = (n − 1/2)π/a. It is seen that the eigenvalues and surface amplitudes, which appear in (3.55), are in close agreement with the reference values for the lower eigenfunctions with 1  n  8. The discrepancies increase as n increases to reach the values shown for n = 14 and 15. In addition, the variational surface

3.5 Variational R-matrix theory

71

Table 3.2 Variational eigenvalues, surface amplitudes and surface derivative for the plane wave test model. n 1 2 3 4 5 6 7 8 14 15

Enref 0.0305 0.2742 0.7615 1.4926 2.4674 3.6859 5.1480 6.8539 22.207 25.618

Envar

[ψn (a)]var

 var ψn (a)

0.0305 0.2742 0.7616 1.4926 2.4674 3.6859 5.1481 6.8539

+0.47140 −0.47141 +0.47141 −0.47141 +0.47141 −0.47140 +0.47138 −0.47136

+0.000174 −0.001538 +0.004125 −0.007631 +0.011551 −0.015100 +0.017077 −0.015627

−0.69581 +3.2109

22.809 87.050

+1.3247 −27.781

[ψn (a)]ref = ±0.47140  ref ψn (a) = 0.00000

derivatives are seen to be small for n  8, but much larger for n = 14 and 15, which confirms that the derivative information in the truncated basis is concentrated in the highest, and least ‘converged’ solutions. Figure 3.10(a) illustrates the importance of including all terms in the variational R-matrix sum, by plotting the fractional difference between the exact inverse log derivative R exact (a) =

tan kE a kE a

(3.74)

and the sum 1  ψn2 (a) , a n=1 kn2 − kE2 N

R(a) =

(3.75)

where kE2 = 2mE/¯h2 . The variable νE = kE a/π is the number variable, which takes integer or half odd integer values when ψE (a) = 0 or ψE (a) = 0, respectively. The dashed line shows the fractional error in R(a) when the sum is truncated to N = 13, which involves omitting terms for which ψn (r) has the highest surface derivative. The resulting errors, which are concentrated around the energies at which ψE (a) = 0, almost exactly replicate those observed for a b = 0 Wigner basis. The solid line shows the dramatic improvement arising from the two final terms in the sum. The blips at half odd integer values of νE are attributable to minor discrepancies between the calculated and reference eigenvalues in Table 3.2.

72

Ab-initio quantum defects

Fractional error

1

(a) 0.5 0 −0.5 −1

3

4

6

5

7

8

νE 0.5

% error

(b)

0

−0.5

Figure 3.10 (a) Comparison between the fractional errors in the complete variational sum (solid line) and the function obtained by omitting the two final terms (dashed line); (b) energy variation of the percentage errors for the variational (solid line) and eigenchannel (dashed line) models. The abscissae are labelled by νE = kE a/π .

Eigenchannel method To illustrate the convergence of the eigenchannel method in a very favourable case, we examine the discrepancies between the exact plane wave inverse logderivative in (3.74) and the approximation obtained by supplementing an N -term closed-channel basis   (3.76) um (r) = 2/a sin km r, km = 2mEm /¯h = mπ/a, with a single open-channel function  un (r) = 2/a sin kn r,

kn = (n − 1/2)π/a,

(3.77)

with n chosen such that kn and kE lie in the same interval between successive km values. The matrix elements required by (3.62) are given by     2 δmm Knn = kE2 − kn2 δnn , Kmm = kE2 − km  2  Kmn = Knm = kE − kn2 Smn ,

nn = nm = 0, nn = 2/a,

(3.78)

3.5 Variational R-matrix theory

73

2kn . Smn = (−1)n−m  2 2 a kn − km

(3.79)

and

Furthermore since the open set is restricted to the single term, un (r), the single solution of (3.70) is given by # N $−1 2 

K 1 nn mn  − Knn . R eigen (a) = − b(E) a m=1 Kmm

(3.80)

It is readily verified that b(E) correctly vanishes for kE = kn and diverges when kE = km . The dashed line in Fig. 3.10(b), which is drawn for N = 15, shows the percentage discrepancy from the exact result as a function of νE , which arises solely from restriction to a finite basis set, because this toy model ensures an exact description at the energies at which either ψE (a) = 0 or ψE (a) = 0. The solid line is a magnification of the corresponding solid line in panel (a). There is close agreement with the model dashed line at background energies, between the special points. Errors of the order of 1% in the immediate vicinity of these points are however inevitable in practice, because the roots of (3.53) in a finite basis will inevitably deviate from the exact eigenvalues.

3.5.4 Spheroidal basis functions Telmini and Jungen and Bezzaouia et al. have applied the variational R-matrix method to the H2 molecule, using basis functions that were expressed in terms of the spheroidal coordinates [51, 52] η=

r a + rb , 2R

ξ=

ra − rb , 2R

(3.81)

where ra and rb are the radial distances between the electron and the two nuclei, and R is the bond length. The R-matrix radius is therefore replaced by a fixed value ξ = ξ0 . In addition, symbols s˜, p˜ etc. are used to distinguish the spheroidal harmonic components from the normal spherical ones. The resulting coordinate dependent R-matrix transforms to a K-matrix, by matching to spheroidal Coulomb functions at a fixed value of ξ . Sets of interacting potential curves, such as those of the 1 g+ states of H2 in Fig. 3.10, may then be derived by solving (3.4) and (3.5). The resulting quantum number functions  ν(R) = Ry /[VX+ (R) − V (R)] for the closed channels, with VX+ (R) > V (R), were found to coincide, to within 2%, with those derived from the ab-initio potential

74

Ab-initio quantum defects 0.0

U (a.u.)

–0.2

–0.4

–0.6

5d 5s O P

HH GK

EF –0.8 1

2

3 R (a.u.)

4

5

Figure 3.11 Comparison between the 1 g+ ab-initio potential curves of H2 (lines) [7] and those determined by the eigenchannel R-matrix method (points and bars) [52]. The points are bound-state eigenvalues and the bars indicate the centres and widths of resonances above the ground-state positive ion potential function. Taken from Bezzaouia et al. [52], with permission.

curves of Wolniewicz and Dressler [7]. There is also resonance information on the rates at which members of the repulsive 2˜pσ n˜pσ series auto-ionize by interaction ˜ channels at energies above the ground-state with the open 1˜sσ ˜sσ and 1˜sσ  dσ + potential curve, VX (R), of the positive ion. The centres and widths of such resonances are indicated by the vertical bars in Fig. 3.11. The centres are seen to follow the dashed ab-initio potentials. In addition, the very strong interaction with the (2˜pσ )2 or ion-pair state, which is responsible for the strongly avoided crossings between the curves in Fig. 3.1, is seen to be matched by very large resonance widths, the largest of which is estimated as 0.1487 Rydbergs at R = 2.5 a0 . The corresponding widths for higher 2˜pσ n˜pσ resonant curves, with n = 4 and 5, are an order of magnitude smaller and those for the 2˜pσ 4˜fσ curve vary between 2 × 10−6 Ry at R = 1.0 a0 and 0.5 × 10−4 Ry at R = 5.0 a0 .

3.6 Rydberg–valence interactions Although the K-matrix is constructed in a strictly Rydberg representation, Fig. 3.7 shows that valence-state interactions can cause localized perturbations to the quantum defects functions. Following Mies, their influence may be parameterized by augmenting the MQDT quantization equations (2.42) with an additional valence

3.6 Rydberg–valence interactions

channel, in the form [53]



Z x KRyd + tan π ψ(E) = 0, T k(E − Ev ) Zv x

75

(3.82)

in which KRyd is the Rydberg–Rydberg part of the MQDT K-matrix. The column vector, x, carries the valence–Rydberg perturbations, Z and Zv are the Rydberg and valence channel amplitudes, respectively, and ψ(E) is either the negative of the eigenphase (if the Rydberg channel is open) or the Coulomb phase  Ry ψ(E) = π −1 β (E) = − (3.83) (E + − E) (if it is closed). Ev is the energy of the unperturbed valence state and the parameter k is taken as constant over the range of the Rydberg–valence interaction. The local form of the effective ab-initio K-matrix may by derived from (3.82) by the usual folding procedure, which casts (3.82) into the form (Keff + tan π ψ (E)) Z = 0,

(3.84)

where A (3.85) , A = 2k −1 xx T , 2(Ev − E) in which the outer product form means that the rank of A is equal to the number of perturbing valence channels. Assuming a single channel for simplicity, the matrix A has a single eigenvalue,  xα2 . (3.86)  = 2k −1 x T x = 2k −1 Keff = KRyd +

α

The corresponding eigenvector is proportional to x, which means that the branching ratios into different Rydberg channels vary as |Zα |2 = α / , where α = 2k −1 |xα |2 =

Aαα  . trace(A)

(3.87)

Assuming that KRyd is block-diagonalized, with elements tan π μ0α , (3.82)–(3.87) predict that the perturbed quantum defects vary as

α 0 (3.88) tan π μα (E) = tan π μα + 2(Ev − E) if the channels are closed. Similarly the eigenphase varies as

α 0 tan π τα (E) = tan π τα + arctan 2(Ev − E) if they are open.

(3.89)

76

Ab-initio quantum defects 0.20 0.4 0.15

τ

0.2 0

0.10

−0.2 −0.4

−129.04

0.05

(c)

(a) −129.02

−129.04

−129.02

0.00

0.02

0.4

μ

0.2 0

0.01

−0.2 −0.4

(d)

(b)

−129.07

−129.06

Etot / a.u.

−129.07

−129.06

0.00

Etot / a.u.

Figure 3.12 Resonant fluctuations (a) in the eigenphase sum and (b) in the quantum defect, due to valence state perturbations above and below the ionization limit, respectively. Panels (c) and (d) show the energies and residues of neighbouring poles. The dashed lines in (a) and (b) were obtained by suppressing the pole with the smallest residue. Taken from Hiyama and Child [54], with permission.

The parameters Ev ,  and α , which characterize the perturbation at any particular bond length are conveniently derived from the R-matrix by identifying the relevant small valence state pole, and diagonalizing the ab-initio K-matrix over a narrow energy range, with and without the perturbing pole [54]. As an example, Fig. 3.12 shows results for the 2  + Rydberg states of NO converging on the 1  + target state owing to perturbations by the I 2  + valence state.10 Panel (a) shows the eigenphase sum, as calculated with (solid line) and without (dashed line) the perturbing pole, at a point on the valence state curve above the ionization limit, while panel (b) shows similar plots for the pσ quantum defect at a point below the  2 for the poles limit. Panels (c) and (d) show the summed surface densities α wαn in the vicinity of the resonance. Finally, it should be noted, for comparison with traditional spectroscopic theory that the partial width α for interactions below the ionization limit may be related to an interaction matrix element Vv,αn between the valence and Rydberg states. It is easily verified, by applying the standard expansion Kαα + tan π ν (E) = π ν sec2 π ν(E − Eαn ), 10

(3.90)

The calculated valence state energies are systematically too high in energy by approximately 15000 cm−1 , owing to deficiencies in the basis set.

3.6 Rydberg–valence interactions

77

Table 3.3 Resonance parameters for perturbations between the npσ Rydberg series and the I valence state. Entries in italics indicate interactions above the ionization limit. R/au

103 /au

[Ev + 129.0] /au

ν or τ

2.175 2.350 2.457 2.500 2.646 2.737 2.385

2.614 2.489 2.664 2.626 2.787 2.310 0.735

−0.02395 −0.05484 −0.06072 −0.06208 −0.06035 −0.05697 −0.05170

0.679 4.785 3.022 2.704 2.196 2.018 1.883

that the relevant 2 × 2 block of (3.82) may be rearranged to the eigenvalue equation

Vv,αn E − Eαn = 0, (3.91) det Vv,αn E − Ev with the interaction matrix element given by " " α Ry cos2 π ν xα2 Vv,αn = . (3.92) =

2 kπ ν sec π ν π ν3   Illustrative results for interaction between the 1  + npσ series and the I 2  + valence state of NO are given in Table 3.3, using italic and roman fonts at energies above and below the ionization limit, respectively [54]. The variation of Ev with R defines the location of the valence-state curve, along which the width parameter  is seen to vary relatively smoothly and slowly apart from a sharp decrease between R = 2.737a0 and 2.385a0 . Similar findings are reported in the paper, for interactions with the B 2  and L 2  valence states, although deficiencies in the ab-initio basis mean that the valence states are all too high in energy by roughly 12000 cm−1 . Table 3.4 extends the picture by listing the partial resonance widths α given by (3.87) for various Rydberg–valence interactions at selected bond lengths. The entries show that the I, B and L valence states interact predominantly with the p Rydberg series, while the B 2  state mainly perturbs the dδ series. The valence– Rydberg matrix elements, given by (3.91), VB,npπ = 1349, 777 and 496 cm−1 for n = 3–5, for interaction between the B 2  state and the 3–5pπ Rydberg states were found to be in good agreement with the values, 1328.6, 803.9 and 594.6 cm−1 , that were extracted from observed spectral perturbations [54, 55]. However, less well-determined experimental interaction elements for the L state, which scale poorly with ν −3/2 , are in relatively poor agreement with estimates derived from Table 3.4.

78

Ab-initio quantum defects

Table 3.4 Partial widths, α /au, for various Rydberg–valence interactions, at the indicated bond lengths. Values in parentheses indicate powers of ten. Valence state

I 2+

B 2

L 2

B 2 

R/a0 E + 129.0/au

2.175 −0.03950

2.268 −0.06421

2.175 −0.00332

2.175 −0.04528

s p d f

2.98(−5) 2.43(−3) 1.27(−4) 1.38(−5)

2.59(−3) 7.39(−7)

2.18(−3) 1.83(−5) 5.17(−7)

1.53(−3) 1.49(−6)

3.7 The influence of positive ion dipoles 3.7.1 General considerations Situations in which the positive ion has a significant dipole moment, Q1 , raise a number of interesting questions. In the first place, the dipole of a body with charge Z can only be defined with respect to a particular origin, although Watson has shown that the combination Q2 − Q21 /Z with the quadrupole, Q2 , is origin independent – an invariant that corresponds to the ion quadrupole, measured from the centre of charge [56].11 It is assumed initially in what follows that Q2 = 0, although a perturbation estimate of the quadrupole contribution to the quantum defect will be added later. The dipole origin is normally taken at the centre of mass in order to simplify the treatment of rotational–electronic coupling. The second important consideration is that the dipole interaction term (in atomic units) V1 (r, θ ) = −

Q1 cos θ , r2

(3.93)

is typically large compared with the rotational energy spacings over a wide range of r. Consequently the electron is effectively coupled to the dipole axis within an 11

For simplicity consider a diatomic ion with charge density ρ (z) along its axis. The charge, dipole and quadrupole, with respect to the origin of z, are given by Z = ρ (z) ,

(0)

Q1 = zρ (z) ,

(0)

Q2 = z2 ρ(z),

where the brackets imply integration. Similarly, if the origin is changed to z1 , (1) (0) Q1 = (z − z1 ) ρ (z) = Q1 − z1 Z (1)

(0)

(0)

Q2 = (z − z1 )2 ρ(z) = Q2 − 2z1 Q1 + z12 Z. It follows by elimination of z1 that

 (1) 2  (0) 2 (1) (0) Q2 − Q1 /Z = Q2 − Q1 /Z.

3.7 The influence of positive ion dipoles

outer ‘dipole core’ region extending to a radius of order  r ∗  Q1 Ry / (2N + + 1) B + ,

79

(3.94)

and bounded at short range by the inequality r  R imposed by the above point dipole form. Two species are of particular interest below. In the case of ArH+ , with Q1 = 1.2 ea0 [57, 58] and B + = 4 cm−1 , this outer core extends to r ∗  170 a0 for N + = 0, while for CaF+ , with Q1 = 3.5 ea0 [59] and B + = 0.37 cm−1 , the figure rises to 1000 a0 . Seen in another way this means that the outer core region covers the full classically range for all Rydberg states with principal quantum number n  13 in the case of ArH and n  30 for CaF. In addition the electronic energy spacing between different principal quantum numbers is of order 50 cm−1 at n = 13 for ArH, which is large compared with B + = 4 cm−1 . The corresponding spacing for CaF at n = 30 is of order 3 cm−1 compared with B + = 0.37 cm−1 . In other words the presence of the dipole sets up a constrained rotational region within which rotational–electronic coupling conforms to Hund’s case (a) or (b) (see Appendix C.1). However, it will be shown that the influence of the dipole field falls off markedly with increasing orbital angular momentum . Thus there is a transition to perturbed case (d) behaviour, with significant Coriolis coupling between members of particular high  manifolds with a narrow spread of quantum defects. The 4f cluster of ArH is a well-studied example [60, 61]. To quantify the distinction between ‘strongly-coupled’ and ‘dipole-perturbed’ behaviour, the common r −2 radial dependence of the dipole interaction and the centrifugal terms  ( + 1) /2r 2 , makes it convenient to transform to a dipole representation in which the operator [56, 62] Pˆ 2 = ˆ2 − 2Q1 cos θ

(3.95)

is diagonal, with eigenvalues p(p + 1); Pˆ 2 = Up(p + 1)U T .

(3.96)

The plot in Fig. 3.13 illustrates the eigenvalue variation with Q1 over a range chosen to include the CaF+ value. The various line types indicate the angular momentum projections onto the dipole axis; σ (solid line), π (dashed line), δ (dot-dashed) φ (dotted). The labels ‘s’, ‘p’, ‘d’, etc. are however schematic because  is not conserved. The shading region defines the ‘strongly coupled’ region within which p(p + 1) lies between ±2Q1 /ea0 . The effective total angular momentum, p(p + 1), is seen to vary roughly linearly with Q21 except for the ‘sσ ’ and ‘pσ ’ solutions, which lie within the strongly coupled region. The remaining curves are well approximated by the perturbation

80

Ab-initio quantum defects 15

f

p(p + 1)

10

d 5

p 0

s −5

0

5

10 2 [Q1 / ea0]

15

Figure 3.13 Eigenvalues of the operator Pˆ 2 as a function of Q1 . The labels s, p, d, f are suggested by the grouping of the eigenvalues around the Q1 = 0 values, 2, 6, 12, etc. Different line types indicate different body fixed projections, σ (solid lines), π (dashed), δ (dot-dashed) and φ (dotted). The shaded region indicates the strongly coupled zone over which p(p + 1) lies between ±2Q1 /ea0 .

formula

    2Q21  ( + 1) − 3λ2 p(p + 1) =  ( + 1) + + O Q41 .  ( + 1) (2 − 1) (2 + 3)

(3.97)

given by Watson [56]. One should also note that p(p + 1) < 0 over the whole ‘sσ ’ branch and the high Q1 end of the ‘pπ ’ branch; and that p takes complex values, p = −1/2 ± iα say, for p(p + 1) < −1/4. A similar perturbation formula for the quantum defect   2Q21  ( + 1) − 3λ2 (dipole) − μλ (3.98)  ( + 1) (2 − 1) (2 + 1) (2 + 3) is applicable to non-penetrating states with approximately hydrogenic wavefunctions.12 The identity    ( + 1) − 3λ2 λ| P2 (cos θ ) |λ = (2 − 1) (2 + 3) 12

The explicit expressions given by Watson for the dipole contributions to s, p and d quantum defects must normally be modified by additional core contributions.

3.7 The influence of positive ion dipoles

81

may be used to provide a revealing difference between the curves inside and outside the coupling zone in Fig. 3.13, because the λ dependence in (3.97) may be attributed to a local second-order effective potential V2 (r, θ ) =

Q21 P2 (cos θ ) . ( + 1) r2

The radial dependence still varies as r −2 , but V2 (r, θ ) is smaller in overall magnitude than the dipole interaction V1 (r, θ ) outside the coupling zone, where ( + 1) > 2Q1 . In view of the rapid fall-off with ( + 1), this means that the high  Rydberg electrons are coupled to the dipole axis over a relatively small range of r. The influence of the dipole on the eigenvectors of Pˆ 2 must also be considered. The angular momentum mixing can become so strong, within the strong coupling region, that their  character becomes meaningless. For example the nominal ‘pσ ’ eigenvector has smaller true pσ amplitudes than the sσ and dσ ones, when Q1 > 2 ea0 . Such mixing has an obvious effect in breaking the normal spectroscopic selection rules. For example emission from the 4f complex of ArH is observed not only to the 3dλ states allowed by  = ±1, but also to the nominal 5sσ state, which has partial d character [60]. Equally importantly for the dynamics, the transformed Coriolis operators, Pˆ± = U T ˆ± U,

(3.99)

modify the strength of the λ = ±1 coupling within a given ‘’ manifold. Offdiagonal Coriolis-induced coupling between different nominal  manifolds, within the strong coupling zone, is the main source of spectroscopic perturbations in the observed spectra of the strongly dipolar species such as ArH, CaF [61, 63]. However, the contamination with non-diagonal orbitals decreases markedly with decreasing Q1 and increasing  and λ. Thus the small dipole of NO+ , Q1 = 0.2 ea0 leads to minimal  mixing, with over 95% true  character, for all  and λ (except for the sσ –dσ mixing arising from core polarization). The following perturbation formula applies for such weakly coupled species [56]:

2 Q 1 Pˆ± = 1 − 2 (3.100) ˆ± . 2 ( + 1)2 In summary, the presence of an andipole can create an outer core region, spanning possibly hundreds of atomic units, within which the Rydberg electrons with low orbital angular momenta are tied to the dipole axis, a situation which can apply for principal quantum numbers up to n ∼ 10–30, depending on the relative magnitudes of the dipole moment and the positive ion rotational constant. The corresponding body-fixed wavefunctions are conveniently transformed to the

82

Ab-initio quantum defects

dipole representation defined by (3.95), which is characterized by an effective noninteger or even complex angular momentum label p, in place of the normal index . The associated strong angular momentum mixing breaks the  = ±1 optical selection rules and leads to Coriolis perturbations between, as well as within, the nominal ‘’ Rydberg manifolds. The high  states, lying outside the strong coupling zone, are, however, well described by the perturbation formulae in (3.97), (3.99) and (3.100), except that additional core contributions to the quantum defect μλ may be required for the penetrating s, p and possibly d orbitals. The following sections describe two special computational procedures for the treatment of the strongly dipolar group II mono-fluorides CaF and BaF. 3.7.2 Group II mono-fluorides As mentioned above the group II mono-fluorides, CaF and BaF, have the largest positive ion dipoles of any diatomic species. The charge is approximately distributed as M2+ F− , which led Arif et al. [64] to devise a special single electron ab-initio procedure, based on the use of elliptic coordinates r 1 + r2 r1 − r2 , η= , (3.101) R R which allow a separable treatment of the electronic problem in the region ξ  ξ0 , with ξ0 chosen sufficiently large that the field is attributable to twin point charges. The model also allows for deep penetration into the M2+ core, while the corresponding interaction with the X− ion is assumed to be sufficiently repulsive that only the charge itself and associated polarization effects need be considered. The physical space is divided into the three regions in Fig. 3.14: (i) the small spherical ‘metal’ zone I, with r1  r1a , where penetration effects are dominant; (ii) a spherical ‘interaction’ zone III, r1a  r1  r1b , with variational R-matrix boundary conditions at the outer edge; and (iii) an ellipsoidal asymptotic zone II, whose inner edge, ξ = ξ0 , lies inside zone III, to allow a transformation of the wavefunction to elliptic coordinates at the surface ξ0 . A pseudo-potential is employed in region I, such that the partial waves at the boundary rI incorporate the known quantum defects of the M+ ion.13 Thus ζ =

ψ(I) (1 , r1 ) = r1−1 Yλ (θ1 , φ) [f (1 , r1 ) cos μ − g (1 , r1 ) sin μ ],

(3.102)

where the base pair [f (1 , r1 ) , g (1 , r1 )] are taken as energy-normalized solutions appropriate to a Coulomb field, augmented by a screened polarization potential. In addition the electron energy, 1 , is taken to include a term arising from the 13

An analagous model for the rare gas hydrides [57] sets the rare gas atom at r1 = 0 and employs electronscattering phase shifts, rather than quantum defects to take account of penetrating into region I.

3.7 The influence of positive ion dipoles III

83

ξ0

r1b

II

1 R 2

r1aI

r2

r1 e–

Figure 3.14 Schematic representation of the electron e− interacting with the M2+ ion at 1 and X− at 2. Zones I and III are spheres around the metal ion, with radii r1a and r1b , respectively, while zone II is an ellipsoidal region, whose inner boundary, ξ = ξ0 , lies outside zone I and inside zone III. Taken from Arif et al. [64], with permission.

mean electrostatic potential at r1 = 0 due to a polarized X− ion. The log-derivative boundary conditions   ∂ r1 ψ(I) /∂r1 ∂ r1 ψ(II) /∂r1   (3.103) = = −b1 (r1a ), r1 ψ(I) r1 ψ(II) at r1 = r1a , serve to connect these inner functions to those in region II. The potential VII (r1 , r2 , R) in region II is taken to arise from the two point charges, modified by a variety of terms arising from the appropriate induced dipoles. Following the lines of Section 3.5, a variational R-matrix calculation is performed by expansion in a basis that includes both positive energy, m = k 2 /2, terms (λ) (r1 ) = r1−1 Yλ (θ1 , φ) [cm sin kr1 + dm cos kr1 ] ψm

and negative energy, m = −κ 2 /2, terms   (λ) (r1 ) = r1−1 Yλ (θ1 , φ) cm exp(κr1 ) + dm exp(−κr1 ) . ψm

(3.104)

(3.105)

The coefficients cm and dm are fixed by the boundary conditions at r1a and r1b . Those at the inner boundary satisfy (3.103), to allow continuity at the boundary with region I. Following the variational R-matrix procedure, those at the outer

84

Ab-initio quantum defects

boundary are taken, for each  value, to allow typically 13 closed radial functions with b1 (r1b ) = 109 and two open functions with b1 (r1b ) = 0. Linear combinations, at a particular energy ,  (λ) am,β () ψm (r1 ), (3.106) β(λ) (, r1 ) = m

are then taken to satisfy the generalized eigenvalue system a = bt a,

(3.107)

where the matrices  and  are given by (3.62). Recall that  includes the test energy  of interest, which is chosen close to that of the open channels. The required Hamiltonian matrix elements are evaluated as    ( + 1) (λ) (λ) (λ) (r1 ) VI I (r1 , r2 , R) + (r1 ) dτ, ψm δm,m  + ψm Hm,m  (R) = m r2 (3.108) taken over the volume r1a ≤ r1 ≤ r1b . The next step is to project β(λ) (r1 ) onto a separable wavefunction of the elliptic coordinate form    ˜ ˜ (, ξ ) Jβ f˜˜ (, ξ ) Iβ ˜ −g ˜ (λ)  β (, ξ, η) = Y˜λ , (3.109) ˜ (, η, φ) 2 1 − ξ ˜ 

  ˜ ˜ (, ξ ) are numeriin which Y˜λ ˜ (, η, φ) are bipolar harmonics and f˜˜ (, ξ ), g cally determined regular and irregular basis functions, with Wronskian π −1 . The computational method, including the calculation of appropriate accumulated phase functions β˜ () is described by Jungen and Texier [65]. Further details are given by Arif et al. [64]. The elliptic coordinate K-matrix elements are obtained in the form  −1 Jβ (3.110) K˜˜‘ (, R) = tan π μ˜˜‘ = ˜ (, R) [I (, R)]β ˜‘ , β

in which the final factor is an element of the inverse of the I matrix in (3.109). The eigen quantum defects of CaF obtained by diagonalizing this K-matrix are given in Table 3.5, together with the  = 0 − 3 components of the eigenvectors [66]. Entries in the ‘fitted’ column were determined by an MQDT analysis of the ν  12–18 members of the highly perturbed Rydberg series, which was later extended to include vibrational perturbations [63]. The  mixing is so strong, particularly for nominal ‘p’ and ‘d’ eigenvectors, that it would be misleading to give even nominal ‘’ labels to the observed

3.7 The influence of positive ion dipoles

85

Table 3.5 Eigen quantum defects of CaF and partial wave components of the eigenvectors. The final column gives spectral labels employed to identify the observed series [59, 63].

Ab-initio [61]

Fit[63]

s

p

d

f

ν

‘s’ ‘p’ ‘d’ ‘f’



0.451 0.191 −0.160 −0.070

0.458 0.131 −0.181 −0.058

+0.86 −0.251 +0.08 +0.05

+0.51 +0.84 −0.17 +0.06

+0.06 +0.17 +0.80 −0.57

−0.05 +0.09 +0.56 +0.82

n.55 n.88 n.19

‘p’ ‘d’ ‘f’



−0.384 0.013 −0.041

−0.352 0.026 −0.032

+0.80 +0.60 +0.06

−0.60 +0.80 −0.09

−0.11 +0.03 +0.99

n.36 n.98

‘d’ ‘f’



−0.139 0.022

−0.135 0.033

+0.99 −0.16

+0.16 +0.99

n.14

‘f’



0.095

0.089

+1.00

‘’ = 0−2 bands [59, 63]. Instead, different series are identified by effective quantum numbers and symmetry labels, e.g. 13.55, as listed in the final column of the table; they are seen to total to approximate integers, when added to the appropriate eigen quantum defect. Despite the pervasive nature of the perturbations between the different ν series, the experimental term values were well reproduced. A total of 491 rovibronic levels were reproduced with a standard deviation of 1.4 cm−1 by a global MQDT calculation [66] based on the  mixed ab-initio quantum defect matrices, a fit that was reduced to a deviation of 0.18 cm−1 , by optimization around the ab-initio values. A later paper [63], from which the data in Table 3.5 are taken, extended the results to include vibrational perturbations, which necessitated knowledge of the diagonal and off-diagonal quantum defect derivatives, with respect to bond length. A total of 621 observed term values were then reproduced with a standard deviation of 0.15 cm−1 .

3.7.3 Dipole modified Coulomb functions As an alternative to the use of elliptic coordinates, Altunata et al. employ the Pˆ representation of (3.95) and (3.96) in the outer dipole-dominated region [67, 68]. The radial Schr¨odinger equation takes the Coulomb form in (2.3), except that  is replaced by non-integer or even complex values of p. The properties of the appropriate dipole-modified base pairs, fp (, r) and gp (, r) are well known [69, 70, 71]. In the case of real non-integer p, they have the forms and properties

86

Ab-initio quantum defects

described by (2.5) and (2.11)–(2.14), with  replaced by p. In particular fp ∼ r p+1 and gp ∼ r −p as r → 0; and the accumulated phase function is given by βp () = π (ν − p).

(3.111)

The situation for negative values of p(p + 1) (specifically p(p + 1) ≤ −1/4 − α 2 ) is more complicated, because the indicial equation requires that the solutions vary as r 1/2±iα close to the origin. Greene et al. [70] show that there are then two independent complex conjugate solutions, which may be defined in terms of confluent hypergeometric functions, ψp,p∗ (, r) = r 1/2±iα e−r/ν F (1/2 ± iα, ±2iα, 2r/ν). (3.112)   The sum and difference, (f, g) = N ψp (, r) ± ψp∗ (, r) , suitably scaled to have Wronskian π −1 , are taken as the base pair. Finally, the general expression for the accumulated phase function in Table I of Greene et al. [70]14 reduces in the case p = −1/2 + iα to the form βp () = π (ν + 1/2) + tan−1 {tanh π α tan [y − ln (2/ν)]} y = arg (2iα) − arg (ν + 1/2 + iα).

(3.113)

It should be emphasized that one is interested in practice only in the behaviour of ψp,p∗ (, r) far from the origin, because the validity condition on the dipole form in (3.93) is that r  R. The theory along these lines was applied by Altunata et al. to the above singleelectron ab-initio model of CaF system except that the variational R-matrix treatment was restricted to a spherical, rather than an ellipsoidal core [67, 68]. In addition, solutions were obtained by an iterative Green’s function method, rather than by diagonalization. The output is an R-matrix in the normal  representation, which is transformed to the p representation, in the form R(p) = UT R() U; The K-matrix is then given by analogy with (3.14), as &−1 % & % K(p) = g − aR(p) g f − aR(p) f ,

(3.114)

(3.115)

where the elements of the diagonal matrices f and g are defined in the p representation. Finally the eigen quantum defects are obtained as the eigenvalues of K(p) . The theoretical results in Fig. 3.15 are seen to be in excellent agreement with the experimental data. Moreover, Altunata et al. find that the quantum defects 14

The earlier paper by Greene et al. [69] employs a slightly different notation.

References

87

Figure 3.15 Comparison between the theoretical eigen quantum defects [68], shown as lines, and experimental values, shown as points. The crosses are values obtained by a global fit to the experimental data. Different panels apply to different electronic symmetries (A) 2 , (B) 2 , (C) 2  and (D) 2 . Taken from Altunata et al. [68], with permission.

calculated in the dipole modified representation are much less dependent on energy than those obtained in the normal spherical, or , representation [68].

References [1] A. Szabo and N. S. Ostlund, Modern Quantum Chemistry (Dover Publishing, 1996). [2] R. J. Buenker and S. D. Peyerimhoff, Theor. Chim. Acta 35, 33 (1974). [3] P. G. Burke and K. A. Berrington, Atomic and Molecular Processes: An R-Matrix Approach (IOP, 1993). [4] W. M. Huo and F. A. Gianturco, eds, Computational Methods for Electron–Molecule Collisions (Plenum, 1955). [5] M. Aymar, C. H. Greene and E. Luc-Koenig, Rev. Mod. Phys. 68, 1015 (1996). [6] P. G. Burke, R-Matrix Theory of Atomic Collisions: Application to Atomic, Molecular and Optical Processes (Springer, 2011). [7] L. Wolniewicz and K. Dressler, J. Chem. Phys. 100, 444 (1994). [8] T. Detmer, P. Schneider and L. S. Cederbaum, J. Chem. Phys. 109, 9694 (1998). [9] G. Staszewska and L. Wolniewicz, J. Mol. Spec. 169, 208 (2002). [10] K. P. Huber and G. Herzberg, Constants of Diatomic Molecules (van Nostrand, 1979). [11] R. S. Mulliken, JACS 88, 1849 (1966). [12] J. T. Lewis, Proc. Phys. Soc. London 68, 632 (1955). [13] L. Wolniewicz and G. Staszewska, J. Mol. Spec. 220, 45 (2003).

88 [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57]

Ab-initio quantum defects H. Wind, J. Chem. Phys. 42, 2371 (1965). C. Jungen and O. Atabek, J. Chem. Phys. 66, 5584 (1977). R. S. Mulliken, Chem. Phys. Lett. 46, 197 (1977). M. Hiyama and M. S. Child, J. Phys. B 35, 1337 (2002). R. de Vivie and S. D. Peyerimhoff, J. Chem. Phys. 89, 3028 (1988). S. C. Ross and C. Jungen, Phys. Rev. A 49, 4353 (1994). A. Kirrander, D Phil thesis, Oxford University (2005). M. S. Child and M. Hiyama, J. Phys. B 40, 1233 (2007). A. U. Hazi, C. Derkits and J. N. Bardsley, Phys. Rev. A 27, 1751 (1983). S. L. Guberman, J. Chem. Phys. 78, 1404 (1983). S. C. Ross and C. Jungen, Phys. Rev. A 49, 4364 (1994). M. S. Child, Phil. Trans. Roy. Soc. London, A 355, 1623 (1997). D. M. Hirst and M. S. Child, Mol. Phys. 77, 463 (1992). G. Theodorakopoulos, I. D. Petsalakis and M. S. Child, J. Phys. B 29, 4543 (1996). B. M. Nestmann and S. D. Peyerimhoff, J. Phys. B 23, 773 (1990). B. M. Nestmann, R. M. Nesbet and S. D. Peyerimhoff, J. Phys. B 24, 5133 (1991). L. A. Morgan, J. Tennyson and C. J. Gillan, Comp. Phys. Commun. 114, 120 (1998). I. Rabad´an and J. Tennyson, J. Phys. B 29, 3747 (1996). I. Rabad´an and J. Tennyson, J. Phys. B 30, 1975 (1997). F. P. Billingsby, Chem. Phys. 3, 864 (1975). D. L. Albritton, A. L. Schmeltekopf and R. N. Zare, J. Chem. Phys. 71, 3271 (1979). C. Bloch, Nucl. Phys. 4, 503 (1957). E. Miescher and K. P. Huber, International Review of Science vol. 3. Physical Chemistry Series 2 (Butterworth, 1976). I. Rabad´an and J. Tennyson, J. Phys. B 31, 4485 (1997). K. Kauffman, J. Phys. B 24, 2277 (1991). E. P. Wigner and L. Eisbud, Phys. Rev. 72, 29 (1947). P. J. A. Buttle, Phys. Rev. 160, 719 (1967). I. S. Gradsteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic Press, 1980). M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1965). J. L. Jackson, Phys. Rev. 83, 301 (1951). D. E. Manolopoulos and R. E. Wyatt, Chem. Phys. Lett. 152, 23 (1988). A. Faure, J. M. Gorfinkel, L. A. Morgan and J. Tennyson, Comp. Commun. Phys. 144, 224 (2002). L. A. Morgan, C. J. Gillan, J. Tennyson and X. Chen, J. Phys. B 30, 4087 (1997). U. Fano and C. M. Lee, Phys. Rev. Lett. 31, 1573 (1973). C. H. Greene, Phys. Rev. A 28, 2209 (1983). H. Le Rouzo and G. Raseev, Phys. Rev. A 29, 1214 (1984). F. Robicheaux, Phys. Rev. A 43, 5946 (1991). M. Telmini and C. Jungen, Phys. Rev. A 68, 062704 (2003). S. Bezzaouia, M. Telmini and C. Jungen, Phys. Rev. A 70, 012713 (2004). F. H. Mies, Phys. Rev. A 20, 1773 (1979). M. Hiyama and M. S. Child, J. Phys. B 36, 4547 (2003). R. Galluser and K. Dressler, J. Chem. Phys. 76, 4311 (1982). J. K. G. Watson, Mol. Phys. 81, 277 (1994). C. Jungen, A. L. Roche and M. Arif, Phil. Trans. Roy. Soc. London, A 355, 1481 (1997).

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[58] K. B. Laughlin, G. A. Blake, R. C. Cohen, D. C. Hoyde and R. J. Saykally, J. Chem. Phys. 90, 1358 (1989). [59] C. M. Gittins, N. A. Harris, M. Hui and R. W. Field, Can. J. Phys. 79, 247 (2001). [60] I. Dabrowski, D. W. Tokaryk, R. H. Lipson and J. K. G. Watson, J. Mol. Spec. 189, 95 (1998). [61] C. Jungen and A. L. Roche, J. Chem. Phys. 110, 10791 (1999). [62] C. W. Clark, Phys. Rev. A 20, 1875 (1979). [63] R. W. Field, C. M. Gittins, N. A. Harris and C. Jungen, J. Chem. Phys. 122, 184314 (2005). [64] M. Arif, C. Jungen and A. L. Roche, J. Chem. Phys. 106, 4102 (1997). [65] C. Jungen and F. Texier, J. Phys. B 33, 2495 (2000). [66] C. Jungen and A. L. Roche, Can. J. Phys. 79, 287 (2001). [67] S. N. Altunata, S. L. Coy and R. W. Field, J. Chem. Phys. 123, 084318 (2005). [68] S. N. Altunata, S. L. Coy and R. W. Field, J. Chem. Phys. 123, 084319 (2005). [69] C. H. Greene, U. Fano and G. Strinati, Phys. Rev. A 19, 1485 (1979). [70] C. H. Greene, A. R. P. Rau and U. Fano, Phys. Rev. A 26, 2441 (1982). [71] A. R. P. Rau, Phys. Rev. A 38, 2255 (1988).

4 Frame transformations and channel interactions

The quantum defect and the frame transformation approximation are the two most important components of the MQDT machinery. This chapter starts by examining the validity of the latter approximation. To put the classical argument in Chapter 1 into a quantum mechanical perspective, Section 4.1 demonstrates the insensitivity of the radial wavefunction accompanying energy changes of the order of typical vibrational and rotational energy intervals. Readers who expect to apply the transformation at the core boundary may be surprised to find that it remains valid over often quite a wide range of radial separations, r, which varies inversely with the magnitude of the rotational or vibrational energy transfer involved. A typical transformation element takes the form of the projection, i|α of an uncoupled state |i onto a coupled Born–Oppenheimer state |α, the form of which varies according to the nature of the relevant motion. For example, in the rotational case |α = |  is a specified body-fixed angular momentum projection, while |i = |N +  is the positive ion angular momentum after the Rydberg electron has been uncoupled from the molecular frame. Section 4.2 restricts attention to the simplest angular momentum coupling case, with applications chosen to illustrate the quantum defect description of topics in the spectroscopic literature, such as

-doubling and -uncoupling [1, 2, 3]. The angular momentum manipulations required to handle more complicated coupling cases are treated in Appendix C, which also includes an account of the relevant parity and symmetry considerations. Turning to vibrational channel coupling, the transformation is from the Born– Oppenheimer picture, with a fixed bond length, R, to an uncoupled regime with well-defined vibrational states, |v + . Hence the frame transformation amplitude α |i is the positive ion wavefunction R|v +  and the strength of vibrational channel interactions depends on the nuclear coordinate dependence of the quantum defects, via the matrix elements v + | tan π μ |v + . Applications in Section 4.3 to the discrete level structure and to vibrationally induced auto-ionization in the np series of H2 demonstrate a necessary interconnection between vibrational and 90

4.1 Physical assumptions

91

n = 15

f

n = 10

d

s p

n=5

Figure 4.1 Semiclassical p orbits with  = 1.5 and n = 5, 10 and 15, compared with a core of radius 10 a0 . The inset shows that the two latter orbits are indistinguishable in the core region, with a small but discernible difference from the inner n = 5 orbit. The dotted curves in the inset follow the s, d and f orbits for n = 10.

rotational interactions. Finally, Section 4.4 covers vibronic channel coupling, by either configuration interaction or Jahn–Teller coupling, of which the latter serves as an introduction to the theory of dissociative recombination of H+ 3 in the following chapter. Space constraints preclude the inclusion of other topics. The most important omissions include quantum defect treatments of the Stark and Zeeman effects [4, 5, 6]. Readers are also referred to work on spin-orbit [7], nuclear-hyperfine [8] and Renner–Teller-induced [9] channel coupling.

4.1 Physical assumptions 4.1.1 The classical picture The most important simplifying approximations of the theory are illustrated in Fig. 4.1, which is taken from Chapter 1. It shows that the fractional length of each Rydberg orbit within the core decreases rapidly with the principal quantum number. In addition, the inset shows that the transit path varies very little with n, for orbits with a given angular momentum. When expressed in terms of time, it was concluded in Chapter 1 that although the total orbit time increases as n3 , the core transit time is essentially independent of n. It is also very short (τc ∼ 10−16 s) compared with the periods of molecular vibrations and rotations. Hence the Rydberg electron may be visualized as moving slowly and independently around its orbit |i until

92

Frame transformations and channel interactions

it approaches the core, where it jumps rapidly into and out of an allowed Born– Oppenheimer-like state |α, to emerge into a different slow Coulombic orbit |j , with jumping amplitudes i |α and α |j  and a quantum mechanical phase change determined by the quantum defect μα . This process may continue forever, in which case the energies, amplitudes and phases combine to determine the discrete boundstate energy levels. Alternatively the orbit velocity, after a final jump, may exceed the escape velocity, in which case the system auto-ionizes. To see how the history of such events is reflected in the nature of the wavefunction, we follow Fano [10, 11] in expressing the real K-matrix in the eigenchannel form  T Kij = Uiα tan π μα Uαj , (4.1) α

where the orthogonality of the frame transformation, with elements Uiα = i|α, ensures the symmetry of the matrix K. It is useful later to work with frame transformation elements that are diagonal in the Rydberg angular momentum . Hence angular momentum mixing within the state |α is taken into account by the transformation  |λ λ |α, |α = (4.2) λ

in which case (4.1) goes over to  (el)

i |λ Kλ, Kij =

λ  λ |i,

(4.3)

λ, λ

in which the frame transformation elements i |λ are then diagonal in , and the electronic K-matrix elements are then given by  (el) λ |α tan π μα α| λ . (4.4) Kλ,

λ = α

Chapter 3 is devoted to the ab-initio determination of this K (el) matrix. The implications of (4.1) for the form of the short-range wavefunction may be understood by referring to Section 2.4, where the total wavefunction is written at radii outside the core in terms of the functions    |i fi (r)δij − gi (r)Kij , r (j ) = r>a (4.5) i

in which the superscript j and the symbol δij indicate that  (j ) has regular character in channel |j . An alternative form, which is implied by combining (4.1) with the

4.1 Physical assumptions

orthogonality of U, is  |i i |α [fi (r) cos π μα − gi (r) sin π μα ] sec π μα α |j . r (j ) =

93

(4.6)

iα

The validity of the frame transformation lies in the extent to which the base pairs, fi (r) and gi (r), appropriate to different fragmentation channels may be replaced ¯ by a common average pair, f¯(r) and g(r), at radii r close to but outside the core region. Equation (4.6) then reduces to  r (α) sec π μα α |j , ra (4.7) r (j )  α

where   ¯ sin π μα . r (α) = |α f¯(r) cos π μα − g(r)

(4.8)

The crucial implication of this approximation is that the inner part of wavefunction depends only   on the quantum defects, μα , and the properties of a mean base pair ¯ f¯(r), g(r) , common to all fragmentation channels |i. In addition, (4.7) shows that the wavefunction  (j ) picks up  (α) character in proportion to the projection, or frame transformation element, α |j . To assess the validity of the approximation and to relate it to the physical picture in the introductory paragraph we note that the radial components [fi (r), gi (r)] depend on the reduced energy  and angular momentum  in channel |i in the form [f (i , r), g (i , r)] . Thus the scattering from channel |i to channel |j  will involve an energy transfer ij equal to the energy difference between rotational, vibrational or electronic levels of the positive ion. As argued above,  is held constant, because the frame transformation is applied in the uncoupled region outside the core, where  is a good quantum number. It is revealing to examine the energy transfer question within the JWKB approximation [12] f (, r)  Ck (, r) sin φ (, r) ,  r φ (, r) = k (, r) dr + π/4 rt

√ k (, r) =

2m [E − V (r)]1/2 h¯

(4.9)

where rt is the classical turning point. The predominant energy dependence comes from the phase factor, φ (, r), and it is easily verified that  ∂φ 1 r dr t (r) = = , (4.10) ∂ h¯ rt u (r) h¯

94

Frame transformations and channel interactions −1

−1

ΔE = 3.6 cm

f (ε,r)

ΔE = 36 cm

(b) n = 15 0

100

200

300

−1

ΔE = 360 cm

500 −1

f (ε,r)

ΔE = 3600 cm

400

(a) n = 10 0

10

20 r / a0

30

40

Figure 4.2 Comparison between the forms of the regular wavefunctions, f (, r), with  = 1 (a) at n = 10 and (b) at n = 15, with those appropriate to energy increases, E, of the stated magnitudes. Wavefunctions at the integer quantum numbers are indicated by solid lines. Those with the energy increases are marked by circles, which are filled for the larger and open for the smaller increase.

where u (r) is the classical velocity and t (r) is the time (in atomic units) to pass from the turning point, rt , to the radius r of interest. The validity of (4.7) therefore depends on the inequality Eij τ (r) = ωij τ (r) = 2π cν¯ ij t (r)  1, h¯

(4.11)

where ωij = Eij /¯h is the angular frequency associated with Eij and ¯νij is the corresponding wavenumber transfer in cm−1 , while t (r) is equal to half the core transit time in atomic units. Given the atomic time unit τ ∼ 2.42 × 10−17 s, the frame transfer approximation is justified for energy transfers equivalent to t  10 atomic units and ¯ν  105 cm−1 . Thus even for the transfer of a vibrational energy quantum in H2 , for which ¯ν = 4400 cm−1 the core transit time is one and a half orders of magnitude smaller than the vibrational period. Similar conclusions are reached by examining the energy dependence of the internal part of the wavefunction in Fig. 4.2, which plots the regular basis functions, fl (, r) with  = 1, for different E values. Panel (a) compares the n = 10 wavefunction (solid line) with those for additional energies of E = 3600 cm−1 (filled circles) and E = 360 cm−1 (open circles). The former shift is seen to cause a graphically negligible change in the wavefunction over the range r  8 a0 , while graphical agreement is seen for E = 360 cm−1 over the larger range r  28 a0 .

4.1 Physical assumptions

95

Similar behaviour is seen in Fig. 4.2(b), for smaller energy increments. A higher reference quantum number, n = 15 is now employed, because graphical agreement between the solid curve and the open circles for E = 3.6 cm−1 extends almost to r = 400 a0 , which is beyond the outer turning point for n = 10. The corresponding range for E = 36 cm−1 (closed circles) reaches r  200 a0 . Similar results are obtained for the irregular Coulomb functions. The conclusion is that, although one normally thinks of applying the frame transformation close to the surface of the short-range ion core, it may in fact be validly applied over wide range of r, especially when considering purely rotational channel interactions. It is interesting to return to the question of strongly dipolar ion cores at this point (see Section 3.7), bearing in mind that a significant cosine barrier arising from the dipole interaction perturbs the free rotations of the ion core, but has little influence on the vibrations of the ion core. Hence only the validity of the rotational frame transformation comes seriously into question. The discussion follows a line presented by Jungen and Roche, but with a slightly different emphasis [13]. The aim is to establish conditions under which (i) the potential is Coulomb dominated; (ii) the electron is coupled to the rotating dipole axis and (iii) the radial basis functions are independent of the rotational energy transfer Erot . Put in the terminology of Section 3.7.1, the first two conditions require that Z |Q1 |  ∗2 , ∗ r r

(4.12)

where r∗ =

 Q1 Ry /Erot

(4.13)

denotes the outer limit of the dipole coupling zone. Taken together (4.12) and (4.13) require that  N1 = Z

Ry |Q1 |Erot

1/2  1,

(4.14)

in which N1 is the suddenness parameter of Jungen and Roche [13]. The question of whether or not the radial basis functions at r ∗ are independent of Erot can only be established on a case-to-case basis. In the case of ArH+ , with Q1 = 1.2 ea0 , assume that Erot = 36 cm−1 , which corresponds to the N + = 1 spacing at N + = 5. Equations (4.13) and (4.14) show that N1  50, which is indeed large, while r ∗  60 a0 , which is well within the range r < 200 a0 over which the basis functions in Fig. 4.2(b) for E = 36 cm−1 are graphically indistinguishable. A similar comparison may be made for CaF+ , for which Q1 = 3.5 ea0 . The rotational constant is ten times smaller than for ArH+ , so we assume Erot = 36

96

Frame transformations and channel interactions

cm−1 , in which case N1  90  1. In addition r ∗  320 which is close to but inside the radius at which the E = 36 cm−1 curves in Fig. 4.2(b) become distinguishable. These considerations show that cases involving even very large ion dipoles can permit a valid rotational frame transformation by (4.3) provided that the electronic K-matrix, K (el) , is derived from an R-matrix at the outer boundary of the dipole coupling zone. One should also remember that there may be several or indeed many members of the Rydberg series for which a transformation to the uncoupled representation is irrelevant, because their outer radial turning points lie within the dipole-coupling region. Coriolis effects within this region may be taken into account by use of a dipole-modified frame transformation (see Appendix C.4), including the off-diagonal Coriolis perturbations that are characteristic of strongly dipolar species [13, 14].

4.1.2 Frame transformation types The following sections deal with the nature and computation of various types of frame transformation matrix element, between body-fixed |αλ  components of the Born–Oppenheimer states |α and uncoupled or space-fixed states |i, which include all degrees of freedom except the radial motion of the Rydberg electron. A typical |αλ  component is therefore labelled by a core electronic state |γ + , a rotational state |J M, a body-fixed state |λsσ  of the Rydberg electron, and a set of nuclear coordinates Q. By contrast the uncoupled states |i, which specify the various fragmentation channels, are labelled by perhaps a different core state |γ + , rotational and vibrational states |J +  + M +  and |v +  of the positive ion and states |m sms  or |j mj  of the Rydberg electron, now with well-defined space-fixed rather than body-fixed projections. Thus |α = γ + ; λsσ ; J M; Q

!

! |i = γ + ; m sms ; J +  + M + ; v + ,

(4.15)

to which one can also add nuclear spin states of both the molecule and the positive ion. As a result the frame transformation elements factorize into three main types of term, Uiα = i |α = γ + |γ + m sms ; J +  + M + |λsσ ; J M v + |Q, because the Rydberg electron terms cannot be separated from the rotation. The following section deals with the simplest type of rotational–electronic frame transformation, leaving other more complicated angular momentum coupling situations to be treated in Appendix C. Section 4.3 contains a discussion of the vibrational

4.2 Rotational channel interactions

97

transformation. Finally, Section 4.4 concerns vibronic channel coupling arising from off-diagonal elements of the electronic K-matrix in (4.3) and (4.4). 4.2 Rotational channel interactions The rotationally induced Coriolis interactions between different channels are introduced by means of angular momentum frame transformations. The theory required for commonly occurring cases in diatomic molecules, and symmetric and asymmetric tops, is given in Appendix C. Here the essential ideas are introduced in the simplest context of a Hund’s case (b) to case (d) transformation in a diatomic molecule. The resulting transformation matrix is first related to the familiar -doubling phenomenon in traditional spectroscopy [3, 15]. Other applications include the MQDT theory of -uncoupling in a single high  Rydberg cluster [16] and an interesting stroboscopic effect, as the time period of the electronic motion tunes through integer multiples of the rotational period [17]. Details of the angular momentum notation are given in Appendix F. 4.2.1 Elementary case (b) → case (d) transformation Reference to any spectroscopic text [1, 2] shows that the rotational–electronic structure of even a diatomic molecule is complicated by the interplay between the electrostatic binding energy, the strength of the spin–orbit coupling and the magnitude of the rotational energy separations. The relative values of these three quantities allow six possible angular momentum coupling cases, which are discussed in detail in Appendix C. Here we concentrate on the simplest situation of a transformation between Hund’s [18] cases (b) and (d) in a species with a closed shell ion core and negligible spin–orbit coupling. The relevant angular momentum coupling schemes are illustrated in Fig. 4.3, where the angular momenta L and N+ apply to the electronic motion and the nuclear rotation of the positive ion, respectively, and N = L + N+ is their resultant. Spin is ignored.1 The difference between the two cases is that L is coupled to the molecular axis in case (b), with a well-defined body-fixed projection , whereas in case (d) it is coupled to N+ in space. It is assumed in this initial exposition that N  L. Since the transformation is performed for a fixed magnitude of the resultant, N, we are interested in the transformation between case (b) states |α = | , with

= −L, . . . , L, and case (d) states |i = |N + , with allowed values N + = N − L, . . . , N + L, which satisfies the obvious requirement, for orthogonality, that the two types of basis each have the dimension 2L + 1. An alternative specification for 1

Case (b) and case (d) states have common space-fixed spin components |SMS .

98

Frame transformations and channel interactions L +

N

N

+

N

N

L

Λ Case (b)

Case (d)

Figure 4.3 Vector diagrams for the angular momentum coupling in Hund’s cases (b) and (d).

the case (d) state, which will become relevant later, employs the projection, LN , of L onto the total angular momentum N [19], which tends in the limit of large N to L = N − N + in the earlier notation of Jungen and Miescher [16]. The forms of appropriate parity-adapted case (b) and case (d) states, |α and |i, which are determined in Appendix C, are given by   1 |α = √ |L  |N M + (−1)p+N |L −  |N − M 2(1 + δ 0 )

(4.16)

and |i =



−L+N + −M

(−1)

√ 2N + 1



ML M +

L ML

N+ M+

N −M



|LML  |N + 0M + . (4.17)

Following Fano [10], the transformation between the states |i and |α is obtained by using [3], |LML  =

 ∗ L DM (R) |L , L

(4.18)

to transform |LML  to the body frame and expressing |N M in the Wigner rotation representation [3] " R |N M =

∗ 2N + 1  N D (R) . M 8π 2

(4.19)

4.2 Rotational channel interactions

99

The essential integral is given by ' LML ; N + 0M + L ; N M √   N ∗ (2N + + 1) (2N + 1) L N+ (R)D dR D = + 0 (R) DM (R) M M 8π 2

 L N+ N L N+ M− + (2N + 1) (2N + 1) = (−1) ML M + −M

0

(4.20)

N . −

The orthogonality of the space-fixed Wigner coefficients in (4.17) and (4.20) then determines the overlap i| L ; N M. Finally the integral over the second term in (4.16) is related to that over the first, by the symmetry of the 3j coefficients under sign reversal of . The resulting frame transformation matrix element, with fixed N , L and p, is found to be given by [20, 21, 22]2 i |α = N + | NLp  # $ + + 1) 1 + (−1)p+L+N + 2(2N N+ = (−1)N− 0 (1 + δ 0 ) 2

L N .

− (4.21)

This transformation factorizes into the well-known e and f type ‘total parity’ blocks, for which p + N is even and odd, respectively [3]. For example, (4.16) allows case (b) states of either parity, unless = 0, in which case p + N must be even. By a similar argument the term in square brackets in (4.21) vanishes unless p + L + N + is even, which means that a given parity block is restricted to either even or odd values of N + . In addition the Wigner coefficient vanishes in the special case = 0 unless N + + L + N is even, which requires that N + has the same parity as N + L. Consequently the e type block includes states with the labels

= 0, 1 . . . , min(L, N);

N + = |N − L|, N − L + 2, . . . , N + L,

while the f type block, with dimension L × L, includes the states

= 1, 2 . . . , min(L, N);

N + = |N − L| + 1, N − L + 3, . . . , N + L − 1.

Equation (4.21), which was first given by Fano [10], has been widely used in molecular applications of MQDT. As a simple, almost trivial example, we consider the np series in H2 , which was used with N = 1 in Section 2.4 to illustrate the 2

Note that for comparison with these authors the first two columns of the Wigner coefficient have been permuted and N + has been recognized as an integer in passing from (4.20) to (4.21).

100

Frame transformations and channel interactions

connection between spectral perturbations and auto-ionization. Taking L = 1, the e and f blocks have dimension 2 × 2 and 1 × 1, respectively. With labels ordered as = 0, 1 and N + = N − 1, N + 1 for e states and = 1 with N + = 1 for the f state, the transformation matrix U with components N + | NLp is given by ⎛  ⎞  N N+1 0 ⎜ 2N+1  2N+1 ⎟ ⎜ ⎟ U = ⎜ − N+1 (4.22) ⎟, N 0 ⎝ ⎠ 2N+1 2N+1 0 0 1 except that U collapses to a unit matrix (with e parity) for N = 0, because only

= 0 and N + = L are allowed. The upper block U agrees, for N = 1, with the form in Section 2.4.

4.2.2 -doubling The phenomenon of -doubling is concerned with changes in the rotational– electronic level structure as the energy increases [3, 15]. In the low-energy case (b) limit there are L distinct energy levels, because diatomic states for = 0 are doubly degenerate, whereas they evolve to form series converging on 2L + 1 rotational energy levels of the ion. Consequently the degeneracy must be lifted. The resulting -doubling arises in physical terms because, for example, the |pe  and |p components for L = 1, which lie in the plane perpendicular to N+ , can be mixed by the nuclear rotation, whereas the |pf  component lies parallel to N+ and therefore experiences no mixing. The magnitude of this mixing and the associated energy splitting, are normally treated by spectroscopic perturbation theory [15]. However, the frame transformation treatment takes into account not only the increased magnitude of the doubling as the principal quantum number increases but also the accidental perturbations between the high n series, and the auto-ionization above the lower convergence limit, which were discussed in Sections 2.4.2 and 2.4.3. Figure 4.4 provides an illustration by plotting the theoretical 1 u+ and 1 u eigenvalues of the p series of H2 , with total angular momentum N = 1, against the reduced energy, E − EnH , where EnH = I −

Ry . n2

(4.23)

The level positions are derived from (2.58) from the data given in Table 2.2. Filled and open circles are used for the e and f parity levels, respectively. The diagram shows how the doubling increases in magnitude with increasing n, until the e and

4.2 Rotational channel interactions

101

0

100 * (E − I) / Ry

−1

n 8

−2

7 6

−3 −4 −5

n 8 7 6

5 5

1

Σu+

1

Πu

−300 −200 −100 0 100 200 300 −1 (E − E n) / cm

Figure 4.4 A graphical representation of the case (b) to case (d) transformation for the np series of H2 with total angular momentum N = 1. Filled and open circles apply to the e and f parity levels, respectively. The reference energy En is defined by (4.23).

f branches of successive 1 u states break apart to form separate series, terminating on the N + = 2 and N + = 1 levels of H+ 2 , respectively. The irregularities, which first occur at an energy coincidence between n = 11 of the 1 u+ series and n = 10 of 1 u , are symptoms of the perturbations that were discussed in Chapter 2.

4.2.3 -uncoupling in non-penetrating series As a second illustration, which brings out the physical significance of the pseudoquantum number L = N − N + , we consider the uncoupling of  from the internuclear axis due to increasing angular momentum N in a single Rydberg cluster. The 4f-complex of NO is an interesting example in which the electronic energy varies between the components in the form [16] EL = TnL + C L | 3 cos2 θ − 1 |L    2C 3 2 − L(L + 1) = TnL − (2L − 1)(2L + 3)

L L 2 L− , = TnL + D(−1)

− 0

(4.24)

where C depends on the quadrupole moment, the anisotropy of the core polarizability and the principal quantum number, while D contains additional terms in L.

102

Frame transformations and channel interactions

The rotational–electronic Hamiltonian therefore contains the leading terms  |L  EL L | + B Nˆ +2 , (4.25) Hˆ =

from which the eigenvalues are normally derived in the case (b) representation, by substituting Nˆ + = Nˆ − Lˆ and using angular momentum shift operators Lˆ ± and Nˆ ± to handle the Coriolis effects [3]. Here we work in the case (d) representation, with matrix elements  N + |  E  | N +  + BN + (N + + 1)δN + N + , N + |Hˆ |N +  = TnL +

(4.26) where the superscripts NL are suppressed to simplify the notation. It is convenient to neglect parity in this situation, by allowing to take both positive and negative values and employing the unsymmetrized frame transformation matrix elements

+ √ L N N + N− +  | N  = (−1) . (4.27) 2N + 1 0 − One finds that by combining the triple products of 3j symbols [3]   N + |Hˆ |N +  = TnL + BN + (N + + 1) + D(−1)N 2N + + 1 ) +

( N N+ 2 L L 2 (4.28) × 0 0 0 N+ N+ N   C 3L2 − L(L + 1) + +  TnL + BN (N + 1)δN + N + + , (2L − 1)(2L + 3) where the final line, in which L = N − N + , follows from an asymptotic approximation for the 6j coefficient [23], ) 

( 3 L L 2 L L 2  , (4.29) N+ N+ N 2(2N + + N) −L L 0 which is valid for (N, N + )  (L, 2). It is seen by comparison with (4.24) that the electronic energy contributions to the case (b) and case (d) eigenvalues differ in effect by the substitution 1 L | 3 cos2 θ − 1 |L  =⇒ − LL| 3 cos2 θ − 1 |LL . (4.30) 2 The physical explanation for this change is that the body-fixed projection , which arises from the axial symmetry of the stationary molecule, becomes progressively ill-defined as N increases because the electronic motion fails to adjust to the increasingly rapid nuclear rotation. At very high N , however, the rapid rotation

4.2 Rotational channel interactions N ″= 20 2

3

35

40

4

5

Intensity

k=1

103

25

30

45

Intensity

nd N + = 21

39700

39750

39800

E (cm–1)

Figure 4.5 Stroboscopic fringes in the theoretical (upper) and experimental (lower) spectra obtained by double-resonance excitation of the nd series of Na2 , via the N

= 20 rotational level of the A 1 u+ state. Taken, with permission from Labastie et al. [17], in which J corresponds to the present N

.

sets up an average axially symmetric field, around the nuclear rotation vector N+ in Fig. 4.3, which is perpendicular to the molecular axis. Moreover, the total angular momentum N becomes parallel to N+ as its magnitude increases. Thus the difference L = N − N + may be properly quantized in the direction of the new axial field [16]. Finally the factor of −1/2 in (4.30) is associated with a change in the magnitude of the appropriate quadrupole component, from Qzz in case (b) to Qxx = −Qzz /2 in case (d), and similarly for the anisotropy of the polarizability.

4.2.4 Stroboscopic beats between electronic and nuclear motion The nd← A 1 u+ spectrum of Na2 , shown in Fig. 4.5 displays an interesting stroboscopic effect arising from the relative frequencies of the electronic and rotational motions [17]. Both the upper (MQDT simulation) and the lower (experimental) spectra show a recurrent ‘fringe’ pattern, which is interpreted here as arising from a periodic switch between case (b) and case (d) dynamics as the principal quantum number increases. The frame transformation in (4.21) again applies, because

+ = 0 in the 2 g+ state of Na+ 2 . Hence the five electronic channels for a given excited total angular momentum, N , divide into three e type and two f type channels with even and odd values of N + p , respectively, of which the latter, which actually provide the dominant intensity in the high n spectrum, give rise to series that converge on the N + = N ± 1 levels of the positive ion [24]. The two-channel

104

Frame transformations and channel interactions

theory is therefore analogous to that in Section 2.4.3, except that the rotational + constant of Na+ 2 is very much smaller than that of H2 . The clue to the stroboscopic behaviour is that the two-channel MQDT quantization condition   U tan π μU T + tan π ν (E) Z = 0 (4.31) uncouples to the diagonal form [tan π μ + tan π ν (E)] Y = 0,

(4.32)

where Y = U T Z, at energies such that tan π ν1 (E) = tan π ν2 (E), which means that the two numbers functions ν1 (E) and ν2 (E) differ by an integer [25]. Accidental occurrences of this nature therefore lead to coincident case (b) energy levels of the form E = E1+ − = E1+ −

Ry Ry = E2+ − 2 2 ν1 ν2 Ry Ry = E2+ −  2 , 2 [n1 − μα ] n2 − μβ

(4.33)

which are responsible for the stroboscopic fringes in Fig. 4.5. The lengths of these fringes may be estimated by expressing the ionization energy difference in terms of the rotational constant, B + . Bearing in mind that N + = N ± 1, E2+ − E1+ = 4B + (N + 1/2) =

Ry 2Ry (ν2 − ν1 ) Ry − 2  , 2 ν¯ 3 ν1 ν2

(4.34)

where ν¯ a suitable average of ν1 and ν2 . It follows that ν = (ν2 − ν1 ) 

2B + (N + 1/2) , R/¯ν 3

(4.35)

When applied to Na2 for which, B + = 0.11 cm−1 [24], the above classical estimates of ν¯ at N = 20 for k = ν = 1, 2, 3 and 4 take the values ν¯ = 29, 36, 42 and 46, broadly in line with the observations in Fig. 4.5. Similar integer recurrences occur in any two-channel system, at ν¯ intervals determined by (N + 1/2)B + . However, the persistence length of the fringes decreases with the magnitude of the rotational constant. Thus in the case of H2 , which was discussed in Section 2.4.3, with N = 1 and B + = 30 cm−1 , one finds that ν¯ = 13, 17, 19 and 21 at k = 1, 2, 3 and 4, which allows an interval of only four ν¯ values between k = 1 and k = 2.

4.3 Vibrational channel interactions

105

Thus the recurrences are detected as isolated perturbations to the np series of H2 , rather than persistent fringes.

4.3 Vibrational channel interactions Vibrational frame transformations are used to handle vibronic non-adiabatic interactions within a given Rydberg series. In conventional spectroscopic theory such interactions are introduced via the matrix elements of nuclear coordinate derivative operators between the Born–Oppenheimer electronic wavefunctions [26]. In MQDT theory, however, the response of the Rydberg wavefunction to changes in the nuclear geometry is carried by the quantum defect functions, written as μ (R) for a diatomic molecule, as given by the Rydberg formula3 Vn (R) = V + (R) −

Ry . [n − μ (R)]2

(4.36)

Since vibrational and rotational interactions cannot normally be separated, it is convenient to include a weak rotational dependence in the vibrational frame transformation, by defining the transformation elements in terms of the eigenfunctions of the centrifugally modified vibrational Hamiltonian. Thus v + N + |R = χv+ N + (R), where   h¯ 2 d2 N + (N + + 1) + + + χvN+ (R) = Ev+ N + χvN+ (R). − + V (R) + 2 2 2μN dR 2μN R

(4.37)

(4.38)

in which μN is the reduced mass of the nuclei. The resulting eigenvalues Ev+ N + , which constitute the channel thresholds, may often be replaced by experimental values, and the transformation elements may be approximated as v + 0 |R, at least for low N + values. The K-matrix responsible for the rotational and vibrational 3

Stolyarov et al. [27] relate the two approaches by using Rydberg scaling arguments to obtain the following approximations for the radial derivative matrix elements between members of a given Rydberg series, in terms of radial derivatives of μ (R): √ , ! '  2μ  νn νm  ∂R  n ∂m 2 νn2 − νm # $  2  2 √ '  2  , 2 ! μ 3νm + νn2 2 νn νm

  n ∂ ∂m ∂R  2 − μ ,  2 2 νn − νm νm νn2 − νm where νn2 (R) =

Ry . V + (R) − Vn (R)

106

Frame transformations and channel interactions

interactions then has elements ' !' ! !' Ni+ vi+ tan π μ vj+ Nj+ , Kij =

(4.39)

Ni+

|  is given by (4.21) in the simplest case, and the vibrational matrix where element is given by  ! ' + + (4.40) vi tan π μ vj = χvi+ (R) tan π μ (R)χvj+ (R)dR. Elementary techniques may be used to evaluate the low v + matrix elements, within the harmonic approximation. For example the expansion   2

(4.41) tan π μ (R)  tan π μ e + π sec π μe μ (R) − μe · · · leads to '

! v + tan π μ (R) v + = π αv + ,v+ sec2 π μe  (v + + 1) h¯ dμ e

αv+ +1,v+ = dR 2μN ωe √ + (v + 1)(v + + 2) h¯ 1 d2 μ e

αv+ +2,v+ = . 2 dR 2 2μN ωe

(4.42)

Complications can, however, occur in evaluating the integrals, owing to the divergence of tan π μ (R) when μ  ±1/2. For example the p quantum defect function of H2 in Fig. 3.2 increases almost by unity over the range 0 < R < 8a0 . Du and Greene [28] overcome such problems by first evaluating the matrix M of the smooth function μ (R), within a complete basis.4 The symmetry of M leads by an orthogonal transformation X to the diagonal form, λ = XMXT . After taking tangent functions and reversing the transformation, the matrix elements of tan π μ in (4.39) are given by !  ' + Xvi+ ,k (tan π λk ) XTk,v+ . (4.43) vi tan π μ vj+ = j

k

An alternative is to employ the sin–cos version of the theory in Appendix B.2, which requires evaluation of the non-divergent integrals v + | sin π μ (R)|v +  and v + | cos π μ (R)|v + . The following examples apply to the np series of H2 . The first illustrates the determination of bound state eigenvalues, by the equivalent of diagonalizing the 4

In practice it is necessary to perform convergence tests on the vibrational K-matrix elements for the problem in hand.

4.3 Vibrational channel interactions

107

vibrational–rotational Hamiltonian of the nuclear Hamiltonian in the positive ion basis. The difference is however that the MQDT procedure includes the bound vibrational–rotational levels in successive electronic states, as well as any accidental perturbations between them. The second illustration concerns the line-widths and branching ratios arising from vibrational auto-ionization. Before proceeding to these applications, a word is in order about the influence of nuclear mass within MQDT. Leaving aside the familiar reduced nuclear mass term in (4.38), we note that the accumulated phase function in the outer region # $ Ry  +  − , βi (E) = π (4.44) Ei − E depends on a Rydberg constant Ry = (m/me )R∞ , which is scaled by the reduced electron mass m = me M/(Me + M), where M is the mass of the positive ion.5 On the other hand the quantum defect function, μ (R), or electronic K-matrix elements are either extracted from fixed nucleus ab-initio potential curves, or computed by one of the fixed nucleus R-matrix procedures of Chapter 3, which assume infinite nuclear mass. While there is no strict resolution of this contradiction, it is handled in practical terms by recognizing the very different timescales between the motions inside and outside the ion core. The transit time through the core is so short that the nuclei are effectively stationary; hence the quantum defects are related to the potential functions in (4.36) by the infinite mass Rydberg constant, R∞ = 109737.318 cm−1 . There is, however, no question that the reduced mass m applies in the outer region, where the electron spends most of its time, and the resulting ‘normal’ mass correction may be shown to be equivalent to the inclusion of diagonal matrix elements of the one-electron operator −(m/8μ)∇e2 in ab-initio theory [26]. There are also matrix elements of the two-electron operators, −(m/4μ)∇1 · ∇k , involving the Rydberg electron at r1 and core electrons at rk , which contribute a small ‘specific’ term to the quantum defect specific

μtotal  (R) = μ (R) + (m/μ)μ

(R),

(4.45)

but detailed investigation, for the p series of H2 and D2 shows that its influence is very small, except for the lowest states [29]. 4.3.1 Bound-state eigenvalues In view of the substantial differences between the shapes of the potential energy curves in Fig. 4.6, the np series of H2 provides a severe test of the vibrational channel 5

The values Ry = 109707.42 and 109722.36 cm−1 apply, for example, to H2 and D2 , respectively.

108

Frame transformations and channel interactions

H 2+

E / au

−0.5

−0.6

Bʺ

D

Bʹ −0.7

1

Σu+

C

1

Πu

B −0.8 0

1

2

3

4 5 R /a 0

6

7

8

Figure 4.6 Potential curves for the low lying 1 u+ (solid lines) and 1 u (dashed lines) states of H2 and for the ground 2 u+ state of H+ 2.

coupling theory, in regard to both the bound vibrational–rotational eigenvalues and the rates of auto-ionization. It is appropriate to start by reviewing the early achievements of Jungen and Atabek in reproducing the vibrational–rotational level structure for the B, B , B

, C and D states of both H2 and D2 , which correspond to the n = 3 − 5 members of the np 1 u+ and n = 3 − 4 members of the 1 u series [29].6 The forms of the relevant quantum defect functions, as indicated in Fig. 3.2, were derived from accurate ab-initio potential curves [30]. A small additional ab-initio-determined isotope specific term was included for the B and C states [29]. It is essential to include the rotational frame transformation terms N + | , as given by (4.22), to conserve the total angular momentum N = N+ + ; and to account for the doubling, which increases markedly from the n = 3 in the C state to n = 4 in the D state. The associated -e mixing actually becomes so strong that the e-parity vibrational levels of the D state are completely predissociated for v  4 – a topic which will be covered in the next chapter. The case of the 2pσ B 1 u+ state is particularly challenging in view of the marked difference between the parameters ωe = 1358 cm−1 and Be = 20 cm−1 and the corresponding values, 2322 cm−1 and 30 cm−1 , for the H+ 2 ion. Expansions involvwere required for convergence. Table ing typically 10–15 vibrational states of H+ 2 1 + 4.1 lists results for the B u state of H2 , derived from the μp (R) quantum defect function for n = 2, which is seen from Fig. 3.2 to deviate significantly from the 6

Jungen and Atabek employed the sin–cos variant of the theory, as described in Appendix B.2.

4.3 Vibrational channel interactions

109

+

Table 4.1 Levels of the B 1 u state of H2 in cm−1 [29]. N =0

N =1

N =3

v

Expt

o-c

Expt

o-c

Expt

0 1 2 3 4 5 6 7 8 9 10

90203.51 91521.84 92803.31 94049.98 95263.18 96443.12 97590.8 98705.34 99789.14 100842.38 101864.65

−0.6 −0.1 −0.2 −0.1 +0.4 +0.4 −0.4 −2.3 −3.0 −3.0 −4.8

90242.32 91558.82 92838.61 94083.72 95295.81 96474.27 97621.55 98733.92 99817.74 100869.39 –

−0.7 0.0 −0.2 −0.2 +0.5 +0.3 +0.3 −2.6 −4.2 −3.7 –

90434.73 91741.67 93013.43 94251.52 95456.86 96629.47 97769.74 98878.75 99957.11 101003.19 102022.27

o-c −0.7 −0.1 −0.3 −0.2 +0.4 +0.2 −1.1 −1.6 −2.6 −3.0 −3.7

higher n functions as R increases. The agreement with experiment for both H2 and D2 is comparable to, or slightly better than, that obtained by conventional quantization within the B state potential, including the ab-initio adiabatic correction term n |∂ 2 ∂n /∂R 2  [30]. The modest decrease in accuracy for v  7 in both types of calculation is attributed to the neglect of configuration mixing for R > 4.5 a0 (see Fig. 3.1). Comparable results were also obtained for the for the higher electronic states, using small uniform energy corrections to μp (E, R), for n  3, and μp (E, R), for all n, which resulted in energy corrections of order 2–10 cm−1 . As further evidence of the detailed level of theoretical and experimental agreement, Table 4.2 compares the experimental doubling splittings for the C 1 u and D 1 u states with the measured values, which depend not only on the magnitude of the rotational coupling, but also on the vibrational energies of the B 1 u+ and the B 1 + u states, respectively, to which they are coupled. Even the sign reversal for the (v, N ) = (1, 5) level of the C state, which arises from an accidental perturbation by a higher-lying vibrational level of the B state is well reproduced.

4.3.2 Vibrationally induced auto-ionization and rotational branching ratios The presence of the rotational transformation factors N+ |  in (4.39) means that the fragments arising from vibrationally induced auto-ionization may be distributed between different rotational channels. We assume for the purpose of illustration that the auto-ionization occurs from a single bound state, denoted as |nv + N +  or |nv +  according to the relevant coupling scheme (see below) and that there is a single open vibrational channel v + = 0. The resulting rotational branching ratios

110

Frame transformations and channel interactions

Table 4.2 doubling, in cm−1 , in the C 1 u and D 1 u states of H2 [29]. C 1 u

D 1 u

v

N

MQDT

Expt

MQDT

Expt

0 0 0 1 1 1 2 2 2

1 3 5 1 3 5 1 3 5

1.1 5.9 11.6 1.1 4.9 −16.8 1.8 − 18.4

1.20 6.07 13.04 1.42 5.55 −26.49 1.18 −3.4 24.29

3.6 18.7 38.9 1.3 5.3 −0.5 3.8 15.8 18.4

3.7 19.6 42.6 1.4 6.2 +0.7 4.9 16.6 12.7

are proportional to the squares of the open-channel amplitudes, Zio , of the folded MQDT eigenvalue equation    −1 co K − tan π τ (E) I oo Z o = 0. (4.46) K oo − K oc K cc + tan β(E) The resonance is centred in the present single closed-channel approximation at the energy, En , at which K cc + tan β(En ) = 0. The behaviour at the resonance is clearly dominated by the folded part of the matrix, which is conveniently written as  −1 co K =− K oc K cc + tan β(E)

xxT , (En − E)

(4.47)

where x is a column vector with elements  −1/2 xi = cos βn βn Kic .

(4.48)

Such matrices have a single non-zero eigenvalue given by tan π τ (E) = K¯ oo +

 ; 2(En − E)

 = 2xT x = 2



xi2 ,

(4.49)

i

with eigenvalue components Zi = cxi , where c is a constant. Hence the branching ratios are given by x2 i ρi =  i 2 = .  i xi

(4.50)

4.3 Vibrational channel interactions

111

Explicit expressions for these partial and total widths, in the case of vibrationally induced auto-ionization, depends on whether or not the principal quantum number of the auto-ionizing state is large enough to assume an uncoupled, case (d), description. If so, the partial line-widths for auto-ionization from a well-defined closed channel |v + N +  to the open channel |v + N +  is given by # $2 2  cos β 4R y n + N + |  π αv+ ,v+ sec2 π μ , N(n)+ v+ →N + v+ = e  |N  π νn3

(4.51) where αv+ ,v+ is given by (4.42). Hence after summing over |N + v + , the total width becomes7 2  4Ry cos2 βn  + + v(n) N |  π αv+ ,v+ sec4 π μ (4.52) +N + = e  |N . 3 π νn +

v

On the other hand the quantum number change may be so large that the autoionization occuring from such low principal quantum numbers is so low that case (b) coupling applies. The partial and total line-widths are then given by (n)  v + →N + v + =

 4Ry cos2 βn   2

+ 2 α , + ,v + π sec π μe  |N  v π νn3

(4.53)

4Ry cos2 βn   2 4 π αv+ ,v+ sec π μ e . π νn3

(4.54)

and v(n) + =

The treatment of vibrational auto-ionization by MQDT methods dates from the early papers of Jungen and Dill [32]. More recently, Jungen and Pratt have given a comprehensive review of experimental and theoretical developments [33]. Here, attention is restricted to double-resonance experiments on the np series of H2 , which provide an opportunity to test both the full MQDT theory and the above perturbation expressions. The following account is taken from a theoretical study by Jungen et al. [34]. The relevant experiments by Dehmer et al. involved excitation via the N = 1, v = 1 level of the E 1 g+ double minimum state to the converging on + the |v + , N +  = |v + , 1 and |v + , 3 levels of H+ 2 (mainly for v = 1), which autoionize to the |0, 1 and |0, 3 open channels [35]. The theory therefore employed 7

Equation (4.52) generalizes the early form given by Herzberg and Jungen [31]   ! 2 4π Ry 1 dm μ ' + (n) m + v + +m,v+ = v + m ) , v (R − R e νn3 m! dR m by including the rotational channel coupling terms plus trigonometric terms arising from the MQDT formulation.

112

Frame transformations and channel interactions (a)

R(1)7pπ, v = 3 14 21 21 21

20 20 20

Intensity

Q(1)5pσ, v = 3 15 23 22 23 22 23 22

R(1)np3, R(1)np1, P(1)npσ , Q(1)npπ,

v =1 v =1 v =1 v =1

(b)

126340

126300 (2ω 1 +ω 2 ) (cm –1)

126260

Figure 4.7 (a) Experimental and (b) theoretical segments of the double resonance auto-ionization spectrum of H2 , excited via N = 1, v = 1 level of the E 1 g+ double minimum state. Assignments are given for the v = 1 series, unless otherwise stated. Taken from Jungen et al. [34].

two open channels and as many closed channels as were required to converge the resonance positions to within 0.3–0.9 cm−1 . Figure 4.7 compares a segment of the experimental spectrum with the MQDT model [34, 35]. The theoretical lines have been convoluted with an experimental line-width function, because the experimental ones are broadened by instrumental factors. Four series converging on the v + = 1 limits are assigned. Those marked R(1)np3 and R(1)np1 belong to series converging on N + = 3 and N + = 1, analogous to those shown in Fig. 4.4, except that they arise from the two e rotational channels with N = 2. Each has partial = 0 and = 1 character. The other two series P (1)npσ and Q(1)npπ involve, respectively, excitation to the single e rotational channel with |N,  = |0, 0 and the single f state with |N,  = |1, 1. As indicated in (4.22), both have purely N + = 1 character. There are also two isolated transitions labelled Q(1)7pπ v = 3 and R(1)npσ v = 2, which belong to higher vibrational series. The agreement between the upright experimental and inverted theoretical spectra is excellent, although there are some differences in intensity.

4.3 Vibrational channel interactions

113

Table 4.3 Line-widths, /cm−1 , and branching ratios ρ1 = (σN + =1 /σtot ) in the double resonance photo-ioinization spectrum of H2 . Blended lines are marked a b , . Entries labelled ‘exp’ and ‘MQDT’ are taken from [34]. Those marked ‘pert’ are given by equations (4.52)–(4.54). pert

exp

pert

Series

n

E MQDT

MQDT tot

tot

ρ1

ρ1MQDT

ρ1

R(1)p3 v = 1

14 15 20 21 22 23 20 21 22 23 20 21 22 23 7 5

126251.0 126329.7 126272.3 126293.9 126318.5 126339.1 126266.2 126291.9 126314.5 126334.2 126273.4 126298.5 126320.8 126339.5 126303.6 126319.1

0.19 0.41 0.05 0.16 0.22 0.02 0.41 0.34 0.03 0.25 0.014 0.012 0.005 0.009 0.006 0.007

0.63 0.51 0.33 0.29 0.25 0.21 0.45 0.39 0.34 0.29 0.014 0.012 0.011 0.009 0.007 0.0002

0.53 0.38 0.21 0.43 0.28 − 0.39a 0.39 − 0.70b 0.99 0.99 0.99 1.00 0.62 0.99 0.63 0.39 − 0.70b 0.40 0.28 − 0.39a

0.53 − 0.59 0.32 − 0.40 0.21 0.24 − 0.27 0.24a 0.24b 1.00 0.99 0.97 0.99 0.99 1.00 0.90 − 0.95 1.00b 0.82 1.00a

0.61 0.61 0.31 0.31 0.31 0.31 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.40 1.00

R(1)p1 v = 1

P (1)pσ v = 1

Q(1)pπ v = 1

R(1)pσ v = 2 Q(1)pπ v = 3

In considering information on the total line-widths and rotational branching ratios in Table 4.3, it is useful to recognize that the first two series, R(1)np3 and R(1)np3 for v = 1, are subject to perturbations of the types shown in Fig. 2.9. There are also two overlapping lines marked a ,b . It is seen that there is generally excellent agreement for the remaining series, between the MQDT and perturbative total linewidths, given by (4.51)–(4.54),8 except that the scaling with ν −3 is interrupted for two of the MQDT values. The marked difference between tot for the P (1)npσ and Q(1)npσ v = 1 series arises because |μ σ (Re )|  |μ π (Re )| in Fig. 3.2. The lines in the two final series, which are observed at energies below the v + = 1, 2 ionization thresholds, are also very narrow because they depend on the second and third derivatives of the quantum defect functions, respectively. The final point, with regard to the total line-widths of the P (1)npσ and Q(1)npσ v = 1 series is that there is satisfactory order-of-magnitude agreement between the MQDT and perturbative estimates, but the absence of any clear trend with ν −3 points to the influence of the rotational channel interactions in the full MDQT calculation, which are neglected in (4.51)–(4.54). 8

The quantum defect derivatives are taken from Fig. 3.2 and the rotational frame transformation elements N + |  are given by (4.22).

114

Frame transformations and channel interactions

Turning to the branching ratios, the experimental ρ1 values were found to vary across the blended features marked a ,b . Similar variations were also found by convoluting the two contributing peaks in the MQDT calculation, but the values in the table are given for the unconvoluted peaks [34]. The agreement between experiment and both types of calculation is seen to be good, except in the case of the long-lived Q(1)npπ series, for which the discrepancies are attributed to field effects in the magnetic bottle analyzer [34, 35]. Similar considerations may also apply to the two final experimental branching ratios, but discrepancy between pert ρ1MQDT = 0.8 and ρ1 = 0.4 for the R(1)pσ resonance suggests that the case (b) designation is oversimplified, because (4.53) implies that ρ1 = N + | 2 = 2/5 for N + = 1 and = 0. 4.4 Vibronic channel interactions 4.4.1 Configuration interaction Configuration interactions between different core electronic states |γi  and |γj  are handled by evaluating K-matrices of the form ' !' ! !' γi Ni+ γi vi+ Kij( ) (R) vj+ γj γj Nj+ , (4.55) Kij =

involving an electronic K-matrix function K( ) (R) in place of the tangent function in (4.39). The algorithm in (4.43) for evaluating vibrational matrix elements of μ (R) has been extended to the present situation by Ross and Jungen [36]. The first step is to diagonalize K( ) (R) by an orthogonal transformation K( ) (R) = V(R) tan π μ( ) (R)VT (R), which leads to the smooth matrix M( ) (R) = VT (R)μ( ) (R)V(R).

(4.56)

The matrix elements of the various blocks are then combined in the form ' ! ' + ! !' !+ γi Ni+ vi+ M γj Nj+ vj+ = Ni γi Ni+ ; vi+ |M ( ) γj Nj+ ; vj+ Nj .

(4.57) Finally the full vibronic K-matrix is constructed by (i) determining the matrix U, such that UT MU is diagonal, (ii) taking tangents of the diagonal result, and (iii) reversing the U transformation. Thus   K = U tan π UT MU UT , (4.58) provided that the vibrational basis is sufficiently large to converge the K-matrix elements relevant to the physical situation in hand.

4.4 Vibronic channel interactions

115

The first application of this result was to determine the bound vibrational levels of the double minimum, 1 g+ states of H2 , which are seen from Figs 3.1 and 3.3 to arise from interaction between the s and d channels converging on the bound 2 + 2 + g state of H+ 2 and the p channel that converges on the repulsive u state. The forms of the relevant quantum defect functions are shown in Fig. 3.4. The vibrational matrix elements in (4.57) were evaluated by quadrature over the range 0.1  R  12.0 a0 . A suitable vibrational basis for the repulsive 2 u+ state of H+ 2 was constructed by imposing the condition χv+ (R) = 0 at the outer integration boundary, and a similar construction was used to extend the bound 2 g+ vibrational progression into the dissociative continuum. The total number of vibrational states required for convergence involved levels with v + = 0−40 for the bound s and d channels, and v + = 0−109 for the repulsive p channel. The MQDT quantization equation9 reproduced 39 vibrational eigenvalues over the range 99100–17300 cm−1 with a mean deviation of 6.0 cm−1 compared with 3.0 cm−1 for a non-adiabatic coupled channel study [37]. They include the lowest 29 EF, lowest 7 GK and the lowest 3 H state levels, in a single calculation. The resulting model is extended in Chapter 5 to the treatment of competitive predissociation and auto-ionization and to the related process of dissociative recombination.

4.4.2 Jahn–Teller channel coupling Primitive model It is useful at the outset to distinguish between Jahn–Teller effects arising from symmetry-determined electronic degeneracies in the positive ion core and those arising from Rydberg orbital degeneracies. A striking example of the former is provided by CH+ 4 [38], but Staib et al. [39] conclude, albeit for a simpler system, that core Jahn–Teller coupling merely complicates the Rydberg-level structure without adding any new principles. The theory of the more interesting Rydbergbased effect has been formulated from slightly different standpoints by Staib et al. [39, 40] and Stephens and Greene [41]. Attention is restricted here to the simplest E × e case in which two Rydberg electronic states, with a symmetry-determined degeneracy, are linearly and weakly coupled by a doubly degenerate vibrational

9

Written

  tan π ν (E) det K + =0 A (ν)

because Yu et al. actually work in the Ham representation, as discussed in Section 2.2.2, by using η( ) (R) functions rather than the Seaton–Fano defect functions μ( ) (R).

116

Frame transformations and channel interactions

coordinate [42], with components Q2a = ρ cos φ,

Q2b = ρ sin φ.

(4.59)

The linearized electronic K-matrix takes the form [39, 40, 41]

K=

+1| −1|



|+1

|−1

tan π μe

λρe−iφ

λρeiφ

tan π μe

 ,

(4.60)

which diagonalizes to

where

K = U(φ) tan π μ(ρ)U† (φ),

(4.61)

iφ/2 1 e−iφ/2 e U(φ) = √ iφ/2 −e−iφ/2 2 e tan π μ± (ρ) = tan π μe ± λρ.

(4.62)

Notice that U(φ + 2π) = −U(φ), in agreement with the expected Berry (or Longuet–Higgins) phase [42]. The aim in what follows is to relate the spectroscopic Jahn–Teller theory [42] to the MQDT description of vibronic channel coupling close to and above the ionization limit. Note first that (4.61)–(4.62) imply a Rydberg sequence of potential matrices of the form Vn (ρ, φ) = V + (ρ) I − U† (φ)

Ry U(φ), [n − μ(ρ)]2

(4.63)

which simplify for small λρ, to a series of Jahn–Teller Hamiltonians at successive principal quantum numbers

kn ρe−iφ Tnα + Hˆ 0 (ρ, φ) ˆ , (4.64) Hn (ρ, φ) = kn ρeiφ Tnα + Hˆ 0 (ρ, φ) where Tnα = I −

Ry , ν2

kn =

2λRy cos2 π ν , π ν3

and

ν = n − μe .

(4.65)

Examples of such series are given by Staib et al. [39, 40]. Each such Hamiltonian has a dimensionless stabilization parameter

2 2 Ry 1 kn ρ0 2 λρ0 cos2 π ν (n) =2 , (4.66) D = 3 2 h¯ ω πν h¯ ω

4.4 Vibronic channel interactions

117

where ρ0 is related to the reduced mass, μρ and frequency ω by the formula  ρ02 = h¯ /μρ ω . In physical terms, D (n) , which falls off rapidly with increasing n, measures the energy difference between Tn and the potential minimum in units of h¯ ω. To understand the vibronic energy level structure, it is revealing to express the Hamiltonian in (4.64) in terms of the Pauli spin matrices, σx and σy . The resulting Hamiltonian     ˆ n (ρ, φ) = Tnα + Hˆ 0 (ρ, φ) I + 2kn ρ cos φσx + sin φσy , H (4.67) is readily confirmed to commute with the operator 1 1 ∂ jˆz = lˆz + σz = −i I + ∂φ 2 0

0 , −1

(4.68)

which means that the two-component vibrational eigenstates may be assigned a half-odd quantum number j , such that the upper and lower components vary as exp [(j − 1/2)φ] and exp [(j + 1/2)φ], respectively [42]. Furthermore, the energy ˆ n (ρ, φ) is independent of σz and levels are degenerate in the sign of j, because H 0 ˆ ˆ H (ρ, φ) is quadratic in lz . Perturbation formulae for the energy levels are readily derived. Using the notation |v, j, ± for an upper or lower degenerate harmonic oscillator component |v, , with  = j ∓ 1/2, the relevant vibronic matrix elements take the forms ρ0  v + 1, j, ∓| ρe±iφ |v, j, ± = √ v ± j + 3/2 2 ρ 0  v − 1, j, ∓| ρe±iφ |v, j, ± = √ v ∓ j + 1/2, (4.69) 2 from which the second-order energy levels are given by [42] Evj ± = (v + 1)¯hω ∓

kn2 ρ02 (2j ± 1) = [v + 1 ∓ (2j ± 1) Dn ] h¯ ω, 2¯hω

(4.70)

subject to the constraint that v  j − 1/2. The magnitudes of the resulting level separations, which are shown in Fig. 4.8, are the most obvious spectral signatures of the Jahn–Teller effect. Note the symmetry between the states |v, j, ± and |v, −j, ∓. It is also important for understanding the auto-ionization selection rules to see that, for example, v = 1 levels are coupled to v = 0 only for |j | = 1/2, because the |j | = 3/2 components with v = 1 couple to v = 2 and higher levels. It is outside the scope of his chapter to discuss the bound state structure at greater ˆ n (ρ, φ) is length, but the reader should be aware that a proper diagonalization of H (n) required for D > 0.1, and that quadratic and cubic corrections to the K-matrix,

118

Frame transformations and channel interactions

j =

−7/2 −5/2 −3/2 −1/2 1/2 3/2 5/2 7/2

3

+

+

+ +

1 0

+

+ +

υ

2

+

+ +

Figure 4.8 Vibronic level structure for the linear E × e Jahn–Teller effect.

that invalidate j as a constant of the motion, may be required as the coupling strength increases [43]. Turning to other aspects of the Rydberg spectrum, it may be seen that the Kmatrix in (4.60) commutes with the operator jˆz in (4.68). Hence the matrix elements in (4.69) also govern the magnitude of perturbations between interpenetrating Jahn– Teller manifolds and the vibronically induced auto-ionization rates at energies above the ionization limit. Following (2.64) and (4.60) the v → v − 1 perturbation matrix elements are given, for j > 0, by Hnv,n v−1,j

2Ry λρ0 cos π ν cos π ν =− π (νν )3/2

"

v − j + 1/2 . 2

(4.71)

Figure 4.9, which is taken from a full MQDT study by Staib and Domke [40], shows an interesting example in the spectrum of H3 , where close coincidences occur between the (n, v) = (5, v) and (7, v − 1) levels. Notice first that the small lines at E  −3000 cm−1 in panels (a) and (b) belong to v = 1 for which |j | = 1/2 or 3/2. The matrix element therefore allows a perturbation between the the (5, 1) and (7, 0) levels for j = 1/2, but not for j = 3/2. Similar arguments apply to the (5, 2) and (7, 1) levels, for which v = 2 → 1 coupling is allowed for j = 1/2 and j = 3/2, but not for j = 5/2. It is also interesting to observe broadening of the (5, 2) and (7, 1) peaks for j = 1/2 compared with those for j = 3/2 and 5/2, which is attributable to selective auto-ionization. The matrix elements governing this auto-ionization are again given by (4.60), subject to the selection rules v = −1 and j = 0. Following (2.51), the resulting

4.4 Vibronic channel interactions 1000

(a)

119

4 v=2 l=0 j = 1/2

500

5,2 7,1

0

1000

5,2 7,1

0 –4000 –3000 –2000 –1000 1000 (b) 4 v=2 l=2 j = 3/2

5,1

Absorption cross section (arb. units)

5,1 7,0

500

0 –4000 –3000 –2000 –1000 1000 (c) 4 v=2 l=2 j = 5/2

0

1000

5

500

0 –4000 –3000 –2000 –1000 E (cm–1)

0

1000

Figure 4.9 Vibronic perturbations between the (n, v) = (5, v) and (7, v − 1) levels of H3 for v = 1 and 2. Energies in cm−1 are measured from the ionization limit. The different panels apply for (a) j = 1/2, (b) j = 3/2 and (c) j = 5/2, and  = 0 and 2. Taken from Staib and Domke [40], with permission.

line-width is given by nvj

4Ry λ2 ρ02 cos2 π ν v − j + 1/2 = π ν3 2

3 2 2π ν (¯hω) v − j + 1/2 D (n) , = cos2 π ν Ry 2

(4.72)

where D (n) is given by (4.66). Hence auto-ionization between the v = 0 and v = 1 limits can occur, by the above argument, only for j = 1/2, which accounts for the line-width differences in Fig. 4.9. As written, the formula for nvj applies to the pure |n, v, j  = |7, 1, 1/2 resonance, but the perturbation with the |5, 2, 1/2 state transfers the width to both peaks.

120

Frame transformations and channel interactions

Equations (4.66), (4.71) and (4.72) provide various experimental signatures of the Jahn–Teller effect in the Rydberg spectrum, and it is interesting to compare their magnitudes, in the case of the npE Rydberg system of H3 , for which μe = 0.366 and D (3) = 0.0301, corresponding to an energy stabilization of E = 87 cm−1 [44]. The ν −6 scaling law implies D (4) = 0.0044 and E = 12 cm−1 at n = 4,10 falling to D (6) = 0.0003 and E < 1 cm−1 at n = 6. Hence the intra-channel second-order level splittings rapidly become too small to detect. On the other hand, the inter-channel perturbations, described by (4.71), arise from first-order interactions, which fall off only as (νν )−3/2 . An estimate of the matrix element for the (5, 2)−(7, 1) perturbation, derived from the D (3) stabilization parameter, yields 15| H |07 ∼ 60 cm−1 at j = 1/2. Similarly the auto-ionization line-width given by (4.72) is estimated as  ∼ 18 cm−1 for n = 7, falling to 6 cm−1 at n = 10. A further simplicity, which is exploited by Jungen and Pratt, in estimating the dissociative recombination cross-section (see Chapter 5), is that the mean contribution of a given resonance, averaged over the resonance spacing is independent of the principal quantum number [33]. Extended model The above discussion is intended to bring out specifically Jahn–Teller aspects of the vibronic channel coupling. Stephens and Greene extended the model to allow for simultaneous rotational channel coupling to the npA

2 states, abbreviated here to = 0, and to include all three nuclear degrees of freedom within the appropriate D3h symmetry [41]. Aspects of these extensions are outlined next, and the reader ˜ is referred to Watson and Spirko et al. for technical details of the symmetry and parity arguments required, for example, to handle the nuclear spin statistics [46, 47]. With regard to the latter, the experimental information which is discussed here, is available for the ortho (I = 3/2) nuclear spin states [48]. The extension to include rotational coupling requires the frame transformation [49] 

(2N + + 1) (1 + δK0 δK + 0 ) (1 + δK + 0 )

 N+  N p+N + + , × 1 + δK + 0 (−1) K + −K

N + K + | KN = (−1)N−

10

In agreement with an ab-initio estimate [45].

(4.73)

4.4 Vibronic channel interactions

121

which is analogous to the form in (C.15), plus a 3 × 3 coordinate dependent K-matrix

K(el)

|+1 ⎛ +1| tan π μ + δρ 2 e = −1| ⎝ λρeiφ 0 0|

|−1 λρe−iφ tan π μe + δρ 2 0

|0

⎞ 0 ⎠, 0 tan π μa (Q)

(4.74)

in which |±1 denote the components of the np(E ) electronic states while |0 denotes the np(A

2 ) component. The vibrational matrix elements of K(el) were evaluated in a properly symmetrized triple-product bond–oscillator basis states |v + ;  [47], whose degenerate components are conveniently taken in complex ! forms v + ; E± , which transform under the elements of D3h (M) as [1] ! ! ! ! (123) v + ; E± = ω±1 v + ; E± and (12) v + ; E± = v + ; E∓ , (4.75) where ω = e2iπ/3 . Stephens and Greene applied the theory to the ortho (I = 3/2) nuclear spin states of the np series of H3 , for which the positive-ion basis included states |N + K + ; v +  +  = |10; v + A 1 , |30; v + A 1 , |22; v + E  and |32; v + E  [41]. The inclusion of 10 A 1 and 20 E target vibrational states gave rise to 40 rovibrational channels for an MQDT calculation at N = 2. Three types of auto-ionization were identified. In the first place, there is well-defined Beutler–Fano resonance structure between the |N + K +  = |10 and |30 rotational limits of the ground vibrational state, |v +  +  = |0A 1  of H+ 3 . Secondly, vibrational auto-ionization in the series + converging on higher |10; v A1  and |30; v + A 1  limits can arise from the nuclear coordinate dependence of the diagonal terms in (4.74). Finally, the present Jahn– Teller mechanism, which involves a simultaneous change of electronic state, |  ⇒ | − , applies to series converging on the |22; v + E  and |32; v + E  limits. Comparison with experiment The H3 species is notorious in having only a single long-lived, B˜ 2p 2 A

2 , electronic state, the lowest vibrational–rotational state of which allows an ortho (I = 3/2) nuclear spin state. The np series was accessed by Bordas et al. [48] in a double resonance experiment involving the intermediate 3s 2 A 1 (N = 1) state, which resulted in final total angular momenta (apart from spin), N = 0, 1, 2. As discussed in Chapter 5, these possibilities may be distinguished experimentally by differences in the selection rules according to the relative polarizations of the two lasers. In the present case the states accessed with parallel and perpendicular geometries are N = 0, 2 and N = 1, 2, respectively.

122

Frame transformations and channel interactions 10 (a)

Intensity (arb. units)

c

a b 5 h f m

o n

k

l

i

e

dp

j

g r

s

0 0.60 Nf = 0 + Nf = 2

(b)

df /de(eV–1)

0.40

f 0.20 m

0.00 12900

a o

n

e d

g

h

j i

p

13300 13700 Photon energy (cm –1 )

r

s

l 14100

Figure 4.10 Comparison between the (a) experimental and (b) calculated photoabsorption intensity observed by two-photon double resonance excitation of H3 with parallel polarization. Taken from Stephens and Greene [41], with permission.

Bordas et al. [48] gave a convincing MQDT rotational channel analysis of the discrete spectrum and of the Beutler–Fano resonance structure between the |v +  + ; N + K +  = |0A 1 ; 10 and |0A 1 ; 30 limits (see Fig. 2.6). Note that K + = 0, because these series terminate on a non-degenerate vibrational state. Hence the analysis mirrors the discussion of the np series of H2 in Sections 2.4.2 and 2.4.3. In addition, Fig. 2.6 shows a perturbation in the Beutler–Fano region caused by an interloping member of a higher series. While no attempt was made to model the vibronic coupling at higher energies, tentative assignments were given for the auto-ionizing resonances shown in Fig. 4.10. Stephens and Greene extended the theory to include Jahn–Teller-induced autoionization at energies up to the |v +  + ; N + K +  = |1E ; 32 ionization limit [41]. It was noticeable that a calculation for final angular momentum N = 0 gave rise to many fewer resonances than for N = 1, 2, which is clear evidence of the importance of the Jahn–Teller mechanism, because the nuclear spin constraints allow auto-ionization at N = 0 only via the nuclear coordinate dependence of diagonal terms in (4.74). Figure 4.10 shows that there is also good agreement between the

References

123

experimental and theoretical resonance structure, bearing in mind the complexity of the system. Moreover, some of the experimental peaks, including those marked ‘b,c’, are assigned [48] to nd series, which were excluded from the calculation. References [1] P. R. Bunker and P. Jensen, Molecular Symmetry and Spectroscopy, 2nd edn (NRC Press, 1998). [2] J. M. Brown and A. Carrington, Rotational Spectroscopy of Diatomic Molecules (Cambridge Molecular Science, 2003). [3] R. N. Zare, Angular Momentum (Wiley-Interscience, 1988). [4] U. Fano, Phys. Rev. A 24, 619 (1981). [5] K. Sakimoto, J. Phys. B 22, 3011 (1986). [6] P. F. O. Mahony and K. T. Taylor, Phys. Rev. Lett. 57, 2931 (1986). [7] A. Mank, M. Drescher, T. Huth-Fehre et al., J. Chem. Phys. 95, 1676 (1991). [8] A. Osterwalder, A. W¨uest, F. Merkt and C. Jungen, J. Chem. Phys. 121, 11810 (2004). [9] C. Jungen and S. T. Pratt, J. Chem. Phys. 129, 16430 (2008). [10] U. Fano, Phys. Rev. A 2, 353 (1970). [11] U. Fano, J. Opt. Soc. Am. 65, 979 (1975). [12] M. S. Child, Semiclassical Mechanics, with Molecular Applications (Oxford University Press, 1991). [13] C. Jungen and A. L. Roche, J. Chem. Phys. 110, 10791 (1999). [14] R. W. Field, C. M. Gittins, N. A. Harris and C. Jungen, J. Chem. Phys. 122, 184314 (1979). [15] H. Lefebvre-Brion and R. W. Field, The Spectra and Dynamics of Diatomic Molecules. (Academic Press, 2005). [16] C. Jungen and E. Miescher, Can. J. Phys. 47, 1769 (1969). [17] P. Labastie, M. C. Bordas, B. Tribollet and M. Broyer, Phys. Rev. Lett. 52, 1681 (1984). [18] F. Hund, Handb. der Physik, Band I. 24, 561 (1933). [19] E. E. Nikitin and R. N. Zare, Mol. Phys. 82, 85 (1994). [20] E. S. Chang and U. Fano, Phys. Rev. A 6, 173 (1972). [21] M. S. Child and C. Jungen, J. Chem. Phys. 93, 7756 (1990). [22] C. Jungen and G. Raseev, Phys. Rev. A 57, 2407 (1998). [23] G. Ponzano and T. Regge. In Spectroscopic and Group Theoretical Methods in Phyics, ed. F. Bloch, S. G. Cohen, A. de-Shalit, S. Sambursky and I. Talmi (North Holland, 1968). [24] S. Martin, J. Chevaleyre, M. C. Bordas et al., J. Chem. Phys. 79, 4132 (1983). [25] K. T. Lu, Phys. Rev. A 4, 579 (1971). [26] W. Kołos and L. Wolniewicz, Rev. Mod. Phys. 35, 473 (1963). [27] A. V. Stolyarov, V. I. Pupyshev and M. S. Child, J. Phys. B 30, 3077 (1997). [28] N. Y. Du and C. H. Greene, J. Chem. Phys. 85, 5430 (1986). [29] C. Jungen and O. Atabek, J. Chem. Phys. 66, 5584 (1977). [30] W. Kołos and L. Wolniewicz, Can. J. Phys. 45, 2189 (1975). [31] G. Herzberg and C. Jungen, J. Mol. Spec. 41, 425 (1972). [32] D. Dill and C. Jungen, J. Phys. Chem. 84, 2116 (1980). [33] C. Jungen and S. T. Pratt, Phys. Rev. Lett. 102, 023201 (2009). [34] C. Jungen, S. T. Pratt and S. C. Ross, J. Phys. Chem. 99, 1700 (1995). [35] J. L. Dehmer, P. L. Dehmer, S. T. Pratt, F. S. Tomkins and M. A. O’Halloran, J. Chem. Phys. 90, 6243 (1989).

124 [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49]

Frame transformations and channel interactions S. C. Ross and C. Jungen, Phys. Rev. A 49, 4364 (1994). S. Yu, K. Dressler and L. Wolniewicz, J. Chem. Phys. 101, 7692 (1994). H. J. Worner, X. Qian and F. Merkt, J. Chem. Phys. 126, 144305 (2007). A. Staib, W. Domke and A. L. Sobolewski, Z. Physik D 16, 49 (1990). A. Staib and W. Domke, Z. Physik D 16, 275 (1990). J. A. Stephens and C. H. Greene, J. Chem. Phys. 102, 1579 (1995). H. C. Longet-Higgins. In Advances in Spectroscopy II, ed. H. W. Thompson (Interscience, 1961). R. Englman, The Jahn–Teller Effect in Molecules and Crystals (Wiley, 1972). G. Herzberg and J. K. G. Watson, Can. J. Phys. 58, 1250 (1980). H. F. King and K. Morokuma, J. Chem. Phys. 71, 3213 (1979). J. K. G. Watson, J. Mol. Spec. 103, 350 (1984). ˜ ˜ V. Spirko, P. Jensen, P. R. Bunker and A. Cejchan, J. Mol. Spec. 112, 183 (1985). M. C. Bordas, L. J. Lembo and H. Helm, Phys. Rev. A 44, 1817 (1991). S. Pan and K. T. Lu, Phys. Rev. A 37, 299 (1988).

5 Competitive fragmentation

This chapter extends the theory to include molecular dissociation into neutral fragments, as well molecular ionization. The first application is to predissociation, which is the dissociative analogue of auto-ionization, in the sense that interaction with a lower-lying open channel leads to slow dissociation at an energy below the adiabatic dissociation limit for the state in question. Situations can also occur in which a particular state lies above both the lowest ionization and dissociation limits, in which case there is competitive fragmentation into the two types of product. Dissociative recombination is a closely related reverse process, whereby an electron is temporarily captured by a positive ion, via reverse auto-ionization, after which the captured intermediate dissociates into neutral fragments. The connections between these processes are indicated below. ( A+B predissociation † AB + hν → AB → + AB + e auto-ionization AB+ + e → AB† → A + B

dissociative recombination

The theory proceeds by extending the K-matrix to include dissociation as well as ionization channels, a procedure that also allows for the treatment of vibrational perturbations between different electronic states. The ideas are first introduced in a diatomic curve-crossing model by using continuum perturbation theory to handle the predissociation rates from different members of a Rydberg series. Extension of this perturbation model to include auto-ionization leads in Section 5.3.1 to an account of the general connection between auto-ionization, predissociation and dissociative recombination, including a distinction between direct and indirect contributions to the recombination. Applications of a specially devised two-step theory of dissociative recombination are then described. Subsequent sections replace the perturbation model by a more general R-matrix treatment of the dissociation channels, which is closely allied to the eigenchannel 125

126

Competitive fragmentation +

2



E / 1000 cm–1

140

120

100 0

(a) 2

4 6 R / a0

(b) 8

10

Figure 5.1 (a) Interacting potential curves roughly modelled on interactions with the (E, F ), (G, H ), etc., 1 g+ states of H2 . Positions of the v = 3 vibrational levels are indicated. (b) Energy dependence of the squared overlap v + |Fd 2 between the v + = 3 bound state and the ‘energy normalized continuum’, Fd (E, R). The text describes how the dashed part of the curve is continued into the bound region.

R-matrix theory in Section 3.5.2. Applications to competitive auto-ionization and predissociation in H2 and to the Jahn–Teller induced dissociation recombination of an electron with H+ 3 are described.

5.1 Perturbation model for diatomic species Standard texts typically treat predissociation and auto-ionization as separate decay mechanisms, each with its own golden rule formula – see for example LefebvreBrion and Field [1]. Here, the aim is to provide a unified MQDT-based theory of vibrational perturbations, predissociation and auto-ionization, which can be extended to dissociative recombination. The present section is devoted to a perturbative curve-crossing model for diatomic species, with emphasis on purely vibrational channel coupling. To set the scene, Fig. 5.1(a) illustrates a simplified potential model relevant to the familiar E,F 1 g+ states of H2 , in which the horizontal lines indicate the energies of the v = 3 vibrational states in successive members of the Rydberg series. As a measure of the energy variation of the valence–Rydberg coupling strength, Fig. 5.1(b) plots the squared overlap integral v + |Fd (E)2 between the unit

5.1 Perturbation model for diatomic species

127

normalized v + = 3 state of H+ 2 and the energy normalized dissociative continuum function Fd (E, R), which largely determines the predissociation matrix element. The dashed section extends this latter curve into the purely bound region, by ‘Wronskian normalization’ of the bound states, along lines that were first applied to valence–Rydberg perturbations in NO by Raoult [2].1 Note, for future reference, that the qualitative form of v + |Fd (E) function ‘reflects’ the nodal structure of the v + = 3 wavefunction, because the overlap samples regions dominated by the monotonically decreasing valence potential [3]. In other words, there are two further lobes of v + |Fd (E)2 at higher energies. Moreover the v + = 0 overlap function consists of a single lobe. The theory proceeds by extending the K matrix to include not only the Rydberg– Rydberg channel coupling elements ! ' (5.1) Kv(I+)v+ = v + tan π μα (R) v + , but also the Rydberg–valence elements, Kv(I+)d (E), which are written as explicitly energy dependent, to account for the variation in Fig. 5.1(b). As a complement to this discussion, the generalized theory in Appendix E shows how MQDT arguments can be extended to the dissociative motion. In particular, each valence state, with potential V (R), has regular and irregular solutions of the nuclear Schr¨odinger equation  ) ( ) ( h¯ 2 d2 Fd (E, R) Fd (E, R) − =E , (5.2) + V (R) Gd (E, R) Gd (E, R) 2μN dR 2 the continuum versions of which are energy normalized by the Wronskian condition W [Fd , Gd ] = 2μ/πh¯ 2 . Coupling with the Rydberg states is introduced at a perturbative level by considering the inhomogeneous nuclear equation  2  d2 h¯ h¯ 2 + E − V (R) χ (E, R) = X(E, R), (5.3) d 2μN dR 2 2μN 1

In practice the regular dissociative wavefunction Fd (E, R), which was required to compute the overlap v + |Fd (E), was integrated outwards from a suitably small R value and scaled by a factor N −1 designed to ensure a smooth JWKB-like [3] behaviour at Re .  R

1 sin k(R)dR + π/4 Fd (E, R) = √ π k(R) a The scaling factor is given by

 N 2 = π Fd2 (Re ) + Fd 2 (Re )/k 2 (Re ) × k(Re ),

where k 2 (R) = 2μN [E − V (R)]/¯h2 .

128

Competitive fragmentation

in which X(E, R) is product of the Rydberg–valence interaction potential Vd (E, R) and a unit-normalized vibrational wavefunction of the positive ion;2 X(E, R) =

2μN Vd (E, R)χv+ (R). h¯ 2

The Green’s function solution of (5.3) takes the form [4]  ∞ G(R, R )X(E, R )dR , χd (E, R) = Fd (E, R) +

(5.4)

(5.5)

0

where 1 G(R, R ) = − W

(

Fd (E, R)Gd (E, R ) R < R Gd (E, R)Fd (E, R ) R > R .

(5.6)

It follows by substituting in (5.5) and taking the limit R → ∞ that χd (E, R) ∼ Fd (E, R) − Gd (E, R)Kv(I+)d (E),

(5.7)

where Kv(I+)d (E) is the first-order approximation to the Rydberg valence K-matrix element  ∞ 1 (I ) Kv+ d (E) = Fd (E, R )X(E, R )dR (5.8) W 0  = π Vv+ d (E) = π χv+ (R)Vd (E, R)Fd (E, R)dR. Factors of 2μN /¯h2 in X(E, R) and in the Wronskian have cancelled in reaching this result. In addition Vv+ d (E) like Kv(I+)d (E) is strictly dimensionless because it is implicitly measured in the units employed for the energy normalization. The presence of the first-order bound-continuum interaction, Kv(I+)d (E), also implies the following second-order or off-shell contribution to the Rydberg– Rydberg K-matrix elements   Vv+ d (E )Vv+ d (E ) P dE (5.9) Kv(I+Iv)+ (E) = −π

E − E d which was found to be essential to account for large valence–Rydberg perturbations at energies below curve crossings, such as those in Fig. 5.1(a), because the firstorder terms die rapidly away in this region [2]. Investigations by Guberman and Giusti-Suzor also improved earlier treatments of predissociation, auto-ionization and dissociative recombination by incorporating the second-order term [5, 6, 7]. 2

The factors involving h¯ 2 /2μN are introduced to simplify the form of (5.5).

5.2 Diatomic predissociation

129

5.2 Diatomic predissociation In considering the theory of pure predissociation, it is useful to examine the energy dependence of the matrix elements Kv(I+)d (E), and to see how it relates to the variation in predissociation rates from one member of a Rydberg series to the next. Notice that a large part of the energy dependence comes from the nodal structure of the valence state wavefunction, which is responsible for the oscillatory variation of the nuclear overlap integral. In addition, the integral in (5.8) is known to be dominated by the value of the integrand at the ‘crossing point’ at which the kinetic energies in the two potential functions are equal [3]. Now the crossings between the valence state and the Rydberg potentials in Fig. 5.1(a), Vnn (R) = V + (R) −

Ry , [n − μ(R)]2

(5.10)

move to smaller R as the principal quantum number increases. Hence there is a possible contribution to Kv(I+)d (E) from the radial and energy dependence of the interaction potential Vd (E, R) in (5.8). However, the extensive pattern of 2  Rydberg-valence perturbations in NO were found to be adequately described by taking Vd as constant [2]. Finally it should be noted that the resonance energies Env = Ev+ −

Ry , (n − μ¯ v )2

(5.11)

which are drawn for v = 3 and n = 2 − 5 in Fig. 5.1(a), span such a wide energy range that the oscillatory contribution to the integral plays a significant role in determining the scaling of predissociation rates with n, at least for relatively low n values. Turning to the MQDT analysis, the working equations take the familiar form  o  oo Z K oc (E) K − tan π τ (E)I oo = 0, (5.12) co cc K + tan β(E) Zc K (E) or

   −1 co K (E) − tan π τ (E) Z o = 0, K oo − K oc (E) K cc + tan β(E)

(5.13)

in which K cc are the Rydberg–Rydberg elements, given by (5.1), K oc (E) is the Rydberg–valence interaction element and K oo = 0, to the extent that Fd (E, R) in (5.6) is unperturbed by valence–valence interactions. The resonances typically occur close to isolated zeros of det [K cc + tan β(E)], at which one of the eigenvalues vanishes. Thus  −1 co K oc (E) K cc + tan β(E) K (E) 

[K oc (Env )]2 sec2 βv βv (E − Env )

(5.14)

130

Competitive fragmentation

over the energy range of interest, where βv (E) = π νv (E), as implied by (5.11). It follows with, K oo = 0, that tan π τ (E) 

 (nv) , 2(Env − E)

(5.15)

where with the help of (5.8) and the identity βv = π ν 3 /2Ry ,  (nv) 4π [Vv+ d (Env )]2 cos2 βv = = 2π [Vnv+ d (Env )]2 , Ry ν3

(5.16)

in which Vnv+ d (Env ), at a given quantum number, n, differs from the interaction strength, Vv+ d (Env ), by the scaling factor, N 2 = π cos2 β/β , between unitnormalized and energy-normalized states in (A.29). As emphasized, the scaling of the actual predissociation rate is governed by the energy variation of Vv+ d (Env ), as well as the expected ν −3 factor arising from the amplitude of the Rydberg orbital in the core region. However the rapid decrease in energy spacings as n increases means that the energy variation of Vv+ d (Env ) may be ignored for high-lying states. The residual scaling with ν −3 is particularly interesting in the context of pulsed-field ZEKE spectroscopy, which requires that highly excited Rydberg states with n  150 should be stable over periods of order 400 ns between the excitation and ionization pulses – a condition that is often inconsistent with (5.16). The NO molecule is a good example because it has strong, np and nf, Rydberg series, which are known to decay predominantly by predissociation and accurate line-width measurements for n = 10–20 conform to n  0 /ν 3 with p0  4000 cm−1 and f0  1500 cm−1 for the v = 0 series [8, 9]. Further lifetime measurements at intermediate principal quantum numbers closely confirm the ν 3 scaling law for the f series, while the p series follow an nk dependence with k  2.7 [10]. The implied lifetimes are however too short to allow ZEKE-PFI detection, but there is a marked change to much longer lifetimes at n = 115 and n = 65 for the p and f series, respectively. It is therefore inferred that the remarkable stability of the ZEKE states arises from -mixing with much higher non-penetrating angular momentum states, under the influence of stray or deliberately applied electric fields [11].

5.3 Dissociative recombination and related phenomena 5.3.1 Competing processes The theory of dissociative recombination is conveniently couched in terms of a composite K-matrix, of the type encountered in Section 5.1, which is open to fragmentation into both ionization and dissociation channels. The system is again

5.3 Dissociative recombination and related phenomena

subject to the familiar folded MQDT equations,  phys  K − tan π τ (E)I oo Z o = 0,

131

(5.17)

where   K phys = K oo − K oc K cc + tan β(E)−1 K co .

(5.18)

The difference from (5.13) is that the physical K-matrix, K phys now has both open ionization and open dissociation channels, which will be labelled as |i and |d, respectively. In addition the scattering aspect of the dissociative recombination process must be recognized by transforming K phys to the S-matrix form [12] S phys = (I − iK phys )−1 (I + iK phys ),

(5.19)

in terms of which the partial integral cross-section for scattering from an incident ionization channel |i to a dissociative channel |d is given by [12, 13, 14] π  phys 2 σid = 2 g Sid , (5.20) ki where ki2 is related to the initial kinetic energy of the electron by ki2 = 2μe (E − Ei+ )/¯h2 . The sum is taken over the orbital angular momenta, , of the incident electron, and g denotes the degeneracy factor g=

(2 − δ 0 )(2S + 1) , 2(2S + + 1)

(5.21)

where and S are the electronic angular momentum projection and spin of the relevant dissociation channel, and S + is the spin of the incident positive ion. There is also the possibility of vibrational energy transfer between states |i = |v +  and |i  = |v +  of the positive ion, with cross-sections given by π  phys 2 σii = 2 , (5.22) Sii ki with a degeneracy factor of unity, because there is no change in multiplicity. In considering the sums in (5.20) and (5.22) it is useful to recognize the selective nature of these processes. Strong valence–Rydberg interactions configuration interactions will normally be limited to low, strongly penetrating, values of . In addition there is often marked selectivity with regard to the symmetry of the channel in question. For example the strong 1 g+ interaction in Fig. 5.1 is only accessible to the pσ component of the  = 1 partial wave. Two types of mechanism are commonly recognized. The first, which is illustrated in Fig. 5.2(a), involves direct coupling between the ionization and dissocioo in the open–open part of the physical ation continua via off-diagonal terms Kid

132

Competitive fragmentation

V

E

(a)

E

(b) R

R

Figure 5.2 Potential curves to illustrate (a) the direct and (b) the indirect dissociative recombination mechanisms. The dashed curves belong to the positive ion, while the solid bound curves belong to a Rydberg state.

K-matrix, which is passed on to the S-matrix by (5.19). This mechanism will be most effective if the crossing between the bound and dissociative potential curves occurs within the classically accessible range of the initial vibrational state |v + , which is drawn in the diagram for v + = 0. By contrast the indirect mechanism, illustrated in Fig. 5.2(b), involves temporary capture into a Rydberg vibronic state, via inverse auto-ionization, followed by predissociation on the repulsive potential curve. Such processes are induced by off-diagonal elements of the folded matrix in (5.18). Both mechanisms operate in practice, but panel (b) illustrates a case for which energy-conserving crossings to the repulsive curve from the ionic and Rydberg curves are classically forbidden and classically allowed, respectively, which will favour the indirect route. As shown in more detail below, the direct contribution to the cross-section is expected to vary smoothly with energy, while the indirect terms contribute additional resonance structure. Direct process According to the perturbation model of Section 5.1, the K-matrix for direct coupling between the ionization channels |i and a single dissociation channel |d has an open – open block of the form ⎞ ⎛ 0 ξ1 · · ξ n ⎜ ξ1 0 0 0 0 ⎟ ⎟ ⎜ oo ⎟ (5.23) K =⎜ ⎜ · 0 0 0 0 ⎟, ⎝ · 0 0 0 0⎠ ξn 0 0 0 0 where ξi = Kid = π Vvi+ d , as given by (5.8). Such matrices have two non-zero  corresponding noreigenvalues λ = ±ξ , with ξ 2 = i ξi2 and the elements √ of the(±) √ (±) (±) are given by ζd = ±1/ 2 and ζi = ξi / 2ξ . The malized eigenvectors ζ

5.3 Dissociative recombination and related phenomena

133

remaining eigenvectors ζ (k) are chosen to be mutually orthonormal and orthogonal to ζd(±) . The resulting S-matrix S = U e2iπμ U T ,

(5.24)

where π μ± = ± tan−1 ξ, and π μk = 0, has direct coupling elements Sdi = 2i sin 2π μ+ (ξi /ξ ),

(5.25)

from which [6, 14] σdi(dir) =

4ξi2 gπ   . ki2 1 + ξ 2 2

(5.26)

There is a universal energy dependence from the factor ki−2 ∝ (E − Ei )−1 , and a possible further smooth energy variation from the elements ξi . Indirect process Purely indirect processes are handled by neglecting the direct term K oo in (5.18). Resonant contributions to the cross-section then typically arise close to isolated zeros of (E) = det [K cc + tan β(E)], at which one of the eigenvalues vanishes. Hence if xn denotes the normalized eigenvector   cc (5.27) K + tan β(En ) xn = 0, one finds that 

−1 K cc + tan β(E) ≈

xn xTn ,  n (E − En )

(5.28)

where  n = (d/dE), evaluated at En . It follows that  −1 co K Phys = K oc K cc + tan β(E) K 

γ n γ Tn , 2(E − En )

(5.29)

where γ n = (2/ n )K oc (En )xn is a column vector with N o components. Matrices of this form have a single non-zero eigenvalue tan π τn (E)  where  = γTγ =

 , 2(Env − E)

 i

γi2 =

√ with the normalized eigenvector γ˜ = γ / .

 i

i ,

(5.30)

(5.31)

134

Competitive fragmentation

Written in another way, this means that K phys  GPn GT ,

(5.32)

where Pn is a matrix with dimension N o × N o , all of whose elements are zero except that Pnn = tan π τn , and G is an N o × N o orthogonal matrix, whose nth row is the vector GT = γ˜ T . It follows that   S phys = I + G e2iπτn (E) − 1 GT = I + 2iGeiπτn (E) sin π τn (E)GT , (5.33) where I is the N o × N o unit matrix. Typical off-diagonal elements are given by phys

Sid

= 2ieiτn (E) sin π τn (E)γ˜i γ˜d .

(5.34)

Taken together with (5.30) and (5.32) this means that the purely indirect contribution to the DR cross-section for E  En takes the Breit–Wigner form [13] σid(ind) (E) = = 

πg phys S ki2 id

2

4πg 2 sin π τn (E)γ˜i2 γ˜d2 ki2 ki2



(5.35)

πgi d . (En − E)2 +  2 /4

Fano line-shape Equations (5.26) and (5.35) are, of course, limiting forms. The full cross-section involves interference between the two types of contribution, which is expected to lead to a Fano line-shape profile instead of the Lorentzian form in (5.35) [6]. There is no easy general analysis, but the essence of the situation is caught by a simplified model involving directly coupled open dissociation and ionization channels, both of which also interact with a closed channel. Using the notation of (5.23) and (5.29), the relevant folded physical K-matrix takes the form

ξ + aγi γd aγd2 , (5.36) K phys = ξ + aγi γd aγi2 where a = [2(E − En )]−1 . Hence, after some manipulation, σid =

gπ phys S ki2 id

= σid(dir)

2

=

( + q)2 , 1 + 2

4(γi γd + a −1 ξ )2 gπ   ki2 2ξ γi γd + a −1 (1 + ξ 2 ) 2 + (γi2 + γd2 )2 (5.37)

5.3 Dissociative recombination and related phenomena

135

in which σid(dir) is given by (5.26) with ξi = ξ , because the present model allows only a single open ionization channel. In addition √   (1 + ξ 2 )(E − En ) + ξ i d , =2  √ (1 − ξ 2 ) i d q= . (5.38) ξ i + d Equation (5.37) confirms that resonant indirect contributions to the cross-section take the expected Fano form [6]. The ξ dependence of the asymmetry parameter is open to question in view of the restriction to a single ionization channel, but the  dependence is fully in accord with physical intuition. In the case of optical excitation the line-shapes in Fig. 2.7 depend on the relative amplitudes for transitions to the open and closed channels. Now the asymmetry parameter depends on the ratio of fragmentation rates, i / d , from the resonant intermediate to the ionization and dissociation channels. The dissociative recombination of NO+ provides an interesting example. In a competitive study of the auto-ionization and predissociation from the 2  states of NO, Giusti-Suzor and Jungen find that auto-ionization accounts for less than 5% of the fragmentation cross-section, at energies between the v + = 0 and v + = 1 ionization thresholds, rising to 10–15% below the v + = 2 threshold [5]. Correspondingly, Sun and Nakamura find that the v + = 0 dissociative recombination cross-section in Fig. 5.3 appears as a sequence of window resonances corresponding to q  1, whereas the v + = 1 and 2 cross-sections show oscillatory indirect structure appropriate to larger values of the Fano q parameter [15]. The same authors find that v + = 1 cross-section via the L 2  state shows almost symmetrical up–down indirect structure appropriate to q  1. The other notable feature of Fig. 5.3 is the sharp dip in the direct part of the v + = 1 cross-section at E  0.75 eV, which is a typical consequence of the oscillatory overlap function in Fig. 5.1. A similar dip for v + = 2 occurs at E  2.0 eV [15].

5.3.2 Two-step theory The theory of dissociative recombination is frequently handled by a two-step procedure of Giusti-Suzor, which employs quantum defect functions in place of Kmatrix elements [6]. Details are also given by Giusti-Suzor and Jungen [5], Greene and Jungen [16] and Nakashima et al. [17]. The essence of the method is to divide the K-matrix into two parts K = K (1) + K (2) ,

(5.39)

136

Competitive fragmentation 10–14

10–10

v+ = 2

Cross section (cm2)

10–15

10–11

v+ = 1

10–16

10–12

10–17

10–13

10–18

10–14 v+ = 0

10–19

10–15

10–20

10–16

10–21

10–17

10–22 0.02

0.1

0.1

1.0

10–18 2.0

Electron energy (eV)

Figure 5.3 Contributions to the NO+ + e dissociation recombination crosssections for v + = 0–2, via the B 2  valence state. To avoid congestion, the right-hand vertical scale applies to the v + = 0 cross-section. Taken from Sun and Nakamura [15], with permission.

with K (1) taken in the present application as the Rydberg–Rydberg K-matrix, given by the combination of (5.1) and (5.9), while K (2) provides the Rydberg–valence coupling, with elements given by (5.8). The two sets of quantum defects are defined, in the notation of Giusti-Suzor and Jungen, by the transformations [5]   Kii(1) = i |β tan π μβ β|i , Kii(2) = i |α tan π μα α|i . (5.40) α

β

Subsequent manipulations follow the lines outlined in Appendix B. Thus, following (B.15), the outer part of the wavefunction under the influence of K (1) is given by   (1) |i fi (r)δii − gi (r)Kii(1) Zi(1) = (5.41) rout

ii

=

 iα

  |i i |β fi (r) cos π μβ − gi (r) sin π μβ A(1). β

5.3 Dissociative recombination and related phenomena

137

One then defines new basis functions3 fi(1) (r; β) = fi (r) cos π μβ − gi (r) sin π μβ gi(1) (r; β)

(5.42)

= fi (r) sin π μβ + gi (r) cos π μβ ,

which are modified by the influence of the eigenphase shifts of K (1) . The influence of K (2) in this new basis then yields the final wavefunction 

 (2) (2) |i i |β fi(1) (r; β)δii − gi(1) (r; β)Kββ = (5.43) rout Zβ(2) , iββ (2) in which Kββ are the matrix elements in the |β representation   (2) Kββ = β|iKii(2) i |β  = β|α tan π μα α|β . ii

(5.44)

α

Thus, again following (B.15), 

 (2) |i i |β β |α fi(1) (r; β) cos π μα − gi(1) (r; β) sin π μα A(2) = rout α . iαβ

(5.45) Finally, on reverting to the original basis functions  (2) |i [fi (r)Ciα − gi (r)Siα ] Aα , rout =

(5.46)

iα

where Ciα =

 i|β cos π(μα + μβ )β|α β

 i|β cos π(μα + μβ )β|i i |α, =

(5.47)

βi

and similarly for Siα with sine instead of cosine function. Standard manipulations lead to the following alternative S-matrix form of (5.46),    (2) + + rout = |i fi− (r)J− iα − fi (r)Jiα Aα iα

=



  |i fi− (r)δii − fi+ (r)Sii Xi ,

(5.48)

ii

3

At this stage fi (r) and gi (r) formally include both ionization and dissociation channels, but both the amplitude (1) coefficients and phase shifts associated with the dissociation channels vanish, μβ(d) = Zd = 0. Similarly the (i) (1) dissociative ‘modified’ basis functions are actually unchanged by K . Thus fd (r) ≡ Fd (R), etc.

138

Competitive fragmentation

√ where fi± (r) = [−gi (r) ± ifi (r)] / 2π are outgoing and incoming basis functions and4 J± iα = Ciα ± iSiα , i   − −1 J αi Xi . Aα = √ 2π i   − −1 J+ Sii = iα J αi .

(5.49)

α

Practical implementation In practical applications K (1) and K (2) are taken as the Rydberg–Rydberg and Rydberg–valence parts of the K-matrix. Thus the sum over β contributes only to the Rydberg terms, for which i |β = N + |  v + |R. Hence   CN + v+ ,α = N + | v + | cos π[μα + μ (R)]|v +  |N + v + |α v + N +

Cdα = d|α cos π μα ,

(5.50)

with similar expressions for Sv+ α and Sdα . In addition K (2) has the form K (2) = ξ ξ T in (5.23), with two non-zero eigenvalues tan π μα = ±ξ and corresponding eigenvector components √ √ ' + ! vi α = ξi / 2|ξ |. (5.51) d|α = ±1/ 2, Similar expressions apply for each dissociation channel, to the extent that interactions between the dissociation continua may be ignored. Finally the matrix S, as written in (5.49), includes both open and closed channels. The physical matrix, required in (5.20), takes the folded form [17]  −1 co S . (5.52) S = S oo − S oc S cc − e−iπν(E)

4

Following Greene and Jungen, the elements of the S-matrix may also be expressed as [16]  1/2 (2)  1/2 S (1) ij Sjj S (1) j i , Sii = jj

where



S (1)

1/2 ij

=



i|βeiμβ β|j 

β (2)

Sjj =



j |αe2iμα α|j .

α

These full S-matrix elements (including closed channels) are sometimes denoted by Xii [17].

5.3 Dissociative recombination and related phenomena

139

Applications One of the most important applications of the theory is to the dissociative recombination of H+ 2 and its isotopic variants. It was recognized from the outset that the dominant contribution to the cross-section must come from the 1 g+ states, in view of the strongly avoided crossing between the dissociative (2pσu )2 state and the (1sσg nsσg ) and (1sσg ndσg ) Rydberg systems (see Figs. 3.1 and 3.3). The early calculations by Giusti-Suzor et al. and Nakashima et al., including only vibrational channel coupling, reproduced the main qualitative features of the cross-section [17, 18]. Improvements to the theory, by including the second-order perturbation term in (5.9) coincided with experimental developments, which allowed the first experimental confirmation of the calculated resonance structure (see [7, 19, 20]). A careful comparison between theory and experiment showed good agreement with regard to the order of magnitude of the cross-section, although the detailed fluctuations with energy were not well reproduced [21]. Moreover, the theoretical cross-sections for super-elastic (SEC) vibrational energy transfer + + + + H+ 2 (vi ) + e → H2 (vf < vi ) + e, phys

which are proportional to |Sv+ ,v+ |2 , were significantly underestimated. These defii f ciencies were largely removed by the inclusion of the rotational frame transformation terms N + |  in (5.50) that take account of Coriolis interactions between the dominant 1 g+ states and those of 1 g symmetry [22]. As a consequence, the incoherent contributions from the two electronic symmetry species go over to a coherent superposition, which account for the improved level of agreement, shown in Fig. 5.4, while also correcting the SEC cross-sections. Another species of interest is NO+ , owing to its high concentration in the terrestrial ionosphere, and also in the present context because the two-step theory was first applied to the competitive predissociation and auto-ionization of NO [5]. Following early work by Lee, Sun and Nakamura used a variety of ab-initio and spectroscopic information to make a detailed study of the relative importance of five dissociation channels with symmetries A 2  + , I 2  + , B 2 , L 2  and B 2  [14, 15]. The dominant contributions over the energy range 0.02–2.0 eV were found to come from the B and B channels, with σ (B)  10σ (B ), for the dissociative recombination of the v+ = 0 state of NO+ . It is also noticeable from Fig. 5.3 that the dominant B channel shows mainly window resonances, because the resonant intermediates are more strongly coupled to the dissociation than the ionization channels. By contrast, the cross-sections in the weaker A 2  + and L 2  channels show strongly asymmetric Fano resonance profiles appropriate to q  1 in (5.37), which indicate roughly equal dissociation and ionization rates. An improved model of the NO potential functions based on ab-initio R-matrix

140

Competitive fragmentation 2.0 Rate coefficients (10–8 cm3s–1)

1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0

0

20 40 60 80 100 120 140 160 180 Energy of the incident electron (meV)

Figure 5.4 Comparison between the experimental (dots) and theoretical (solid line) dissociative recombination rate constants for HD+ . Supression of the rotational coupling terms led to the dashed theoretical line. Taken from Motapon et al. [22], with permission.

information extended the theory to higher energies, in order to account for a strong peak in the impact energy range 4–7 eV, which was attributed to the opening of a higher 3 2  dissociation channel [23].

5.4 R-matrix formulation The previous perturbation procedure is limited to situations in which the Rydberg series interacts with a dissociative valence state, via a curve-crossing interaction. The following R-matrix methods are designed to handle more general situations. To motivate the discussion, Fig. 5.5 is drawn to illustrate the possibility of competitive predissociation and auto-ionization in the 1 u system of H2 . The dominant predissociation mechanism involves Coriolis coupling between the bound 3pπ D state and the 3pσ B continuum, whose potential energy function is strongly distorted from that of the positive ion by the strong R dependence of the quantum defect function μ3pσ (R) in Fig. 3.2. Auto-ionization starts to compete above the H+ 2 (v = 0) ionization limit. The R-matrix procedures are similar in principle to those discussed in Section 3.5.2 except that they apply to nuclear rather than electronic motion, and that the internal dynamics are determined by MQDT equations rather than matrix diagonalization. The theory also builds on the treatment of vibrational–rotational channel coupling described in Section 4.3, except that the vibrational frame transformation components v + |R = χv+ (R) are truncated at a particular bond length,

5.4 R-matrix formulation –0.55

H 2 + (X)

25

3p πD 4p σ B

Potential energy (a.u.)

–0.57

20

–0.59

141

20

10

5

15

0

H 2 + (v + = 0)+e 5

–0.61 10

15

–0.63

5

0

10

H(1s)+H(2s,2p)

0 3p σ B

–0.65

0 5

2p πC

–0.67 0

2

4

R0

6

8

10

R (a.u.)

Figure 5.5 Potential curves relevant to the predissociaton the 3pπ D state of H2 by interaction with the 3pσ B continuum. Taken from Gao et al. [24], with permission.

taken as R0 = 7 a0 in Fig. 5.5, subject to a specified common logarithmic derivative condition,

d +b χ =0 (5.53) dR at R0 . One might proceed by a nuclear variant of the Wigner R-matrix method, which requires a single log-derivative boundary condition, at the cost of including a Buttle correction (see Section 3.4). It is however more convenient to adopt the variational or eigenchannel method, which employs a combination of two distinct internal basis sets, denoted globally as χv(x) + (R). The first is a large ‘closed’ set (1) χv+ (R) designed to converge the form of the internal wavefunction, by the choice b(1) = −∞, which means that χv(1) + (R0 ) = 0. The second smaller ‘open’ set, with   /dR = 0, is required to connect the internal wavefunction to its b(2) = dχv(2) + R0 proper asymptotic form in the dissociation channel. The energy levels drawn in Fig. 5.2 are those derived from the closed set, whose continuum levels interleave those of the open set, owing to the difference in log-derivatives.

142

Competitive fragmentation

Two methods have been given in the literature. The first, from Gao et al., is very close in spirit to the ab-initio eigenchannel method of Section 3.5.2, in the sense that the required energy-dependent surface log-derivatives are derived from a discrete set of internal energy levels, surface amplitude coefficients and surface derivatives, although these latter quantities are obtained by MQDT techniques, rather than matrix diagonalization [24]. This approach is however restricted to systems undergoing pure predissociation, because the MQDT equations have no discrete solutions in the auto-ionizing resonance region. The second method, given by Jungen and Ross is applicable to systems with competing fragmentation into ionization and dissociation continua [25, 26]. Care is however required in handling the auto-ionization, because the nature of the true MQDT solutions fluctuates wildly as the energy crosses a resonance, which would invalidate any attempt to interpolate across a reasonable energy range. Fortunately it proves possible to smooth out such fluctuations by using false asymptotic boundary conditions at an intermediate step in the calculation, while still retaining the ability to handle the auto-ionization correctly. The account here follows this latter approach. The aim is to obtain a wavefunction with open-channel components of the form   ( −1  |i fi (r) cos π τβ − gi (r) sin π τβ T iβ for r → ∞, R  R0 r (β) i   = −1 |d Fd (R) cos π τβ − Gd (R) sin π τβ Tdβ for r < r2 , R → ∞, R (5.54) with a common eigenphase for the ionization and dissociation waves. The state |i = |v + N +  has the wavefunction el NM ˆ r, R), φi (ˆr, R) = ϕ + χv + (R)N + (ˆ

(5.55)

in which the final factor is an electronic–rotational eigenfunction with defined angular momenta |NM, arising from the coupling of N + and . Similarly |d = |nN  is a Born–Oppenheimer electronic state with the wavefunction el NM ˆ = ϕ Nn [f (r) cos π μ (R) − g(r) sin π μ (R)] , φn (r, R) + X

(5.56)

J M ˆ is related to NM ˆ (ˆr, R) r, R) where the angular momentum function X N + (ˆ  J M J M + in (5.55) by X = N +  |N N + , and the scaling constant Nn = [n − μ (R)]−3/2 is required to ensure normalization to unity. The functions Fd (R) and Gd (R) are out-of-phase energy normalized dissociative wavefunctions. Both the Coulomb and dissociative basis functions vary with energy, but the radius r2 > rcore is chosen to be sufficiently small that the energy dependence of fi (r) and gi (r) may be ignored over ranges comparable to vibrational and rotational intervals. Notice that the nuclear radial part of φi (ˆr, R) must be matched at R0 to the [Fd (R), Gd (R)] combination in the lower of (5.54), while the electronic radial

5.4 R-matrix formulation

143

ˆ must correspond to the [fi (r),gi (r)] combination in the upper part of φn (r, R) equation, over the range r2 > r > rcore . The first step in the R-matrix procedure is to set up the MQDT equations in each of the two truncated bases χv(x) + (R). The functions in the ionization channels are written in the standard MQDT form    |i; x fj (r)δij − gj (r)Kij(x) Zj(x) , (5.57) r (x) = ij

where x = 1 or 2, Kij(x) =

 ! ' Ni+ |  vi+(x) tan π μ (R) vj+(x)  |Nj+ ,

(5.58)

and |i; x denotes the truncated channel function channel function φi(x) (ˆr, R) given 5 by (5.55) with χv+ (R) = χv(x) + (R). To allow for auto-ionization, the MQDT equations are taken in the familiar resonance form

(x)

(x) (x) Zo Koc Koo − tan π τ (x) (E) = 0, (5.59) (x) (x) (x) Kco Kcc + tan β (E) Zc(x) but rapid energy fluctuations in the solutions are eliminated at this stage by artificially ‘opening’ closed channels that could cause resonances in the energy range of interest, by substituting −tan π τ (x) (E) for tan βi(x) (E) [25, 26]. The strongly closed channels are treated by the normal folding procedure. The two sets of MQDT solutions are characterized by eigenphases τρ(x) and expansion coefficients (x) Ziρ , where ρ = 1, 2, . . . , N (x) , in which N (x) is the number of open (and artificially opened) ionization channels. The open-channel coefficients are related as usual (see (x) = cos π τρ(x) Tiρ(x) . At Appendix A.1) to elements of the orthogonal matrix T by Ziρ this stage, (5.57) reduces to    |i; x fi (r) cos π τρ(x) − gi (r) sin π τρ(x) Tiρ(x) , (5.60) rρ(x) ∼ i⊂P

in the open ionization channels. Moreover the artificial removal of slightly closed channels in (5.59) ensures that τρ(x) and Tiρ(x) vary slowly and smoothly with energy. (x) The next step is to deduce the analogous quantities τd(x) and Tdρ in the dissociation component of the wavefunction    (x) |d Fd (R) cos π τd(x) − Gd (R) sin π τd(x) Tdρ . (5.61) Rρ(x) ∼

5

Cases involving configuration interaction are handled by replacing the terms involving tan π μ (R) by appropriate matrices Kel (R).

144

Competitive fragmentation

Since the log-derivative of the truncated vibrational basis must match that of the dissociative continuum wave at R0 , it follows that −b

(x)

=

Fd (R0 ) cos π τd(x) − G d (R0 ) sin π τd(x) Fd (R0 ) cos π τd(x) − Gd (R0 ) sin π τd(x)

,

(5.62)

which is readily solved for the dissociative eigenphase τd(x) . The latter is, of course, (x) are obtained unrelated to the ionization phase τρ(x) . The associated amplitudes Tdρ by recognizing that the wavefunction of state |d in (5.61) is the electronic part of the function ˆr, R |i; x = φi(x) (ˆr, R) multiplied by the dissociative radial factor in (5.57), apart from the difference between unit normalization of |d and Wronskian normalization of the MQDT product function.6 Hence, to the extent that there is only one dissociative state in each Rydberg channel, !  (x) ' +(x) !'

Ni+ (x) i Zi R0 v i (5.63) Tdρ = .  Nd (R0 ) Fd (R0 ) cos π τd(x) − Gd (R0 ) sin π τd(x) Notice that the Zi(x) coefficients necessarily belong to channels that are closed to ionization. If there are several such channels within a given Rydberg series, it is necessary to retain only the coefficients Zi(x) for which νi (E) = [Ry /(Ei(x)+ − E)]1/2 is closest to the desired n value. We next recognize that the combined sets of truncated vibrational basis functions allow sufficient flexibility in the solutions ρ(x) , with x = 1 or 2, that suitable linear combinations    (β) = ρ(1) cρ(1) + ρ(2) cρ(2) , (5.64) ρ

may be chosen to have a common eigenphase τβ at both asymptotic limits, as required by (5.54). Experience shows that it is sufficient to combine the entire closed set ρ(1) with a single member of the open set ρ(2) for each dissociation channel, provided that the associated Rydberg energy (2) En = E (2)+ −

Ry [n − μ (R0 )]2

(5.65)

is close to the energy of interest. To simplify the notation, the coefficients of the resulting set of N (1) type-1 solutions and N (2) type-2 solutions, will be labelled as cq , with q = 1, 2, . . . , N (1) 6

The assumed identity between these two electronic functions rests on the approximation that fi (r) and gi (r) in (5.54) are independent of energy over the range of interest.

5.4 R-matrix formulation

145

for the former set and q = N (1) + 1, N (1) + 2, . . . , N (1) + N (2) for the latter. It is important to retain the distinction between the eigenphases in the ionization and dissociation channels, now labelled τρ(x) ⇒ τq and τd(x) ⇒ τqd , respectively (recall from (5.62) that τd(x) is independent of ρ). One finds by substituting from (5.60) and (5.61) in (5.64) and comparing coefficients with (5.54) that   cos π τβ Tiβ = cos π τq Tiq cq ; sin π τβ Tiβ = sin π τq Tiq cq q

cos π τβ Tdβ =



q

cos π τqd Tdq cq ;

cos π τβ Tdβ =

q



cos π τqd Tdq cq ,

(5.66)

q

which rearrange to the generalized eigenvalue form S cβ = tan π τβ C cβ ,

(5.67)

in which the elements of the matrices C and S are given by Ciq = cos π τq Tiq ,

Siq = sin π τq Tiq

Cdq = cos π τqd Tdq ,

Sdq = sin π τqd Tdq .

(5.68)

In addition the system (5.66) implies that  Tiβ = cos π (τβ − τq )Tiq cqβ q

Tdβ =



  cos π τβ − τqd Tdq cqβ .

(5.69)

q

The desired composite K- and S-matrices may be expressed in terms of the solutions of (5.67)–(5.69) in the forms.   T T Tmβ tan π τβ Tβm and Smm = Tmβ e2iπτβ Tβm (5.70) Kmm =

, β

β

in which m and m stand for i or d. The picture is not however quite complete. In practice it is usually found necessary to subject the matrix T to the transformation T ⇒ V T , such that K in (5.70) is symmetric. Jungen and Ross find that V  I if the boundary radius is chosen (d) in (5.65) is close to the test energy [26]. There is also the more such that En serious problem that the matrices have been constructed to vary smoothly with energy, to allow interpolation over a reasonable energy range; but this precludes any resonance behaviour. The final step is therefore to close the channels that were artificially opened in setting up (5.58). The resulting MQDT system takes the

146

Competitive fragmentation

form ⎛

⎞⎛ i ⎞ K id K ic Z K ii − tan π τ (E) ⎠ ⎝ Z d ⎠ = 0, ⎝ K di K dd − tan π τ (E) K dc K ci K cd K cc + tan β(E) Zc (5.71)

which reduces to an eigenvalue equation for tan π τ (E) by the normal folding procedure. The physical S-matrix required for modelling dissociative recombination may be expressed in the equivalent folded form S phys = S oo − S oc [S cc − e−2iπν(E) ]−1 S oc ,

(5.72)

where the label ‘o’ includes both open ionization and dissociation channels. 1

u states of H2

Applications of the theory to the 1 u states of H2 are particularly interesting. As discussed in Section 4.2.1, they fall under the influence of rotational channel coupling into e-type (or + ) and f-type (or − ) components. The former are Coriolis-coupled to the rotational levels of the 1 u system, the 3pσ member of which is seen from Fig. 5.6 to be open to dissociation above the H(1s) + H(n = 2) limit. Competition can therefore occur between auto-ionization and the resulting Coriolis-induced predissociation [25, 26]. The total line-widths and competitive auto-ionization and predissociation branching ratios for selected resonances are given in Table 5.1. The 3pπ(v = 8) resonance is a classic example of Coriolisassisted heterogeneous predissociation [1]. The line-width increases proportional to N(N + 1), and the very low ionization branching ratio shows that there is negligible auto-ionization. By contrast, the predissociation contribution to the 4pπ (v = 5) decay rate is at most only a few percent, because the resonance lies below the 4pσ dissociation threshold, and there is presumably very little v + = 5 character in the 3pσ continuum wavefunction. The high ionization yields and the insensitivity to the angular momentum N show that both the 4pπ (v = 5) and 5pσ (v = 5) resonances decay by vibrationally induced auto-ionization, which is much faster for the latter because |dμpσ /dr| > |dμpπ /dr|. The rotational–vibrational levels of the − system are much more stable, to the extent that fluorescence may even be observed above the ionization limit. Careful recent measurements of the absolute cross-sections for absorption, ionization and dissociation are shown in Fig. 5.6 (a), (b) and (c), respectively [27]. The difference σabs − σion − σdiss in Fig. 5.6 (d) may be compared with the shape of the fluorescence spectrum in Fig. 5.6 (e). The solid and dashed arrows indicate absorption peaks to states with dominant ionization and dissociation decay channels,

5.4 R-matrix formulation

147

Table 5.1 Energies, widths and ionization branching ratios for selected singlet ungerade states of H2 . Values in parenthesis are estimated from Fig. 1 of [26]. N =1 nλ

v

3pπ

8

4pπ

5

5pσ

4

E/cm−1 Obs Calc Obs Calc Obs Calc

127248.2 127246.9 127667.6 127665.4 127599.4 127602.2

N =2

/cm−1 Ion(%) 3.4 3.3 − 0.046 − 0.38

3.4 7.0 (98) 98 77 96

E/cm−1

/cm−1

Ion(%)

127321.6 127321.0 127758.7 127758.4 127666.9 127669.7

10.2 11.4 − 0.024 − 0.48

(5) (5) (20) (20) (75) (80)

Figure 5.6 Excitation spectra of H2 above the ionization limit showing absolute cross-sections for (a) photo-absorption, (b) photo-ionization and (c) photodissociation. The difference (a) − (b) − (c) in panel (d) may be compared with the fluorescence excitation spectrum in panel (e). Arrows indicate aborption peaks that are absent in dissociation (solid) or ionization (dashed). Taken from GlassMaujean et al. [27], with permission.

respectively. The typical line-widths are of order 10−3 cm−1 , corresponding to lifetimes on the scale of > 0.01 μs, because auto-ionization and predissociation arise only from the weak vibrational dependence of the pπ quantum defect. The variational R-matrix procedure was used to assign the complex spectrum and to compute the fragmentation branching ratios for the N = 1 rotational levels

Competitive fragmentation

γ

γ

γ

148

Figure 5.7 Experimental (filled symbols) and theoretical (open symbols) fragmentation branching ratios for N = 1 levels of selected vibrational states of (a) the 3pπ D state (b) the 4pπ D state and (c) the 3pπ D

state. Taken from Glass-Maujean et al. [27], with permission.

of selected vibrational–electronic states, which are compared with experiment in Fig. 5.7. The level of agreement is excellent. Fluorescence and ionization are seen to dominate for the 3pπD and 5pπD

states respectively, while the 4pπ D state is an intermediate case. In addition, dissociative decay is faster than ionization for n = 3, for which the calculated partial line-widths were shown to be in good order of magnitude agreement with the measured values. 5.5 Vibronically induced dissociative recombination of H+ 3 5.5.1 The Jahn–Teller mechanism The tri-hydrogen ion H+ 3 is not only the simplest polyatomic species. It is also the most abundant molecular ion in diffuse interstellar clouds, where the dissociative recombination reactions ( H2 + H † + H3 + e → H 3 → H+H+H

5.5 Vibronically induced dissociative recombination of H+ 3

149

act to control the charge balance. These latter processes have been subject to numerous, often contradictory, experimental investigations by ion storage ring and flowing afterglow methods [28]. The theoretical work of Kokoouline and Greene, which starts this section, was vital in identifying the fundamental mechanism and designing an efficient scheme for implementing the theory [29]. The section concludes with an elegant connection with related spectroscopic studies, given by Jungen and Pratt [30]. † The electronic states of the intermediate H3 species in the above reaction are quite different from those of typical diatomic molecules, because there is no potential surface crossing between the the dissociative ground state potential surface and that of the positive ion. Hence little or no ‘direct’ dissociative recombination is expected. Instead the ground state potential surface, which controls the exchange reaction, H + H2 → H2 + H, is connected to the first excited 2p(E) potential surface by a conical intersection at equilateral triangular D3h geometries, which was seen in Section 4.4.2 to persist throughout the np Rydberg series. Moreover we have seen in Section 4.4 that the dominant auto-ionization mechanism is via Jahn–Teller coupling. The outlines of the resulting dissociation recombination theory are clear. The † electron capture to form the intermediate H3 proceeds by Jahn–Teller mediated reverse auto-ionization into the np(E) states, for which the necessary K-matrix elements are given in Section 4.4. Details of the theory required to couple the capture states to the dissociative ground state are, however, quite complicated [29]. In the first place, the ‘dissociation coordinate’, analogous to the diatomic bond length is taken as the radial variable R of the hyperspherical system (R, θ, φ) defined by the equations  r1 = R23 = 3−1/4 R 1 + sin θ sin(φ + 2π/3)  r2 = R13 = 3−1/4 R 1 + sin θ sin(φ − 2π/3) (5.73)  r3 = R12 = 3−1/4 R 1 + sin θ sin φ. They are related to the normal coordinates in Section 4.4 by f Q1 = √ (r1 + r2 + r3 ) 3 f Q2x = √ (2r1 − r2 − r3 ) 3 Q2y = f (r2 − r3 ) , where f = 2.639255 a0−1 and ri = ri − rref , with rref = 1.6504 a0 . It is useful to remember that the hyperspherical radius coincides with the symmetric stretching

150

Competitive fragmentation

Energy (a.i.)

0.2

0.0

−0.2

1.5

2.0 2.5 Hyperradius (a.i.)

3.0

Figure 5.8 Adiabatic potential curves as a function of the hyperspherical radius, R. The heavy line is the positive ion potential, with the marked zero-point energy. The lighter continuous and dashed lines correlate with dissociation fragments H + H2 and H + H + H, respectively. Taken from Kokoouline and Greene [29], with permission.

coordinate, while changes in the angular variables (θ, φ) correspond to motions in the degenerate coordinates (Q2x , Q2y ). Kokoouline and Greene treat these latter motions adiabatically by quantizing with respect to the (θ, φ) variables at fixed values of R, for both the neutral species H3 and the ion H+ 3 [29]. Considerable care is required in handling the symmetry of the vibrational, rotational and nuclear spin states. The forms of the resulting adiabatic potential curves in the radial variable are shown in Fig. 5.8. The heavy line belongs to the lowest energy state of the ion. The mesh of thinner curves apply to the neutral H3 , using continuous and dashed lines for those that correlate with H + H2 and H + H + H, respectively. As a measure of the accuracy of the adiabatic approximation, subsequent quantization of the R motion in these hyperspherical adiabatic curves was found to reproduce selected vibrational eigenvalues of H+ 3 to within 0.5%.

5.5 Vibronically induced dissociative recombination of H+ 3

151

Proceeding to the MQDT calculation, the Jahn–Teller K-matrix in (4.74) was expressed in the quantum defect form7 K = U tan π μU T .

(5.74)

with elements K

, and the equivalent S-matrix is given by S = U e2iμ U T . A vibrational contraction over the angular variables then yields ' ! + + S˜v2 2 ;v2  2

(R) = v2+ + 2 ; R S

(R, θ, φ) v2 2 ; R ,

(5.75)

(5.76)

where the tilde indicates that the integral over R remains to be performed. At this point, one could revert to the diatomic variational R-matrix formalism of ! + Section 5.4, by integrating over truncated states v1 in the R variable, subject to ‘open’ and ‘closed’ boundary conditions at a suitable point R0 . Instead Kokoouline and Greene employ Siegert pseudo-states, which satisfy the complex outgoing boundary condition

d − ik χv1+ (R) = 0 (5.77) dR at R0 [29, 31]. The properties of such states require some discussion. Following Hamilton and Greene they are expanded in a non-orthogonal B-spline basis [32]  cj yj (R), 0  R  R0 , (5.78) χv1+ (R) = with the coefficients subject to the matrix equation   ˆ c = 0, ˆ − ik L ˆ − k2O H

(5.79)

ˆ is the overˆ jj = 2μN Hjj + yj (R0 )y (R0 ), Lˆ jj = yj (R0 )yj (R0 ) and O in which H j lap matrix. The difficulty that (5.79) is quadratic in k has been overcome by Tolˆ = ik Oc, ˆ stikhin et al. by introducing auxiliary coefficients dj = ikcj , so that Od which lead to the linear system [33]







ˆ ˆ H 0 c Lˆ −O c = ik (5.80) ˆ ˆ d d 0 −O −O 0 in the doubled space. Solutions for which the energy has a negative imaginary part are the required outgoing pseudo-states [32]. 7

Kokoouline and Greene [29] include an additional term μ0 equivalent to μe in (4.74).

152

Competitive fragmentation

Given the set of such states, the remaining integrals over R are evaluated with an additional surface term. Thus ! ' + + +

v1 v2 2 S v1+ v2+ + 2  R0 χv+ (R0 )S˜v2 2 ;v2  2

(R0 )χv1+ (R0 ) = χv1+ (R)S˜v2 2 ;v2  2

(R)χv1+ (R)dR + i 1 . kv1+ + kv1+ 0 (5.81) Note that the complex functions χv1+ (R) are not conjugated. The final S-matrix elements are then obtained by the rotational frame transformation ! ' + + + + + + + v1 v2 2 N K S v1+ v2+ + 2 N K ' !' ! !' + + + + + N + K + K v1+ v2+ + , (5.82) = 2 S v1 v2 2 K N K



where the elements N + K + | K are given by (4.73), and K is conserved by virtue of the D3h symmetry. The resulting matrix may be divided as usual into open and closed channel blocks and folded into the physical form  −1 co S phys (E) = S oo − S oc S cc − e−2iβ(E) S .

(5.83)

The reader may be surprised to see no reference to dissociation channels in labelling the S-matrix, because there has been no explicit matching to dissociating wavefunctions. The boundary condition (5.77) merely specified ‘outgoing character’ at R0 , which plays the same role as a ‘complex absorbing potential’ in scattering calculations by destroying the normal unitarity of the S-matrix [34]. The elastic and inelastic scattering between the ionization channels is correctly described, but the total probability of scattering from any particular state no longer sums to unity, because flux into the dissociation channels has been removed by the outgoing boundary condition. The dissociative recombination cross-section for a given initial ionization channel |i may therefore be evaluated from the ‘unitary defect’ of the appropriate S-matrix column. ⎤ ⎡     2N + 1 ⎣ π phys phys † ⎦ Sij Sj i σid = 2 , (5.84) 1− + ki N 2N + 1 j where (S phys )† is derived by taking the Hermitian conjugate of S˜ in (5.81), but not of the pseudo-state wavefunctions χv1+ (R). The open and closed blocks of the resulting matrix, say S † , then define (S phys!)† via (5.83). The ! index |i denotes the + + + + + 0 + + initial ro-vibrational state v1 v2 2 N K = 00 N K .

5.5 Vibronically induced dissociative recombination of H+ 3

153

DR rate coefficient, αtor(cm3s–1)

10 –6

10 –7

10 –8

10 –9

CRYRING experiment TSR experiment Present study

10 –4

10 –3 10 –2 10 –1 Electron energy, EII(eV)

10 0

Figure 5.9 Comparison between the experimental (dots) and theoretical (lines) dissociation recombination rate coefficient for H+ 3 . Taken from dos Santos et al. [35], with permission.

A test calculation for a simplified model of H+ 2 dissociative recombination reproduced the correct direct background contribution, with appropriate indirect resonant fluctuations [32]. The full calculation for H+ 3 is a considerable computational task, because the unfolded S-matrix has typical dimension 2000 × 2000 and S phys (E) must be scanned over a very fine energy grid to map out the detailed shapes and positions of all the resonances. Moreover, the calculation must be repeated for all total |NK values required for initial rotational states up to |N + , K +  = |5, 1 because the storage ring experiments run at 1000K. Figure 5.9 shows excellent agreement between the most recent theoretical and experimental results [35, 36].

5.5.2 A spectroscopic related model The Jahn–Teller theory in Section 4.4.2 is partly intended as a preparation for an elegant analytical model that was used by Jungen and Pratt to relate the dissociative recombination of H+ 3 to the spectroscopy of H3 [30]. The argument rests in part on the fact that the Jahn–Teller-assisted auto-ionization rate depends, via (4.72) on a dimensionless Jahn–Teller stabilization parameter D (n) , which scales with principal quantum number as ν −6 , where ν = n − μ. Secondly, the Jahn–Tellerinduced vibrational level splitting, which may be determined spectroscopically,

154

Competitive fragmentation

perhaps at a different quantum number n , depends on the product of D (n ) and the vibrational energy quantum h¯ ω. Rydberg scaling arguments therefore allow prediction of the auto-ionization rate from the Jahn–Teller level spitting, or vice ˜ such versa. Jungen and Pratt find it convenient to define a scaled quantity D, that  ν 3 √ (n) D˜ = D , (5.85) 2Ry in terms of which the auto-ionization width at n is given by (4.72) in the form8 ˜ hω)2 v − j + 1/2

8π Ry D(¯ (a) . (5.86) vj n = ν 3 cos2 π ν 2 We also note that the Jahn–Teller vibronic coupling rules allows auto-ionization to ! the vibrational ground state only from series that terminate on v1+ v2+ = |01, with j = 1/2. The second part of the argument is that the dissociative recombination of H+ 3 is an entirely indirect process and that auto-ionization is the rate determining step (although neither of these is stated explicitly). Hence the contribution to the crosssection from a single resonance in (5.35) may be approximated as σidres (E) 

i /2 2π  . ki2 (Eres − E)2 +  2 /4

(5.87)

Finally, since these resonances are not resolved in a typical DR experiment, it is permissible to integrate the contributions from individual resonances and average over the local resonance spacing   2Ry /ν 3 . Thus, following Mikhailov et al. [37]

2π 2 i σid (E)  2 . (5.88)  ki Since i and  both scale as ν −3 , the energy variation comes from the factor ki−2 ∝ E −1 , with a cut-off at the v2+ = 1 threshold energy of H+ 2. The input required for application of this simple model comprise the Jahn– Teller stabilization parameter D (3)h¯ ω = 75.89 cm−1 , for the 3p 2 E state of H3 and the vibrational quantum, h¯ ω2 = 2521.2 cm−1 , for Q2 mode of H+ 3 , [38, 39]. Figure 5.10, in which αid (E) = uσid (E) where u is the collision speed, shows excellent agreement between the experimental results [36], the full MQDT calculation 8

The golden-rule formula used by Jungen and Pratt omits the factor cos2 π ν and these authors use the spectroscopic symbol ω for the present vibrational quantum h¯ ω.

References

155

10 –6 Kreckel et al.

Rate coefficient (cm3/s–1)

Present, unconvolved Present, convolved Fonseca dos Santos et al.

10 –7 H 3 + ,v 2 + =1 threshold

10 –8

10 –9 0.0001

0.001

0.01 Energy (eV)

0.1

1.0

Figure 5.10 Comparison between the experimental rate coefficient (circles) for the dissociative recombination of H+ 3 [36], the full MQDT calculation (dot–dash) [35] and the model of Jungen and Pratt (solid and dashed lines) [30]. Taken from [30], with permission.

[35], and the model of Jungen and Pratt [30]. A similar level of agreement is found + for the isotopic variants, H2 D+ , HD+ 2 and D3 .

References [1] H. Lefebvre and R. W. Field, The Spectra and Dynamics of Diatomic Molecules (Academic Press, 2005). [2] M. Raoult, J. Chem. Phys. 87, 4736 (1987). [3] M. S. Child, Semiclassical Mechanics with Molecular Applications (Oxford Univeristy Press, 1991). [4] G. B. Arfken, H. J. Weber and F. Harris, Mathematical Methods for Physicists 5th edn (Academic Press, 2001). [5] A. Giusti-Suzor and C. Jungen, J. Chem. Phys. 80, 986 (1984). [6] A. Giusti, J. Phys. B 13, 3867 (1980). [7] S. L. Guberman and A. Giusti-Suzor, J. Chem. Phys. 95, 2602 (1991). [8] A. Fujii and N. Morita, J. Chem. Phys. 98, 4581 (1993). [9] A. L. Goodgame, PhD thesis, Oxford University (2001). [10] M. J. J. Vrakking and Y. T. Lee, Phys. Rev. A 51, R894 (1995). [11] M. J. J. Vrakking and Y. T. Lee, J. Chem. Phys. 102, 8818 (1995). [12] M. S. Child, Molecular Collision Theory (Dover, 1996). [13] J. N. Bardsley, J. Phys. B 1, 349, 365 (1968). [14] C. M. Lee, Phys. Rev. A 16, 109 (1977).

156 [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39]

Competitive fragmentation H. Sun and H. Nakamura, J. Chem. Phys. 93, 6491 (1990). C. H. Greene and C. Jungen, Adv. At. Mol. Phys. 21, 51 (1985). K. Nakashima, H. Takagi and H. Nakamura, J. Chem. Phys. 86, 726 (1987). A. Giusti-Suzor, J. N. Bardsley and C. Derkis, Phys. Rev. A 28, 682 (1983). I. F. Schneider, O. Dulieu and A. Giusti-Suzor, J. Phys. B 24, L289 (1991). H. Hus, F. Yousif, A. Sen and J. B. Mitchell, Phys. Rev. Lett. 60, 1006 (1988). V. Ngassam, O. Motapon, A. Floescu et al. Phys. Rev. A 68, 032704 (2003). O. Motapon, F. O. W. Tamo, X. Urbain and I. F. Schneider, Phys. Rev. A 77, 052711 (2008). I. F. Schneider, I. Rabad´an, L. Carata et al., J. Phys. B 32, 4849 (2000). H. Gao, C. Jungen and C. H. Greene, Phys. Rev. A 47, 4877 (1993). C. Jungen, Phys. Rev. Let. 53, 2394 (1984). C. Jungen and S. C. Ross, Phys. Rev. A 55, R2503 (1997). M. Glass-Maujean, C. Jungen and H. Schmoranzer, Phys. Rev. Lett. 104, 183002 (2010). M. Larsson and A. E. Orel, Dissociative Recombination of Molecular Ions (Cambridge University Press, 2008). V. Kokoouline and C. H. Greene, Phys. Rev. A, 68, 012703 (2003). C. Jungen and S. T. Pratt, Phys. Rev. Lett. 102, 023201 (2009). A. J. F. Siegert, Phys. Rev. 56, 750 (1939). E. L. Hamilton and C. H. Greene, Phys. Rev. Lett. 89, 263003 (2002). O. L. Tolstikhin, V. N. Ostrokov and H. Nakamura, Phys. Rev. A 58, 2007 (1998). R. Kosloff, J. Comput. Phys. 63, 363 (1986). S. F. dos Santos, V. Kokoouline and C. H. Greene, J. Chem. Phys. 127, 124309 (2007). H. Kreckel, M. Motsch, J. Mikosch et al., Phys. Rev. Lett. 95, 263201 (2005). I. A. Mikhailov, V. Kokoouline, A. Larson, S. Tonanzi and C. H. Greene, Phys. Rev. A 78, 032707 (2006). G. Herzberg and J. K. G. Watson, Can. J. Phys. 58, 1250 (1980). R. Jaquet, W. Cencek, W. Kutzelnigg and J. Rychlewski, J. Chem. Phys. 108, 2837 (1998).

6 Photo-excitation

6.1 Introduction The properties of molecular Rydberg states are most commonly observed experimentally by photo-excitation and photo-ionization, and it is impossible to ignore the explosion of interest in multiphoton phenomena over the past twenty years. It is, however, beyond the scope of this book to attempt anything like a comprehensive treatment. Attention is therefore restricted to the weak field theory, in which light acts as a perturbation. Readers are referred to Lambropoulos and Smith for a fuller discussion [1]. Explicit results are restricted to one and two photons, leaving the reader to consult the literature for extension to n photons. This chapter concerns excitation between discrete bound states, either by single-photon absorption or n + 1 resonant multiphoton ionization (REMPI) [2, 3, 4]. The first of the following sections outlines the perturbation theory of oneand two-photon absorption, as initiated by G¨oppert-Mayer, and the extension to three-photon processes is indicated [5]. Aspects of the theory, such as the point group symmetries of the resulting dipole (n = 1), polarizability (n = 2) and hyperpolarizability (n = 3) operators are readily deduced in a Cartesian formulation [6]. However, the relevant angular momentum manipulations including the selection rules for the various n-photon linear and circular polarization possibilities are often most easily performed in a complex spherical tensor representation, which is outlined in Section 6.3 [6, 7]. There are also advantages, for resonant processes, in employing an alternative density matrix, which focuses on spatial characteristics of the excited angular momentum, rather than the overall excitation probability. The case of resonant two-photon excitation provides an illustration in Section 6.4. The remaining sections cover applications to specific situations. The symmetry and angular momentum selection rules for two- and three-photon processes in Section 6.6 lead to a discussion of the band structure of multiphoton spectra, with extensive tables for both symmetric and asymmetric tops [6, 7]. Two interesting

157

158

Photo-excitation

features of the spectrum at energies approaching the ionization limit are covered in Sections 6.7 and 6.8. The first concerns the spectral signature of the electron as it starts to decouple from the nuclear framework [8]. The second relates to intensity anomalies in the zero-kinetic energy (ZEKE) spectrum, which arise from the extreme polarizability of the Rydberg states as n → ∞ [9]. 6.2 n-photon discrete absorption We start by deriving expressions for the rates of absorption of successive photons, on the assumption that the laser fields are sufficiently weak to allow a time-dependent perturbation treatment. This account follows Cohen-Tannoudji et al., with regard to the semiclassical electromagnetic formulation and McClain and Harris with regard to the multiphoton iteration [7, 10]. Details of the notation are taken to conform with Dixon et al., for ease of comparison with the spectroscopic literature [6]. The properties of the radiation are defined by the vector potential [10] A(r, t) = A0 sin(ky − ωt)ˆz,

(6.1)

which corresponds to radiation propagating in the y direction with angular frequency, ω, wavevector k = ω/c, and linear polarization in the zˆ direction.1 The associated electric and magnetic fields are given by ∂A = ωA0 cos(ky − ωt)ˆz ∂t B(r, t) = ∇ × A = kA0 cos(ky − ωt)ˆx.

E(r, t) = −

(6.2)

It will be convenient later to relate the laser power density, P , and photon flux, F , to the coefficient A0 , by the equation ! ' (6.3) F = P /¯hω = ε 0 c2 E × B /¯hω = ε 0 ωA20 c/2¯h, where the bracket indicates a long time average. The following alternative form is used below to simplify the perturbation results A20 ω2 /¯h2 = 8παωF /e2 , 1

(6.4)

The corresponding forms for left and right and circularly polarized light, propagating in the z direction would be 

A(r, t) = Re −iA0 ei(kz−ωt) 2−1/2 (1, ±i, 0) = 21/2 A0 [sin(kz − ωt), ±cos(kz − ωt), 0] giving rise to E(r, t) = 21/2 ωA0 [ cos(kz − ωt), ∓sin(kz − ωt), 0] B(r, t) = 21/2 kA0 [± sin(kz − ωt), −cos(kz − ωt), 0] , with the flux again given by (6.3).

6.2 n-photon discrete absorption

159

where e is the electronic charge and α = (e2 /4πh¯ c)  1/137 is the fine structure constant. Finally, within the dipole approximation, the perturbation Hamiltonian for interaction with monochromatic radiation takes the general form ˆ · ˆ ) , H pert (r, t) = μ · E(r,t) = ωA0 cos(k · r − ωt) (μ

(6.5)

where ˆ is the polarization vector. It is illuminating to include a twist due to Manolopoulos eliminates the spatial part of the travelling wave by evaluating the matrix element [11] ! ' pf | cos(k · r − ωt)|pi   1 = 3 e−i(pf −pk −pi ).r/¯h−iωt + e−i(pf +pk −pi ).r/¯h+iωt d3 r 2h    1  −iωt  e (6.6) δ pf − pi − pk − eiωt δ pf − pi + pk , = 2 where pk = k¯h is the de Broglie momentum of the photon. The first and second terms show an increase and a decrease in the centre of mass momentum, pf = pi ± pk , attributable to absorption and emission, respectively. Consequently, the perturbation operator for absorption is given by H (t) =

ˆ  −iωt A0 ωμ . e 2

(6.7)

6.2.1 Single-photon absorption The amplitude for single-photon absorption from state |0 to state |1, both of which are normalized to unity, is determined, at the perturbation level, by   (1) ωA0 μ 10 (ω) M10 da1 (1) i(ω10 −ω)t , M01 (ω) = = a0 (0)e , (6.8) dt i¯h 2 ˆ  |0 is the dipole matrix element and ω10 = (E1 − E0 )/¯h. The where (μ )10 = 1|μ solution is

i(ω10 −ω)t (1) (ω) M10 −1 e a0 (0) . (6.9) a1 (t) = − h¯ ω10 − ω The next step follows McClain and Harris in accepting that the laser frequency, ω, and the frequency, ω10 of the molecule are typically distributed over ranges defined by distribution functions, gM (ω10 ) and gL (ω), which are taken to be normalized to unity [7];   (6.10) gM (ω10 )dω21 = gL (ω)dω = 1.

160

Photo-excitation

It follows from (6.9) that the number of molecules in state |2 evolves as   N1 (t) = N0 (0) |a1 (t)|2 gM (ω10 )gL (ω)dω10 dω    = N0 (0)

(1) (ω) M10 h¯

2

sin2 δt gM (ω10 )gL (ω)dω10 dω, δ2

(6.11)

where δ = (ω10 − ω)/2. Simple analysis shows that the term sin2 δt/δ 2 peaks ever more sharply, as time increases, around δ = 0, with an area [12]  ∞ sin2 δt dδ = πt. (6.12) δ2 −∞ One may also assume that the laser frequency distribution gL (ω) is narrower than gM (ω10 ), in which case the introduction of new integration variables, δ and ω¯ = (ω10 + ω)/2, with Jacobian ∂(ω10 , ω)/∂(δ, ω) ¯ = 2, readily leads, via (6.4), to an expression of the form N1 (t) = N0 (0)F tσ 10 (ω). Thus the number of excited molecules increases linearly with N0 (0), with the photon flux F and with time. The molecular specific coefficient, σ 10 (ω), is identified as the absorption cross-section  2 σ 10 (ω) = 4π 2 α (ˆ · μ)10 /e ωgM (ω), (6.13) which is readily verified to have the dimensions of area, because α and ωgM (ω) are dimensionless, while the bound state matrix element (ˆ · μ)10 /q is a length. 6.2.2 Coherent two-photon absorption It is assumed for the sake of generality that the two photons have distinct frequencies, ωa and ωb , and distinct polarization projections, μα and μβ . An ambiguity with respect to the order of photon absorption must also be accepted. To simplify the exposition, we first consider the equations relating population transfer from |0 to an intermediate state |i by photon A, followed by excitation to |1 by photon B in situations where ωa + ωb  ω10 . There is also an allowance for depopulation of the intermediate with a phenomenological rate constant i , arising from both radiative and non-radiative processes. The analogue of (6.8) is therefore modified to read M (1) (ω) dai = −i ai (t) + i0 a0 (0)ei(ω10 −ω)t , dt i¯h with solution M (1) (ω) ai (t) = − i0 h¯



ei(ωi0 −ω)t − 1 ωi0 − ωa − ii

(6.14)

.

(6.15)

6.2 n-photon discrete absorption

161

When combined with the driving equation for the second step, namely M (1) (ωb ) da1 = 1i ai (t)ei(ω1i −ωb )t , dt i¯h

(6.16)

this means that (1)   M (1) (ωb )Mi0 (ωa ) da1 = 2 1i a1 (0) ei(ω10 −ωa −ωb )t − ei(ω1i −ωb )t , dt i¯h (ωi0 − ωa − ii )

(6.17)

where ω10 = ω1i + ωi0 . Note that the first term in the final bracket is independent of |i, while the non-resonant assumption, ωi0 = ωb , means that the second term oscillates so rapidly that it makes a negligible contribution to af (t). After allowing for multiple intermediate states and for ambiguity in the order of the two photons, the amplitude for two photon absorption is given by (2) (ωa + ωb ) M10 da1 = a0 (0)ei(ω10 −ωa −ωb )t , dt i¯h

where (2) M10 (ωa

+ ωb ) =



# Pab

i

$ (1) M1i(1) (ωb )Mi0 (ωa ) , h¯ (ωi0 − ωa − ii )

(6.18)

(6.19)

in which the symbol Pab implies permutation with respect to a and b. In view of the similarity between the structure of (6.8) and (6.18), it follows that

i(ω10 −ωa −ωb )t (2) (ωa + ωb ) M10 −1 e (6.20) a1 (t) = − a0 (0) h¯ ω10 − ωa − ωb and



(2) (ωa + ωb ) M10 N1 (t) = 2πtN0 (0) h¯

2 gM (ωa + ωb )

= N0 (0)tFa Fb σ [2] 10 ,

(6.21)

where 2 3 2  a ˆ b ) · T (Oˆ [2] )10 /e2 , σ [2] 10 = 8π α ωa ωb gM (ωa + ωb ) T(ˆ

(6.22)

in which T(ˆ a ˆ b ) = ˆ a ˆ b is a two-component tensor specified by the polarization vectors of the light beams and T (Oˆ [2] )10 is the 1 ← 0 matrix element of the molecular polarizability operator, with elements    [2]   μβ |i i| μα μα |i i| μβ , (6.23) Tαβ Oˆ = + ωi0 − ωa − ii ωi0 − ωb − ii i

162

Photo-excitation

in which the square bracket indicates the number of photons. The notation Oˆ [2] is employed to indicate the molecular component of the two-photon operator – as distinct from a property of the light beams. The relaxation rate constants i can usually be ignored, except in cases of resonant two-photon excitation. The symbol σ [2] 10 is termed the absorptivity, δ 10 , by McClain and Harris [7]. The dimensions are L4 T rather than area, because (μ/e) is a length, α and ωa g(ωa + ωb ) are dimensionless, and the ωb combines with the square of the frequency denominator to contribute a factor with the dimensions of time. Looked at in another way, the fluxes Fa and Fb are expressed as photons per unit area, per 4 unit time (L−2 T −1 ) which again means, via (6.21), that σ (2) 10 varies as L T . Notice that although the transition amplitudes have been symmetrized with respect to the order of the driving frequencies, ωa and ωb , the operator Tab is not itself symmetric with respect to a and b, because the two denominators in (6.23) differ, unless ωa = ωb . The magnitude of the asymmetry therefore depends on the difference |ωa − ωb |, compared with ωi0 − ω, ¯ where ω¯ is the mean. Resonant excitation The case of resonant absorption, with ωa = ωi0 , merits special attention, because the matrix element of Tαβ is dominated by the term '   ! i 1| μβ |i i| μα |0 1 Tαβ Oˆ [2] 0 = . i

(6.24)

We also assume initially that the dominant decay mode is radiative emission at frequency ωa , in which case (6.4), (6.8) and (6.14) show that √ 8παωFa (rad) i = (6.25) i|μα |0. e Consequently, (6.21) goes over to N1 (t) = N0 (0)tFb σ (res) 10

(6.26) 2

2 σ [res] 10 = π αωb gM (ωb ) 1| μβ |i /e ,

subject to the requirement that i| μα |0 = 0. In other words σ [res] 10 reduces to a single-photon cross-section for excitation from the intermediate state |i at frequency ωb . Two points should, however, be noted. The first, which is explored in detail in Section 6.4, is that the distribution over the magnetic quantum number Mi of the resonant intermediate is determined by the hidden transition moment i| μα |0. Secondly, there may, of course, be additional non-radiative decay modes, such as predissociation, which may diminish or completely eliminate the ionization current.

6.3 Spherical tensor representation

163

The argument for three-photon absorption via successive intermediate levels, |i and |l, proceeds along similar lines. with the result    2 4 3  a ˆ b ˆ c ) · Tabc Oˆ [3] 10 /e3 , σ [3] 10 = 16π α ωa ωb ωc gM (ωa + ωb + ωc ) T(ˆ (6.27) in which T (ˆ a ˆ b ˆ c ) specifies the polarization field, with elements ˆ aα ˆ bβ ˆ cγ and [3] ˆ [3] (O )10 is the 1 ← 0 matrix element of the hyperpolarizability tensor, with Tabc elements    [3]   |l l| |i i| μ μ μ c b a . (6.28) = Pabc Tabc Oˆ (ω ) (ω ) − ω − ω − i − ω − i l0 a b l i0 a i il The dot in (6.27) implies a sum over abc. 6.3 Spherical tensor representation The scalar product in (6.22) is most conveniently evaluated in a spherical tensor representation by referring the polarization vectors to the complex base vectors [13]   eˆ0 = ˆ z , eˆ±1 = ∓2−1/2 ˆ x ±iˆ y , (6.29) and similarly for the dipole components, μq . Thus   k  † ˆ T(ˆ a ˆ b ) · T Oˆ [2] = Tq (ˆ a ˆ b ) Tq(k) (O) kq

=



ˆ (−1)q Tk−q (ˆ a ˆ b )Tq(k) (O),

(6.30)

kq

where Tkq (ˆ a ˆ b ) =



1a1β|kq ˆ aα ˆ bβ

ab

ˆ = Tq(k) (O)



[2] ˆ 1α1β|kq Tαβ (O).

(6.31)

αβ

Notice the difference in notation between the coefficient Tkq (ˆ a ˆ b ) which depends ˆ on the laser polarization and the body-fixed operator Tq(k) (O). Equations (6.30) and (6.31) are used to determine the weighting of the body-fixed ˆ appropriate to the polarization characteristics spherical tensor components Tq(k) (O) of the light beams. In the two-photon case, the components T(ˆ a ˆ b ) of the photon field transform as the direct product T (1) × T (1) , which decomposes as T [2] = T (1) × T (1) = T (0) + T (1) + T (2) ;

(6.32)

164

Photo-excitation

Table 6.1 The transformation from two-photon products to irreducible spherical tensor components. k

q

Symmetric tensors 0 0 2 2 2 1 2 0 −1 −2

2 2

Antisymmetric tensors 1 1 1 0 −1

1

T(k) q   ¯ − (00) + (11) ¯ 3−1/2 (11) (11) 2−1/2 [(10) + (01)]   ¯ + 2(00) + (11) ¯ 6−1/2 (11)   ¯ + (10) ¯ 2−1/2 (01) ¯ (1¯ 1) 2−1/2 [(10) − (01)]   ¯ − (11) ¯ 2−1/2 (11)   ¯ − (10) ¯ 2−1/2 (01)

and the coefficients Tkq (ˆ a ˆ b ) in (6.31) are listed in Table 6.1, in which the nota¯ indicate linear, left-circular and right-circular polarization, tions (0), (1) and (1) respectively. Similar tables for three-photon excitation are given by Dixon et al. [6]. The appropriate weightings for an isotropic molecular sample require a sum over the space fixed angular momentum projections of the molecular wavefunction. The appropriate matrix element is given by the Wigner–Eckart theorem [13]

-  ! ' (k) k J  J J −M γ J -T (k) -γ J , (6.33) γ J M Tq γ J M = (−1)

−M q M where |γ  specifies the internal - molecular -  state. The use of square brackets for the  reduced matrix element γ J -T (k) -γ J in the total angular momentum representation is discussed in Appendix E. The orthogonality equation [13]



 k J

k J

J J = δ kk δ qq , (6.34) (2k + 1) −M q M

−M q M



MM

shows that  ' !2 T(ˆ a ˆ b ) · γ J M T (Oˆ [2] ) γ J M M M

   Tk−q (ˆ a ˆ b ) 2  - 2 = γ J -T (k) (Oˆ [2] -γ J . (2k + 1) kq

(6.35)

6.3 Spherical tensor representation

165

Table 6.2 Weighting factors [Tkq ]2 /(2k + 1) for various two-photon polarization excitation schemes. Note that the reduced matrix elements of the corresponding Tq(1) operators vanish if the two frequencies are equal. k Polarization Lz Lz C+ C− C+ Lz Lx Lz

q

2

1

0

0 0 1 ±1

2/15 1/30 1/10 1/10

0 1/6 1/6 1/6

1/3 1/3 0 0

Thus each spherical tensor component makes an independent contribution to the sum, weighted by a reduced matrix element of the form discussed in Appendix D. The parity constraints on these reduced matrix elements are obtained by noting that the matrix elements of the polarizability tensor in (6.23) involve products such as 1| μa |i i| μb |0, each term of which carries a parity factor of (−1). The parity change for the overall transition |0 ⇒ |1 is therefore (−1)2 for two photons and (−1)n in the general n-photon case. Appropriate weightings for various polarization situations are given in Table 6.2, in which Lz and C+ , for example, denote linear polarization along z and left circular polarization, respectively. To see how the weightings arise, first consider the simple case in which the two beams are linearly polarized along a common axis. √ √ (0) 2/3. Taken The (00) entries in Table 6.1 show that T0 = −1/ 3 and T(2) 0 = together with the factors of (2k + 1)−1 in (6.35), this implies weightings of 1/3 and 2/15 for the squared reduced matrix elements with k = 0 and 2, respectively. The more complicated situation involving crossed linear polarization, along √ say ˆ 1 )/ 2. The the x and z axes, is handled by noting from (6.29) that ˆ x = (ˆ√ −1 −  ¯ − (10)]/ 2 is obtained by entries in Table 6.1 show that the relevant term [(10) taking (1) (2) (1) T(2) −1 = −T−1 = −T1 = −T−1 = 1/2.

(6.36)

Moreover, the terms T(k) ±1 combine with a common squared reduced matrix element

(k) ˆ [2] 2 [γ J T (O γ J ] , which accounts for the Lx Lz entry in Table 6.2. The discussion so far has focused on the influence of polarization, via the 2 k squared coefficients T−q (ˆ a ˆ b ) , in (6.35). The relative frequencies of the two beams must also be considered, because there is an important distinction between the T (0) and T (2) terms on the one hand and the T (1) terms on the other. The first

166

Photo-excitation

ωb

ωb

ωa Ei

ωa

ωa

ωa

ωb

ωb

1

E0 0 m

S1

Ei

E0 −1

3

E1

E1

+1

−1

0

S0

+1

m

Figure 6.1 Level diagrams for two-photon Hg 3 S1 ← 1 S0 excitation with opposite circular polarizations, with ωa > ωb (left) and ωa = ωb (right). E0 , E1 and Ei are the energies of the initial, final and intermediate states, respectively.

pair arise from the symmetric part of the polarizability tensor, with components # $   μβ |i i| μα + μα |i i| μβ [s]  ˆ [2]  Tαβ O (ωi0 − ω) = ¯ , (6.37) (ω ) (ω ) − ω − ω i0 a i0 b i where ω¯ = (ωa + ωb ) /2, while T (1) arises from the antisymmetric term # $ (ωa − ωb )  μβ |i i| μα − μα |i i| μβ [a]  ˆ [2]  Tαβ O =− . (ωi0 − ωa ) (ωi0 − ωb ) 2 i

(6.38)

The latter clearly vanishes whenever either the two polarizations α and β, or the two frequencies ωa and ωb are equal. A beautiful experimental consequence of this difference is demonstrated by the two-photon absorption from Hg 1 S0 to Hg 3 S1 [14]. Figure 6.1 gives a schematic level diagram for the experiment, which involves two counter-propagating beams with opposite circular polarizations. Spin-orbit assisted transitions to the M = 0 level of 3 S1 are parity allowed with intensity borrowing from either the M = 1 or M = −1 level of the intermediate state, I , depending on the polarization of the 1 S0 → I transition. As drawn on the left side of the diagram, the M = −1 route has the larger contribution because ωa is more nearly resonant with ωi0 . However, although the T−11 operator in Table 6.1 contains contributions from all three T0(k) , components, the intensity falls to zero if ωa = ωb , because the relevant matrix element, between the |J M = |10 and |00 components of 3 S1 and 3 S0 ,

-  '3 1 k 0  (k) 1 ! 1-T (k) -0 , S1 T0 S0 = − (6.39) 0 0 0

6.4 Spatial selectivity

167

vanishes for k = 0, 2. Hence, only the antisymmetric T0(1) spherical tensor operator, which has zero weight when ωa = ωb , can contribute to the transition. Such striking effects are rarely observed, but it is always useful to establish the relative importance of the symmetric and antisymmetric terms, which depends, ¯ according to (6.37) and (6.38), on the smallest ratio (ωi0 − ω)/(ω a − ωb ) in the sum over i. Hence the antisymmetric terms vanish in any single-colour experiment and the symmetric terms will predominate whenever all possible single photon transitions are far from resonance. On the other hand, the antisymmetric terms formally predominate in resonant or near-resonant situations, although (6.24) shows that it is usually possible to treat resonant two-photon absorption as a single-photon transition from the intermediate state. Note also that (6.33) carries immediate implications for transitions from rotationless states, with J

= 0. For example, the two-photon absorption intensity with linear polarization has contributions from the T0(2) and T0(0) operators if the (2) (1) two polarization axes are parallel, or from T±1 and T±1 if they are perpendicular. (1) Moreover the T terms vanish if the two frequencies are equal. The selection rules on the Wigner coefficient therefore allow transitions only to J = 2 or 0 in parallel polarization and only to J = 2 or 1 with perpendicular polarization, with the J ← J

= 1 ← 0 transition forbidden in a single colour experiment. The resonant two-photon np 2 A

2 , 2 E ← 2p 2 A

2 spectrum of H3 via the 3s 2 A1 state is one of many possible examples [15]. 6.4 Spatial selectivity The usual averaged intensity expressions, given for example by (6.35), disguise the fact that even single-photon absorption has spatial characteristics, which are important for the interpretation of multiphoton excitation and photo-ionization. It is convenient to employ the notation |J

M

⇒|Ji Mi  to allow for subsequent excitation to a further state |J M . As an illustration of the spatial behaviour, Fig. 6.2 shows marked differences in the distribution

2  Ji 1 J

P (Mi ) = 3 , (6.40) −Mi q M

M

according to the choice of the laser polarization and the total angular momentum transfer J = Ji − J

. The factor of three ensures unit normalization with respect to the sum over Mi . A systematic discussion of the consequences of such distributions is discussed under the headings of orientation and alignment in Sections 7.6 and 7.7. Here we simply note that a P branch (Ji = J

− 1) transition with linear polarization favours low |Mi | values, which is interpreted as creating a preferred (negative)

168

Photo-excitation

alignment of the intermediate angular momentum vector Ji perpendicular to the polarization axis, while a Q branch (Ji = J

) transition with the same polarization creates a preferred (positive) alignment of Ji parallel to the axis. These two distributions are strictly symmetric in the sign of Mi . However excitation with circular polarization gives a twist to the system, which yields a strongly asymmetric distribution when excited via a P (or R) branch transition. The resulting strong preference, in the P (6) distribution, for positive over negative Mi values is termed positive orientation of the Ji vector. The Q(5) distribution for linear polarization also has a slight asymmetry and therefore a weak orientation, by virtue of a displacement of its centre from Mi = 0 to Mi = −1/2. Insight into the forms of these distributions is provided by an illuminating asymptotic approximation for the Wigner coefficient, which yields the expression [16]2 P (Mi , qi ; q) 

 (1) 2 3 dq,qi (θ ) , 2Ji + 1

(6.41)

(1) where qi = Ji − J

and dq,q (θ ) is the reduced Wigner rotation matrix for the axis i rotation given by [13]

cos θ = −

Mi . Ji + 1/2

(6.42)

The accuracy of this result is well attested, even for Ji = 5, by the agreement between the histograms and their dashed envelopes in Fig. 6.2. The squares of the (1) (θ ) are given for convenient reference in Table 6.3. factors dq,q i Equation (6.41) is convenient for evaluating the classical limit of any average over Mi because (6.42) and (6.41) imply that  Ji  F  = P (Mi , qi ; q)F (Mi )  P (Mi , qi ; q)F (Mi )dMi −Ji

Mi

 2

3 2



π



0

2 (1) dq,q (θ ) F [−(Ji + 1/2) cos θ ] sin θ dθ, i

(6.43)

To see the physical origin of (6.41), θ is the angle between the polarization axis and the Ji vector [17]. Moreover the axis transformation  (1) Dqq (0, θ , 0)rq(1) rq(1) = i i qi

leads to



Ji −Mi

1 q

J

M



=

 qi

(1) Dqq (0, θ, 0) i

Ji −Ji

1 qi

J

M



,

in which Mi takes the extreme value −Ji on the right-hand side. Finally, inspection of the explicit forms shows that the magnitudes of the extreme squared Wigner coefficients for large Ji and J

vary approximately as (2Ji + 1)−1 if J

= Ji − qi , while the remaining terms have much smaller magnitudes, of order (2Ji + 1/2)−2 [13].

6.4 Spatial selectivity

169

Table 6.3 Forms of the squared reduced rotation matrix elements (1) (θ )]2 in (6.41). The index qi takes values (−1, 0, 1) for the [dq,q i (P , Q, R) transitions, respectively. q

P 1 0

−1

1 4 1 2 1 4

Q

[1 − cos θ]2

1 2

sin2 θ

cos2 θ

[1 + cos θ]2

1 2

Linear

R 1 4 1 2 1 4

sin2 θ sin2 θ

[1 + cos θ]2 sin2 θ [1 + cos θ]2

Circular

P(6)

−6 −4 −2

0

2

4

6 −6 −4 −2

0

2

4

6 −6 −4 −2

0

2

4

6

0

2

4

6

Q(5)

−6 −4 −2

Mi

Mi

Figure 6.2 Probability distributions, P (Mi ; q, J ) obtained by excitation to Ji

= 5 from J

= 6 and 5, via P (6) and Q(5) transitions, respectively, using either linear (q = 0) or left circular (q = 1) polarization. The dashed lines are the asymptotic approximations, given by (6.41).

where the scaling factor of (3/2) ensures unit normalization. Particular examples, with regard to the distributions in Fig. 6.2 are the orientation and alignment coefficients [13] / .  2 3 π  (1) Mi

− dq,qi (θ ) cos θ sin θdθ (6.44) O(J , Ji ; q) = √ 2 0 J (J + 1) / .  2 3Mi2 − J (J + 1) 3 π  (1)

 dq,qi (θ ) (3 cos2 θ − 1) sin θ dθ . A(J , Ji ; q) = J (J + 1) 4 0

170

Photo-excitation

Table 6.4 Classical limits of the orientation and alignment parameters for single-photon excitation in with polarization index q via P , Q and R branch transitions.

P Q R

O(J

, Ji ; q)

A(J

, Ji ; q)

q/2 0 −q/2

(3q 2 − 2)/10 −(3q 2 − 2)/5 (3q 2 − 2)/10

The resulting limiting parameter values, given in Table 6.4, are readily verified to correspond to the J → ∞ limits of the exact quantum mechanical expressions, which are given in Table 7.4. The orientation parameters for linear polarization (q = 0) vanish identically. Those for P and R branch excitation reach half the maximum allowed values for q = ±1 but the orientation arising from the slight asymmetry of the Q(5) distribution for q = 1 in Fig. 6.2 vanishes in the classical limit. We also see from Table 6.4 that the limiting alignment parameters for P , R excitation with q = 0 and Q excitation with q = ±1 all give values A = −1/5, because they all arise from the same sin2 θ distribution.

6.5 Resonant two-photon excitation As a prelude to the density matrix treatment of resonant two-photon excitation in Section 7.6, the intensity of a transition from |γ

J

 to |γ J  via a state |γ i Ji  may be expressed in the form I∝

 ' !' ! γ J M rq(1) γ i Ji Mi γ i Ji Mi rq(1) γ

J

M

b a

2

M M

∝

' ! ' ! ¯ i rq(1) γ J M , M γ J M rq(1) ρ γ J M J γ ¯ i i i M M i i i i b b

(6.45)

Mi M¯ i

where the density matrix elements are off-diagonal generalizations of the probabilities in (6.40); ρ Mi M¯ i

 Ji =3 −Mi

M

1 qa

J

M



Ji −M¯ i

1 qa

J

M



.

(6.46)

6.6 Multiphoton band structure

171

Table 6.5 Limiting enhancement factors for different two-photon transitions and polarization combinations. The entry (qa , qb ) = (0, 1) includes both linear or left-circular and crossed linear polarizations. qa qb

Polarization Transition

0, 0

1, −1

1, 1

0, 1

P P , RR P Q, QR P R, RP QP , RQ QQ

1.2 0.6 1.2 0.6 1.8

0.3 0.9 1.8 0.9 1.2

1.8 0.9 0.3 0.9 1.2

0.9 1.2 0.9 1.2 0.6

Formulations along these lines are usually preferable to the spherical tensor approach in resonant situations, because they provide specific insight into the angular momentum orientation at different steps. The off-diagonal elements of the density operator necessarily vanish if the two light beams have a common polarization axis, and only the above diagonal probabilities need be considered. As a simple illustration, the high J analogue of (6.45) is used to calculate the single-step probabilities required to estimate the relative rotational branch intensities for resonant two step J

⇒ Ji ⇒ J excitation between discrete states. It is convenient to adopt the notation qi = Ji − J

and qi = J − Ji , in which case (6.41) and (6.45) imply the following expression for the various combined relative transition intensities Q(J

, Ji , J ; qa qb ) =



P (Mi , qi ; qa )P (Mi , −qi ; qb )

Mi

9 ≈ 4



π 0



2  (1) 2 dq(1) d (θ ) sin θ dθ ,

(θ ) ,q ,−q q a i b i

(6.47)

which we term enhancement factors. The limiting values of these factors, as given by the second line, are given in Table 6.5.

6.6 Multiphoton band structure The appearance of the spectroscopic band structure depends on how the transition amplitudes vary with the body-fixed projections of the angular momenta. The factorization |γ J M = |J MK |η, in which case the body-fixed transformation

172

Photo-excitation

in (D.4) yields the following expression for the reduced matrix element in (6.33)3  - (k)  [n]  -



 -γ J γ J -T Oˆ (6.48)

   ! J ' k J

, [J ][J

] η Tν(k) Oˆ [n] η

= (−1)J −K −K ν K

ν

where [J ] = 2J + 1. 6.6.1 Symmetric tops Equation (6.48) applies for linear and symmetric top molecules to a single spherical tensor component of the appropriate multiphoton operator. Taken in conjunction with (6.35) this means that the transition strength is proportional to a sum of the form  k  [n] 2



2  T−q ˆ ' (k)  [n] 

!2 k J J

I= η , (6.49) [J ][J ] η Tν Oˆ −K ν K

(2k + 1) kqν

where the initial factor depends on the photon field, as given by Table 6.2 in the two-photon case. A corresponding table for three photons is given by Dixon et al. [6]. Experimental implications of this result have been reviewed in detail by k (ˆ [n] ) always includes a non-zero k = n term, Ashfold and Howe [3, 4]. Since T−q the J selection rule implies a total of 2n + 1 rotation branches for every vibronic transition. By analogy with the P QR labels for J = −1, 0, +1 for single-photon transitions, they carry the labels OP QRS for n = 2 and NOP QRST for n = 3. Figure 6.3 illustrates the resulting calculated branch structure associated with the (k)  ˆ [n]  spherical operator components for n = k = 1 − 4. Data appropriate to T1 O the origin bands of the A2  + −X2  vibronic transitions in the NO were employed [3]. Each panel shows two superimposed spectra, arising from the 2 1/2 and 2 3/2 components of the X state. The ν = 1 spherical tensor components were chosen because the angular momentum projections, which are written as , rather than 3

In cases with well-defined electronic angular momenta, the further factorization |η = |v| + λp, leads via (C.8) to # $



'

' (k)

! ' (k)







! 1 + (−1)p +p + + v v  η Tν η = v  λ p Tν v  λ p = √ 2 (1 + δ

0 δ + 0 )(1 + δ

0 δ + 0 )

- 

 k 

 × (−1) −λ  - T (k) -

.

−λ ν λ One also knows in the single photon case that [13]  - (1) -



  - T - = (−1) 0



1 0



0





d

λλ

,

where d

λλ

is the radial term (see Appendix D.3). However, the required sum over intermediate states severely complicates the situation in the general multiphoton case.

6.6 Multiphoton band structure

173

  Figure 6.3 Calculated rotational structure associated with the T1n Oˆ [n] spherical tensor components for (a) one-photon, (b) two-photon, (c) three-photon and (d) four-photon A2  + −X  vibronic transitions in the NO molecule at 300 K. Taken from Ashfold [3], with permission.

K for a diatomic molecule in Hund’s case (a), take values = 0 and = ±1 for the  and  states, respectively. The computational package PGOPHER [18] is available for the computation and interpretation of such spectra.  theoretical 

! selection rules for non-vanishing vibronic matrix elements ' Group (k) ˆ [n] η may be deduced from Table 6.6, which gives the representations η Tν O spanned by the Tν(k) spherical tensor components appropriate for single-colour experiments with one, two and three photons [3]. Suppose for simplicity that the transition is from a totally symmetric closed-shell ground state. Table 6.6 shows that single-photon allowed transitions are restricted to u+ and u states, but that g+ , g and g states become accessible with two photons. Thus the E,F 1 g+ double minimum states of H2 are directly accessible by 2 + 1 MPI spectroscopy [19]. Such transitions may also be observed with Doppler free resolution by using counter-propagating beams with the same frequency [3].

174

Photo-excitation

Table 6.6 Representations of the spherical tensor components ˆ of the one colour, n photon (n = 1 − 3) transition Tq(k) (O) operator for some common dihedral point groups. n

k

ν

1

1 1 0 2 2 2 1 1 3 3 3 3

0 ±1 0 0 ±1 ±2 0 ±1 0 ±1 ±2 ±3

2

3

a b

a

D∞h

D6h

D3h

u+ u g+ g+ g g u+ u u+ u u u

A2u E1u A1g A1g E2g E1g A2u E1u A2u E1u E2u B1u , B2u

A

2 E A 1 A 1 E

E A

2 E A

2 E E

A 1 , A 2

b

C3v

A1 E A1 A1 E E A1 E A1 E E A1 , A2

Assuming Hund’s case (a) or (b) Ignore g/u labels in C∞v symmetry

The NH3 molecule, which was an early focus of attention [20], has the interesting property that it has pyramidal, C3v , symmetry in its ground electronic state, while its Rydberg states are planar, with D3h symmetry. It is therefore simplest to use the D3h selection rules appropriate to an A 1 ground state in planar geometries. Table 6.6 shows that only A

2 and E states are accessible by single-photon spectroscopy, but that A 1 and E

states become accessible with two photons. Similarly the three(3) (3) photon operators T±2 and T±3 allow transitions to states with E

, A 1 and A 2 symmetries. We also see that the single-photon allowed transitions A

2 − A 1 or E − A 1 give rise to both T (1) and T (3) contributions in the three-photon version of (6.49). By contrast there is only a T (3) contribution for transitions to the E

, A 1 and A 2 states, because the Wigner coefficient carries the restriction ν = 0, ±1 for k = 1. Moreover, Table 2 of Dixon and Ashfold [6] shows that the intensities of these transitions increases by a characteristic factor of 2.5 in changing from linear, Lz Lz Lz to circular C+ C+ C+ polarization. Observations such as these have led to a major reassignment of the Rydberg series of NH3 [21].

6.6.2 Asymmetric tops Asymmetric tops differ from symmetric ones in the nature of their rotational states, which are no longer individual |J KM terms but linear combinations of the form

6.6 Multiphoton band structure

175

(see Appendix C.5) |J KMκ =

 K

(κ)   aK |J KM + (−1)τ |J − KM , √ 2(1 + δ K0 )

(6.50)

where J , K and M may be taken as integers, because spin–orbit coupling is normally quenched in such species. The sum is taken over either even or odd (κ) , which are obtained by diagonalization of the K values, and the coefficients aK rotational Hamiltonian, depend on the asymmetry index, κ, of the top [22] 2B − A − C . (6.51) A−C The transition amplitude in (6.48) therefore goes over to a combination of four terms  - (k) -



 γ J - T -γ J   (κ

) (κ )  !  ' = aK

aK [J ][J

] η Tν(k) η

/ 2 (1 + δ K 0 )(1 + δ K

0 ) κ=

νK K



J

J k × + (−1) K ν −K





k J

J k J

J τ

+ (−1) . (6.52) + (−1)τ K ν K

−K ν −K

' ! In addition the vibronic matrix elements η Tν(k) η

for positive and negative values of ν are related by ' (k)

! ' ! (k) η Tν η = (−1)ρ η T−ν (6.53) η

, J −K

k ν

J

K



τ +τ



where ρ depends on the relative symmetries of |η  and |q

 with respect to a plane containing the quantization axis. It follows from 'the symmetry - of the ! Wigner

- (k) -



coefficients under sign reversal of the second line that γ J T γ J vanishes unless J + k + J

+ ρ + τ + τ

(6.54)

is even. In addition the Wigner coefficients again carry the selection rule J = 0, ±1, ±k, which again gives rise to 2n + 1 rotational branches in any n-photon vibronic spectroscopic band. The states in (6.50), which are conventionally labelled JKa Kc , fall into four classes according to the evenness or oddness of the angular momentum projections Ka and Kc onto the a and c inertial axes, respectively [22]. Moreover the changes in Ka and Kc in single photon cases give rise to very useful a, b or c-type symmetry

176

Photo-excitation

selection rules according to whether the transition moment lies along the a, b or c inertial axis. To generalize these rules for n photons it is convenient to follow Dixon et al. in defining the following real combinations of the spherical tensor components [6]

  1 (k)  [n]  (k)  [n]  O =√ O + (−1)n−k+ν Tν(k) O [n] T−ν Tν+ 2(1 + δ 0ν )

  i (k)  [n]  (k)  [n]  Tν− O =√ O − (−1)n−k+ν Tν(k) O [n] , (6.55) T−ν 2(1 + δ 0ν ) which are constructed to be respectively symmetric and antisymmetric with respect to reflection in the xz plane. This choice corresponds to ρ = n − k + ν + σ in (6.54), where σ = 0 or 1 for the ν+ or ν− components. It is usual but not essential to choose the asymmetric top axes according to the sign of κ in (6.51) [13, 22]. Here we adopt the Ir convention, appropriate to a prolate (κ > 0) top like H2 O, such that (x, y, z) ≡ (b, c, a). Consequently Ka = K in (6.50)–(6.52), which means that Ka = (even or odd) for ν = (even, odd).

(6.56)

The corresponding restriction on Kc may be derived from (6.54) with the help of the symmetry discussion in Appendix C. In particular, comparison between (6.50) and (C.24) shows that τ = J + pr , where (−1)pr is the character of |J KMκ under a (12) permutation of the H atoms. Moreover the character under E ∗ , to rotation about the c-axis, shows that Kc = K + pr (mod 2). It therefore follows from (6.54) that Kc = (even or odd) for n + σ = (even, odd).

(6.57)

Table 6.7 lists the representations  of the C2v point group spanned by the symmetry-adapted spherical tensor components for n = 1–3, as defined in the Ir axis convention. The associated parity selection rules on Ka and Kc for the four possible band types, a, b, c and d are also given. Note that the operator symmetries A1 , A2 , B1 and B2 give rise to band types b, d, c and a, respectively, when the number of photons is odd, but to types d, b, a and c, respectively when the number is even. Table 6.7 also shows that the two- and three-photon spherical tensors Tν(k) include operators belonging to the A2 representation, which are excluded for one photon. The characterization of hitherto unobservable A2 states of both H2 O and H2 S by 3 + 1 MPI spectroscopy provided valuable information on the organization of the Rydberg states [3]. A limitation of MPI spectroscopy, with regard to the competition with predissociation, has also been turned to advantage. Although it proved impossible to detect the elusive 3p 1 A2 state of H2 O, owing to very rapid predissociation, the ˜ 1 A1 spectrum is seen much more clearly in the 3 + 1 MPI than in singleC˜ 1 B1 ← X ˜ 1 A1 ← X ˜ 1 A1 bands in the same spectral photon absorption, because the intense D

6.7 Angular momentum decoupling in high Rydberg states

177

Table 6.7 Representations under C2v of the symmetry-adapted body-fixed (k) of the n-photon transition operators in the Ir spherical tensor components Tq± axis convention. The associated parity selection rules on Ka and Kc yield four possible bands types a, b, c and d. n

k

ν±



Ka

Kc

Type

n

k

ν±



Ka

Kc

Type

1

1 1 1 0 2 2 2 2 2

0 1+ 1− 0 0 1+ 1− 2+ 2−

B2 A1 B1 A1 A1 B2 A2 A1 B1

Even Odd Odd Even Even Odd Odd Even Even

Odd Odd Even Even Even Even Odd Even Odd

a b c d d c b d a

3

1 1 1 3 3 3 3 3 3 3

0 1+ 1− 0 1+ 1− 2+ 2− 3+ 3−

B2 A1 B1 B2 A1 B1 B2 A2 A1 B1

Even Odd Odd Even Odd Odd Even Even Odd Odd

Odd Odd Even Odd Odd Even Odd Even Odd Even

a b c a b c a d b c

2

region are removed by rapid predissociation [23]. Moreover, a much slower rotationally selective predissociation mechanism in the C˜ 1 B1 state, which competes with the MPI rate, was identified. The high frequency lines in the 3 + 1 MPI spectrum were found to be systematically reduced by a factor proportional to 1 + 0.35Ja 2 , assuming a predissociation line-width of 2 cm−1 . The Ja depen˜ 1 A1 state because the Ja dence is consistent with Coriolis coupling to the 3p D ˜ 1 B1 state is well rotational component has B1 symmetry. Furthermore, the 3p D known to be strongly predissociated by coupling to the directly dissociative B˜ 1 B1 state [24]. It also proved possible to analyze the strengths of different transitions (3) (3) for evidence of' interference the ! 'between ! T3 and T1 contributions to the sum (3) (3)



in (6.52). The η T3 η : η T1 η amplitude ratios for H2 O and D2 O were found to be 1.0 : −0.4 and 1.0 : −0.2, respectively [23]. 6.7 Angular momentum decoupling in high Rydberg states ! ' While the symmetry restrictions on the electronic transition amplitudes η Tν(k) η

in (6.49) and (6.52) are well established, it is seldom possible, at least for the valence states, to assess the detailed composition of the electronic wavefunctions, without a detailed ab-initio calculation. There are, however, small hydride molecules, such as H2 O and NH3 , in which the outermost electron is well approximated as occupying a single atomic orbital; and the Rydberg orbitals to which it is excited become more and more atomic in character as the principal quantum number increases. Moreover, the advent of double-resonance spectroscopy allows transitions between sequences of states with well-defined electronic angular momenta, which are often subject to a strong Franck–Condon selection rule, which conserves the level of

178

Photo-excitation

vibrational excitation during the transition, in view of the close similarity between the potential functions terminating on a common electronic state of the positive ion.4 There are also ‘Rydberg molecules’, such as the rare gas hydrides H3 and NH4 , which are completely unbound in their ground states, but which have sufficiently stable Rydberg states to allow spectroscopic investigation. It is also natural to assume that the excited states will become progressively more atomic in character as they approach the ionization limit. This section describes systematic changes in the transition amplitudes associated with uncoupling such well-defined electronic angular momenta from the internuclear axis, as the principal quantum number increases – the condition for ‘coupling’ or ‘uncoupling’ being that the energy separations between all |λ components are large or small compared with the Coriolis coupling matrix elements [13]. Case (b) transition amplitudes The coupled states are assumed for simplicity to conform with Hund’s case (b), with state functions given by (C.8) and Table C.2. Thus ! |b = + ; λ; N MN ; p (6.58)   + | |λ|N MN  + (−1)τ |− + | − λ|N − MN  , = √ 2(1 + δ 0 δ + 0 ) where τ = p − q + + N, and +  0. The relevant reduced matrix element for transitions from a lower coupled state |b

 is then given by  - (k) -

 b - T -b # $ p +p

+ +

 1 + (−1)



 +J −K −λ = [N ][N

](−1) √ 2 (1 + δ

0 δ + 0 )(1 + δ

0 δ + 0 ) ν



  - (k) -

 k N

k 

 N 

λ

(6.59)  ×  T

d λ

− ν −λ ν λ





N k N

 k 



λ

d + (−1)τ δ + 0  −λ ,

ν

λ ν λ

-   where [N ] = (2N + 1),  - T (k) -

is a reduced angular momentum matrix



element, with a parity equal to the number of absorbed photons, and d λλ is the radial matrix integral. In addition, the first square bracket term requires that 4

Since electronic transition moments are much larger than vibrational ones, the transition amplitude is normally well approximated as !' ! ' k

! '  Tp   ψ el ψ rot Tpk ψ

el ψ

rot ψ vib ψ

vib , in which the final vibrational overlap factor carries the Franck–Condon selection rule [2].

6.7 Angular momentum decoupling in high Rydberg states

N

+

N′

dσ

+

N′

N

179

+

3

5

2

4 2 0

1

3 2 1

dπ

3

3 1

3 2

2

Even parity

3 2 1

dδ

Odd parity

Figure 6.4 A schematic correlation diagram relevant to nd(N ) ← 3p(N

= 2) transitions in a diatomic molecule. The central and outer parts of the diagram apply to the case (b) and case (d) limits, respectively. Lower and upper members of each doublet correspond to e and f parities, respectively.

p + p

+  + 

is even. Written in terms of (6.48), this means that $ #

p +p

+ +

- 



' (k)

! 1 + (−1)  k 



 - T (k) -

d λλ . η Tν η = C C

−λ ν λ 2 (6.60) To follow the changes as the low-energy case (b) transition amplitudes in (6.59) go over to case (d) amplitudes at higher energies, Fig. 6.4 shows a correlation diagram for the nd rotational–electronic states accessible by single-photon excitation from the N

= 2 rotational level of an n

p electronic manifold. It is assumed that spin doubling can be ignored and that + = q + = 0. We first consider transitions from the initial npσ component, for which the initial parity is required to be even when N

is even. Hence transitions occur only to the right-hand (odd parity) side of Fig. 6.4. Moreover transitions to the n δ electronic component are forbidden in the case (b) limit, because −1  ν  1. Hence the low-energy spectrum consists of (N = 0, ±1) and (N = ±1) transitions to the dπ electronic component and dσ components, respectively – otherwise termed (P , Q, R) and (P , R) transitions, respectively. The situation with regard to the initial n

pπ electronic component is less restrictive because both positive and negative parities are allowed, of which the former satisfy the same selection rules as for the n

pσ component, except that

180

Photo-excitation

(N = 0, ±1) transitions to the n δ component are no longer forbidden. Those from the odd initial parity level satisfy the same N selection rules, with regard to the dδ and dπ components, but with only a N = 0 transition to the n σ electronic component. Case (d) propensity rule Spectral changes occur as the principal quantum number increases, first because the dσ , dπ and dδ components become increasingly mixed by Coriolis coupling. The resulting intensity borrowing leads to the appearance of the previously forbidden nominal dδ ← pσ transitions. Eventually, the convergence of the three electronic origins to within a separation that is small compared with the rotational one leads to the case (d) limit, in which the energies are given, to a first approximation, by EnN + = I −

Ry + B + N + (N + + 1), (n − μ) ¯ 2

(6.61)

where N + is constrained by the vector coupling rule J = N+ +  to lie in the range, |N −  |  N +  N +  . Moreover, Table C.2 requires that p +  + N + is even for + = 0, which explains why the even and odd parity terms in Fig. 6.4 converge on even and odd N + limits. There is also an interesting case (d) propensity rule, whereby the intensity concentrates in the series converging on selected ionization limits – an effect that was first observed and explained in the spectrum of Na2 by Martin et al. [8]. The origin of this intensity pooling may be understood by using the frame transformation in (C.14) to transform the transition amplitude in (6.59) from the case (b) to the case (d) representation. Within the approximations that  is well defined, and that the radial matrix elements are independent of λ , one finds5 that  - (k) -

   - (k) -

 d |b  b - T -b d - T -b = b

- !

'





= (−1)N +N + +k−M + B  - T (k) -

d )

( +  N N+ 

N N , × k N





− + −λ

5

(6.62)

The sum over b includes a sum over parity p , which allows one to drop the symmetrization of the |b  states. Equation (6.62) follows with the help of the recoupling identity [13] )

+

( + 

N N N+ 

 N  N N (−1)τ =





+ +

k N 

− −λ − −λ λ

ν



× where τ = N + +  − N + k + N

+ 

− ν + + .

k ν

N



N −



 −λ

k ν



λ



,

6.7 Angular momentum decoupling in high Rydberg states

181

where  B=

[N

][N + ][N] (1 + δ

0 δ + 0 )(1 + δ + 0 )  



1 + (−1)p +p + + 

+ +

1 + δ + 0 (−1)p +q −N + . × 2

(6.63)

This propensity rule shows that the optically accessible positive ion states |N + +  are dictated by the initial rotational electronic quantum numbers, |N





λ

. It generalizes an earlier form given by Martin et al., from single-photon to multiphoton processes, and to situations in which the positive ion may have + = 0 or may be in a  − state with q + = 1 (see Martin et al. [8], Fredin et al. [25], Child and Jungen [26]). The restrictions implied by (6.62) and (6.63), namely |N

− k|  N  N

+ k, |N

− 

|  N +  N

+ 

, p + q + − N + +  = even

for + = 0,

|

− k|    

+ k

+ =

= λ

(6.64)

shows that the energy of the final N + level in (6.61) depends only on the initial angular momenta 

and N

, while the degeneracy of the final state depends on the order k of the relevant spherical tensor operator. The term propensity rule, is preferred to selection rule because the derivation rests on the assumption that the spectrum arises from excitation of a single electron with well-defined initial and final electronic angular momenta, | λ  [25]. In addition, the sum over |b  in (6.62) is performed on the assumption that the λ dependence in (6.60) lies only in the 3j symbol. Consequently, - the radial matrix  elements contained in the reduced matrix element  - T (k) -

are taken to be independent of λ . On the other hand there is an alternative route to this rule, arising from the necessarily close connection between the selection rules for high n excitation and those for photo-ionization that are discussed in Section 7.3.2. In fact the total peak intensity given (6.62) and summed over N , may be shown to be proportional to the Q(t ) factor for photo-ionization in Table 7.1, where the contributing t terms depend on the composition of the initial rather than the final orbital, with t = 

when the latter is well defined. Thus the rules in (6.64) appear to require only that  should be well defined in either the initial or the final state. As an example, Fig. 6.5 shows fragments of the computed nd ← D 3p 2 − (N

= 2) spectrum of NO, which takes account of significant λ dependence of

182

Photo-excitation d4 d2 d0

(c) 26 35

74600

74580

(b)

74200

74640 d4 d2 d0

19

25

74450

(a)

74620

74500

74550

15

74250

18

74300

E / cm

74350

d4 d2 d0

74400

1

Figure 6.5 Fragments of the computed nd ← D 3p 2 − (N

= 2) spectrum of NO for (a) n = 14 − 17, (b) n = 18 − 24 and (c) n = 25 − 34, as numbered for the d2 series. The computed spectrum is convoluted with a 1 cm−1 wide Lorentzian profile. Data employed in [25] were kindly recomputed by Jungen (private communication).

the radial matrix elements, as well as strong sσ − dσ orbital mixing in the final state [25]. The initial orbital was however taken to have pure p character, with 

= 1, giving rise to the prediction of a single series of high n peaks with N + = 2. The left-hand side of Fig. 6.4 may be used to see how this series emerges from the low n spectrum, because the initial parity is odd. Groups of six nd transitions are predicted at low n, of which five are apparent in the first three groups in Fig. 6.5(a).6 The groups merge to form well-defined d0, d2 and d4 series in Fig. 6.5(b), of which the d2 series is much the strongest. Finally the intensities of the remaining series die away in Fig. 6.5(c), in accordance with the restrictions on (6.64). A similar confirmation of the case (d) propensity rule as found for 6

The calculation also predicts a single ns series terminating on N + = 0, which is negligibly weak on the scale of the diagram.

6.8 ZEKE intensities

183

an asymmetric top by Child and Glab in the resonant photo-ionization spectrum of H2 O [27]. 6.8 ZEKE intensities ZEKE-PFI spectroscopy is a technique for determining very highly resolved photoelectron spectra, by exploiting the convergence of the Rydberg series at successive vibrational and rotational ionization limits [28]. The experiment involves single or multiple photon absorption to reach Rydberg levels with n  100, which lie within a few cm−1 of the desired ionization limit. A subsequent delay of 0.5–5 μs allows the dispersion of electrons arising from lower auto-ionization limits, after which an electric field pulse of a few V cm−1 is applied to lower the chosen ionization threshold, thereby creating a photo-electron pulse with an energy determined by the exciting laser. Details of the molecular mechanism are discussed in Section 8.2. Finally, a correction for the influence of the pulse voltage7 gives the field-free ionization limit. In addition to this energy information, the high density of states in this near ionization region allows the identification of subtle intramolecular interactions, via the intensities of the spectral lines [9]. Before considering the complications arising from the very high density of the ZEKE states, we may employ the selection rules in (6.64) to predict the rotational branch structure of the very high n ZEKE spectra, provided that there are no significant spin considerations. Thus for example the σ + g orbital from which the electron is excited in N2 may be assumed to have predominantly s character, with 

= 0, which explains the occurrence of very strong Q branches, with N + = N

in the ZEKE spectrum [29]. The presence of additional branches, with both positive and negative N values remains however to be explained later. More complicated cases, involving significant spin recoupling are most conveniently handled by reference to the photo-ionization Q(t ) factors in Table 7.1. For example, the photo-ionization of O2 X 3 g− in case (b) involves removal of a π g orbital, with predominantly d or 

= 2 character to produce a case (a) molecular ion state 2 +

O+ 2 X g . The allowed values of J –N depend on the possible values of Jt in the fourth section of Table 7.1, which are given, within the approximation t  

= 2 by Jt = 3/2, 5/2. Now the rotational levels of a case (a) 2  state vary as N + (N + + 1), with N + = J + − 1/2 for the + = 1/2 component and 7

In a classical picture, the application of a Stark field, E, in the z direction Coulomb potential √ creates the modifield √ V (z) = Fz − 1/z (in atomic units) with a maximum at z = −1/ F and V (z) = −2 F. When converted to laboratory units, this corresponds to a reduction of the ionization limit given by √ E  0.6 F cm−1 , if the field F is measured in V cm−1 .

184

Photo-excitation

Figure 6.6 Asymmetric profiles of the (v + , v

) = (1, 0) and (0, 0) vibrational bands in the ZEKE-PFI spectrum of N2 . Taken from Merkt and Softley [29], with permission.

N + = J + + 1/2 for the + = 3/2 component [30].8 Hence the final 3j symbol allows branches with N + − N

= 0, ±1, ±2 and ±3, whereas the observed ZEKE spectrum contains branches up to N + − N

= ±4 and ±5 [31]. 6.8.1 Asymmetric band profiles One of the first pieces of evidence for systematic discrepancies between observed intensities and those implied by the orbital approximation model comes from the presence of anomalous a-type rotational lines in the ZEKE-PFI spectrum of H2 O, instead of the purely c-type structure expected on the basis of (6.56) and (6.57) for excitation from a 2pb1 orbital [32].9 This behaviour was rationalized by Gilbert and Child on the basis of a frustrated rotational auto-ionization mechanism, whereby the positive ion dipole couples relatively low-lying (n ∼ 20–60) levels of the d series with the p-wave quasi-continua at a lower ionization limit than that allowed by the normal selection rules [33]. Similar frustrated rotational, vibrational and electronic auto-ionization mechanisms apply for other molecules. The typical effect is to enhance the intensities of certain rotational branches of the ZEKE-PFI spectrum, as illustrated for the N2 molecule in Fig. 6.6. The presence of a strong Q branch, with N + − N

= 0, is explained above but an explanation 8 9

To avoid confusion with the body-fixed projecion of N + in a polyatomic molecule, the symbol N + is used in preference to the conventional K + [30]. ! Table C.6 designates pb1 orbitals as 

, λ

, pe

= |1, 1, 1, from which (6.56) and (6.57) predict Ka+ − Ka

= +

odd and Kc − Kc = even, whereas the a-type lines correspond to Ka+ − Ka

= even and Kc+ − Kc

= odd.

6.8 ZEKE intensities

185 I0(N + = 12)

n 40

Rotational (Quadrupolar) Auto-ionization + I0(N = 10)

36

I0 – 6 f1

34 32

I0 – 6 f2

N + = 12

Detection band N + = 10 Direct O(12) transition J = 12

Figure 6.7 The rotational auto-ionization mechanism responsible for asymmetric rotational branch structure in the ZEKE-PFI spectra of diatomic molecules.

is required for the strength of some of the other branches, and the disparities in intensity between the branches S,U on one hand and Q,M on the other, which have positive and negative values of N + − N

, respectively. The enhancement of the negative N branches in the (0, 0) band in Fig. 6.6(a) is attributed below a pervasive rotational auto-ionization effect. However, the ‘anomalous’ intensity of the S and M branches of the (1, 0) band in Fig. 6.6(b) will be attributed to intensity borrowing between the quasi-continua belonging to rotational levels at the v + = 1 limit and levels of a series converging on the excited A 2 u state of N+ 2 that accidently fall into the same energy region. The essence of the auto-ionization mechanism is illustrated in Fig. 6.7, by reference to the spectrum at the v + = 0 ionization limit of N2 . This particular experiment employed a discrimination field, F1 , which was followed by a stronger extraction field F2 to give an extraction window   (6.65) ν = 0.6 F2 − 0.6 F1 ≈ 3 cm−1 , which is marked by the shaded box at the two lower ionization limits, which belong to the weakly allowed nd series. The illustration assumes an initial rotational level J

= 12 from which transitions to the series terminating on N + = 12 and 10 are strongly and weakly allowed, respectively. The fields are adjusted to ensure that the electronic-level separation Ry /n3 in the allowed series is of the same

186

Photo-excitation

order or smaller than the detection band ν at the two lower ionization limits, which allows intensity borrowing into the N + = 10 quasi-continuum, thereby enhancing the intensity of the O(12) transition. This rotational auto-ionization mechanism necessarily enhances rotational branches of the ZEKE-PFI bands with negative values of N + − J

, thereby giving the band an asymmetric intensity profile. Moreover, the number of branches that can be enhanced in this way depends on the magnitude of the rotational constant, because the limiting condition on the level density of the allowed series, Ry /n3  ν is independent of the molecule in question, but the number of ZEKE continua over the corresponding energy range, Ry /n2 , increases inversely with the magnitude of the rotational constant. Thus, while the enhancement of the N2 spectrum, for which B + = 1.9 cm−1 is limited to the N + − N

= −2 and −4 branches, the corresponding enhancement for Na2 , for which B + = 0.11 cm−1 extends at least to the N + − N

= −12 branch [9]. A similar mechanism applies to vibrationally and electronically induced autoionization, provided that an appropriate optically accessible state lies within the extraction window of the ZEKE quasi-continuum. An interesting example is found in the (v + = 1) ← (v

= 0) band in the ZEKE spectrum of N2 , because there are well-known perturbations between the f series converging on the X 2 g+ of 2 N+ 2 and the d series of the low-lying A u states [29, 34, 35]. The ZEKE-PFI experiments show enhanced intensity in the S and U rotational branches of the (v + = 1) ← (v

= 0) state, involving positive angular momentum differences, N + − J

= 2 and 4, instead of the negative differences in the (0 ← 0) band. Such enhancement may be explained by an accidental coincidence between the rotational levels of the A(v + = 0) 4d 1 u series and f wave quasi-continua at successive ionization limits of the X(v + = 1) state with N + > J [29]. To take the general context, we know that an  series converging on an N + limit allows total angular momenta N = N + , N + ± 1, . . . , N + ± , and that N is conserved in any interaction. Moreover the selection rule for single photon optical excitation requires N = N

± 1 for a homonuclear diatomic. Two extreme cases may be considered (a) N = N + −  = N

± 1 → N + − N

=  ± 1 (b) N = N + +  = N

± 1 → N + − N

= − ± 1.

(6.66)

For  = 3, the former case, which is assumed to apply for the (1 ← 0) ZEKE band of N2 , leads to enhancement of the S and U rotational branches, whereas case (b) would cause enhancement of the M and O branches. Notice that both these examples give rise to asymmetric intensity enhancement, 2 either with positive or negative values of N + − N

. Cases such as O+ 2 X g ← O2

6.8 ZEKE intensities

187

Table 6.8 Character table for the molecular symmetry group Td (M). Td (M)

E

(123)

(14)(23)

(1423)∗

(23)∗

A1 A2 E F1 F2

1 1 2 3 3

1 1 −1 0 0

1 1 1 −1 −1

1 −1 0 1 −1

1 −1 0 −1 1

 (s)  (∗)

X 3 g− , in which both positive and negative N branches are roughly equally enhanced, suggest an addition angular momentum component (here 

= 4) in the initial orbital [36].

6.8.2 Symmetry selection rules Symmetry selection rules are particularly useful in analyzing the photo-ionization spectra of polyatomic molecules. The general symmetry selection rule for a dipoleallowed single-photon transition may be expressed as [37] 

⊗  =  (∗) ,

(6.67)

where 

and  are the total representations spanned by the initial and final states and  (∗) is the totally antisymmetric representation, which means antisymmetric under all operations containing the space-fixed inversion E ∗ in the appropriate molecular symmetry group. Complications arise, however, in a group such as Td (M) because E ∗ , which reverses the handedness of the tetrahedron, is not itself an operation of the group. However, the handedness may be restored by subsequent or prior (1423) or (23) permutations. Thus the composite operations (1423)∗ and (23)∗ do belong to the Td (M) group, and Table 6.8 identifies  (∗) as A2 . Signorell and Merkt apply (6.67) to photo-ionization by recognizing that the wavefunction of the ejected electron transforms as the totally symmetric representation,  (0) , if  is even and as  (∗) if  is odd [38]. In other words ( (s)  for  even (el) . (6.68)  =  (∗) for  odd Thus, if  (+) denotes the symmetry of the positive ion, the chain 

⊗  = 

⊗  (+) ⊗  (el) ⊃  (∗) ,

(6.69)

188

Photo-excitation

requires that

 ⊗

( (+)

⊃

 (∗) for  even ,  (s) for  odd

(6.70)

which is the desired group theoretical selection rule for photo-ionization. In addition the  = ±1 selection rule for single-photon absorption requires the following restriction on the difference between the total angular momenta, J

and J + , of the initial neutral state and the positive ion, respectively; J + − J

=  − 3/2,  − 1/2, . . . ,  + 3/2,

(6.71)

N + − J

=  − 1, , . . . ,  + 1,

(6.72)

or

if the spin–orbit splitting in the positive ion is unresolved. The case of CH4 is particularly interesting, because the electron is ionized from what is predominantly a p orbital, leaving the positive ion in an F2 electronic state in tetrahedral symmetry. The resulting strong quadratic Jahn–Teller effect gives rise to twelve equivalent equilibrium structures, with C2v symmetry, which divide into two essentially non-interacting enantiomeric sets of six [39]. The feasible operations within each set comprise the elements of the Td (M) point group, for which (6.70) carries the selection rules [38] A1 ⇐⇒ A2 , E ⇐⇒ E and F1 ⇐⇒ F2 for  even and A1 ⇐⇒ A1 , A2 ⇐⇒ A2 , E ⇐⇒ E F1 ⇐⇒ F1 , F2 ⇐⇒ F2 for  odd. The observed high-resolution single-photon ZEKE-PFI spectrum contains 24 lines that were assigned to transitions to the lowest six-degenerate tunnelling manifold [40]. The rotational and symmetry assignments of the upper and lower states are however complicated by the low temperature of the sample, which restricts the nuclear spin states to the lowest initial rotational states allowed by symmetry, namely J

= 0, 1 and 2 for nuclear spin symmetries A, F and E, respectively [22]. The initial J

value and the nuclear spin assignment for a given transition were established by a double-resonance depletion, or ZEKE dip, experiment involving saturation of a ν 3 band transition by a laser tuned to different known J

= 0 − 2 lines in the IR spectrum. A second IR + VUV double resonance experiment simplified the assignments of the Jahn–Teller upper state by recording the

References

189

ZEKE-PFI spectrum via selected intermediate states of the ν 3 fundamental. The expected A ⇐⇒ A, E ⇐⇒ E and F ⇐⇒ F selection rules were confirmed, but the symmetry assignments based on nuclear spin cannot distinguish between A1,2 and F1,2 symmetries, because the total wavefunction evr ⊗ ns may contain either A1 or A2 [22]. Plausible A1,2 and F1,2 assignments are however available from a sophisticated tunnelling–rotation model, which involves combining the asymmetric top rotational states of the six C2v structures [40]. The matrix elements, appropriate to the feasible tunnelling paths between these structures, had signs adjusted to take account of the geometric phase [41].

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P. Lambropoulos and S. Smith, Multiphoton Processes (Springer, 1984). G. Herzberg, Electronic Spectra of Polyatomic Molecules (van Nostrand, 1967). M. N. R. Ashfold, Mol. Phys. 58, 1 (1986). M. N. R. Ashfold and J. D. Howe, Ann. Rev. Phys. Chem. 45, 57 (1994). M. Goppert-Mayer, Ann. Physik 9, 273 (1931). R. N. Dixon, J. M. Bayley and M. N. R. Ashfold, Chem. Phys. 84, 21 (1984). W. McClain and R. A. Harris. In Excited States, vol. 3, Chapter 1, ed. E. C. Lim (Academic Press, 1977). S. Martin, J. Chevaleyre, M. C. Bordas et al., J. Chem. Phys. 79, 4132 (1983). F. Merkt and T. P. Softley, Int. Rev. Phys. Chem. 12, 205 (1993). C. Cohen-Tannoudji, B Diu and F Lalo¨e, Quantum Mechanics (Wiley Interscience, 1977). D. E. Manolopoulos. Summer school lecture notes Quantum Theory of Molecular Photodissociation at http://physchem.ox.ac.uk/∼Emano/. I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products, 5th edn (Academic Press, 1994). R. N. Zare, Angular Momentum (Wiley Interscience, 1988). G. Grynberg and B. Cagnac, Rep. Prog. Phys. 40, 791 (1977). M. C. Bordas, L. J. Lembo and H. Helm, Phys. Rev. A 44, 1817 (1991). P. J. Brussard and H. A. Tolhoek, Physica 23, 955 (1957). M. S. Child, Semiclassical Mechanics with Molecular Applications (Oxford University Press, 1991). http://pgopher.chm.bris.ac.uk/index.html. E. E. Marinero, R. Vasudev and R. N. Zare, J. Chem. Phys. 78, 692 (1983). G. C. Nieman and S. D. Coalson, J. Chem. Phys. 68, 5656 (1978). J. H. Glownia, G. C. Nieman, S. J. Riley and S. D. Coalson, J. Chem. Phys. 73, 4296 (1980). G. Herzberg, Infra-red and Raman Spectra (van Nostrand, 1945). R. N. Dixon, J. M. Bayley and M. N. R. Ashfold, Chem. Phys. 84, 235 (1984). S. Bell, J. Mol. Spec. 16, 205 (1965). S. Fredin, D. Gauyacq, M. Horani et al., Mol. Phys. 60, 825 (1987).

190 [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41]

Photo-excitation M. S. Child and C. Jungen, J. Chem. Phys. 93, 7756 (1990). M. S. Child and W. L. Glab, J. Chem. Phys. 112, 3754 (2000). K. M¨uller-Dethlefs and E. W. Schlag, Ann. Rev. Phys. Chem. 42, 109 (1991). F. Merkt and T. P. Softley, Phys. Rev. A 46, 302 (1992). G. Herzberg, Spectra of Diatomic Molecules (van Nostrand, 1950). R. G. Tonkyn, J. W. Winniczek and M. G. White, Chem. Phys. Lett. 164, 137 (1989). R. G. Tonkyn, R. T. Wiedmann, E. R. Grant and M. G. White, J. Chem. Phys. 95, 7033 (1991). R. D. Gilbert and M. S. Child, Chem. Phys. Lett. 187, 153 (1991). K. P. Huber, C. Jungen, K. Yoshino, K. Ito and G. Stark, J. Chem. Phys. 100, 7957 (1994). C. Jungen, K. P. Huber, M. Jungen and G. Stark, J. Chem. Phys. 118, 4517 (2003). M. Braunstein, V. McKoy, S. N. Dixit, R. G. Tonkyn and M. G. White, J. Chem. Phys. 93, 5354 (1990). P. R. Bunker and P. Jensen, Molecular Symmetry and Spectroscopy, 2nd edn (NRC Press, 1998). R. Signorell and F. Merkt, Mol. Phys. 92, 793 (1997). R. F. Frey and E. R. Davidson, J. Chem. Phys. 88, 1775 (1988). H. J. W¨orner, X. Qian and F. Merkt, J. Chem. Phys. 126, 144305 (2007). I. B. Bersuker, The Jahn–Teller Effect (Cambridge University Press, 2006).

7 Photo-ionization

The theory of photo-ionization owes much to the treatment of photo-excitation in the previous chapter, but there are significant differences. Most importantly the species is excited to a final state in which the electron is detached from the positive ion. The necessary boundary conditions resemble those for a scattering event, except that the partial waves are combined to produce outgoing plane wave motion in a particular target channel, instead of an incoming plane wave in the incident scattering channel. Confusingly the former are referred to as ‘incoming’ and the latter as ‘outgoing’ boundary conditions, because the amplitudes and phases are adjusted to ensure only incoming spherical waves in photo-fragmentation and outgoing spherical waves in scattering. Details are given in this chapter for the simple case of a single open ionization channel, leaving the multichannel boundary conditions to be treated in Appendix D.2. It is also shown in Section 7.1 how the spherical tensor machinery in Chapter 6 can be adapted to handle multiphoton ionization. The theory is presented for a bulk sample with a random distribution of magnetic sub-levels, but the averaging over fragment sub-levels is more awkward than for a final bound state. Further complications come from possible changes in the angular momentum coupling regime between the parent neutral molecule and the resulting positive ion, details of which are covered in Appendix D.2. The following presentation is intended to combine the early results of Buckingham et al. with the formal ‘angular momentum transfer’ theory of Fano and Dill [1, 2, 3]. Expressions for the integrated photo-ionization cross-sections are combined with a table appropriate to different angular momentum coupling schemes plus the appropriate selection rules. The subsequent treatment of photo-electron angular distributions, or differential cross-sections, includes a discussion of factors that determine the photo-electron asymmetry parameters for single- and two-photon processes, with particular reference to H2 .

191

192

Photo-ionization

These introductory sections are taken from the well-established literature. Recent experiments have opened up new possibilities, by showing how the combined angular distributions of photo-electrons and dissociating positive ion fragments can provide information on the photo-electron angular distribution with respect to the axes of ‘molecules fixed in space’, which provide direct information of the nature of the orbital from which the electron is detached [4]. The discussion in Section 7.5 follows the theory given by Dill and Lucchese et al. [5, 6]. Another relatively recent application is to the theory of resonant multiphoton ionization (REMPI), which is included in Section 7.6, firstly to demonstrate the advantage of a density matrix, rather than spherical tensor formulation of the theory, and secondly to illustrate how the magnitudes and phases of the photo-ionization matrix elements can be experimentally determined by a careful choice of relative laser polarizations and quantization axes [7]. The chapter also includes a discussion of ‘orientation and alignment’ of the angular momentum of the positive ion, which is an additional source of information on the underlying matrix elements [8]. Finally, there is a section on spin polarization along the lines expounded by Raseev and Cherepkov [9].

7.1 Boundary conditions and cross-sections The theory of photo-ionization is a variant of scattering theory in which the wavefunction consists of an outgoing electron plane wave associated with a particular target state of the positive ion plus a linear combination of incoming spherical waves, whereas the normal scattering states comprise an incident plane wave in a particular initial channel plus a combination of outgoing spherical waves. The ‘outgoing’ boundary conditions on the scattering spherical waves are therefore replaced by ‘incoming’ conditions on the spherical parts of the photo-ionization wavefunction. Detailed treatments of the two processes also differ to the extent that photo-ionization involves an additional photo-excitation step, which probes the form of the wavefunction at relatively short electron separations, whereas scattering theory concentrates on the behaviour as r → ∞. To understand the photo-ionization theory in its simplest form, we assume + + + a single possible target state1 |χ + 1  = |γ N M , and a free electron, with an outward plane wave component with wavevector k. By the argument above, the wavefunction must be chosen to satisfy the incoming boundary conditions [10, 11]

f (θ k.r ) −ikr − −3/2 ik.r e + χ+ 1k ∼ (2π) (7.1) e 1, r 1

The generalization to include multiple target states is given in Appendix D.2.

7.1 Boundary conditions and cross-sections

193

where θ k.r is the angle between the electron displacement vector r and the wavevector k. Such wavefunctions have the partial wave expansion [11]2  1 − ∗ 1k = χ+ e−iσ  Ym (θ , φ)Ym (ϑ, ϕ) Rk (r), (7.2) kr 1 m where the angles (θ , φ) fix the direction of k, (r, ϑ, ϕ) are the electron coordinates and " 2 sin[kr + δ  ], Rk (r) ∼ (7.3) π in which (2.5), (2.6) and (2.39) show that the phase term is given by δ  = arg ( + 1 − i/k) − π /2 + τ  (E).

(7.4)

The normalization conditions, at this stage, ensure that  Rk (r)Rk  (r)dr = δ(k − k ) ' − − ! γ k |γ k = δ(k − k ).

(7.5)

The theory differs from that for discrete absorption in Chapter 6 only to the extent that the final step in the multiphoton chain involves excitation into the continuum, |1 = |1k−, with outgoing wavevectors in the interval [k, k + dk]. The golden rule formula [12] for the rate of such transitions is given, in the notation of (6.8), (6.19) and (6.45) by !2 k 2h¯ 2 2π ' . 1|M (k) |0 δ(Ek − E1 )d3 k; Ek = h¯ 2m  where E1 = E0 + h¯ ω. One should also note that   δ(Ek − E1 ) = (dE/dk)−1 δ(k − k1 ) = m/k¯h2 δ(k − k1 ), dW =

2

The choice is achieved by noting the identity 1  (2 + 1) i P (cos θ k.r ) sin (kr − π /2) eik.r ∼ kr 

  2π  ∗ Ym (θ, φ)Ym (ϑ, ϕ) eikr − e−ikr−iπ , ∼ ikr m

and recognizing that all sums of the form 1  (2 + 1) e−iσ  P (cos θ k.r ) sin (kr + δ  ) kr 

have the same outgoing behaviour.

(7.6)

(7.7)

194

Photo-ionization

 which makes it convenient to absorb the factor mk/¯h2 into the energy-normalized (rather than wavevector-normalized) wavefunction "  mk − − ∗ 1k = χ + e−iσ  Ym (θ , φ)Ym (ϑ, ϕ) RE (r) 1E = 1 2 h¯ m " m RE (r) = Rk (r). (7.8) k¯h2 It follows, after integrating with respect to k, that dW =

!2 2π ' 1E − |M (n) |0 d. h¯

(7.9)

Notice for future reference that  1 − ∗ ∼ √ χ+ e−iδ Ym (θ , φ)Ym (ϑ, ϕ) [sin(kr + δ  )] 1E 1 πk m ∼

   1 ∗ Ym (θ , φ)Ym (ϑ, ϕ) eikr − e−ikr−2iδ . √ χ+ 1 2i π k m

(7.10)

− In other words, the ‘incoming’ character of 1E is ensured by attaching phases to the real radial terms, such that their outgoing components have the simple eikr form required by the plane wave term in (7.1). The rates given by (7.9) for different photon numbers n correspond to the time derivatives of [N1 (t)/N0 ] in (6.11), (6.21) and (6.26). Hence one readily obtains the following photo-fragmentation equivalents of (6.13) and (6.22):  2 2 hω (μα )10 /e , σ (1) 10 (θ k , φ k ) = 4π α¯  2 3 2 σ (2) ¯ ωa ωb (Tαβ )10 /e2 . (7.11) 10 (θ k , φ k ) = 8π α h

The differences from the discrete absorption case are that there is now no lineshape   function gM (ω) and that the energy-normalized final state for the elements in Tαβ 10 couples a discrete state to the energy-normalized continuum, the squared dimensions of which behave as E −1 , which has the inverse dimension of the factor h¯ ω in the numerator. Following the same procedure as for discrete absorption, the operators μα , Tαβ are decomposed into spherical tensor components. Electron spin terms, which were ignored !above, must also be included, in which case the the matrix elements ' 1E − Tq(k) 0 may be expanded as ' ! γ J M − Tq(k) γ

J

M

(7.12)  ! ' + + + Ym (θ , φ) |sms  γ N M m sms eiσ  Tq(k) γ

J

M

, = m sms

7.2 The photo-ionization matrix element

195

where |γ + N + M +  specifies the positive ion state and the |m  term comes from Ym (ϑ, ϕ) in (7.8). Thus, after averaging over the space-fixed orientations   (k) ' ! A0

iσ  (k)





σ (θ , φ) = γ T J M − e T J M γ −q q (2J

+ 1) M

M kp =

2

   (k) A0 T Ym (θ , φ) sms | (2J

+ 1)

+ kp m m −q M M



s

2

' ! × γ + J + M + m sms eiσ  Tq(k) γ

J

M

,

(7.13)

where A0 is the coefficient in (7.11) and T(k) −q are the coefficients of the spherical tensor expansions appropriate to the laser polarization, as given for two photons in Table 6.2. The following sections show how the matrix element and the sums over magnetic quantum numbers in this formal expression are evaluated. 7.2 The photo-ionization matrix element In the formal theory given by Fano and Dill [2, 3], one option is to employ the vector identities j= +s J = J+ + j = k + J

.

(7.14)

to factorize the matrix element;3 ' + + + ! γ J M m sms eiδ Tq(k) γ

J

M

(7.15) ' ! ' = J + M + m sms J M  γ + J (J + , j )M eiδ Tq(k) γ

J

M

, J M +

where γ and γ

describe the internal states of the ion and the neutral molecule. The component matrix elements on the right are given by ' + + ! J M m sms J M (7.16)

+

 j J  s j J J + −j +M +−s+mj

= [J ][j ](−1) m ms −mj M + mj −M j mj

and

3

! ' + + (7.17) γ J (J , j ) eiδ Tq(k) γ

J

M





k J

 + + J - eiδ T (k) (J , j ) -γ

J

, J  = (−1)J −M γ

−M q M

Fano and Dill [2, 3] simplify the argument by omitting the spin factor.

196

Photo-ionization

where the notation [J ] is an abbreviation for 2J + 1. The use of square brackets for reduced matrix elements between total angular momentum states is designed to avoid confusion with those between purely- electronic states. - The connection is illustrated by (7.21). The notation γ + J + - eiδ T (k) (J , j ) -γ

J

is designed defined by (7.15), as distinct from - to designate- the representation   + γ J + - eiδ T (k) (t , Jt ) -γ

J

, which applies to the coupling in (7.18). The resulting expression for the reduced matrix element, with defined J and j , is, however, often conveniently transformed to a ‘transfer angular momentum’ representation, in order to simplify subsequent expressions for the differential cross-section [2, 3] – a transformation that is also convenient for comparison with the Buckingham et al. theory given in Appendix D.2 [1]. The relevant transfer identities Jt = J+ − J

= k − j t = − k = −Jt − s

(7.18)

each have associated recoupling transformations of the forms in (4.16) and (4.17) of [13]. One finds after some manipulation that ' + + + ! γ J M m sms eiδ Tq(k) γ

J

M

(7.19)   +  [t ][Jt ](−1)m +M γ + J + - eiδ T (k) (t , Jt ) -γ

J

= Jt Mt t mt



 × −m where 

k q

t mt



t mt

s ms

Jt Mt



J+ −M +

J

M

Jt Mt

,

 γ + J + - eiδ T (k) (t , Jt ) -γ

J

(7.20)   

[j ][J ][t ][Jt ](−1)−J +Jt +t γ + J + - eiδ T (k) (J , j ) -γ

J

= J j

(

×

 Jt

s k

j t

)(

J

j

k J+

) J . Jt

The similarity between (7.19) and (D.33) implies the following explicit form, if both the neutral and positive ion conform to Hund’s case (a),  + + - iδ (k)  γ J - e T (t , Jt ) -γ

J

(7.21)  ! ' + + +

= [J + ] [J

] [S

][Jt ](−1)J − +S −s+ eiδ Paa (, t ) η+ - T (k) (t ) -η



+

+ s S

J

Jt t s Jt J S , ×  + σ −

λt σ t −+ 

t

7.2 The photo-ionization matrix element

where '

- !  η - T (k) (t ) -η

= [t ](−1)−λ +

λν



 −λ

k ν

t λt



'

197

! η+ + λ Tνk η



,

(7.22) in which |η



 and |η+ +  specify the neutral and positive ion vibronic states, and Paa (, t ) is the parity factor # $

1 + (−1)p +p ++t Paa (, t ) = √ . (7.23) (1 + δ + 0 δ  + 0 δ + 0 )(1 + δ

0 δ 

0 δ 

0 ) The physical significance of the transfer momentum, t becomes apparent in the orbital approximation in which !  ! η



= (7.24) A

η+ + 

λ

, 

where A

is the coefficient of the particular |

λ

 component of the mixed orbital from which the electron is excited. Thus

- !  ' + - (k) !' !  k t ' η - T (t ) -η

= λ Tνk 

λ

v + v

[t ](−1)−λ −λ ν λt λν



   k 

 k t = A

[t ] −λ ν λt −λ ν λ



λν -  

λ

' +

!  × -T (k) -

dλ v v (7.25)  - (k) -

 in which -T - , with a round bracket, is the reduced angular matrix element.4 Notice that the sum in the second line allows simultaneous sign reversal of all 

λ

, are invariant to the projections λ and ν, because the radial matrix elements, dλ sign of λ. The properties of the 3j coefficients therefore generate a common factor

(−1)t − , which must be positive. In other words the sum can only include terms in which the parities of 

and t are equal, and λ

= λt . Moreover, to the extent that the radial matrix elements are independent of λ - !  -   ' + - (k)  !

' η - T (t ) -η

 A

-T (k) -

/ [

]d v + v

δ 

t δ λt λ

. (7.26) 

In other words the transfer angular momentum labels |t λt  may be identified as those of a particular angular momentum component |

λ

 of the orbital from 4

For example [13]

  - (1) -

 -T - = (−1) [][

]

in the case of single-photon ionization.



 1 

0 0 0



198

Photo-ionization

which the electron is excited. However, the discussion in Appendix D.3, which follows Bates and Damgaard, shows that the latter approximation should be used with caution because the matrix elements are sensitive to changes in the quantum defect [14]. Variants of (7.21)–(7.26), for other angular momentum schemes, may be deduced by reference to (D.33)–(D.43) in Appendix D.2. In particular the body-fixed spin coupling term is replaced by a space-fixed one for situations involving case (b) states of either the ion or the neutral molecule. In addition the entries (s, σ ) in the second Wigner coefficient are replaced by (S

, 

) or (S + , − + ) for transitions of the types |a

 =⇒ |b+  or |b

 =⇒ |a + , respectively. Moreover the body-fixed terms for |b

 =⇒ |b+  transitions may be obtained by setting s = 0. The approximation corresponding to (7.26) in the case of |c

 =⇒ |c+  transitions is

- !  ! ' + - (k) j k jt ' + + [jt ](−1)ω η j - T (jt ) -η

= η  j ω Tνk η



−ω ν ωt ων



 j k jt j k j

j  [jt ](−1) −ω ν ωt −ω ν ω

ων

j

ω

× (j T k j

) dj ω v + |v

 j

 (−1)j (j T k j

) dj δ jt j

δ ωt ω

v + |v

 ) (  s j  j ++s+j

+k = (−1) (T k 

) [j, j

] k 

j

j

× dj δ jt j

δ ωt ω

v + |v

.

(7.27)

In the final line j and j

are assumed to arise from (, s) and (

, s) coupling, for which (j T k j

) and (T k 

) are related by (5.72) of Zare [13]. The symmetry under simultaneous sign reversal of + , ω, ν and 

in the first line now requires that j + k + jt is even, and similar considerations with regard to the second line ensure that jt = j

is also even. Finally it should be recognized that the above equations apply to a simplified model involving photo-ionization from an initial state |α

 = |γ

J

M

 with welldefined orbital angular momenta to a single open target state |i = |γ + J + . The multichannel extension of the theory in Appendix D.2 implies a generalization to the form in (D.20), namely - !  (−) ' - k - ! ' (−) - k Sii i - T (t , Jt ) -α

Aα

, (7.28) i -T (t , Jt ) -

= i α

in which Sii(−) are elements of the scattering matrix in (D.18), which takes the

photo-excited state |i  = |γ + J +   to the target state |i = |γ + J + , while the

7.3 Integrated cross-section

199

coefficient Aα

determine the electronic angular momentum composition of the initial molecular orbital. 7.3 Integrated cross-section 7.3.1 General formulation The orthogonality of the spherical harmonics and spin terms in (7.13) implies the following form for the integrated cross-section   σ (θ , φ) sin θ dθdφ (7.29) σ0 = 2

! A0    (k) ' + + + =

T−q γ J M m eiδ Tq(k) γ

J

M

, [J ]

+ m m kq M M



s

in which the sums over magnetic quantum numbers may be evaluated by exploiting the orthogonality properties of the Wigner coefficients in either (7.16) and (7.17) or (7.19). In following the subsequent manipulations it is convenient to use barred ¯ j ) for quantum numbers in the Wigner coefficients belonging to the symbols (j¯, m complex conjugate matrix element in (7.29). Thus the orthogonality under the sums ¯ j ) = (j, mj ) and over (m , ms ), and (mj , M + ) arising from (7.16) require that (j¯, m (J¯ , M¯ ) = (J , M ), respectively, and the final sum over (M , M

) yields σ0 =

(k) A0    T−q [J

]  kq J j [k]

2



2 γ + (J + j )J - T (k) -γ

J

.

(7.30)

Similarly, if the transfer representation in (7.19) is adopted, the sums over (M + , M

) ¯ t ) = (t , mt ) and and (Mt , ms ) require in turn that (J¯t , M¯ t ) = (Jt , Mt ) and (¯t , m the final sum over (m , mt ) yields (k) A0    T−q σ 0 =

[J ]  kq J [k]

2



2 γ + J+ - T (k) (t , Jt ) -γ

J

.

(7.31)

t

t

As derived these expressions apply to systems with a single open ionization channel. However, the generalization in (7.28) may be used to take account of both the multichannel channel interactions in the excited state, and of the electronic angular momentum composition of the initial orbital. Thus (k) A0      T−q σ 0 =

[J ] + +



 kq J  [k] γ J

 λ

t t

2

Sγ + J + ,γ + J + 

2  × γ + J+ - T (k) (t , Jt ) -γ

J

A2

λ

,

2

(7.32)

200

Photo-ionization

where the coefficients A

λ

determine the composition of the initial orbital, and Sγ + J + ,γ + J + is the final state scattering matrix element, given by (D.18). The forms in (7.30)–(7.32) are quite general, but explicit forms, applicable to different angular momentum coupling regimes may be obtained by substituting expressions of the form in (7.21). Thus, assuming a single open channel for simplicity, σ 0 = A0

2   T(k) - !2 ' −q Pγ + γ

(, t ) η+  - T (k) (t ) -η

Qγ + ,γ

(t ), (7.33) [k]  kq t

- ! ' in which (γ + , γ

) designate the coupling scheme. The terms η+  - T (k) (t ) -η

include the electronic transition matrix elements in (7.22). Finally the factors Qγ + ,γ

(t ), which are given for various coupling regimes in Table 7.1, contain the information responsible for the intensities and selection rules for different rotational branches of the photo-electron spectrum. These expressions, which are taken from Appendix D.2, correspond, in a different notation, to those given by Buckingham et al. [1]. Moreover, (7.26) and (7.27) show that different t values may be identified with different 

components of the initial orbital when Hund’s case (a) or (b) applies, while t = j

± 1/2 in case (c). In addition the single sum over t in (7.33) means that the different initial angular momentum components of the orbital give rise to incoherent contributions to the cross-section, σ 0 . 7.3.2 Photo-ionization selection rules Within the approximation that |t λt  may be identified with |

λ

, the Wigner coefficients in Table 7.1 carry approximate selection rules arising from an assumed decomposition of the initial molecular orbital into its angular momentum components. The relevant changes between the total angular momenta of the positive ion and the parent neutral molecule are given in the table. The additional rules  = + −

= −λ

 = + − 

= −λ

± 1/2  = + − 

= −ω

for |a

, b

 ⇒ |a + , b+  transitions for |a

 ⇒ |a +  transitions for |c

 ⇒ |c+  transitions

are readily deduced. There is also a general spin condition S = ±1/2, and a general parity rule obtained in Appendix D.2, that p

+ p + 

+  must be even. Thus in single-photon ionization, the odd angular momentum partial waves of the outgoing electron are generated from even initial angular momentum components and vice versa. Since 

is not strictly defined, these rules are restricted in practice to relating the observed photo-electron band structure to the assumed properties of the parent molecular orbital.

7.3 Integrated cross-section

201

Table 7.1 Coupling factors Q(t ) and Q(jt ) appropriate to different angular mentum coupling schemes in the neutral molecule and the positive ion [1]. To the extent that the radial matrix elements are approximated as independent of λ, the labels |t λt  may be identified with particular components |

λ

 of the initial orbital and similarly for |jt ωt  and |j

ω

. The symbols [J ] and [n · · · m] denote (2J + 1) and n, n + 1, . . . , m, respectively. Transition +

Q(t ) or Q(jt )

2 S + 1/2 S

[J ][S ] ×T  +  −



2 +  t 1/2 Jt Jt J T = Jt [Jt ] −λt  − −+ 



+

a ←a



J + = [|J

− Jt | · · · J

+ Jt ]; +



N+ [N ] − +



+

b ←b

J



2

Jt = t ± 1/2 N



t −λt

2

N + = [|N

− t | · · · N

+ t ] b+ ← a

[N + ]





Jt [Jt ]

t λt

S



Jt

+ − 

N + = [|J

− Jt | · · · J

+ Jt ]; +



a ←b

+

+ −1

[J ][S ]





t Jt [Jt ] −λ t

S+ +

[J + ]

J+ −+

N+ − +

Jt

+ − 

J



2

Jt = [|t − S

| · · · t + S

]

Jt

− +

J + = [|N

− Jt | · · · N

+ Jt ]; c+ ← c

2

2

J+ −+

Jt + −

N



2

Jt = [|t − S + | · · · 

+ S + ] jt −ωt

J



2

J + = [|J

− j

| · · · J

+ j

]

Alternative strict selection rules have been given by Xie and Zare in terms of the exact but unknown angular momentum components |m  of the outgoing electron wave [15]. They are derived from a ‘total’ rather than a ‘transfer’ angular momentum standpoint, with the |J M  components subject to the triangular conditions.5 J = J + + [−j · · · j ] = J

+ [−k · · · k] M = M + + [−j · · · j ] = M

+ [−k · · · k], 5

Readers should note that Xie and Zare use the symbol Jt for total angular momentum, in place of the present symbol J .

202

Photo-ionization

where the notation [−j · · · j ] implies −j, j + 1, . . . , j , etc. Thus J = J + − J

= [− − k − 1/2 · · · l + k + 1/2] M = M + − M

= [−m − k − 1/2 · · · − m + k + 1/2], a result that applies to all coupling cases, except that N = [− − k · · ·  + k] = [−

· · · 

] MN = [−m − k · · · − m + k] = −m

for |b

 ⇒ |b+  transitions. In addition changes in the body-fixed angular momentum components are governed by  = [−λ − k · · · − λ + k]

for |a

, b

 ⇒ |a + , b+  transitions

 = [−λ − k − 1/2 · · · − λ + k + 1/2]

for |a

 ⇒ |a +  transitions

 = [−ω − k · · · − ω + k]

for |c

 ⇒ |c+  transitions.

A more complete table covering other possibilities is given by Xie and Zare [15]. There is also a simple parity selection rule for transitions between the states defined in the notation of Table C.2, because the outgoing electron partial wave and the transition operator T (k) have parity indices equal to  and nphot , respectively, where nphot is the number of photons. Thus p + + p

+  + nphot must be even.6 To see the connection with the earlier rule in terms of p and 

note that t and 

have equal parity, that p

+ p + 

+  = p+ + p

+ 2 + 

and that  + 

+ nphot must be even, which again requires that p + + p

+  + nphot is even. The rule itself means that the outgoing electron angular momenta  must be either all even or all odd, according to the parities of the neutral and ionized species and the number of photons.

7.4 Differential cross-section 7.4.1 General formulation The differential cross-section in (7.13) is treated by similar manipulations to those in ¯ m ¯  ) remain inside the square (7.29)–(7.30), except that the sums over (, m ) and (, modulus in (7.13), because the differential cross-section depends on interference between the spherical harmonics Ym (θ , φ) and Y¯∗m¯  (θ , φ). The advantage of the 6

Xie and Zare present this rule in terms of the the parity factors q + + J − S, etc., which appear in Table C.2, and indices τ

and τ + for each transition (which they write as p

and p + ).

7.4 Differential cross-section

203

  transfer representation now becomes apparent because the sums over M + , M

and (Mt , ms ) lead to incoherent contributions to σ (θ, φ) from the contributing values of Jt and t . Thus, in the single open-channel model σ (θ , φ) =

 (k) A0     ¯ ∗ (−1)−m +m¯  T(k) −q T−q¯

2J + 1 ¯ ¯ J mmM k kq q˜ 

t t



¯

t

¯ k¯ t k t Ym (θ , φ)Y¯∗m¯  (θ , φ) × ¯  −q¯ −mt −q −mt m  ∗  ¯ × γ + J + - eiδ T (k) (t , Jt ) -γ

J

γ + J + ¯- eiδ¯ T (k) (t , Jt ) -γ

J

,

 m



(7.34)

The sum on the right of the equality is performed in two steps. The first employs the Clebsch–Gordan series Ym (θ , φ)Y¯∗m¯  (θ , φ) (7.35)  ¯ ¯ ¯ [][] −m = (−1) Cm (θ , φ)C− ¯ m ¯  (θ , φ) 4π 



¯   ¯ L  ¯ L ¯ ¯ [][] −m CLM (θ , φ), = (−1) [L] ¯  −M 0 0 0 m −m 4π L

where CLM (θ , φ) is a modified spherical harmonic function [16], such that CL0 (θ , φ) = PL (cos θ ). Notice that the second Wigner coefficient vanishes unless L +  + ¯ is even. The second step employs the vector identities t = − k = ¯ − k¯ ¯ L = − ¯ = k − k, to motivate the recoupling transformation [13]

 k  t ¯ k¯ −q−m (−1) ¯  −q¯ q −m mt m m m¯ mt

¯ t +k+k¯ q+

= (−1)

(

k k¯ ¯ 

L t

)

k −q

k¯ q¯

L M

(7.36)

t −mt



 −m

¯ ¯ m

L M



.

(7.37)

It follows, by combining (7.34)–(7.37) that σ (θ , φ) may be expressed as a sum of independent t contributions  σ (θ , φ) = σ (t ; θ , φ). (7.38) t

204

Photo-ionization

After employing appropriate analogues of (7.21) one finds that7  ¯  A0   ¯ ¯ (k)∗ (k) ¯ t ) ∗ × Q (t ) σ (t ; θ , φ) = (−1)t +q+k+k T(k) (, t ) T (k) (, −q T−q¯ ×T 4π ¯ ¯  k kq q¯

¯ k, k, ¯ q, q; ¯ θ , φ), × $(t ; ; in which

(7.39)

- ! ' T (k) (, t ) = η+ - eiδ T (k) (t ) -η

.

(7.40)

The physical interpretation is that σ (θ , φ) takes the form of an incoherent sum over t , each term of which involves a product of three contributions; (i) a factor in T(k) −q appropriate to the laser polarizations; (ii) a term involving electronic transition elements T (k) (, t ), from (7.22), multiplied by the appropriate angular momentum decoupling factor, Q (t ) from Table 7.1; and (iii) a geometrical factor ¯ k, k, ¯ q, q; ¯ θ , φ) $(t ; ; (   k k¯ ¯ = (2t + 1) (2 + 1)(2 + 1) (2L + 1) ¯   L



k k¯ L ¯  L × CLM (θ , φ). −q q¯ M 0 0 0

L t

)

(7.41)

Note that the sum over k and q in (7.39) collapses, in the single-photon case, to a single term with k¯ = k = 1, q = q¯ and M = 0. In generalizing (7.39) to take account of scattering from the photo-excited state

|i  = |γ + J +   to the target state |i = |γ + J + , it is useful to employ the explicit notation γ + J + |Q(t )|γ

J

 for the angular momentum coupling term. The final two lines in (7.39) are then replaced by   ∗ Sγ + J + ,γ + J +  [Sγ + J + ,γ (7.42) ¯ + J + ¯ ] γ + J +  ¯

× T (k) ( , t )[T (k) (¯ , t )]∗ × γ + J + |Q(t )|γ

J

 ¯ q, q; ¯ θ , φ). × $(t ;  ¯ ; k, k, 7

An alternative form in the style employed by Fano and Dill is A0    ¯ ¯ (k) (k)∗ (−1)t +q+k+k T−q T−q¯ σ (t ; θ, φ) = 4π ¯ ¯  k kq q¯ Jt

 ∗ ¯ × γ + J + - eiδ T (k) (t , Jt ) -γ

J

γ + J + ¯- eiδ ¯ T (k) (t , Jt ) -γ

J



¯ k, k, ¯ q, q; ¯ θ, φ), × $(t ; ;

7.4 Differential cross-section

205

In addition, angular momentum mixing in the initial orbital may be taken into account by replacing the sum over t in (7.38) with a sum over 

, weighted by the factor A2

λ

in (7.32). Some general features of the geometrical factor should be noted. As anticipated ¯ L by (7.36), the quantum number L in (7.41) is restricted to the range |k − k| ¯ ( k) k + k¯ for a given pair of spherical tensor components Tq(k) and Tq˜ , regardless of ¯ In particular, 0  L  2 in the single-photon the orbital angular momenta,  and . case. Another condition is that L +  + ¯ must be even. When combined with the parity rule that  and ¯ must be either both even or both odd, this means that L itself must be even. It is also useful to distinguish between cylindrically symmetric and non-symmetric photon fields, according to whether or not the polarization components q and q¯ are equal for all terms in the (7.39). In the cylindrically symmetric case, M = q¯ − q = 0, and σ (t ; θ , φ) takes the canonical form # $  σ 0 (t ) 1+ (7.43) σ (t ; θ , φ) = β L (t ) PL (cos θ ) , 4π L=even where σ 0 (t ) is the t contribution to the integrated cross-section and PL (cos θ ) is a Legendre polynomial. Parity-favoured and parity-unfavoured contributions The geometrical factor in (7.41) has an interesting property for q = q¯ = 0, which was first described for linearly polarized single-photon ionization by Dill and Fano [3]. The effect is that certain so-called ‘parity-unfavoured’ components, ¯ k, k, ¯ q, q; ¯ θ , φ), vanish identically in the forward and backward direc$(t ; ; tions (θ = 0, π), first because PL (cos θ ) = 1 in these special directions, for even L values. Secondly the identity )



(  k¯ k L ¯  L k k¯ L ¯ t k+k+ (−1) [L] ¯ 0 0 0 0 0 0   t L



¯ k¯ t  k t (7.44) = 0 0 0 0 0 0 ¯ k, k, ¯ k, k, ¯ q, q; ¯ q, q; ¯ 0, φ) = $(t ; ; ¯ π , φ) = 0 whenever means that $(t ; ; ¯ ¯ t +  + k or t +  + k is odd. To appreciate the reference to parity, recall that t has the same parity as an orbital angular momentum component 

in the weak spin–orbit coupling situations involving transitions between case (a) or case (b) species. The selection rule that 

+  + 1 is even for a single photon therefore precludes any parity-unfavoured contributions. The situation differs however for strong spin–orbit coupling. The theory is then most naturally formulated in terms of j , jt and j

, subject to

206

Photo-ionization

the restriction, implied by (7.27), that jt + j + 1 and jt − j

must be even for single-photon excitation. Taken together with the rules that t − 

is even, j =  ± 1/2 and j

= 

± 1/2, this means that t +  + 1 may take odd as well as even values. Parity is, of course, conserved during the optical transition, but it can no longer be inferred from 

. It is instructive to examine the atomic case   2  Xe 1 S0 + hν → Xe+ P3/2 ,2 P1/2 + e ( = 0, 2), which was quoted by Dill and Fano [3, 17]. Since J

= 0, the equation Jt = J+ − J

= −t − s allows t = (2, 1) and t = (1, 0) for the J + = 3/2 and 1/2 levels, respectively. However, the overall parity requirement restricts the outgoing wave to even  values, whereas the second identity t =  − k allows only  = 1 for t = 0. The possible contributions to (7.38) are therefore restricted to t = 2 (parity-unfavoured) and t = 1 (parity-favoured) for J + = 3/2 and t = 1 (parity favoured) for J + = 1/2. Similar arguments apply to   2  HI 1 0 + hν → HI+ 3/2 ,2 1/2 + e( = 0, 2). The two-photon situation is more complicated, even for case (a) and case (b) species. For centrosymmetric species t +  is necessarily even and Table 6.2 shows that only the even k spherical tensor components contribute in parallel linear polarization. Thus parity unfavoured contributions are again forbidden, except in cases of strong spin–orbit coupling. However  mixing in the intermediate state would allow both even and odd values of t +  in non-centrosymmetric species, and colinear beams with opposite circular polarizations have non-zero antisymmetric spherical tensor components with k = 1 and q = 0. Hence parity unfavoured contributions in (7.41) are no longer forbidden, even in the absence of spin. 7.4.2 Asymmetry parameters Equations (7.38)–(7.41) imply an expansion for σ (θ, φ) in terms of the modified spherical harmonics CLM (θ, φ) and it is convenient to employ a normalization such that # $ L   σ0 1+ σ (θ , φ) = β LM CLM (θ, φ) , 4π L=even M=−L in which case the coefficients β LM are termed asymmetry parameters. The factor (4π )−1 ensures that σ (θ , φ) integrates to σ 0 . Single-photon excitation The case of single-photon ionization with linearly polarized radiation corresponds (k) ¯ = 0, the first of the to k = k¯ = 1, q = q¯ and T(k) −q = T−q¯ = 1. Hence, with q = q

7.4 Differential cross-section

207

z

y x

(a)

(b)

Figure 7.1 Polar diagrams of σ (θ , φ) for single-photon ionization at the limiting values (a) β = −1 for a sin2 θ distribution and (b) β = 2 for a cos2 θ distribution.

3j symbols in (7.41) carries the selection rules L = 0, 2 and M = 0, which leads to the familiar [18] asymmetry decomposition8 σ0 [1 + βP2 (cos θ )] , 4π   β= σ (t )β (t ) / σ (t ).

σ (θ, φ) =

t

(7.45)

Jt

It is readily verified that β is restricted to the range −1  β  2 because σ (θ, φ) is necessarily positive and P2 (cos θ ) takes maximum and minimum values of 1 and −1/2, respectively. Polar diagrams of σ (θ , φ) are given for the limiting values β = −1 and β = 2 in Fig. 7.1 (a) and (b), respectively. In addition the properties of the 3j coefficient with L = 2 in equation (7.41) mean that β scales by a factor −1/2 on changing from q = q¯ = 0 to q = q¯ = 1, which corresponds to passing from linear to circular polarization. To understand the factors that affect the observed β value in cases with negligible spin–orbit coupling, each t term in (7.45) receives contributions from two partial waves with  = t ± 1, except of course that only  = 1 is allowed for t = 0. Explicit evaluation of the Wigner coefficients in (7.41) then yields the formula [17] % β(t ) = a (t + 2)|T+1 |2 + (t − 1)|T−1 |2  & − 6 t (t + 1)|T+1 T−1 | cos(δ +1 − δ −1 ) % &−1  × (2t + 1) |T+1 |2 + |T−1 |2 , (7.46) 8

In the case of circular polarization, with k = k = 1 and q = q = ±1, the first 3j symbol in (7.41) no longer vanishes for L = 1. Hence the differential cross-section can in principle include a term in P1 (cos θ ), although it is forbidden by the parity constraint for electric dipole absorption that restricts L to even values. However, the situation differs for chiral molecules by allowing simultaneous electric and magnetic transitions, of which the latter conserve parity, rather than changing it, thereby permitting a possible P1 (cos θ ) term.

208

Photo-ionization

where a = −(3q 2 − 2)/2 is a polarization factor and T±1 is a shorthand notation for the reduced matrix element T (1) (, t ) with  = t ± 1. Thus β depends on the relative magnitudes of the reduced matrix elements |T±1 | and on the phase difference between the outgoing waves in the  = t ± 1 channels. Note that the τ  (E) component of the phase δ  in (7.4) can lead to sharp changes in β (t ) at the auto-ionizing resonances. In addition, the scaling with respect to the polarization index, q, means that the t contribution to the angular distribution contains only two observable independent parameters, σ (t ) and a −1 β (t ), which depend on the three theoretical quantities |T±1 | and (δ +1 − δ −1 ). The phase terms can seldom be ignored in practice, but it is interesting to examine the influence of the initial orbital angular momentum on the asymmetry parameters in situations where one of the reduced matrix elements |T±1 | is much larger than the other, perhaps because the experiment is performed at a Cooper minimum for one of the contributing channels [19]. In linear polarization β takes the maximum allowed value, β = 2 for t = 0, because T−1 = 0, which means that σ (t , θ , φ) ∝ cos2 θ – as expected for dipole excitation from a spherically symmetrical orbital. One also sees in an obvious notation, that β + = (t + 2) / (2t + 1) or β − = (t − 1) / (2t + 1), depending on which is the dominant component. In particular β + = 1 and β − = 0 for t = 1, while β + = 0.8 and β − = 0.2 for t = 2, with β for higher t lying between these limits and tending to β + = β − = 1/2 as t → ∞. Thus negative β values can arise only from the interference term in (7.46), in a weak spin–orbit coupling regime. By contrast the behaviour of the geometrical factor for linear polarization means that β(t ) = −1 for all parity unfavoured terms arising from strong spin–orbit coupling, regardless of the initial angular momentum. Hydrogen molecule We see from Table 7.1 and (7.39) and (7.46) that each independent t contribution to the angular distribution has its own characteristic asymmetry factor, β (t ) , and angular momentum decoupling factor, Q (t ), the second of which carries selection rules on the photo-electron angular momentum transfer N + − N

, J + − J

, etc. Moreover t may be identified, for interpretive purposes, with a particular orbital angular momentum component, 

, of the initial molecular orbital. The photoelectron spectrum of H2 provides a simple illustration, for which the initial orbital has predominant s character, with 

= 0, and both the neutral and ionized species conform to Hund’s case (b). The Q(

) factor in Table 7.1 therefore predicts only strong Q branch transitions with N = 0, for which (7.46) requires that β = 2. The experimental results in Table 7.5 and Fig. 7.2 confirm the existence of a strong Q branch, with β  2, but there are also weak S and O branches, corresponding to N = ±2, and measurements of the asymmetry in the S branch indicate β  0.8 for v + = 0, with possibly lower β values for transitions to higher vibrational levels

7.4 Differential cross-section

209

Table 7.2 Experimental [20, 21] and theoretical [22] anisotropy parameters for photo-ionization of H2 by He I ˚ Entries are given for Q and S branches in radiation at 584 A. the photo-electron spectrum for vibrational states v + = 0 − 3. Branch

v+

[20]

[21]

[22]

Q

0 1 2 3

1.86 1.86 1.85 1.85

1.92 1.93 1.94 1.95

1.76 1.79 1.82 1.84

S

0 1 2 3

0.83 0.63 0.59 0.47

0.81 0.71 0.73 0.80

0.75 − 0.69 −

H2+ PES h ν = 21.218 eV 2 + X Σ g cationic state 3

2

1

ν=0 θ = 90°

ΔN = 0 θ = 0° ΔN = 2 16.2

16.0

15.8 15.6 Binding energy (eV)

15.4

Figure 7.2 Rotationally resolved vibrational bands in the He I photo-electron spectrum of H2 , as detected by electron counting in directions parallel (θ = 0◦ ) ¨ and perpendicular (θ = 90◦ ) to the polarization axis. Taken from Ohrwall et al. [20], with permission.

of the ion [20, 21]. The obvious rationalization for the presence and properties of these weaker transitions is that the initial molecular orbital has partial d, or 

= 2, character. The selection rules implied by Table 7.1 then allow N = 0, ±2 (when parity is taken into account), and one has the limiting prediction β = 0.8 if |T+1 |  |T−1 | in (7.46); in other words if the d → f transition moment is much larger than the d → p one. The final column of Table 7.2 gives results from an

210

Photo-ionization 3

C

A

0

1

C

D

0 −1

−1

(a)

B −2 −2

B

2

7 β 4X

β4

1

3

D

2

−1

0

1

2

3

A

−2 4

−2

β2

−1

(b) 0

1

2

β 2X

Figure 7.3 Boundaries of the parameter planes within which σ (θ , φ)  0 for linearly polarized two-photon ionization (a) for linear polarization along the same axis, and (b) for crossed linear polarization. The parameters (β 2 , β 4 ) and (β 2X , β 4X ) are the coefficients in (7.47) and (7.50), respectively. Polar diagrams of σ (θ, φ) corresponding to the points A–D are shown in Figs 7.4 and 7.5. Adapted from Child [23].

ab-initio study, which confirms that f waves are required to account for the presence of the N = ±2 branches and for their associated asymmetry parameters [22]. Two-photon ioization Two-photon ionization allows a richer variety of experimental observables, first because the angular distribution depends on more than one asymmetry parameter, and secondly because the polarizations of the two lasers may differ. In the case of linear polarization along a common axis, Table 6.1 shows that contributions to the ionization come from the spherical tensor coefficients √ √ ¯ 2/3 and T(0) T(2) 0 ( a  b ) = 0 ( a  b ) = − 1/3. Thus k and k in (7.41) can take values 0 and 2, while p = p¯ = M = 0. The first 3j symbol in the latter equation therefore allows spherical harmonic contributions with L = 0, 2, 4, and M = 0. Consequently, σ (θ , φ) =

 σ0  1 + β 2 P2 (cos θ ) + β 4 P4 (cos θ ) , 4π

(7.47)

and it is interesting to examine the conditions on (β 2 , β 4 ) such that σ (θ , φ)  0. Child has applied this positive constraint at θ = 0 and π /2 with the further condition, at intermediate angles, that σ (θ , φ) should touch but not cross zero. The resulting physical region of the (β 2 , β 4 ) plane is enclosed by the contour in Fig. 7.3(a) [23]. Polar diagrams of σ (θ , φ) at the points A–D in this diagram are shown in Fig. 7.4. One should also note that the ‘parity unfavoured’ contributions to σ (θ , φ) lie on the line β 2 + β 4 = 0.

7.4 Differential cross-section

211

z

y x

(a)

(b)

(c)

(d)

Figure 7.4 Polar diagrams of σ (θ , φ) for two photon ionization with parallel linear polarization. The panels (a)–(d) correspond to points A–D in Fig. 7.3(a), respectively. Taken from Child [23] with permission.

It is also found, rather surprisingly, that only two independent asymmetry parameters are required to characterize the angular distribution arising from two beams with orthogonal linear polarizations [23]. Equation (6.36) shows that a combination of an x-polarized and a z-polarized beam are characterized by the coefficients (1) (2) (1) T(2) −1 = −T−1 = −T1 = −T−1 = 1/2.

(7.48)

We assume for simplicity that the two laser frequencies are equal, in which case the (1) vanish. It follows from matrix elements of the odd spherical tensor operators T±1 (7.41) that the model allows diagonal contributions with L = 0, 2, 4 and M = 0, plus crossed contributions with L = 2, 4 and M = ±2. Thus # $   σ (2) 0 1+ σ (θ , φ) = β LM {CLM (θ, φ) + CL−M (θ , φ)} . (7.49) 4π L=2,4 M=0,2 The four β LM parameters are not, however, independent, because the symmetry of the photon field ensures that σ (θ , φ) is symmetric with respect to exchange

212

Photo-ionization

z

y x

(a)

(b)

(c)

(d)

˜ y, ˜ z˜ ) polar projections of σ (θ, φ) arising from crossed linear polarFigure 7.5 (x, ization at selected (β 2X , β 4X ) combinations. The distributions in panels (a)–(d) correspond to points A–D in Fig. 7.3(b). Taken from Child [23], with permission.

between the x and z axes.9 One finds, in effect by transforming to y as the quantization axis, that       σ0  1 + β 2X C2X θ˜ , φ˜ + β 4X C4X θ˜ , φ˜ , (7.50) σ θ˜ , φ˜ = 4π   where θ˜ , φ˜ are the polar angles in the new axis system and    1 2 − 3 sin2 θ˜ , (7.51) C2X θ˜ , φ˜ = 2   1  C4X θ˜ , φ˜ = 4 − 20 sin2 θ˜ + 35 sin4 θ˜ sin2 2φ˜ . 4 An examination of the limits on the physical parameter space shows that (β 2X , β 4X ) must lie within the triangle in Fig. 7.3(b), and polar diagrams of σ θ˜ , φ˜ corresponding to the points A–D in Fig. 7.3(b) are shown in Figs 7.5(a)–(d), respectively [23]. 9

With the implications that β 22 =

 3/35β 20

and

 β 42 = − 5/2β 40 .

7.4 Differential cross-section

213

Table 7.3 Expansion coefficients T(k) −q for different two-photon polarization combinations. k X

Polarization

q

I II III IV

L z Lz C+ C− C+ L z Lx Lz

0 0 1 ±1

2 √ √2/3 √1/6 1/2 ∓1/2

1

0 √ −√1/3 1/3 0 0

√ 0 √1/2 1/2 −1/2

7.4.3 Information content It is interesting to investigate the extent to which the values of the matrix elements, T (k) (, t ), may be determined by combining knowledge of the asymmetry parameters for various possible two-photon ionization experiments. Four possibilities labelled X = I–IV are indicated in Table 7.3, together with the coefficients T(k) −q implied by Table 6.1. In the light of (7.39) and (7.41), the relevant asymmetry parameters may be expressed in the form β (X) LM (t ) =

(−1)t [L, t ]Q(t )  σ (X) 0 (t )

(X) ¯ L (k, k; ¯ t ), BLM (k, k)ς

(7.52)

k k¯

where the coefficient (X) ¯ = BLM (k, k)

 ¯ k¯ ¯ (k)∗ (−1)q−k−k T(k) T −q −q¯ −q¯ q q¯

k q

L M

(7.53)

is fixed by the experimental conditions, while factor )

(  ¯  L k k¯ L ¯ ¯ ς L (k, k; t ) = [,  ¯ 0 0 0   t ¯ 

¯ t )]∗ × T (k) (, t )[T (k) (, ¯

(7.54)

contains universal combinations of the reduced matrix elements T (k) (, t ). In addition σ (X) 0 (t ) = A0 Q(t )

2   |T(k) −q | t

kq

[k]

2

T (k) (, t ) .

(7.55)

Certain restrictions are implied by Table 7.3. For example, the photon fields are cylindrically symmetrical for the first three polarization combinations,

214

Photo-ionization

(X) (X) X = I–III. Hence only three parameters, σ (X) 0 (t ), β 20 (t ) and β 40 (t ), are allowed, because β (X) 00 (t ) = 1 in this representation. The fourth combination, X = IV, with (X) crossed linear polarization allows further parameters, β(X) 22 (t ) and β 42 (t ), of (I V ) (I V ) which the latter is again related to β (X) 40 (t ) by β 42 = − 5/2β 40 (see footnote V) V) and β (I to equation (7.50)), but the corresponding connection between β (I 22 20 no longer applies if the photon frequencies differ, owing to the presence of the non-zero T(1) −q coefficients. Absolute measurements of the angular distribution of the photo-electron, arising from these four polarization combinations, can yield a total of 13 experimental observables; four independent cross-sections and nine independent asymmetry parameters. On the theoretical side, the triangular conditions implied by (7.54) restrict L to the values given by the matrix

2 1 0

⎛

2

1

4, 2, 0 2 ⎝ 2 2, 0 2 −

0

⎞ 2 −⎠. 0

¯ t ) factors associated with diagonal terms, such as ς L (2, 2; t ), are The ς L (k, k; real, while those with k¯ = k are related by the complex conjugate identities ¯ t ) = ς L (k, ¯ k; t )∗ . There is therefore a total of ten such independent ς L (k, k; combinations of the complex matrix elements T (k) (, t ), the values of which may be inferred from ten of the experimental observables. The next step is to recognize that the matrix elements themselves are constrained by the triangular conditions  = t , t ± k provided that t  k. Moreover  and ¯ are constrained to have the same parity in (7.54), which restricts the possibilities to three complex elements T (k) (, t ) with k = 2 and one each with k = 1, 0. This shows that the flexibility allowed by combining the results from experiments with a variety of polarizations  it possible  (k) to extract  values for the ten real and  (k) makes imaginary parts, Re T (, t ) and Im T (, t ) , of these matrix elements from ¯ t ). By contrast, it was shown the ten experimentally determined factors ς L (k, k; in Section 7.4.2 that the two experimental observables, σ 0 and β, were insufficient to determine the magnitudes |T (1) (t ± 1, t )| of the two reduced matrix elements, and their relative phase. Section 7.6.2 below returns to this two-photon inversion problem in a different way by employing resonant 1 + 1 multiphoton ionization (MPI). It then proves more convenient to replace the spherical tensor formulation by a density matrix approach, because the excitation proceeds by via single internal quantum state. In addition the analogues of the reduced matrix elements for the ionization step are expressed in terms of the body fixed transition elements. Thus

7.5 Fixed molecule angular distribution

215

following (7.22) - ! ' T (k) (, t ) = η+ - T (k) (t ) -η

  k [t ](−1)−λt = −λ ν λν

t λt



(7.56) '

! η+ + λ Tνk η



,

and object is to! obtain ‘experimental’ values of the transition amplitudes ' + the + η λ Tνk η



. 7.5 Fixed molecule angular distribution Recent dissociative photo-ionization experiments have been interpreted in terms of the fixed molecule photo-electron angular distribution first derived by Dill many years ago, which provides more information than photo-ionization from a random sample [5]. The essence of the experiment is that ionization is followed by sudden dissociation of the positive ion, which means that coincident detection of the electron and the heavy fragment ion allows one to reconstruct the photo-electron angular distribution with respect to the axes of an effectively stationary molecule [4, 6, 24, 25]. The theory below, which is restricted for simplicity to single photon excitation of a diatomic molecule follows Dill [5] and Lucchese et al. [6]. Related work is given by Cherepkov and Raseev [26]. Since Dill finds it convenient to refer the ionization dynamics to the molecular frame, coordinates r and r are used to denote the electron position in the molecular and laboratory frames, respectively, and (θ k , φ k ) specify the direction of the outgoing electron with respect to the molecular quantization axis. Finally the symbol Rn is used for the Euler angle transformation (φ n , θ n , χ n ) that brings the molecular coordinates into coincidence with the laboratory (or light) frame, with the additional assumption, made by Lucchese et al., that the polarization vector lies in the molecular xz plane, so that φ n = 0 [6]. The relevant transformation    (1) (1) Tν (ˆr) Dνq (Rn ) (7.57) Tq(1) rˆ = ν

means that the transition amplitudes from initial state |ψ 0  to a final state |ψ λ  are given by   (1) λ|Tν(1) |0Dνq (Rn ), (7.58) ψ λ (r)Tν(1) (r )ψ 0 (r)d3 r = ν

where the bracket is evaluated in the molecular frame. Linear polarization is assumed, with q = 0, in which case the rotation matrix element

216

Photo-ionization

reduces to (1) (1) (1) Dν0 (Rn ) = Dν0 (0, θ n , χ n ) = dν0 (θ n ).

The photo-electron angular distribution for polarization at an angle θ n to the molecular axis is then given by       ∗ I θ n , θ k , φ k = 4π 2 αhν Yλ θ k , φ k Y¯λ¯ θ k , φ k ¯ λ¯ ν ν¯ λ

∗ ' (1) (1) × dν0 (θ n ) dν0 (θ n ) ¯λ¯ eiδ¯ Tν(1) |0∗ λ| eiδ Tν(1) |0.

(7.59)

The diatomic molecule assumption implies the selection rules that λ

= λ − ν = λ¯ − ν¯ , where λ

is the body-fixed angular momentum projection of the orbital from which the electron is excited. Thus M = λ − λ¯ = ν − ν¯ is a constant of the motion. Although molecules fixed in space have no well-defined parity, the g ↔ u selection rule for homonuclear species restricts  + ¯ to even values. Moreover it frequently occurs that the photo-ionization, via a particular intermediate positive molecular ion state, proceeds dominantly via the components of a single partial wave , in which case the Coulomb phase terms in the two matrix elements cancel out. Equation (7.59) will later be transformed to a more convenient form for experimental analysis but it is convenient first to consider a variety of special cases [6]. In the first place, it is readily verified that the contribution to the total cross-section from the (assumedly dominant) th partial electron wave is given by    σ 0 = I θ n , θ k , φ k sin θ n sin θ k dθ n dθ k dφ k =

1 8π 2 αhν  ' 2 , λ

+ ν Tν(1) |0 . 3 ν=−1

(7.60)

Secondly, the positive ion distribution with respect to the polarization axis may be obtained by integrating over the electron variables. Thus    I (θ n ) = I θ n , θ k , φ k sin θ k dθ k dφ k =

σ 0 [1 + βP2 (cos θ n )] , 2

(7.61)

where β=

2 2 ' ' ' (1) |0 2 , λ

T0(1) |0 − , λ

+ 1 T1(1) |0 − , λ

− 1 T−1

'

2 2 ' ' (1) |0 , λ

T0(1) |0 + , λ

+ 1 T1(1) |0 + , λ

− 1 T−1

2

2

. (7.62)

7.5 Fixed molecule angular distribution

217

The factor of 1/2 appears in (7.61) instead of the normal 1/4π because the ions are restricted to the xz plane. The third special case arises by setting θ n = 0, in which (1) (1) (1) = 1 and d10 = d−10 = 0, so that ν = ν¯ = 0 and λ = λ¯ = λ

. Thus case d00     I 0, θ k , φ k = 4π 2 αhν Yλ

θ k , φ k

2

2 '



(1)  , λ T0 |0 .

(7.63)

This means that the electron distribution measured in coincidence with the positive ion fragment departing parallel to the polarization axis mirrors the angular distribution of the partial wave obtained by parallel, ν = 0, excitation from the ground state. Similar considerations apply for perpendicular transitions at θ n = π/2. The

gives rise to four contributions most interesting √  case  occurs for λ = 0, which (1) (1) (1) with (ν, ν¯ ) = λ, λ¯ = (±1, ±1). Moreover d00 = 0 and d10 = −d−10 = −1/ 2   and Y±1 θ k , φ k = ±$1 (θ k ) e±iφ k . Finally the matrix elements , 1| T1(1) |0 (1) |0 may be assumed to be related by a factor (−1)τ [6]. For and , −1| T−1

example! τ =  +  + 1 if the parent molecular orbital is dominated by a single 

0 component. The resulting electron angular distribution, measured in coincidence with positive ions perpendicular to the polarization axis is then given by 2    I π/2, θ k , φ k = 4π 2 αhν |$1 (θ k )|2 , 1| T1(1) |0 1 + (−1)τ cos 2φ k .

(7.64) In other words the azimuthal distribution varies as cos2 φ k or sin2 φ k according to whether τ is even or odd. Expressions for intermediate polarization angles θ n are readily derived. Figure 7.6 shows a comparison between experimental and theoretical dis  2 angular 2 2 4 1  4σ 5σ 1π 2π tributions for the inner-shell dissociative ionization of NO X    via fast predissociation of the NO+ c3  4σ 1 5σ 2 1π 4 2π 1 state [6]. The results demonstrate a strong preference for d wave excitation, with readily recognizable dz2 and dxz distributions under parallel and perpendicular excitation, respectively. Similar experiments for different species have been reported by Eland et al. [4] (H2 , N2 , NO, CO and O2 ), Lafosse et al. [25, 27], (NO, H2 ). The results show a pronounced energy dependence in the dominant ionization channel. For example, Lafosse et al. [24] observe pσ and pπ distributions for parallel and perpendicular excitation of H2 at 20 eV. The pσ channel remains dominant for parallel excitation at the higher energy 28.5 eV, but dπ takes over as the dominant perpendicular channel. The g/u mixing mechanism, whereby the positive charge localizes on one of the two equivalent H atoms, has also been discussed [25]. It should be emphasized that the simplicity of these results is crucially dependent on the axial recoil approximation, which would not apply at a rotationally auto-ionizing resonance,

218

Photo-ionization

Figure 7.6 (a) Experimental and (b) calculated molecule-fixed photo-electron angular distributions for the inner-shell ionization of NO [6]. Reading from the left, the arrows indicate parallel, magic angle and perpendicular polarization. Taken from Lucchese et al. [6], with permission.

because rotational structure can only be observed for molecular lifetimes that are long compared with the rotational period. It is also useful to note that (7.59) may be cast into a more convenient form for experimental purposes, by employing Clebsch–Gordan series for the products of spherical harmonics and reduced rotation matrices. One then finds that [6]     I θ n , θ k , φ k = F00 (θ k ) + F20 (θ k )P2 (cos θ n ) + F21 (θ k )P21 (cos θ n ) cos φ k   + F22 (θ k )P22 (cos θ n ) cos 2φ k , (7.65) where FP M (θ k ) =



ALP M PLM (cos θ k ) ,

(7.66)

L

in which PLM (cos θ ) is an associated Legendre polynomial and ALP M =

' 2παhν   (−1)λ+ν−q N × ¯λ¯ Tν¯(1) |0∗ λ| Tν(1) |0 3 ¯ ν¯ ¯ λλν







1 1 P 1 1 P  ¯ L  ¯ L × , −ν ν¯ M 0 0 0 −λ λ¯ M 0 0 0 (7.67)

7.6 Resonant two-photon ionization

in which

219



(2 + 1)(2¯ + 1) (L − M)! (P − M)! . (7.68) (L + M)! (P + M)!   The important point is that I θ n , θ k , φ k may be expressed in terms of the four functions FP M (θ k ), which may be evaluated by integrating over four independent ranges of the angle θ n between the positive ion direction and the polarization axis. For example [6],  π   I θ n , θ k , φ k sin θ n dθ n = 2F00 (θ k ) + 4F22 (θ k ) cos 2φ k N=

0



π /2

  I θ n , θ k , φ k sin θ n dθ n = F00 (θ k ) + F21 (θ k ) cos φ k + 2F22 (θ k ) cos 2φ k

π

  I θ n , θ k , φ k sin θ n dθ n = F00 (θ k ) − F21 (θ k ) cos φ k + 2F22 (θ k ) cos 2φ k

0

 

π /2 2π /3

π /3

   I θ n , θ k , φ k sin θ n dθ n = 8F00 (θ k ) − 3F20 (θ k ) cos φ k

(7.69)

 + 22F22 (θ k ) cos 2φ k /8.

Knowledge of the resulting functions FP M (θ k ) may be used to infer the electron distribution for any arbitrary polarization direction and hence to construct the diagrams in Fig. 7.6. 7.6 Resonant two-photon ionization 7.6.1 The optical density matrix The treatment of resonant two-photon ionization in this section is more conveniently developed from a density matrix standpoint [13, 28] than in terms of spherical tensor operators. The underlying ideas were introduced in Section 6.4 by showing that absorption of photon A creates a non-uniform distribution of intermediate angular momenta, which affects the final photo-electron angular distribution created by photon B. We also allow for a possible difference in the polarization axes of the excitation and probe lasers, by visualizing the intermediate distribution of magnetic quantum numbers Mi in a frame distinct from that of the initial photon, along lines given by Dixit, McKoy et al. [29, 30] and Reid, Allendorf, Leahy et al. [7, 31, 32]. It is assumed for the sake of illustration that the quantization axes of the two frames are separated by an angle χ in a common xz plane, in which case the pure states |J M  in the excitation frame appear as mixed states in the probe frame, whose pure states are denoted as |Ji Mi .

220

Photo-ionization

Now in density matrix language, excitation of the states |J

M

 by a dipole operator μ ˆ q is represented by a density operator  !' μ ˆ q J

M

J

M

μ (7.70) ˆ †q , ρˆ ∝ M

which is independent of the quantization frame, by virtue of the sum over M

. The necessary transformation of the dipole operator to the probe frame is given by   ∗ μ ˆq = μ ˆ p Dp1 a p (0, χ, 0) , (7.71) p

which gives rise to a density matrix in the probe frame with elements [31, 33]  ∗ ¯  1 Ji (0, χ, 0) Dq1p¯ (0, χ, 0) ρ Mi M¯ i (χ , q) = 3 (−1)2Ji −Mi −Mi Dqp p p¯ M



×

J

−M

1 p

Ji Mi



J

−M

1 p¯

Ji M¯ i

(7.72)

,

where q is the polarization index of the excitation laser, p and p¯ are dummy indices in the probe frame, χ is the angle between the polarization axes, (Mi , M¯ i ) are the projected excited space fixed components with respect to the second polarization axis, and the factor of three ensures normalization to T r (ρ) = 1. The matrix ρ is also seen to be Hermitian because the p and p¯ must be interchanged at the same time as Mi and M¯ i . An alternative form, which gives insight into the distributions in Figs 7.6 and 7.7, may obtained by transformations analogous to (7.35) and (7.37). Thus Ji

ρ Mi M¯ i (χ , q) = 3

2 

J

+M¯ i +q

[K](−1)

K=0

(

×

1 Ji

Ji 1

J

K

)



1 1 q −q

K . Q

K D0Q (0, χ, 0)

Ji M¯ i

Ji −Mi

K 0



(7.73)

The diagonal elements of Ji ρ Mi M¯ i (χ ) give the populations of the projected Mi states, examples of which are shown in Figs 7.6 and 7.7, for linear polarization (q = 0) and left circular polarization (q = 1), respectively. Of the three terms in (7.73), the K = 0 term represents the mean value (2Ji + 1)−1 , which is common to all the distributions. The K = 1 term gives the first moment of the distribution, which determines the mean orientation of Ji in the direction parallel to the polarization axis of photon B. The properties of the first Wigner coefficient ensure that such terms change signs under sign reversal of q and vanish identically for q = 0, which

7.6 Resonant two-photon ionization χm

π /6

0

221

π /3

π /2

Q(15) −15 0

15

−15 0

15

−15 0

15

−15 0

15

−15 0

15

−15 0

15

−15 0

15

−15 0

15

−15 0

15

−15 0

15

R(15)

Mi Figure 7.7 Relative populations of the excited MJ states arising from Q(15) and R(15) √excitation with linear polarization, at different angles χ , where χ m = cos−1 (1/ 3) is the magic angle.

χm

π /6

0

π /3

π /2

Q(15) 15 0

15

15 0

15

15 0

15

15 0

15

15 0

15

15 0

15

15 0

15

15 0

15

15 0

15

15 0

15

R(15)

M

Figure 7.8 Relative populations of the excited MJ states arising from Q(15) and R(15) excitation√with right circular polarization (μ0 = 1), at different angles χ. χ m = cos−1 (1/ 3) is the magic angle.

explains the symmetry of the distributions in Fig. 7.6. Finally the K = 2 term gives the second, or quadratic, moment of each distribution, which determines the alignment of the axis of Ji with respect to the final polarization axis, regardless of its sense of direction. It is also useful to note that the asymptotic form of the 6j coefficient for Ji , J

 K [34] (

1 Ji

Ji 1

J

K

)

 +K+2J

∼ (−1)



3 2(2Ji + J

)



1 J

1 −J

K 0

(7.74)

222

Photo-ionization

provides information on the rotational angular momentum change J = Ji − J

. In particular the K = 1 term vanishes asymptotically for Q branch (J = 0) transitions, even for q = ±1, while those for R and P branch transitions, with a given polarization, have equal magnitudes but opposite signs. Moreover the signs of the Q branch quadratic terms are reversed compared with those of the P,R branch transitions, because the 3j coefficient in (7.74) varies as 3(J )2 − 2 [13]. Finally the χ dependence of the diagonal distributions in Figs 7.6 and 7.7 is K determined by the rotation matrix elements D00 (0, χ, 0), because Q = 0 for the diagonal elements. In particular the K = 2 contributions disappear at the magic 2 (0, χ, 0) = (3 cos2 χ − 1)/2. angle, because D00 The same scaling behaviour applies to the off-diagonal matrix elements, or coherences, which arise because the M = 0, ±1 selection rules arising from (7.72) allow coherent excitation from a given initial state |J

, M

 to multiple mixed states |Ji , Mi  and |Ji , M¯ i . Such coherences disappear however when χ = 0 1 (0, 0, 0) = 0 for q = qb , with the associated restriction M = q. because Dqq b Put in another way the various (K, Q) components of Ji ρ Mi M¯ i (χ ) may be expressed in terms of the angular-momentum state multipole operators [35]

 √ Ji K Ji Ji Ji −Mi (7.75) (−1) 2K + 1 ρ Mi M¯ i (χ ) , TKQ (χ ) = Mi −M¯ i Q Mi M¯ i

Ji +J

+q

= (−1)

√

(

1 3 2K + 1 Ji

1 Ji

K J

)

1 −q

1 q

K 0

K D0Q (0, χ, 0)

or by their scaled analogues [13] (1) K OQ (χ ) = O0(1) (Ji , J

; q)D0Q (0, χ, 0) "   2Ji + 1 = Re T1Q (χ) , 3 (2)

K A(2) Q (χ ) = A0 (Ji , J ; q)D0Q (0, χ, 0)    (2Ji − 1) (2Ji + 1) (2Ji + 3) = Re T2Q (χ ) , 5Ji (Ji + 1)

(7.76)

in which the normalization is chosen to ensure that O0(1) (χ ) coincides with the √ scaled mean value, Mi  / J' i (Ji + 1), averaged!over the diagonal elements of (2) Ji ρ Mi M¯ i (χ ), while A0 (χ ) = 3Mi2 − Ji (Ji + 1) /Ji (Ji + 1). The three K = 1 and five K = 2 parameters quantify the angular orientation and alignment respectively, either parallel to the probe laser quantization axis, if Q = 0, or perpendicular to it if Q = 0.

Expressions for the coefficients O0(1) (Ji , J

; q) and A(2) 0 (Ji , J ; q) implied by (7.73)–(7.76) are given in Table 7.4. As expected from the discussion in Section 6.4,

7.6 Resonant two-photon ionization

223

Table 7.4 Orientation and alignment coefficients for single photon excitation. Ji

branch

O0(1) (Ji , J

; q)

A(2) 0 (Ji , J ; q)



 − 12 (3q 2 − 2) − 25 + 5(J

3+1)

 − 12 (3q 2 − 2) 45 − 5J

(J3

+1)   − 12 (3q 2 − 2) − 25 − 5J3

J

J

+1

J

+ 1

R

− 12 q

J

Q

1 √ 1 q J

(J

+1) 2

J

− 1

P

1 q 2



J

+1 J

the orientation coefficients vanish in linear polarization. It is also evident that P, Q and R transitions with left circular polarization (q = 1) cause positive, very small positive and negative orientation, respectively. In addition, the J = ±1 and J = 0 transitions give rise to positive and negative alignment, respectively. 7.6.2 1 + 1 REMPI photo-electron angular distribution of NO As an illustration of the use of the optical density matrix, it is interesting to take the case of 1 + 1 REMPI photo-ionization. A 2  + state of NO by Reid, Allendorf, Leahy et al., for which both the intermediate and fragment states are well-described by Hund’s case (b) [7, 31, 32]. This example is also important in showing how different polarization combinations of the two lasers can be used to extract information on both the magnitudes and phase differences of the excitation matrix elements. The flexibility of the density matrix method is however counterbalanced for exposition purposes by the absence of easy angular momentum contractions, because the sum over M

, which leads to the orthogonality conditions in Section 7.4, has already been employed in constructing the density matrix. Assuming polarizations qa and qb for the excitation and ionization lasers respectively and an angle χ between the polarization axes, the photo-electron differential cross-section is expressed in the form10    ∗ A0 Y σ (θ , φ; χ , qa , qb ) = (θ , φ) Y¯m¯  (θ , φ) m  (2J

+ 1) + ¯ ¯ ¯ M Mi Mi m m

!Ni ' ρ Mi M¯ i (χ , qa ) × γ + N + M + m eiδλ μ(1) qb γ i Ni Mi ' ! + + +¯ ¯ . × γ i Ni M¯ i e−iδ¯λ¯ μ(1) (7.77) qb γ N M m 10

In cases where the initial and intermediate states conform to case (a), with well-defined J

and Ji states, but spin is not resolved in the ionization process, the density matrix in (7.77) can be expressed as [31]



  ¯ Ji Si Ni Si Ni Ji Ni ρ MN M¯ N = (−1)MNi +Mni Ji ρ MN M¯ N . ¯ ¯ ¯ −MJi Msi MNi −MJi Msi MNi i i i i Ji MSi MJ M¯ J i i

224

Photo-ionization

In other words the sum over spherical tensor components in (7.13) is replaced by a two-stage expression in which the excitation step is incorporated in the density matrix Ji ρ Mi M¯ i (χ , qa ), and the ionization step by the matrix elements, which take the form appropriate for a case (bi ) to case (b+ ) transition [31]. Manipulations analogous to those in (7.35)–(7.41) lead to [31] σ (N + , θ, φ; χ , qa , qb ) = β 00 (N + , χ, qa , qb ) +

2  

β LM (N + , χ, qa , qb )YLM (θ, φ) ,

(7.78)

L=0,2 M=−2

where, in the present notation,11 β LM (N + , χ, qa , qb ) =





¯ λ¯ m m  λ

(2 + 1)(2¯ + 1)(2L + 1) 4π

 ¯ L × (−1) 0 0 0  +  ¯ ¯λ; ¯ χ; qa , qb rλ r¯λ¯ cos(δ λ − δ ¯λ¯ ), × γ N ; m λ, m

m

¯ −m¯

 m

L M



(7.79) in which  N   ¯ ¯λ; ¯ χ; qa , qb = [Ni , N + ](−i)−¯ γ N + ; m λ, m ρ Mii M¯ i (χ ; qa ) ¯ t ν ν¯ MN + t ¯t mt m

    ¯ ¯λ¯ ¯t M¯ Ni ν¯ , (7.80) × C m λt MNi ν C m and   C m λt MNi ν; qb = [t ] (−1)MN + +ν ×

t λt

1 ν



1  t −mt qb m

+  Ni N −λ MN + MNi



N + Ni − + i

t . −mt

t −λt



(7.81)

The important difference from single photo-ionization is that the increased number of asymmetry parameters, together with their dependence on the positive ion state N + , the polarizations qa and qb and the angle χ between the polarization

11

  Here (t , mt , λt ) and N

, MN

,

correspond to (Nt , −Mt , − t ) and (Ni , Mi , i ) in the original papers ¯ etc., were primed in the papers. [7, 31]. In addition the barred symbols, ,

7.6 Resonant two-photon ionization

225

axes provides sufficient information to determine the the matrix elements ! ' rλ = η+ + λ rν(1) η



,

(7.82)

in the notation of (D.31), together with the phase differences (δ λ − δ ¯λ¯ ). The asymmetry parameters β LM (N + , χ, qa , qb ) were found to vary strongly with the final positive ion state N + , with the continuously variable angle χ between the laser polarization axes, and with the polarization indices qa and qb . The aim of these studies is to show how these latter quantities may be experimentally determined, by exploiting the known dependence of the γ coefficients on the positive ion state N + , the partial wave characteristics, λ, etc., and the relative polarizations of the exciting and ionization lasers [7, 31, 32]. The experiment was performed by tuning linearly polarized excitation light beam to resonance with the X 2 (v = 0) → A 2  + (v = 0, N = 22) P21 +Q1 (22.5) transition of NO, which has 90% Q character. The assumption that the highest orbital of the A 2  + state has predominant s (or t = 0) character implies, according to Table 7.1, a strong preference for ionization to the N + = 22 state, with an angular dependence attributable to outgoing pσ and pπ waves. There are also weak N = ±2 transitions, which imply partial d (or t = 2) character with asymmetry parameters appropriate to outgoing p and f waves. Finally, some very weak additional N = ±2 transitions imply a partial p (t = 1) component, giving rise to a predominant dπ wave with some additional sσ and dσ character. Initial experiments employed a linearly polarized ionization laser at 314 nm with the polarization axis either parallel, χ = 0 or perpendicular, χ = π/2, to that of the excitation laser [32]. Later, photo-electron angular distributions were obtained with left (q = 1) and right (q = −1) circular polarization of the ionizing laser at an angle χ = π/2 between the polarization axis and that of the excitation laser [7]. The combined results were analyzed for the radial matrix elements, rλ , and phase differences, δ λ − δ ¯λ¯ , in (7.79). The resulting fits to the circular polarization measurements [7] are shown in Fig. 7.9, while the experimental fits and ab-initio parameters computed by Rudolph and McKoy [36] are compared in Table 7.5. The observed preference for excitation to the pσ and pπ waves is in accord with the picture of NO A2  + as an s Rydberg state, although there is also significant dσ and weaker pσ character to account for the excitation of fσ , π , sσ and dσ , π waves. The experimental fits are seen to be in good agreement with the ab-initio results [36]. Evidence of a relatively weak λ dependence of the radial matrix elements may be obtained by comparing the experimental and ‘scaled’ rλ values, with ' the latter taken ! to be proportional to the electronic transition matrix elements  λ Tν(1) 

λ

. Good agreement is also seen between the phase differences, δ pπ − δ pσ and

226

Photo-ionization

Table 7.5 Radial amplitudes and phases obtained by fitting to the experimental data in Fig. 7.9. The values in parentheses represent 1σ uncertainties. Those in the ‘scaled’ column are proportional to the relevant 3j coefficients. 

rλ λ sσ pσ pπ dσ dπ fσ fπ

Photoelectron counts

4000

Fit[7]

Calc. [36]

Scaled

0.204(2) 0.503(11) 0.471(6) 0.166(30) 0.073(15) 0.321(25) 0.244(13)

0.158 0.278 0.537 0.221 0.020 0.358 0.268

0.487 0.487 0.128 0.111 0.311 0.254

ΔN = 0

λ, ¯λ¯

Fit[7]

Calc. [36]

pπ , pσ dπ , dσ fπ, fσ dσ , sσ fσ , pσ

+0.216 −1.187 −0.017 −2.740 −1.030

+0.173 −1.620 −0.028 +3.032 −1.332

ΔN = +1

180

90

125

0

0

0

45 90 135 180 φ (degrees)

0

ΔN = +2

250

2000

0

 δ λ − δ ¯λ¯ /rad

45 90 135 180 φ (degrees)

0

45 90 135 180 φ (degrees)

Figure 7.9 Integrated photo-electron counts as a function of the azimuthal angle φ for three positive ion rotational levels. Circles and crosses denote the signal from ionization by left and right circular polarization, respectively. Squares indicate the circular dichroism in the angular distribution (CDAD), given by the difference between the upper two signals. The lines are predictions of the fit given in Table 7.5. Taken from Leahy et al. [7], with permission.

  δ fπ − δ fσ , and quantum defect differences π μpπ − μpσ =   the corresponding 0.28 and π μfπ − μfσ  0. The remaining phase differences were however less easily interpreted, at least in part because the nominal ‘s’ and ‘d’ series of NO are known to be strongly mixed [37]. 7.7 Orientation and alignment 7.7.1 General formulation Sections 6.4 and 7.6 offered an introduction to optically induced orientation and alignment of the angular momentum of bound intermediate state. Here the theory

7.7 Orientation and alignment

227

is extended to cover the orientation and alignment of the photo-fragment angular momentum, which can in principle be measured by laser-induced fluorescence of the positive ion. One approach is to use the excitation matrix elements to transform from the intermediate density matrix, Ji ρ Mi M¯ i (qa , χ ) to that of the positive ion. Thus [31]  ' ! A0 J+ γ + J + M + m sms rq(1) ρ M + M¯ + = γ i J i Mi b 2Ji + 1 m sm ¯ 

s

Mi Mi

! ' † + + + × ρ Mi M¯ i (qa , χ ) γ i Ji Mi rq(1) . (7.83) J M m sm γ  s b Ji

The angular momentum multipoles are then given by (7.75). However, the angular momentum algebra is less tractable than in spherical tensor representation, even when the two polarization axes have a common polarization axis, χ = 0. The following exposition is a straightforward generalization of the single photon theory given by Greene and Zare, with the same restriction to angle-integrated rather than differential multipoles [8, 13]. We are interested in the orientation and alignment of the positive ion angular momentum vector, J+ . By analogy with (7.75), the relevant angular momentum multipoles are given by

+  √ J+ K J+ J J + −M + TKQ = (−1) 2K + 1 ρ M + M¯ + (χ ) , (7.84) M + −M¯ + Q + + M M¯

in which the the relevant density matrix is taken as the scaled off-diagonal crosssection  (k) ! 1  ' + + + J+ γ J M m sms ρ M + M¯ + = T−q Tq(k) γ

J

M

σ 0 m sm M

kq  s  (k) ' + + !∗ ¯ ¯ × γ J M¯ m sms T−q¯ Tq¯(k) γ

J

M

, (7.85) k¯ q¯

where σ 0 is the integrated cross-section. In evaluating the sums over M + and M¯ + it is convenient to vary the angular momentum transfer theory, by working with the modified transfer vector Js = J+ − k = J

− j,

j = + s,

(7.86)

rather than with Jt in (7.15). The reason for the difference is that the sum over M + and M¯ + in (7.84) implies ambiguity in the direction of J+ and hence also in Jt , whereas the single sums over m , ms and M

in (7.85) imply uniquely defined vectors j, J

and hence Js . To see how this works out, the matrix elements are again given by (7.15)–(7.17) but the 3j coefficients are recoupled to conform with (7.86);

228

thus

Photo-ionization

' + + + ! γ J M m sms Tq(k) γ

J

M

    [j ][Js ](−1)q γ + J + - T (k) (Js , j ) -γ

J

= Js Ms j mj

(7.87)



k Js Js J+ j J

, × M + −q −Ms mj −M

Ms   where the reduced matrix element γ + J + - T (k) (Js , j ) -γ

J

may be shown by the methods in Appendix D.2 to take the form  + + - (k)  γ J - T (Js , j ) -γ

J



+



 k Js Js  s j J j J

(−1)ν = + −ν −s ω −

s λ σ −ω λσ ν

+  ! ' + + s S

S S + −s+

[S ] × (−1) η λ Tνk η



+

 σ − -



  + + - (k) × γ J - T (Js , j ) -γ J ( + )   j J  + + −J + −J

+−s J

[J ][Js ](−1) = γ J -T (k) (J , j )-γ

J

.

J k J s



 m

s ms

j −mj



J

(7.87a) The right-hand side of (7.85) involves a product of two such matrix elements ¯ j , Ms , M¯ s ), all of summed over (m , ms , M

) with internal sums over (mj , m which are subject to orthogonality constraints. Thus the sum over (m , ms ) requires ¯ j ) = (j, mj ), while that over (M

, mj ) requires that (J¯s , M¯ s ) = (Js , Ms ). that (j¯, m Finally, after substituting into the J + analogue of (7.75), the remaining sum over (Ms , M + , M¯ + ) may be evaluated by (4.15) of Zare to yield [13]

1  + ¯  k k¯ K ¯ (k) TKQ = (−1)J +Js +k−k+q T(k) T [K] −q −q¯ −q q¯ Q σ0 ¯ k kq q¯ Js

( ×

k J+

k¯ J+

K Js

)

  ¯ Js , σ k, k;

(7.88)

where   ¯ Js σ k, k; - ¯    A0   + + γ J -T (k) (Js , j )-γ

J

· γ + J + -T (k) (Js , j )-γ

J

. (7.89) =

[J ] j

7.7 Orientation and alignment

229

√

0 2J + + 1)T00 = 1 therefore shows that

2 (k) T    −q 0 σ (k, k; Js ) σ 0 = 2J + + 1)T00 = [k] kq J

The identity

s

2 (k) 2 A0    T−q  + + =

γ J - T (k) (Js , j ) -γ

J

, [J ] j kq J [k]

(7.90)

s

which may be compared with the forms in (7.30) and (7.31), appropriate to the (J, j ) and (Jt , t ) representations, respectively. The selection rules on the Wigner coefficients in (7.88) allow non-zero angular momentum multipoles in the range 0  K  2k> , where k> is the largest contributing k value, which is actually equal to the number of photons. In addition Q = q − q¯ is restricted by the geometry of the photon field. We have seen from the discussion in Sections 6.3 and 7.4 that light beams with a common polarization ¯ q) ¯ to a common q value; axis restrict the spherical tensor components (k, q) and (k, hence Q = 0 on the obvious physical grounds that a cylindrically symmetric field can only cause orientation and alignment with respect to the common polarization axis. However, (6.36) shows that two light beams with crossed linear polarization give rise to (k, q) = (1, ±1) and (2, ±1), which allows additional transverse orientation components with K = 2, 4 and |Q| = 2 plus orientation components with K = 3 and |Q| = 2. Notice that the sum over q and q¯ in (7.88) ensures 0 0 = (−1)K TKQ . The selection rules on the 6j coefficient in (7.88) also that TK−Q allow values of Js in the range |J + − k> |  Js  J + − k> , while changes in the polarizations  of therelevant light beams alter the weights given to the partial cross¯ Js . Measurements of the orientation and alignment coefficients sections σ k, k; (see - therefore-offer  + below) potential information on the reduced matrix elements + - (k)



γ J  T (Js , j ) γ J in (7.89). The implication in the case of a single photon is that there are three independent partial cross-sections, σ (Js ), for Js = J + , J + ± 1, which jointly determine the three multipole moments TK0 (J + ) for K = 0−2, which are related in turn to the integrated cross-section σ 0 and the orientation and alignment coefficients, O0(1) (J + ) + and A(2) 0 (J ) by (7.76) and (7.90). The situation for two photons is complicated by ¯ J +  -2, there interference between the various Tq(k) and Tq¯(k) in (7.88).  +Assuming will typically be nine independent matrix elements γ J + - T (k) (Js , j ) -γ

J

for the values J + − k  Js  J + + k, and k = 0−1. On the experimental side, there will be up to five possible parallel angular momentum multipoles, TK0 with K = 0−4, the odd members of which vanish if q = q¯ = 0 in (7.88) plus possibly three independent perpendicular multipoles, TK2 for K = 2−4. In addition,

230

Photo-ionization

their magnitudes vary according to the relative polarizations of the light sources. It must also be remembered that matrix elements of the antisymmetric Tq(1) operators vanish if either the two polarizations or the two frequencies are equal. Equation (7.48) shows that a two-colour crossed linear polarization (Lz Lx ) experiment is (2) (1) and T±1 spherical tensor operators. The represented by combinations of the T±1 considerations in the previous paragraph show that it would yield eight of the nine required experimental quantities, which might be complemented by knowledge of the three even parallel multipoles created by parallel linear polarization. Alternatively the five multipoles produced by two colour C+ Lz polarization could be combined with two sets of three even multipoles excited by Lz Lz and C+ C− polarization. To see how this knowledge of the orientation and alignment parameters the- information on the alternative reduced matrix elements   + complements γ J + - T (k) (t , Jt ) -γ

J

in the (t , Jt ) representation, which may be derived from the photo-electron angular distribution, it may be shown that  + + - iδ (k)   +

γ J - e T (Js , j ) -γ

J

= [Js ] [j ] [Jt ] [t ](−1)J +J −−s−Js +t J t t

  × γ + J + - eiδ T (k) (t , Jt ) -γ

J

)( ) ( + k Js  s j J . ×  J

t J t k t

(7.91)

Moreover in the simplest single photon case, such that spin can be ignored and that the photo-ionization arises from a single t contribution, the discussion in Section 7.4.2 shows that there are two outgoing waves with  = j = t ± 1, while (7.21) reduces to  + + - (k)  γ J - T (t , Jt ) -γ

J

 - ! ' + + = [J + ] [J

](−1)J − +t +λt Paa (, t ) η+ - T (k) (t ) -η

+ J

Jt J δ J t t . (7.92) × −+ 

t   Thus the three matrix elements γ + J + - eiδ T (k) (Js , j ) -γ

J

derived from knowledge of the angular momentum -reduce ! in this simple case to ' multipoles combinations of the matrix elements η+ - T (1) (t ) -η

for  = t ± 1. Reference to (7.40) and (7.46) shows that the photo-electron asymmetry parameter β depends on the magnitudes of the same matrix elements, together with the phase difference between the outgoing waves. Experimental measurements of the integrated crosssection, the orientation and alignment parameters and the asymmetry parameter may therefore be combined to determine both the matrix elements and the phase difference δ  .

7.7 Orientation and alignment

231

7.7.2 Experimental determination It is appropriate to close the section by reference to techniques for experimental measurement of the orientation and alignment parameters. If the photo-ionization fragments are electronically excited, it is simplest to analyze the polarization characteristics of fluorescence to a lower excited state [8]. Such fragments are, however, normally in their ground electronic states, in which case the argument in Section 7.6.1, may be reversed, in order to extract the angular momentum multipoles TK0 (J + ) from the relative intensities of the various J transitions in a spectrum excited from the nascent photo-fragment. To see the argument in its simplest form we assume the single-photon probe laser, with polarization qp , which excites the |J + M +  fragments to states |J , M . The starting point is to reverse (7.75) by using the multipoles TK0 (J + ) to define an associated density matrix

+   J+ K J J+ Ji+ −M + + ρ M + M¯ + = (−1) [K] (7.93) + ¯ + 0 TK0 (J ). − M M + + M M¯

The line strengths of the probe J + → J transitions are therefore given by



 J 1 J+ 1 J + J+ J

ρ Mi M¯ i (J + ) F (J ) = N

+

¯+ M −M q M −M q p p

+ ¯+ MM M



=N

J + −Mi

(−1)



M M + M¯ +



J+ [K] M+

J+ −M¯ +

1 J+ 1 J J × −M qp M + −M qp   +

=N [J ] [K](−1)J +J −K+qp

K

(

1 × J+

J+ 1

J K

)

1 −qp

1 qp

K 0

J+ M¯ +



K 0



TKQ

TK0 (J + ),

(7.94)

where N is a normalization factor12 chosen to ensure that T00 = [J + ]−1/2 . It follows from the orthogonality of 6j symbols that

1 1 K 1+qp TK0 (J + ) (−1) −qp qp 0 ) (   1 J+ J −1 −J +J + −K+1 F (J ). (−1) (7.95) [K] =N + J 1 K

J

12

One finds by setting K = 0 that N =



J

F (J ).

232

Photo-ionization

Notice, of course, that the K = 1 orientation multipole can only be determined by probing with circular polarization, qp = ±1. Equation (7.95) demonstrates the essential physical connection between the orientation and alignment parameters and the relative intensities of the P , Q and R branch transitions from the nascent J + state. It is not, however, the end of the story because so few ions are produced that the ‘intensity’ cannot be measured by monitoring the absorption. Instead it is measured by the means of laser-induced fluorescence, which involves further angular momentum manipulations. Details are given by Greene and Zare [38]. The extension of (7.93)–(7.95) to several photons is also straightforward in principle. For example two-photon ionization fragments, which are characterized by up to five rotational multipoles TK0 parallel to any chosen polarization axis, may be monitored by probing the intensities of the five two-photon transitions with −2  J  2. The simplest probe, consisting of a combination of linearly and circularly polarized lasers, with a common frequency and a common polarization axis, is represented by a single spherical tensor operator, Tq(2) , which would therefore entail simply replacing k = 1 in the above equations by k = 2. The reader is referred to Kummel et al. and Mo and Suzuki for further details [38, 40]. 7.8 Spin polarization Spin polarization is a subtle effect, which arises in heavy atoms from a differential relativistic contraction of the electronic core, such that the radial matrix element for excitation to the j =  + 1/2 spin-coupled partial wave differs from that for j =  − 1/2. The example 6s → p excitation in Cs by left circular polarization was chosen by Fano [41] as an illustration, for which . / . / 1 1

iφ

j m r sin θe 00 ms = j m  m ms  m | sin θ eiφ |00 dj 2 2  m  " . / 2 1 j m 11 ms dj , = (7.96) 3 2 in which !dj is the radial matrix element. Hence the amplitude of the outgoing  m sms partial wave is governed by the matrix element / " . / . / . 1 2 1 1 1 1m  m s j m dj j m 11 ms , (7.97) 1m ms r1 00 ms = 2 2 3 j 2 2 which may be written in the compact matrix form 1/2 1/2 r3/2 −1/2 0

−1/2 √

2(r3/2 − r1/2 )/3 , (r3/2 + 2r1/2 )/3

(7.98)

7.8 Spin polarization

233

in which the rows and columns are labelled by m s and ms , respectively and rj = √ 2/3dj . Differences in the diagonal elements when r3/2 = r1/3 correspond to different transition amplitudes for α and β spins, while the non-zero off-diagonal element implies an associated β → α spin change, under left circular polarization – with a corresponding α → β change under right circular polarization. The effect is particularly marked in the vicinity of any Cooper minima [42] (see Section D.3) at which the dipole matrix element for one of the j channels vanishes (the Fano effect). The theory of such atomic processes, which was first comprehensively reviewed by Cherepkov [43], has been extended to molecules by Raseev and Cherepkov [9], from whom the following account is taken. The required extension of the theory of the photo-electron angular distribution in Section 7.4 proceeds by calculating the mean value of the spin polarization operator 12 [1 + sˆ · σ ], where σ is the Pauli matrix vector and sˆ is the (arbitrary) quantization axis of the spin polarization detector. The required matrix elements are given by [9]

√  1 1/2 1/2 S ¯s 1/2−m sms | [1 + sˆ · σ ] |s m ¯ s  = 2π . (−1) YS−MS (ˆs) ¯ s −MS ms −m 2 SMS

(7.99) Hence the spin-polarized angular distribution (ADSP) may be expressed as ˆ sˆ) = I (k,

   A0 ˆ m ˆ ∗ Ym (k)[Y ¯ ¯ (k)] (2J

+ 1)

+ ¯ m m¯ m m¯ M M







s

s

1 ¯ s × sms | [1 + sˆ · σ ] |s m 2 ' !





× γ + J + M + m sms eiδ μ(1) q γ J M ! '





∗ ¯ s eiδ¯ μ(1) ¯ s m . × γ + J + M + ¯m q γ J M

(7.100)

The sums over magnetic quantum numbers are performed by extension of the treatment of the differential cross-section in Section 7.4. On the assumption of strong spin–orbit coupling the matrix element is written, with the help of (D.43) as ! ' + + + J η  ; m sms ; p Tq(k) J

η



; p

  - ! '



(−1)−s+k+jt +M − P(, jt ) η+ j - T (k) (jt ) -η

= [J + ] [J

]

 s × m  ms + jt J × + ωt

jt

j −mj J

−



,

j mj

k −q

jt −mt



J+ M+

jt mt

J

−M



(7.101)

234

Photo-ionization

where P(, jt ) is the parity factor



1 + (−1)p +p ++jt −s . P(, jt ) = √ 2 (1 + δ + 0 )(1 + δ 

0 )

(7.102)

Lengthy but straightforward further manipulations13 lead to the expression

  1 1 K ˆ sˆ) = σ 0 I (k, 3[L][S](−1)1+q C (K, L, S, ML ) q −q 0 KLSML

× CLML (θ k , φ k )CS−ML (θ s , φ s ),

(7.103)

where CLML (θ , φ) etc. are modified spherical harmonics [16], (θ k , φ k ) and (θ s , φ s ) ˆ and the final are the polar angles of kˆ and sˆ with respect to the polarization axis, q, coefficient is given by √

 6[L][K] K L S C (K, L, S, ML ) = Q(jt )c(K, L, S; jt ), 0 ML −ML B jt

(7.104) where B = 3σ 0 /NE , NE = 4π 2 αa02 (hν/Ry ), α is the fine structure constant, hν is the energy of the ionizing photon and   ¯ +Jt +1 +j ¯ j, j¯]P(, jt )P(, ¯ jt )Q(jt ) c(K, L, S; jt ) = (−1) [, , j j¯

¯

⎫ ⎧ ) ⎨ j j¯ K ⎬ 1 1 K (7.105) ×  ¯ L j j¯ jt ⎩ ⎭ s s¯ S - !' - ! ' ¯ s)- eiδ¯ T (k) (jt ) -η

. × η+ j (, s)- eiδ T (k) (jt ) -η

η+ j¯(,

 ¯ L 0 0 0

(

in table! (7.1), while the properties of the reduced matrix The factor ' +Q(jt ) is-given (k) element η j (, s) T (jt ) η

are given in (7.27). There are five possible types of term in (7.103) according to the allowed values of K, L and S, because the Wigner coefficients in (7.99) and (7.103) carry the restrictions S = 0, 1 and K = 0, 1, 2, respectively. In addition the 3j symbol in (7.104) formally allows L = 0, 1, 2, 3 but we know that parity arguments restrict L to even values for any dipole induced excitation. Finally K = L for S = 0. The terms with K = 1 can occur only with a circularly polarized light beam, whereas those with K = 2 may be observed with either linear or circular polarization. Each 13

The steps involve (a) the orthogonality of the (J

, J + , Jt ) Wigner coefficients, (b) a 6j recoupling transformation of the (j, 1, Jt ) coefficients, (c) the Clebsch–Gordan transformation in (7.35) and (d) a sum over five 3j coefficients involving the 9j symbol [16] in (7.105).

7.8 Spin polarization

235

Table 7.6 Asymmetry and spin polarization parameters in terms of the coefficients C (K, L, S , ML ) and their associated angular dependences. ˆ sˆ) g(k,

Parameter √ β=−  2 C(2, 2, 0, 0) A = 23 C(1, 0, 1, 0)  γ = − 15 C(1, 2, 1, 0) 2 ξ=

3i √ [C(2, 2, 1, −1) 2 2

ˆ 2 − 1) = P2 (cos θ k ) (3(kˆ · q)

− C(2, 2, 1, −1)]

ˆ = cos θ s (ˆs · q)

 ˆ − (ˆs · q) ˆ 3(kˆ · sˆ)(kˆ · q) ˆ kˆ · q) ˆ [ˆs · (kˆ ∧ q)](

ˆ and sˆ, as indicated in the has a characteristic dependence on the unit vectors k, expansion [9] (

 σ0 ˆ ˆ sˆ) + γ gγ (k, ˆ sˆ) 1 + q AgP¯ (k, I (k, sˆ) = (7.106) 8π ) (2 − 3q 2 ) ˆ sˆ) − 2ξ gξ (k, ˆ sˆ) . βgβ (k, + 2 Expressions for the various parameters and the associated angle dependences, ˆ g(k, sˆ), are given in Table 7.6, in which qˆ is the polarization axis of the excitation laser. Here β is the familiar asymmetry parameter, with no spin polarization connotations because it arises for S = 0. The second parameter P¯ is the total spin polarization, integrated over the photo-electron angular distribution, which is observable only with circular polarization, q = 0. The remaining parameters provide correlations between the kˆ and sˆ distributions. The parameter γ , which is observable only by circularly polarized excitation, gives the spin polarization in the ˆ 14 while ξ relates to the spin polarization perpendicular plane containing kˆ and q, to this plane. Separate measurements must therefore be be performed with sˆ either ˆ q) ˆ plane. in or perpendicular to the (k, It is interesting, in the light of (7.96)–(7.98), to confirm15 that the singlephoton integrated spin polarization vanishes when the reduced matrix elements are 14 15

Note that although γ is expressed in terms of the ML = 0 coefficient C(1, 2, 1, 0), the angular dependence ˆ sˆ) has contributions from ML = 0, ±1 in (7.103). gγ (k, With the help of the identities [16] ⎧ ⎫ ( )( ) ⎨ L ¯  ⎬ )(     k 

K k¯ k ¯ k¯ 

τ +j ¯ ¯ K j j j, j (−1) j

s j j

s¯ j ⎩ j

j j¯ S s¯ s ⎭ j j¯

( =

s j



s



s¯

⎧ )⎨ L K ⎩S

¯ k¯ 

⎫  ⎬ k , 

⎭

236

Photo-ionization

approximated by the final line of (7.27), the radial matrix elements are independent of both j and ω, and

  - (1) -

  1 



. (7.107)  T  = (−1) [,  ] 0 0 0 Thus measurements of the P¯ parameter in Table 7.4 provide information on the j and ω dependence of the radial matrix elements. In addition the angle resolved spin polarization coefficients have non-zero contributions only from the off-diagonal interference terms in 1 − cos (δ  − δ ¯) . be noted that the reduced !matrix elements - iδ (k) it -should ' +In conclusion ! '

 ¯ η j (, s) e T (jt ) η correspond to c , 1 j1 T (jt ) 

in the notation of Raseev and Cherepkov [9]. However, these elements are taken as combinations transition elements rather than the present spin–orbital elements ! ' + +of orbital η  j ω Tνk η



. The former would be written in the present notation as ! ' tλ = η+ + λ Tνk η



. (7.108) As a specific example Raseev and Cherepkor consider the photo-ionization of HI HI(1 0+ ) + hν → HI+ (2 3/2 ) + e (λσ ) ,

(7.109)

including both parity forms for the final state, and final angular momenta  = 0, 1, 2. Explicit expressions for the coefficients c(K, L, S; jt ) are given in terms of the transition elements tλ . An MQDT-based calculation, which incorporated these results was reported by B¨uchner et al. [44], who used ab-initio transition elements given by Lefebvre-Brion et al. [45]. The calculated integrated photo-ionization cross-section, σ 0 () , at energies between the 2 3/2 and 2 1/2 ionization thresholds, shows a broad 2 3/2 continuum mainly attributable to p→(s,d) excitation, with superimposed 2 1/2 auto-ionization resonance peaks. The energy dependence of the corresponding angular asymmetry and spin polarization parameters also shows broad 6sσ and 6dσ features, but there are also some sharp 6pπ and 6fπ peaks. Later papers obtained good agreement with experimental integrated where τ =  + ¯ − K − k − j − S − 

+ 2j

+ s¯ ; and 

 , ¯

¯

=

L 0



L 0

 ¯ 0 0

K 0

S 0





K 0

¯ k¯ 0 0 k 0

k¯ 0



0





S 0

 0 

0



0

k 0 

0

⎧

⎨ L K ⎩S

¯ k¯ 

⎫  ⎬ k 

⎭

.

Note that the spin-polarization terms require S = 1, which means that the final Wigner coefficient in the second equation necessarily vanishes. However, the angle resolved spin polarization coefficients contain nonzero contributions from the interference terms in cos(δ  − δ ¯ ).

References

237

spin polarization measurements for HBr at various points in the resonance region between the 2  fine-structure thresholds [46, 47, 48].

References [1] A. D. Buckingham, B. J. Orr and J. M. Sichel, Phil. Trans. Roy. Soc London, A 268, 147 (1970). [2] U. Fano and D. Dill, Phys. Rev. A 6, 185 (1972). [3] D. Dill and U. Fano, Phys. Rev. Lett. 29, 1203 (1972). [4] J. H. D. Eland, M. Takahashi and Y. Hikosaka, Faraday Discuss. 115, 119 (2000). [5] D. Dill, J. Chem. Phys. 65, 1130 (1976). [6] R. R. Lucchese, A. Lafosse, J. C. Brenot et al., Phys. Rev. A 65, 020702 (2002). [7] D. J. Leahy, K. L. Reid, H. Park and R. N. Zare, J. Chem. Phys. 97, 4948 (1992). [8] C. H. Greene and R. N. Zare, Phys. Rev. A 25, 2031 (1982). [9] G. Raseev and N. A. Cherepkov, Phys. Rev. A 49, 3948 (1990). [10] H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One and Two Electron Atoms (Springer-Verlag, 1957). [11] I. I. Sobel man, Atomic Spectra and Radiative Transitions (Springer-Verlag, 1979). [12] L. D. Landau and E. M. Lifshitz, Non-Relativistic Quantum Mechanics (Pergamon Press, 1965). [13] R. N. Zare, Angular Momentum (Wiley Interscience, 1988). [14] D. R. Bates and A. Damgaard, Phil. Trans. Roy. Soc. London A 242, 101 (1949). [15] J. Xie and R. N. Zare, J. Chem. Phys. 93, 3033 (1990). [16] D. M. Brink and G. R. Satchler, Angular Momentum, 2nd edn (Oxford University Press, 1979). [17] D. Dill, Phys. Rev. A 7, 1976 (1973). [18] J. Cooper and R. N. Zare. In Lectures in Theoretical Physics: Atomic Collision Processes, ed. S. Geltman, K. T. Mahanthappa and W. E. Britten (Gordon and Breach, 1969). [19] T. A. Carlson, M. O. Krause, W. A. Svensson et al., Z. Physik D 2, 309 (1986). ¨ [20] G. Ohrwall and P. Balzer, Phys. Rev. A 58, 1960 (1998). [21] M. W. Ruf, T. Bregel and H. Hotop, Phys. Rev. B 16, 1549 (1983). [22] Y. Itikawa, Chem. Phys. 37, 401 (1979) . [23] M. S. Child, PCCP 10, 6169 (2008). [24] A. Lafosse, M. Lebech, J. C. Benot et al., J. Phys. B 36, 4683 (2003). [25] F. Martin, J. Fernandez, T. Havermeier et al., Science 315, 629 (2007). [26] N. A. Cherepkov and G. Raseev, J. Chem. Phys. 103, 8283 (1995). [27] A. Lafosse, M. Lebech, J. C. Benot et al., Phys. Rev. Lett. 84, 5987 (2000). [28] K. Blum, Density Matrix Theory and Applications (Plenum Press, 1981). [29] S. N. Dixit and V. McKoy, J. Chem. Phys. 82, 3546 (1985). [30] S. N. Dixit, D. L. Lynch, V. McKoy and W. M. Huo, Phys. Rev. A 32, 1267 (1985). [31] K. L. Reid, D. J. Leahy and R. N. Zare, J. Chem. Phys. 95, 1746 (1991). [32] S. W. Allendorf, D. J. Leahy, D. C Jacods and R. N. Zare, J. Chem. Phys. 91, 2216 (1989). [33] R. L. Dubs and V. McKoy, J. Chem. Phys. 91, 5208 (1989). [34] G. Ponzano and T. Regge, Spectroscopic and Group Theoretical Methods in Physics, ed. F. Bloch et al. (North-Holland, 1968). [35] U. Fano, Rev. Mod Phys. 29, 74 (1975). [36] H. Rudolf and V. McKoy, J. Chem. Phys. 91, 2235 (1989).

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Photo-ionization

C. Jungen, J. Chem. Phys. 53, 4168 (1970). C. H. Greene and R. N. Zare, J. Chem. Phys. 78, 6741 (1983). A. C. Kummel, G. O. Sitz and R. N. Zare, J. Chem. Phys. 85, 6874 (1986). Y. Mo and T. Suzuki, J. Chem. Phys. 109, 4691 (1998). U. Fano, Phys. Rev. 178, 131 (1969). J. W. Cooper, Phys. Rev. 128, 681 (1962). N. A. Cherepkov, Adv. At. Mol. Phys. 19, 395 (1983). M. B¨uchner, G. Raseev and N. A. Cherepkov, J. Chem. Phys. 96, 2691 (1992). H. Lefebvre-Brion, A. Giusti-Suzor and G. Raseev, J. Chem. Phys. 83, 1557 (1985). V. V. Kuznetsov and N. A. Cherepkov, J. Chem. Phys. 110, 9997 (1999). R. Irrgang, M. Drescher, M. Spieweck, U. Heinzmann and N. A. Cherpkov, J. Chem. Phys. 108, 10070 (1998). [48] M. Drescher, R. Irrgang, M. Spieweck et al., J. Chem. Phys. 111, 10883 (1999).

8 Manipulating Rydberg states

The huge spatial extension of atomic and molecular Rydberg states makes them amenable to manipulation in a variety of ways. One type of experiment involves the creation of a time-dependent wavepacket, which may be manipulated by subsequent light pulses to control the outcome of the fragmentation products [1]. Interesting intensity recurrences and revivals are also observed as leading and trailing elements of the wavepacket interfere with each other. The response to electric fields is also experimentally important in the field-ionization detection of highly excited species and in the technique of high-resolution pulsed-field zero-kinetic energy (ZEKE-PFI) spectroscopy [2, 3]. This chapter concentrates on these two topics, but the reader should be aware of the quasi-Landau response to magnetic fields, particularly at field strengths such that the Landau frequencies are comparable to those of hydrogenic orbits, because the Rydberg scaling properties make them ideal candidates for investigating ‘quantum chaos’ [2, 4]. 8.1 Rydberg wavepackets Despite the well-known equivalence between the time-dependent and timeindependent pictures for conservative systems (i.e. those with time-independent Hamiltonians), the ability to create and manipulate Rydberg wavepackets offers novel insights into the underlying dynamics. Here we concentrate on three aspects of the time-dependent theory. The first shows that the familiar level structure of the hydrogen atom leads to a surprisingly intricate pattern of recurrences and revivals arising from interference between different components of the spreading wavepacket. Revivals of a different type are seen to occur in molecules as a result of the stroboscopic beats between the frequencies of rotational and electronic motion that were described in Section 4.2.4. The second application shows how knowledge of the phases responsible for these recurrences can be exploited to ‘control’ the outcome of a picosecond experiment, by varying the properties of an appropriate 239

240

Manipulating Rydberg states

pulse sequence. Finally it is shown how the MQDT techniques described in the previous chapters can be modified to treat the time-dependent dynamics. 8.1.1 Recurrence and revival As an introduction to the theory of recurrences and revivals it is appropriate to start with the hydrogen atom, along the lines pioneered by Parker and Stroud [5] and Averbukh and Perelman [6]. It is convenient to reference the energies En = I −

Ry n2

(8.1)

of the wavepacket components to the energy EL = E0 + h¯ ωL of the exciting laser pulse. The wavepacket is therefore expressed as  an (t)φ n (r) exp (−in t) , (8.2) (r, t) = n

where n = (En − E0 )/¯h − ωL , in which the coefficients for weak-field excitation by a pulse, E(t) = ezE0 f (t) sin ωL t,

(8.3)

are governed by the perturbation equation dan i = − n f (t) exp (in t) , ddt 2

(8.4)

where n = (E0 /¯h) n| ez |0. By the usual scaling arguments the matrix element for transitions from the ground may be assumed to vary as n−3/2 . The assumption of a normalized Gaussian pulse, with width parameter α yields the solution1    t  n 3/2 1 t2 0 an (t) ∝ −i an 0 √ exp in t − 2 dt n 4α 2πα −∞ 

  n 3/2  2 2 1 t 0 ∝ −i an0 exp −α n 1 − erf √ , (8.5) − in τ n 2 2α where an0 = n0 /2 is the coefficient for En0  EL . Hence at the end of the pulse  n 3/2   0 an (∞) ∝ −i (8.6) an0 exp −α 2 2n . n It is interesting for the following discussion to compare the properties of the wavefunction given by (8.2) and (8.6) with those of a ‘classical’ or ‘harmonic’ 1

Equation (8.4) employs the rotating-wave approximation by neglecting the rapidly oscillatory √ term exp[i(En + EL )t/¯h]. The parameter α is related to the full width at half maximum (FWHM) by τ = 4α ln 2.

8.1 Rydberg wavepackets

Quantum wavepacket

241

Classical wavepacket

2

(a)

2

(b)

2

(c)

2

(d)

|Ψ|

|Ψ|

|Ψ|

|Ψ|

0

1000

2000

r / au

3000

4000 0

1000

2000

3000

4000

r / au

Figure 8.1 Steps in the time evolution of the quantum and classical wavepackets, given by (8.2) and (8.7) at times (a) t = 0 (shaded) and t = Tcl /2; (b) t = Tcl ; (c) t = 7Tcl /4; and (d) t = Trev /4. The wavepacket was excited by a 0.5 ps pulse centred at n0 = 39 for which Tcl = 9 ps and Trev = 117 ps.

wavepacket cl (r, t) =



an (t)φ n (r) exp [2π i(n − n0 )t/Tcl ] ,

(8.7)

n

in which Tcl is the period of classical motion at energy E0 ,

2π d 1 , ω0 = = 3 au. Tcl = ω0 dn n0 n0

(8.8)

Figure 8.1 illustrates some steps in the time evolution of these two wavepackets, both of which were initiated by a 0.5 ps pulse centred on a transition to the (n0 , ) = (39, 1) state of hydrogen, which spans a range n  n0 ± 5 corresponding to a mean energy spread E  0.2 meV  1.6 cm−1 . The times appropriate to the various panels, which are measured from completion of the excitation pulse, are expressed in terms of the classical period Tcl = 9 ps. The initial wavepacket at time t = 0, which is shown as shaded in Fig. 8.1(a), is seen to peak at r  750 au, which is the distance at which the various radial components begin to dephase (see Fig. 4.2). The solid curves in the same panel, which are drawn for t = Tcl /2, show that the mean positions of the quantum and classical packets are almost identical, although there is already some difference in shape. More substantial changes are evident at the end of the first classical period in Fig. 8.1(b). The spread

242

Manipulating Rydberg states

of the quantum packet has approximately doubled, compared with t = 0 while the classical one has returned precisely to its initial form. Further spreading of the quantum packet at time t = 7Tcl /4 in Fig. 8.1(b) contrasts with the compactness of the classical packet. The final panels at the much longer time Trev = 13Tcl , in Fig. 8.1(d) show something more remarkable, which is explored in detail below. The classical packet has again reverted to its initial form after an integral multiple of classical periods, while the quantum one has concentrated itself into a single peak, which is actually narrower than the peaks at all three earlier times. This ‘revival’ phenomenon will be shown below to be preceded by ‘partial revivals’ at times t = Trev /k, where k is an integer, at which the wavepacket may be decomposed into k classical sub-packets, uniformly distributed around the orbit. The origin of this striking effect lies in the fact that the atomic energy level spacings or classical frequencies, ωn = (dE/dn)/¯h, decrease as n increases. It is assumed for illustrative purposes that the linear approximation

dω ωn = ω0 + (n − n0 ) (8.9) dn 0 is adequate at least for the central members of the packet. Alternatively the two parameters in (8.9) may be expressed in terms of characteristic times

2π dω 2π ω0 = , =− , (8.10) Tcl dn 0 Trev so that Tcl = 2πn30 au and Trev = 2πn40 /3 au for the H atom, while n0 would be replaced by n0 − μ for species with a finite quantum defect. The negative sign in (8.10) ensures that Trev is positive, in contrast with the notation of Averbukh and Perelman [6]. Before addressing the quantum mechanical behaviour, it is convenient to set the scene by following the evolution of a quasi-classical ensemble, with the same anharmonicity, in a formal canonical classical phase space with components [7] √ √ pn = − 2n sin θ n , (8.11) qn = 2n cos θ n , where θ n is the angle variable, which varies linearly with time with frequency ωn . Thus in the light of (8.9) and (8.10)   t t , (8.12) − (n − n0 ) θ n = −π + ωn t = −π + 2π Tcl Trev where the choice of the initial angle θ n = −π at t = 0 ensures that each component starts from its inner classical turning point, which corresponds in Coulomb systems to the closest approach to the positive ion core.

8.1 Rydberg wavepackets

243

The most illuminating consequences of this equation are illustrated in Fig. 8.2, which is drawn for n = 35–45. The ensemble starts as a compact group on the line p = 0 at the inner turning point a. Thereafter each bead moves along its own orbit in a clockwise sense at its own characteristic frequency ωn . Thus at time Tcl /4 the central bead, with n = n0 has advanced to θ = π/2, while those on the inner orbits with n < n0 have advanced further because they have higher frequencies. Similarly, those with n > n0 lag behind because ωn < ω0 . The resulting overall spread is already of order θ  π/3. The next snapshot at the ‘recurrence’ time t = Tcl shows that the central bead has returned to the turning point a, while the spread has increased to the extent that the head of the chain has almost caught up with the tail. Thereafter a quantum mechanical wavepacket would begin to interfere with itself. The striking ‘revival’ at time t = Trev , which is the classical analogue of the quantum revival in Fig. 8.1(d), may be understood by reference to (8.12). The second term in the bracket shows that bead n has performed n − n0 perfect cycles compared with the central one, and the first term shows that the whole ensemble has advanced by 2π(Trev /Tcl ) = 2π(n0 /3) (mod 1), which corresponds to 2π /3 for n0 = 40. The final snapshot at t = Trev /3 illustrates a three-fold ‘partial revival’, which is more clearly illustrated in this classical picture than in a simple plot of the quantum wavepacket. The angular separation of successive beads is now governed by [2π (n − n0 )/3] (mod 2π) which can take the three values 0, ±2π/3. The same argument can be applied at times t = (p/q)Trev for other rational fractions, (p/q). For a sufficiently long string there will be q distinct partial revivals, separated by angles 2π/q. The corresponding quantum mechanical analysis, which is taken from Averbukh and Perelman, centres on the phases, n t, in (8.2), which are expanded in the form [6]

 1 d2 E 2 t n t  (n − n0 ) + (n − n0 ) 2 dn2 0 h¯ 0  

dω = ω0 (n − n0 ) + (n − n0 )2 t dn 0   t 2 t . = 2π (n − n0 ) − (n − n0 ) Tcl Trev 

dE dn



(8.13)

Within the validity of this quadratic approximation, the quantum wavepacket is represented by (r, t) =

 n

  2πi(n − n0 )t 2π i(n − n0 )2 t , an φ n (r) exp − + Tcl Trev

(8.14)

244

Manipulating Rydberg states

p

15

15

10

10

5

5

0

a

0

−5

−5

−10

−10

−15 −15 −10

p

T cl / 4

−5

0

5

10

15

−15 −15 −10

15

15

10

10

5

5

0

T rev

a

0

−5

−5

−10

−10

−15 −15 −10

−5

0

q

5

10

15

T cl

a

a

−15 −15 −10

−5

0

5

10

15

10

15

T rev / 3

−5

0

5

q

Figure 8.2 Steps in the time evolution of the classical wavepacket, given by (8.11) and (8.12). The motion starts in a clockwise direction, from the inner classical turning point, q = a, with all components on the line p = 0.

in which the linear term drives the underlying classical motion while the quadratic term causes not only the initial spreading in Fig. 8.1, but also the revivals and partial revivals. Note that the sign of the quadratic term differs from Averbukh and Perelman, owing to the sign choice in (8.10). The revivals themselves, at t = Trev , are readily understood because the factors (n − n0 )2 are all integers, which means that (r, t) reverts to the non-spreading classical form cl (r, t).2 Analysis of the partial quantum revivals, analogous to the classical revivals in Fig. 8.2 at Trev /3, is more complicated. The general case of partial revival at rational fractions, t = (p/q)Trev , of the full revival time is treated by Averbukh and Perelman [6]. Here we restrict attention to the specific case q = 3, by noting that sequences of terms n, n + 3, n + 6, etc., with periodic 2

The displacement between the ‘revived’ quantum and classical wavepackets in Fig. 8.1(d) is attributable to an additional cubic contribution to the phase in (8.13).

8.1 Rydberg wavepackets

245

length three, have a common phase because (k + 3)2 k2 = + 2k + 9, 3 3

(8.15)

where k = n − n0 . There are three such sequences, analogous to the three groups in the final panel of Fig. 8.1, which may be labelled by an integer s = k(mod 1) = 0, ±1. The key point made by Averbukh and Perelman is that the three-fold periodicity of the quadratic part of the phase term in (8.14) may be expanded in the three fundamental sequences exp(2π iks/3) [6]. Thus exp(2πik /3) = 2

1 

cs exp(2πiks/3).

(8.16)

s=−1

The orthogonality of the factors exp(2πiks/3) allows the following inversion for the coefficients. cs =

1 1  2π i(k 2 − ks) exp . 3 k=−1 3

(8.17)

Thus   c1 = c−1 = e−2πi/3 c0 = e−2πi/3 1 + 2e2πi/3 /3.

(8.18)

The important implication of (8.16)–(8.18), for times close to Trev /3, is that full wavepacket behaves locally as a combination of three partial classical packets with different time shifts,   (r, t) = c0 cl (r, t) + e−2πi/3 cl (r, t + Tcl /3) + e−2πi/3 cl (r, t − Tcl /3) , (8.19) in which the present sign choice for Trev means that the phases differ in sign from those of Averbukh and Perelman. There is an obvious similarity between the three classical components of (r, t) and the three components of the classical ensemble at t = Trev /3 in Fig. 8.2. Extension of the theory to partial revivals at rational fractions t = (p/q)Trev of the full revival period shows that the three-fold sequence is typically replaced by q sequences of length q (with the additional possibility of a sequence of length l = q/2 when q/4 is an integer) [6]. Hence (8.19) generalizes to a combination of q classical components, separated in time by multiples of Trev /q. The physical consequence, within the present quadratic phase approximation is that a succession of classical components refocuses close to the origin at intervals of Tcl /q at times of order t ≈ Trev /q.

246

Manipulating Rydberg states

1/6 1/4 1/3

1/2

|C(t)|

2

(b)

|Ccl(t)|

2

(a)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

t/Trev Figure 8.3 Squared auto-correlation functions of (a) the classical and (b) the quantum wavepacket created by a 2 ps pulse with the laser frequency tuned to excite the (n, ) = (69, 1) state of the H atom.

Parker and Stroud suggested a scheme for experimental observation of such revivals by monitoring the time variation of spontaneous emission from the wavepacket that passes through a peak whenever the wavepacket returns close to the positive ion core [5]. However, it proves more practical to employ a pump probe technique, which monitors the auto-correlation function [8]  C(t) = (0) |(t) = an2 exp (−in t) . (8.20) n

The example illustrated in Fig. 8.3 was computed by (8.20) for a wavepacket created by a 2 ps pulse from a laser tuned to excite the (n, ) = (69, 1) state of the H atom. The comb of classical recurrences at intervals of Tcl = 75 ps in panel (a) is included to allow an easy comparison with the pattern of quantum revivals in panel (b), for which Trev = 2 ns. The first three peaks show the increasing breadth of the quantum wavepacket at intervals of the classical recurrence time, while the subsequent, much sharper, peaks clearly demonstrate the quantum revival pattern. In particular, it is easy to pick out two, three and four revival peaks over the comb interval, Tcl , at times Trev /2, Trev /3, and Trev /4, respectively.3 3

Although the occurrence of these revival sequences was well predicted by Averbukh and Perelman, their expressions for the coefficients cs show the limitations of the quadratic phase approximation. In particular alternate coefficients for even q revivals are found to vanish, which means that wavepacket contains half the expected number of classical components, cl [r, t + s(p/q)Tcl ], which would double the revival interval Trev /q compared with that in Fig. 8.3. Such doubling is indeed observed if an expansion for the full phase n t is truncated after the quadratic term.

8.1 Rydberg wavepackets

247

hν

hν E0

Figure 8.4 Schematic illustration of the level scheme for pulse-probe determination of Rydberg auto-correlation functions

Figure 8.4 illustrates the essence of the experiment, which was first performed by Yeazel and Stroud [9]. Pairs of coherent picosecond pulses, with mean energy hν, are generated in a split-beam arrangement, which allows a time delay between the two arms of a Michelson interferometer. Conditions are adjusted so that the first pulse creates a wavepacket by excitation to a sufficiently dense spectral region, from a single quantum state, with energy E0 . The second pulse, which arrives with a time delay, ionizes or dissociates the molecule with a probability determined by its overlap with the evolving wavepacket. Hence the detected signal provides a direct measure of |C(t)|2 . The conditions employed by Yeazel and Stroud excited potassium atoms directly into the ionization continuum, as indicated in the upper left of the diagram. It may, however, be more interesting or more convenient, particularly in molecules, to excite to an energy below the direct ionization limit (as on the upper right hand side), in which case the signal may be detected by pulsed-field ionization [10], auto-ionization [11] or predissociation [12]. Stroboscopic revivals Partial revivals are also observed in molecules, where they are taken as an indication that the wavepacket has accessed a single Rydberg series [12]. We know, however, from the discussion in Sections 2.4 and 4.2 that the situation is complicated by interactions between series that converge on different vibrational and rotational levels of the positive ion. Particular attention has been given to the influence of the electronic–rotational stroboscopic effect that was discussed in Section 4.2.4

248

Manipulating Rydberg states

[10, 13]. When seen in the energy domain, the uncoupled case (d) coupling scheme reverts to case (b) coupling at energies such that the rotational energy spacing of the positive ion is an integral multiple, k, of the local electronic energy spacing in the lower series. Hence, in classical terms the positive ion performs k exact rotations (or half rotations in the case of a homonuclear molecule) during the period of a single electronic orbit, which means that the electron sees what looks like a stationary ion when it returns to the core. Since the wavepacket itself is actually monitored in the case (d) picture (because only members of the lower series are ionized), reversion to case (b) leads to systematic amplitude borrowing from the higher series. The resulting changes in the recurrence pattern are described next, along lines described by Smith et al. [10]. Readers are also referred to Altunata et al. for a semiclassical perspective [13]. Labels ν 0 = n0 − μ0 and ν 2 = n2 − μ2 , etc., are employed for the two series, in view of the subsequent application to NO. The connection between the energy and time domains is established by expressing the coincidence condition of (4.33) in the form Erot =

Ry Ry 2Ry (ν 0 − ν 2 ) 2Ry (k + μ) − 2  = , 2 3 νs ν 3s ν2 ν0

(8.21)

√ where ν s  ν 0 ν 2  (ν 0 + ν 2 ) is defined as the stroboscopic quantum number, k = n0 − n2 and μ = μ2 − μ0 . The rotational beat period Trot = 2πh¯ /Erot and the stroboscopic period TS = 2πh¯ ν 3s /2Ry are therefore related by TS = (k + μ)Trot .

(8.22)

Experimental work has concentrated on the np(0) and nf(2) series NO molecule, which are coupled by the positive-ion quadrupole, rather than the rotational frame transformation. They have quantum defects μp(0) = 0.7286 and μf(2) = 0.0101 and the rotational period for v + = 1 is Trot = 2.82 ps [14]. Hence the stroboscopic periods for k = 1−3 are 0.79 ps, 3.61 ps and 6.43 ps, respectively. The experimental auto-correlation measurements, for excitation via the A 2  + (v = 1, N = 0, J = 1/2) level of NO in Fig. 8.5, were obtained by tuning the excitation frequencies over the n0 = 25–37 members of the np(0) series, using a pulse width of approximately 0.7 ps [10]. The stroboscopic and rotational periods are marked in the diagram by solid and dashed lines, while the curved lines follow the classical periods of hydrogenic series terminating on the N + = 0 and N + = 2 rotational limits. The filled points mark strong ‘classical first recurrence peaks’ in the auto-correlation spectrum, in the sense that they broadly follow the upper curved line. The open circles correspond to weaker peaks at earlier times, which demonstrate a clear trend in the overview provided by Fig. 2 of Smith et al., whereby the wavepacket clings to the stroboscopic time period TS , rather than the classical hydrogenic period tcl (N + = 0) [10]. Smith et al. also demonstrate that modifying

8.1 Rydberg wavepackets

249

10 tcl (N+ = 0)

9 8

Time / ps

7 6

TS (k = 3) tcl (N+ = 2)

5 4 TS (k = 2) 3 2 TS (k = 1)

1 0

20

25 30 35 Principal quantum number n0

40

Figure 8.5 Variation of recurrence periods in the experimental auto-correlation function of NO, plotted against the principal quantum number at the centre of the exciting pulse. Filled circles mark the most intense peaks; open ones indicate less intense but still prominent peaks, at lower frequencies. Solid horizontal lines indicate the stroboscopic periods; dashed ones are the rotational periods. The two curved lines indicate the lassical periods of unperturbed hydrogenic series. Taken from Smith et al. [10], with permission.

(8.11), to include additional wavepacket components from the higher series, leads to similar stroboscopic plateaus in the analogue of Fig. 8.5. Moreover the positions of the plateaus were shown to vary with the quantum defect according to (8.22). 8.1.2 Coherent control The relatively new field of coherent control relies on the influence of optical phase relations in determining the outcome of physical and photochemical processes [15, 16]. Illustrations based on the use of Rydberg wavepackets have been reviewed by Fielding [1]. The element of control is obtained by modifying the scheme in Fig. 8.4 by replacing the initial excitation pulse by a pair of time- and phaserelated pulses, by the so-called optical Ramsey method [17]. The two arms of the Michelson interferometer employed for the Alber pump-probe experiment are extended to three, the first two of which are designed to create a composite excitation pulse, the second component of which arrives with a variable time delay, td , and phase difference φ. The third arm contains the detection pulse.

250

Manipulating Rydberg states

By analogy with (8.3)–(8.6), the composite laser excitation field E(t) = ezE0 [f (t) sin ωL t + f (t − td ) sin(ωL (t − td )]

(8.23)

generates wavepacket coefficients of the form [17]  n 3/2    0 an0 exp −α 2 2n 1 + ein td eiωL td cn (td ) = −i n   = an (∞) 1 + ein td eiωL td ,

(8.24)

where an (∞) is given by (8.5). Notice that the final term oscillates rapidly with the delay time td – at the frequency of the exiting laser, ωL . The auto-correlation function with the third pulse, which creates the detection signal, is therefore given by  C(t) = |cn (td )|2 e−in t , (8.25) n

where |cn (td )|2 = 2an2 (∞)[1 + cos(n + ωL )td ],

(8.26)

which oscillates with τ d , between zero and four, at the very rapid laser frequency, ωL . The influence of the dephasing terms n terms becomes apparent, however, in what is called the total Rydberg population  A(td ) = |cn (td )|2 , (8.27) n

which is plotted in Fig. 8.6 for excitation in the vicinity of n = 40 (Tcl = 10 ps) in the H atom by a pair of identical 2 ps pulses, at a notional laser frequency, which has been reduced by a factor of 200 for display purposes. It can be seen that the population now oscillates at the laser frequency within an envelope that follows the recurrence and spreading of the wavepacket. To see how such twin-pulse excitations can be used for control purposes, consider the excitation of two independent wavepackets, within say the np(0) and nf(2) series of NO that were discussed in Section 8.1.1. The total wavepacket will evolve as       |i (r, t, td ) = ani φ ni (r) 1 + eini td eiωL td exp −ini t . (8.28) i

ni

Each series will have its own Rydberg population of the form in Fig. 8.6, but with a relative phase that depends on the level structure within the laser bandwidth. In particular, Minns et al. find a marked variation between excitation at a frequency corresponding to one of the stroboscopic plateaus in Fig. 8.5, compared with excitation between the plateaus [12]. Thus excitation around n0 = 30 of the np(0)

8.1 Rydberg wavepackets

251

4

A(td)

3 2 1 0

0

0.5

1

1.5

2

2.5

3

td / Tcl Figure 8.6 The total Rydberg population A(td ) given by (8.27) arising from excitation of an H atom around n = 40 (Tcl = 10 ps) by a pair of identical ps pulses with bandwidths σ = 10 cm−1 and time delay td .

series, which is in near coincidence with n2 = 28 of the nf(2) series leads to inphase rapid Rabi oscillations in the two total Rydberg populations. On the other hand, the Rabi oscillations of the two total populations are found to be out of phase when the laser is tuned to the n0 = 25 level of the p(0) series, which lies midway between n2 = 23 and 25 of the f(2) series. In addition of course the envelope of the nf(2) population in the analogue of Fig. 8.6 oscillates slightly more rapidly than that of the np(0) population, because the quantum numbers are systematically lower. Finally it is clear from the rapidity of the Rabi oscillations that a very small change in the time delay in the vicinity of the first recurrence could cause one or other of the two total populations to fall from a maximum value close to four to a minimum value close to zero. This is the basis of the control scheme in a second paper by Minns et al. [18]. Figure 8.7 shows a comparison between (a) experimental and (b) calculated recurrence spectra obtained by exciting the two series of NO at around n0 = 33, which leads to out-of-phase excitation, because n0 = 33 lies between n2 = 30 and 31. The experimental signal was determined by the pulse-field ionization arising from overlap with a third, detector, pulse. The calculated one is |C(t)|2 , as given by (8.25). The lowest plot in the two parts of the diagram, which was obtained by using a single excitation pulse, shows a broad double-peaked structure over the recurrence period 4–6 ps. The middle traces are obtained using a pulse pair with a delay designed to eliminate the f(2) series, with a recurrence peak at 5.6 ps, appropriate to n0 = 33, while the upper trace, which was obtained by eliminating the p(0) series shows a peak at Tcl = 4.6 ps, appropriate to n2 = 31.2. Since the essence of this scheme lies in the rotational energy separation, responsible for the smaller Kepler period in the higher energy series, it is appropriately claimed as method for rotational angular momentum control [18].

252

Manipulating Rydberg states (b)

Intensity (arb. units)

(a)

0

2

4

0 6 2 Time (ps)

4

6

8

Figure 8.7 (a) Experimental and (b) calculated recurrence spectra derived from wavepacket in NO, which was excited at n¯ 0 = 33 with a laser bandwidth of 14 cm−1 . The bottom trace in each panel was derived from a single pulse. The middle and upper traces were obtained by double-pulse excitation with time delays designed to excite wavepackets restricted to the np(0) and nf(2) series, respectively. Taken from Minns et al. [18], with permission.

8.1.3 Time-dependent photo-fragmentation The following section turns from wavepacket recurrences and revivals to an MQDT-based time-dependent theory of auto-ionization along the lines laid out by Texier and Jungen [19]. Minor changes in notation are used to conform with Appendix D.2.1. Thus the auto-ionizing wavepacket in channel |i is given by the integral  ! ' (i−) (8.29)  (r, t) = f (t, E)  (i−) r 0  (i−) (r, E)e−iEt dE, where  (i−) (r, E) may be expressed in terms of the eigenchannel solutions  (ρ) (r, E) of the usual time-independent MQDT system oo

o

K − tan πτ (E)I oo Z (E) K oc = 0, (8.30) K co K cc + tan β(E) Z c (E) which are characterized by N o eigenphases πτ ρ (E) and an orthogonal N o × N o matrix T (E), whose columns are related to the open-channel amplitudes by (o) Ziρ (E) = cos πτ ρ Tiρ (E), while the closed-channel amplitudes are related to the open ones by (2.44). The connection between  (i−) and  (ρ) is given, according

8.1 Rydberg wavepackets

253

to (D.21), by N  o

r

(i−)

(r, E) =

r (ρ) (r, E)e−iπτ ρ (E) TρiT (E)e−iδi ,

(8.31)

ρ=1

and the corresponding expression for the matrix element is !  ' (ρ) ! ' (i−)  (E) r 0 eiπ τ ρ (E) TρiT (E)eiδi  (E) r 0 =

(8.32)

ρ

r

(ρ)

(E) =

N 

(ρ)

j ψ j (E),

ρ = 1 . . . N o.

(8.33)

j =1

Finally, the temporal evolution in any open channel |j  may be followed by monitoring the projection  ! ! ' ' (i−) (8.34) j  (t) = f (t, E)  (i−) r 0 ψ j(i−) (r, E)e−iEt dE, where according to (D.17) ψ j(i−) (E) ∼



1 eiki r δ ij − e−iki r Sij(−) (E) , √ 2i πki

(8.35)

in which Sij(−) (E) is given by (D.18). In particular the auto-ionizing radial flux is given by   ' !d' !∗ j (i−) . (8.36) F = Im j (i−) dr Texier and Jungen assumed the creation of a wavepacket by excitation from the v

= 0, N

= 1 level of the ground X 1 g+ electronic state of H2 to the Beutler– Fano region between the N + = 0 and N + = 2 limits of the np series converging on the v + = 2 state of the positive ion. The frame transformation elements are given by (4.22) and the quantum defects were taken as μpσ = 0.191 and μpπ = −0.078 [19]. Equation (8.36) was used to calculate auto-ionization fluxes in the single open channel, assuming Gaussian laser pulses with a duration of 5 ps (FWHM = 12.3 cm−1 ) centred on the auto-ionization resonances at n∗ = n − μ = 33, 50 and 60. Figure 8.8 shows the temporal variation of the auto-ionization flux at r = 104 au, generated by these three pulses. Each excitation pulse is followed by a sequence of ionization peaks, which decrease in intensity as the wavepacket drains away. Moreover the intervals between these peaks, of 5.5, 19.6 and 33.6 ps, in the three panels are in excellent agreement with the classical recurrence times, Tcl = 1.52 ∗ 10−4 n∗3 ps, in the closed channel. In addition, the exponential decay times of 9, 32 and 36 ps correlate well with the values 8, 29 and 50 ps inferred

254

Manipulating Rydberg states

Figure 8.8 Time variation of the ionization flux, monitored at r = 2 × 104 au, for laser pulses centred at n∗ = n − μ  33, 50 and 60 in the auto-ionizing series converging on the v + = 2, N + = 2 level of H+ 2 . Taken from Texier and Jungen [19], with permission.

from the widths of jumps in τ (E) auto-ionization lifetimes, bearing in mind the ill-defined nature of the third peak in the final panel of Fig. 8.8. A subsequent paper extended this treatment to the more demanding case of competitive auto-ionization and predissociation from a ‘complex resonance’ in H2 [20]. The interaction involves coupling between auto-ionizing np states in the Beutler–Fano region between the (v + , N + ) = (0, 0) and (0, 2) of H+ 2 and two very

8.2 The Stark effect

255

close-lying Born–Oppenheimer levels, 5pπ (v = 2) and 7pπ (v = 1), which predissociate to the H(1s) + H(n = 2) dissociation threshold. The energy-independent components of the wavepacket were constructed by the eigenchannel R-matrix method of Section 5.4, and fluctuations in the temporal evolution of the fragmentation fluxes were interpreted in terms of the classical Kepler times, Tcl , for the three types of state and the quantum beat periods, τ ij = h¯ /[2π (Ei − Ej )], given by the relevant energy separations. 8.2 The Stark effect The Stark effect, which describes the response of atoms and molecules to a uniform electric field, is particularly relevant to the field ionization of such species, either for detection purposes or for high resolution pulsed field zero kinetic energy (ZEKE-PFI) spectroscopy [2, 3]. The theory is of particular interest in showing how the super-symmetry of the hydrogen atom, lies behind the separability of the Stark Hamiltonian in parabolic coordinates. The first section below, which outlines this symmetry, is followed by a description of the parabolic states of the hydrogen atom. Consequences of the symmetry in relation to the differences between the H atom Stark spectrum and the spectra of other species are discussed in the following sections. Finally, it is shown how the symmetry can be built into an MQDT-based theory of the Stark effect for general species. The discussion is given throughout in atomic units,4 for ease of comparison with the literature. 8.2.1 Symmetry considerations The super-symmetry of the H atom is associated in classical theory with the restriction to motion on a fixed ellipse. The additional constants of the motion are the components of the Runge–Lenz vector

r 1 r − (r · p)p, (8.37) A = p × L − = p2 − r r which lies in the direction of the principal axis, with a magnitude equal to the product of the energy and the separation between the two foci of the ellipse.5 It is 4 5

The atomic unit of electric field is equivalent to 5.1422 × 109 Vcm−1 . Since dr/dt = p and dp/dt = −r/r 3 it is easy to verify that r × (r × p) d r × (r × dr/dt) (p × L) = − =− dt r3 r3  dr/dt d r (dr/dt)r , = = − 2 r r dt r

256

Manipulating Rydberg states

also readily shown that A · L = (p × L) · L − r −1 (r · L) = p · (L × L) − r −1 (r × r) · p = 0,

(8.38)

which confirms that A lies in the plane perpendicular to L. These conclusions are equally applicable in quantum mechanics, provided that ˆ is properly symmetrized, the operator A   ˆ = 1 pˆ × L ˆ −L ˆ × Pˆ − r . A (8.39) 2 r Full accounts of the resulting symmetry are given by Hughes and Wybourne [21, 22]. In brief, it can be shown that   ˆ 2 + 1) + 1 , ˆ 2 = − 1 2H (L (8.40) A 2Hˆ and that the scaled vector B = (−2H )−1/2 A and the angular momentum L satisfy the SO(4) commutation relations [Li , Lj ] = i ij k Lk [Li , Bj ] = i ij k Bk [Bi , Bj ] = i ij k Lk .

(8.41)

Consequently the two composite vectors 1 1 J1 = (L + B), J2 = (L − B) 2 2 each obey the normal angular momentum rules

(8.42)

[J1i , J1j ] = i ij k J1k [J2i , J2j ] = i ij k J2k [J1i , J2j ] = 0, which confirms that dA/dt = 0. To find the direction and magnitude it is sufficient to consider the apogee (r = rmin ) and perigee (r = rmax ) on the principal axis, rp , at which (r · p) vanishes. Thus



1 1 A = p2 − rp = 2H + rp , r r which fixes the direction. Finally, in the absence of radial motion H =

L2 1 L2 1 − = 2 − , 2 2rmax rmax 2rmin rmin

which rearranges to H (rmax + rmin ) = −1, so that



1 [H (rmax − rmin )] rmax A = 2H + . rmax = rmax rmax

8.2 The Stark effect

257

which lead to total angular momenta Ji2 = ji (j1 + 1), with ji = 1/2, 1, 3/2, etc., for i = 1, 2. There are also two quadratic Casimir operators that commute with the Hamiltonian 1 G = J12 − J22 = L · B = 0. (8.43) F = J12 + J22 = (L2 + B 2 ), 2 The second requires that j1 = j2 = j , while the first may be combined with (8.40) to show that 1 (8.44) H = − 2, 2n where n = 2j + 1. The resulting states |j k1 k2 m satisfy Jˆ12 |j k1 k2 m = J22 |j k1 k2 m = j (j + 1) |j k1 k2 m Jˆ1z |j k1 k2 m = k1 |j k1 k2 m Jˆ2z |j k1 k2 m = k2 |j k1 k2 m Lˆ z |j k1 k2 m = (J1z + J2z ) |j k1 k2 m = (k1 + k2 ) |j k1 k2 m Bˆ z |j k1 k2 m = (J1z − J2z ) |j k1 k2 m = (k1 − k2 ) |j k1 k2 m,

(8.45)

where k1 and k2 may be chosen such that k1  −j and k2  j . The connection with the parabolic states |n, n1 , n2 , m described in the following section depends on the substitutions [23, 24] 1 k1 = (m + n1 − n2 ), 2

1 k2 = (m − n1 + n2 ), 2

(8.46)

so that (8.47) Lˆ z |n, n1 , n2 , m = m |n, n1 , n2 , m Aˆ z |n, n1 , n2 , m = nBˆ z |n, n1 , n2 , m = n(n1 − n2 ) |n, n1 , n2 , m. This shows that the parabolic states |n, n1 , n2 , m are simultaneous eigenstates of Lˆ z and Aˆ z but no longer of Lˆ 2 . Their connection with the angular momentum states follows however from the identity L = J 1 + J1 .

(8.48)

Consequently, the familiar |nm states may be expressed as vector-coupled combinations of |j1 , k1  and |j2 , k2 , subject to the restriction j1 = j2 = j = (n − 1)/2;  |j, k1 , j, k2  j, k1 , j, k2 |, m |n, , m = n1 n2

|n, n1 , n2 , m = |j, k1 , j, k2  =

 m

|n, , m , m |j, k1 , j, k2 ,

(8.49)

258

Manipulating Rydberg states  Table 8.1 Reduced rotation matrices dm0 (θ ) for  = 0−2.



m=0

m=1

m=2

0 1 2

1 cos θ (3 cos2 θ − 1)/2

– √ −√1/2 sin θ − 3/2 sin θ cos θ

– √ – 2 3/8 sin θ

where j = (n − 1)/2 and the quantum numbers ki are given by (8.46). Written out explicitly as a 3j symbol, this means that n, n1 , n2 , m |m √  (n − 1)/2 = (−1)m 2 + 1 m −(m + n1 − n2 )/2

(n − 1)/2 −(m − n1 + n2 )/2)

(8.50)

.

Further insight is obtained by employing the asymptotic approximation of Brussard and Tolhoek to the 3j symbol [25]; " n2 (2 + 1)  dm0 (θ ) n, n1 , n2 , m |m ∼ (−1) n n 1 − n2 , (8.51) cos θ = n  where dm0 (θ ) is a reduced Wigner rotation matrix element [26]. An equivalent form in terms of the normalized associated Legendre polynomial Pm (cos θ ) is also  available [2, 27]. One sees from the expressions for dm0 (θ ) for  = 0–2 in Table 8.1 that the s ( = 0) angular momentum state is uniformly distributed over all the parabolic states. In addition, the extreme parabolic states with n1 − n2 = ±n have the highest p and d character for m = 0, while the coefficients of the |11 and |22 angular momentum components vanish for at the limits n1 − n2 = ±n and take their largest values when n1 = n2 . One should also note that , m |j, k1 , j, k2  = 1 for  = n − 1, j = (n − 1)/2 and k1 = k2 = m/2. Consequently, there is a unique pair of so-called circular states with m = ± = ±(n − 1) and n1 = n2 for which |n, n1 , n2 , m = |n, , m. They correspond in classical terms to states of clockwise or anticlockwise circular motion around the z axis.

8.2.2 Parabolic states of the H atom The term ‘parabolic state’ stems from the separability of the Schr¨odinger equation in parabolic coordinates (ξ , η, φ), which are related to the Cartesian variables by   (8.52) x = ξ η cos φ, y = ξ η sin φ, z = (ξ − η)/2,

8.2 The Stark effect

259

with the Wronskian ∂(ξ , η, φ)/∂(x, y, z) = (ξ + η)/4.

(8.53)

It is shown by Landau and Lifshitz that the Hamiltonian is given by [28] 



 ∂ ∂ ∂ ∂ 1 ∂2 2 1 ξ + η − Hˆ = − + . 2 ξ + η ∂ξ ξ ∂η ∂η 2ξ η ∂φ ξ +η

(8.54)

The assumption of a separable wavefunction ψ = (2π)−1/2 f1 (ξ )f2 (η)eimφ therefore leads to



d d 2 ξ + Eξ − dξ dξ 

d d 2 η + Eη − dη dη

(8.55)

 m2 + Z1 f1 (ξ ) = 0 2ξ  m2 + Z2 f2 (η) = 0, 2η

(8.56)

where Z1 + Z2 = 1. The substitutions E=−

1 , 2ν 2

ξ = νρ 1 ,

η = νρ 2

(8.57)

then allow solutions of the confluent hypergeometric form6 |m|/2

fν i m (ρ i ) = Ni e−ρ ii /2 ρ i Ni = (ν

1/2

−1

m!)



F (−ν i , |m| + 1, ρ i ),

2(νZi + 1/2 + m/2) (νZi + 1/2 − m/2)

1/2 ,

(8.58)

which has properly normalizable solutions provided that the indices 1 ν i = ni = − (|m + 1) + νZi 2

(8.59)

are integers. Bearing in mind that Z1 + Z2 = 1, this means that ν = n = n1 + n2 + |m| + 1. Hence the states may be labelled by n and the difference (n1 − n2 ), which is the eigenvalue of Bˆ z in (8.47), with values between −n + |m| + 1 to n − |m| − 1 in 6

An alternative generalized Laguerre polynomial solution for integer ni is given by [29] F (−n, α + 1, ρ) = M(−n, α + 1, ρ) = where (α + 1)n = (α + 1)(α + 2) · · · (α + n).

n! L(α) (ρ), (α + 1)n n

260

Manipulating Rydberg states

steps of two. The restriction to a fixed principal quantum number means that (n1 − n2 ) fixes the relative extension of the wavefunctions in the ξ and η coordinates. The mean value of z = (ξ − η)/2 is therefore positive or negative according to the sign of (n1 − n2 ). 8.2.3 Stark states of the H atom Moving on to the Stark effect, the introduction of an electric field F modifies the potential to the form 1 V = − + Fz, r

(8.60)

which splits the n2 degeneracy. The analogues of (8.56) are still separable

  d 1 m2 1 2 d ξ + Eξ − − Fξ f1 (ξ ) = −Z1 f1 (ξ ) dξ dξ 2 4ξ 4 

 d d 1 m2 1 2 η + Eη − + Fη f2 (η) = −Z2 f2 (η), (8.61) dη dη 2 4η 4 and the influence of the field may be assessed by perturbation theory. The first-order effect is readily obtained by computing the mean value of z in the unperturbed product state f1(0) (ξ )f2(0) (η) [28]. The quadratic term is more difficult because second-order corrections to the energy involve ‘infinite sums of a complicated form’. Instead Landau and Lifshitz proceed by deriving first- and second-order corrections energy param√ to the separation constants in terms of F and an assumed 2 eter ε = −2E, using the fact that the matrix elements of ξ and η2 are subject to the selection rule ni = 0, 1, 2. An iterative solution to the resulting equations, subject to Z1 + Z2 = 1, leads to [28]  1 3F F2 4  2 2 2 n(n n 17n + − n ) − − 3(n − n ) − 9m + 19 . 1 2 1 2 2n2 2 16 (8.62) Levels with m = 0 are doubly degenerate. The coefficient of the linear term in (8.62) determines the dipole moment (in units of ea0 ) of the parabolic state |n, n1 , n2 , m, which is seen to be proportional to (n1 − n2 ) and independent of the magnetic quantum number m. Similarly the coefficient of the quadratic term determines the polarizability, which is negative in all states and strongly m dependent. It is largest in magnitude for m = 0. It is also seen that the extreme states for a given m, with (n1 − n2 ) = ±(n − |m| − 1), are the least polarizable, and that this smallest polarizability decreases with increasing m. At the same time the range of (n1 − n2 ) decreases, so that the least polarizable E n1 n2 m  −

8.2 The Stark effect

261

n = 12 n = 11 n = 10

0

1

0.5

1.5

2

5

Electric field / 10 au Figure 8.9 Stark eigenvalues for the hydrogen atom as a function of the field strength for n = 10–12 and m = 1. The heavy dashed line follows the energy of the saddle point Vs as the field increases. Heavy segments of the eigenvalue plots indicate field broadening greater than 10−3 times the local eigenvalue spacing.

of all the states are the circular states with n1 = n2 and |m| = (n − 1), which also have zero dipole moment. Figure 8.9 presents a plot of the eigenvalues given by (8.62) as functions of field strength for m = 1 and n = 10–12. As a matter of nomenclature, states whose energies increase or decrease with increasing field are often termed ‘blue’ or ‘red’, respectively. The most important aspect of the diagram is that the highest (blue) eigenstates for principal quantum number n cross the lowest (red) states from the field-free state with n + 1, because they correspond to different separation constants in (8.61). Thus Z1 − Z2 = (n1 − n2 )/n is an additional constant of the motion, and a crossing between the blue n state and the red one for n + 1 implies a change in (n1 − n2 ) of order 2n. Such crossings may be verified to occur at field strengths F

1 , 3n5

(8.63)

which is known as the Inglis–Teller limit, a term that takes on more importance for non-hydrogenic species, for which there is no additional constant of the motion (see Section 8.2.4) Additional information in Fig. 8.9 relates to field ionization at energies above the saddle point of the potential in (8.60), which occurs on the negative z axis (see Fig. 8.10(a)) " √ 1 (8.64) and V = Vs = −2 F. zs = − F

262

Manipulating Rydberg states

Vs

V(z) (a) zs

z

V1 (−ξ) V2 (η) E −b −a 1 1

a2

V2 (η m )

c2

b2

(b) ηm

−ξ

η

Figure 8.10 (a) A section of the field modified potential function at x = y = 0. (b) The Langer modified potentials V1 (ξ ) and V2 (η), respectively. The direction of the ξ axis has been reversed for display purposes and the potentials have been displaced outwards.

The heavy dashed line in Fig. 8.9 follows this latter equation, which is equivalent in laboratory units to  ˜ V˜s = −0.6 F, (8.65) where V˜s and F˜ are measured in cm−1 and Vcm−1 , respectively. To estimate the onset of field ionization in terms of the quantum number n, one might crudely equate Vs with the field free eigenvalue, −1/2n2 , to obtain 1 . (8.66) 16n4 However, the following analysis shows that the motion is divided between the ξ and η degrees of freedom, only the second of which is subject to field ionization in the H atom. Different members of a given Stark manifold therefore ionize at quite different fields, as indicated by the heavy marking at the end of each line. A better estimate of the onset of ionization is therefore obtained by equating Vs to the lowest eigenvalue of the manifold, which is predominantly excited in the η mode. Solutions of the quadratic equation derived from F

Vs = −

1 3 − Fn(n1 − n2 ) 2 2n 2

(8.67)

for (n1 − n2 )  −n yield F

1 . 9n4

(8.68)

8.2 The Stark effect

263

Closer analysis of the state-specific field ionization is facilitated by the substitutions f1 (ξ ) = ξ −1/2 χ 1 (ξ ), and similarly for f2 (η), which cast (8.61) into the forms   d2 χ 1 1 m 2 − 1 Z1 F + E− + + ξ χ1 = 0 2 ξ 4 dξ 2 4ξ 2   2 2 d χ2 1 m − 1 Z2 F + E− + (8.69) − η χ 2 = 0, dη2 2 4η2 η 4 which are convenient for a JWKB treatment [27, 30, 31]. We follow the third of these references in recognizing that the singularity at the origin is most accurately taken into account by use of a Langer correction, which modifies the centrifugal potentials by terms ξ −2 /4 or η−2 /4 [7, 32] For example the quantization condition for the field free ξ equation at energy E = −1/2n2 takes the form 1/2  b1 "  m2 1 −1 2Z1 − 2 (n1 + 1/2)π = + dξ 2 2n2 ξ 2ξ a1  b1  2 −ξ + 4n2 Z1 ξ − m2 1 = dξ 2n a1 ξ

|m| π, (8.70) = nZ1 − 2 which rearranges to 1 n1 + (|m| + 1) = nZ1 , 2

(8.71)

in exact agreement with the strict quantum mechanical result in (8.59). The Langer modified potential functions for the JWKB analysis will therefore be taken as V1 (ξ ) = −

2Z1 Fξ m2 + 2+ ξ 2 2ξ

V2 (η) = −

m2 2Z2 Fη + 2− η 2η 2

(8.72)

where the separation constants control the depths of the two potentials. These effective potentials can take a variety of forms according to the values of Z1 , Z2 and the field strength F [30, 31]. Those relevant to the splitting pattern in Fig. 8.9 are shown in Fig. 8.10 (b), in which the separation constants Z1 and Z2 control the depths of the two potentials. Their minima lie at −2Z12 /m2 and −2Z22 /m2 in the field-free limit. The Stark eigenvalues at a given field strength are

264

Manipulating Rydberg states

derived from the JWKB quantization equations  b1 "  b1 1 β 1 (E, Z1 ) = (n1 + 1/2)π = p1 (ξ )dξ [E − V1 (Z1 , F; ξ )]dξ = 2 a1 a1  b2 " 1 1 β 2 (E, Z2 ) = (n2 + 1/2)π = [E − V2 (Z2 , F; η)]dη + φ(ε) 2 2 a2  b2 1 p2 (η)dη + φ(ε), = (8.73) 2 a2 subject to Z1 + Z2 , in which φ(ε) is a small phase correction arising from the presence of the potential barrier to the η motion. It is given by [7] 1 φ(ε) = arg ( + iε) − ε ln |ε| + ε, 2 where ε is related to the tunnelling integral by  c2 " 1 [V2 (Z2 , F; η) − E]dη. πε = 2 b2

(8.74)

(8.75)

in which b2 and c2 are the outer turning points in Fig. 8.10(b). In addition the resonance line-width arising from this tunnelling may be approximated by [7]7 =

h¯ ω¯ 2 exp[−2π ε], 2π

(8.76)

where h¯ ω¯ 2 = (∂E/∂n2 )Z2 E is the local level spacing in the potential V2 (η). Harmin [31] and Sakimoto [32] denote the second of the above phase integrals as , but the notation β 2 (E, Z2 ) is preferred to emphasize the connection with the accumulated phase of Chapter 2, in preparation for the generalized MQDT treatment in Section 8.2.5. It is also useful, in the latter context, to recognize that the unitnormalized bound state JWKB wavefunction f1 (ξ ) is related to its energy normalized counterpart by a factor C, 1/2    ξ 2 π , (8.77) sin p1 (ξ )dξ + f1 (ξ ) = C πp1 (ξ ) 4 a1 which is given according to Sakimoto by [33] −1/2   b1 dξ 1 C= . π a1 ξ p1 (ξ ) 7

(8.78)

More accurate uniform expressions for  at energies very close to the barrier maximum may be found in (3.119) and (3.120) of [7].

8.2 The Stark effect

265

Analytical expressions for the above phase integrals are given in terms of complete elliptic integrals in table I of Harmin [31].8 The quantization procedure involves solving (8.73) for the energy E and the separation constants Zi , subject to the constraint Z1 + Z2 .9 The field broadening is given by (8.76). As an alternative, at least for illustrative purposes, it may be assumed that E, Z1 and Z2 are adequately represented by the second-order perturbation approximations in Landau and Lifshitz [28]. We also adopt a relatively crude approximation for the tunnelling integral. In view of the wide separation between the maximum and minimum of V2 (η) in Fig. 8.10(c), ηm is estimated by approximating the centrifugal term as a constant. Thus "  Z2 m2 (8.79) ηm  2 and V2 max = V2 (ηm )  −2 Z2 F + 2 . F 2ηm In addition, the tunnelling integral is derived within the quadratic approximation 1 V2 (η)  V2 max + Vm

(η − ηm )2 , 2 where Vm

= −F /ηm . Thus  c2 " 1 π(V2 max − E)  . [V2 (Z2 , F; η) − E]dη  2 4|Vm

| b2

(8.80)

(8.81)

The heavy final segments of the curves in Fig. 8.9 are drawn over the range for which (8.76) and (8.81) yield /¯hω¯ 2 > 10−3 . More accurate JWKB-based treatments are given by Luc-Koenig and Bachelier and Harmin [30, 31]. It is evident from the diagram that the field broadening is most easily induced in the lowest members of each Stark manifold, which are the most excited states of the deepest V2 (η) potentials. Hence the line-widths can vary markedly between broad low-lying (red) levels of an upper manifold and sharp adjacent blue levels from a lower one. 8.2.4 Non-hydrogenic species The presence of a non-Coulombic core alters the picture in two important ways. In the first place, the field-free eigenvalues for low  states break away from the relevant n manifold, under the influence of the quantum defects. Secondly, 8 9

Note that Harmin orders the turning points in the reverse order (a > b > c) to those in Fig. 8.10. A Newton–Raphson iteration may be based on the equations    (r) (r+1) (n)  ∂(β 1 , β 2 ) −1 I1 − (n1 + 1/2)π E E , = − (r+1) (r) (r) Z1 Z1 ∂(E, Z1 ) I2 − (n2 + 1/2)π bearing in mind that (∂β 2 /∂Z1 ) = −(∂β 2 /∂Z2 ).

266

Manipulating Rydberg states Saddle point

16s

15p 14d

15s Inglis−Teller

0

0.5

1

1.5

2

6

Electric field / 10 au

Figure 8.11 Stark splitting near the m = 0 and n  14 levels of the Na atom. The dashed and dot-dashed lines indicate the saddle point and Inglis–Teller limits, given by (8.66) and (8.63), respectively.

the magnitudes of the Stark matrix elements are modified by the contraction of the radial wavefunctions. The Runge–Lenz vector is no longer conserved, which makes it appropriate, at least for relatively low n values, to work in the familiar (r, θ, φ) representation. Although the behaviour of the wavefunction is unknown within the core, except by ab-initio methods, the necessary matrix elements may be computed by the Coulomb approximation described in Appendix D.3, because the dominant contribution to the radial integral comes from well outside the core. The typical pattern of Stark modified eigenvalues is illustrated in Fig. 8.11, which is drawn for levels around n = 14 and m = 0, using the quantum defects for the sodium atom. A similar diagram would apply to any non-hydrogenic species, provided that the electron is sufficiently excited to be uncoupled from the nuclear frame. Each molecular series, converging on a particular vibrational–rotational state of the positive ion has its own Stark splitting pattern, apart from perturbations from neighbouring series, which may in principle be handled by the MQDT theory of the Stark effect that is described later. We should also note that the drastic differences between Figs 8.9 and 8.11 apply only to the lowest m values because there are no significant quantum defect corrections for   m  3. The most significant features of Fig. 8.11 are the breaking away of the s and p states, which therefore have quadratic rather than linear Stark effects. Secondly, and more importantly, there are strong interactions between the adjacent Stark manifolds, which means that the principal quantum number n is no longer defined at field strengths beyond the Inglis–Teller limit in (8.63), which is marked by the dot-dashed curve. The resultant state mixing means that there is no longer wide

8.2 The Stark effect

267

diversity of line-widths. Each state may be assumed to ionize at a significant rate at the field strength that takes it to the heavy dashed saddle point line, which is given by (8.64). For most levels the resulting field ionization occurs via one of the adiabatic curves, such as the heavy line in the diagram. However, levels arising from the s and p field-free states are seen to encounter multiple avoided curve-crossings. The probability of a non-adiabatic transition to the interacting curve depends on the rate of change, dF/dt, of the field strength, according to the Landau–Zener formula [7]  2  −2πV12 , (8.82) P = exp (dV /dt) where V12 is the interaction term responsible for the splitting (equal to half the minimum separation) and (dV /dt) = (dV /dF)(dF/dt), in which (dV /dF) is the difference in slopes of the diabatic (deperturbed) curves responsible for the crossing. As drawn in Fig. 8.11, the ionizing field at 1.75 × 10−6 au corresponds to 9000 Vcm−1 because the effective ionization limit must be reduced by Ry /142 = 600 cm−1 . The technique of pulsed-field zero kinetic energy (ZEKE–PFI) spectroscopy depends on the fact that the minimum ionization field strength diminishes markedly as the quantum number increases. For example states excited to n = 250, which are bound by only ∼ 1.75 cm−1 , are ionized according to (8.65) by a field of only a few volts. Moreover the technique is capable of high spectral resolution even for molecules, with multiple Rydberg series converging on different closelying ionization limits [3]. Barring chance coincidences, such as those discussed in Section 6.8.1, each series √ gives rise to its own +Stark manifold, with a saddle point at E = Ev+ N + − 2 F. For example, the N = 2 ionization limit for N2 lies over 10 cm−1 above the N + = 0 limit, which means that the ionizing field for the N + = 0 series at n = 250 is too low to ionize the upper series by a factor of (12/2)2 = 36. Other aspects of PFI-ZEKE spectroscopy are discussed in Section 6.8. 8.2.5 Generalized MQDT theory The following generalized MQDT theory of the Stark effect is taken from Fano, Harmin and Sakimoto [31, 33, 34]. It recognizes the dominance of non-Coulombic core interactions at small radial distances and of perturbations from the field at long range. There is, however, a significant separation between these two regimes over which purely Coulomb behaviour applies, because the strength of the Coulomb −1 interaction √ is an order of magnitude larger than that of the field term, r > 10Fr, for r < 10F, which implies r < 70 a0 at the strongest field, F = 2 × 10−5 au

268

Manipulating Rydberg states

in Fig. 8.9. Hence it is valid to employ a Coulomb-based frame transformation between the angular momentum and parabolic representations to link the two limiting regimes [34]. The MQDT boundary conditions at arbitrary energies are based on the JWKB phase integrals given by (8.73), with the complication that these vary with both the energy and one of the two separation constants [31, 33]. At any given energy 1 (8.83) E=− 2 2ν the condition β 1 (E, Z1 ) = (n1 + 1/2)π fixes allowed combinations of Z1 , Z2 = 1 − Z2 and, by analogy with (8.59), 1 1 ν 1 = νZ1 − (|m + 1) and ν 1 = νZ1 − (|m + 1). (8.84) 2 2 Moreover the asymptotic behaviour of the regular and irregular JWKB wavefunctions f2 (η; E, Z2 ) and g2 (η; E, Z2 ) provide the analogues of (2.13). Thus, at distances in the range b2 < η < c2 [33]10   f2 (η; E)  cos β 2 (E, Z2 )F (+) (η, E) + sin β 2 (E, Z2 )F (−) (η; E) (8.85)   g2 (η; E)  − sin β 2 (E, Z2 )F (+) (η, E) + cos β 2 (E, Z2 )F (−) (η; E) , where F (±) (η, E) are exponentially increasing and decreasing functions. It remains to establish the frame transformation elements between the states |νm and |ν 1 ν 2 m in the angular momentum and parabolic representations, respectively, bearing in mind that the transformation element for integer n, n1 and n2 is given by the Clebsch–Gordan coefficient in (8.49) n1 n2 m |nm = j k1 j k2 |m ,

(8.86)

in which j = (n − 1)/2, k1 = 12 (m + n1 − n2 ) and k2 = 12 (m − n1 + n2 ). The extension to non-integer ν 1 and ν 2 involves energy normalized solutions of (8.61) which are given, for m  0, at moderate range, r  F−1/2 , by ν 1 ν 2 m (ξ , η, φ) = Nν 1 ν 2 m (ξ η)m/2 e−(ξ +η)/2ν

√ ×F (−ν 1 , m + 1, ξ /ν)F (−ν 2 , m + 1, η/ν)eimφ / 2π, (8.87)

where [33]

  −1 2(ν 1 + m + 1)(ν 2 + m + 1)  Nν 1 ν 2 m = ν m+1/2 m!2 . (ν 1 + 1)(ν 2 + 1)

10

(8.88)

¯ See (15) and (16) of Sakimoto, which employ the notations g(η) and g(η) for f2 (η; E, Z2 ) and g2 (η; E, Z2 ) and  for β 2 (E, Z2 ) [33].

8.2 The Stark effect

269

Similarly, the corresponding angular momentum states are given by √ νm (r, θ, φ) = Nνm r  e−r/ν Pm (cos θ )F (−ν +  + 1, 2 + 2, 2r/ν)eimφ / 2π, (8.89) m where P (cos θ ) is an associated Legendre polynomial [29] and  +1/2 1 2(ν +  + 1)( + m)! 2 . (8.90) Nνm = ν (2 + 1)! (ν − )( − m)! Fano notes at this point that it is sufficient to establish the connection at small r values, at which νm (r, θ, φ) varies as r  [34]. Moreover ξ = r(1 + cos θ ) and η = r(1 − cos θ). One therefore finds, after projecting out the m term from the identity  ν 1 ν 2 m |νm νm (r, θ, φ), (8.91) ν 1 ν 2 m (ξ , η, φ) = ν1ν2

and taking the limit r → 0 that11 ν 1 ν 2 m |νm

(  π Nν 1 ν 2 m −+m = lim r sin θ dθ sinm θPm (cos θ ) Nνm r→0 0

) ×F [−ν 1 , |m| + 1, r(1 + cos θ )/ν]F [−ν 2 , |m| + 1, r(1 − cos θ )/ν] . (8.92)

Note that terms in r t1 +t2 sinm θ (1 + cos θ )t1 (1 − cos θ)t2 arising from expansion of the confluent hypergeometric functions contribute to ν 1 ν 2 m |νm only when t1 + t2 =  − m because Pm (cos θ ) is orthogonal to lower terms and the behaviour as r → 0 eliminates the higher ones. Equation (8.92) provides an integral representation for ν 1 ν 2 m |νm of the same form that would apply for integral values of ν, ν 1 and ν 2 . It is therefore assumed that the generalized frame transformation may be written ν 1 ν 2 m |νm = j˜, k˜1 , j˜, k˜2 |, m,

(8.93)

on the assumption that gamma functions involving the non-integer quantities j˜ = ν/2 − 1,

11

1 k˜1 = (m + ν 1 − ν 2 ), 2

1 k˜2 = (m − ν 1 + ν 2 ) 2

There appears to be an erroneous additional factor of sinm θ in (13) of [34].

(8.94)

270

Manipulating Rydberg states

are substituted for factorials in the usual Racah formula [26]. Alternatively the asymptotic approximation " (2 + 1)  ˜ dm0 (θ ) (8.95) j˜, k˜1 , j˜, k˜2 |, m  (−1)[ν 2 ] ν ν1 − ν2 − m cos θ˜ = ν may be employed, where [ν 2 ] is the closest integer to ν 2 . In applying these results it must be remembered that the exact ‘local’ hydrogenic expression in (8.92) is actually employed in a context where the indices ν 1 and ν 2 given by (8.84) depend on the separation context Z1 , which varies with the field strength, F, in a way that also alters the amplitude of the JWKB approximation to f1 (ξ ) by a factor C/C (0) , where C is given by (8.78) and C (0) = (2ν)−1/2 is the corresponding coefficient in the field-free wavefunction. The frame transformation elements are therefore given by [33] Uν 1  (νmF) = (2ν)1/2 Cν 1 ν 2 m |νm,

(8.96)

as specified by (8.93) and (8.94), where we use ν 1 (or the separation constant Z1 ) to label the Stark channel. Both ν 1 and  then take n − 1 values, where n is the integer part of ν, which increases with increasing energy. In summary, the machinery for generalized MQDT quantization is provided by (8.85) and (8.96). At any given energy, E = −1/2ν 2 , the first of (8.73) is used to determine values of Z1 , and hence ν 1 , ν 2 and Z2 , consistent with 0  n1  ν − 1. Equation (8.84) provides the corresponding values of ν 1 and ν 2 . Equations (8.94)– (8.96) then provide the frame transformation matrix, which cannot however be assumed to be orthogonal except in the field-free limit. Elements of the inverse matrix are therefore required to calculate the K-matrix elements  Uν 1  tan π μ [U −1 ]ν 1 . (8.97) Kν 1 ν 1 = 

The MQDT quantization condition then takes the form   det U tan πμU −1 + cot β 2 (E, Z2 ) = 0,

(8.98)

involving the cot β 2 (E, Z2 ) in place of the normal tan β(E), owing to the difference in the placement of sine and cosine terms in (2.13) and (8.85). The alternative form   (8.99) det tan πμ + U −1 cot β 2 (E, Z2 )U = 0 is more convenient for evaluating the determinant because the elements of the diagonal tan πμ matrix may be taken as zero except for  = 0–2. These MQDT equations play the same role as the JWKB quantization equations (8.73), except

References

271

that the quantum defects responsible for lifting the field-free n2 degeneracy are now taken into account. The resulting values of E and Z2 allow estimates of the line broadening by use of (8.76). As written, (8.98) and (8.99) apply to Stark interactions between the series converging on a single ionization limit and calculations have been reported for the sodium atom [31, 33]. The theory can also be extended to include rotational and vibrational channel interactions by adding terms to the K-matrix of the forms described in Chapter 4, and by determining different sets of indices, ν, ν 1 and ν 2 for the series converging on different ionization limits.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]

H. H. Fielding, Ann. Rev. Phys. Chem. 56, 91 (2005). T. Gallagher, Rydberg Atoms (Cambridge University Press, 1994). K. M¨uller-Dethlefs and E. W. Schlag, Ann. Rev. Phys. Chem. 42, 109 (1991). H. Friedrich and D. Wintgen, Physics Reports 183, 37 (1989). J. Parker and C. R. Stroud, Phys. Rev. Lett. 56, 716 (1986). I. S. Averbukh and N. F. Perelman, Phys. Lett. A 139, 449 (1989). M. S. Child, Semiclassical Mechanics with Molecular Applications (Oxford University Press, 1991). G. Alber, H. Ritsch and P. Zoller, Phys. Rev. A 34, 1058 (1986). J. A. Yeazel and C. R. Stroud, Phys. Rev. A 43, 5153 (1991). R. A. L. Smith, V. G. Stravros, J. R. R. Verlet and H. H. Fielding, J. Chem. Phys. 119, 3085 (2003). V. G. Stravros, J. A. Ramswell, R. A. L. Smith et al., Phys. Rev. Lett. 83, 2552 (1999). R. S. Minns, J. R. R. Verlet, L. J. Watkins and H. H. Fielding, J. Chem. Phys. 119, 5842 (2003). S. N. Altunata, J. Cao and R. W. Field, Phys. Rev. A 65, 053415 (2002). M. J. J. Vrakking and Y. T. Lee, J. Chem. Phys. 105, 7336 (1996). S. A. Rice and M. Zhao, Optical Control of Molecular Dynamics (Wiley, 2000). M. Shapiro and P. Brumer, Principles of the Quantum Control of Molecular Processes (Wiley, 2003). L. D. Noordam, D. I. Duncan and T. F. Gallagher, Phys. Rev. A 45, 4734 (1992). R. S. Minns, R. Patel, J. R. R. Verlet and H. H. Fielding, Phys. Rev. Lett. 91, 243601 (2003). F. Texier and C. Jungen, Phys. Rev. A 59, 412 (1999). F. Texier, C. Jungen and S. C. Ross, Faraday Discuss. 115, 71 (2000). J. W. B. Hughes, Proc. Phys. Soc. 91, 810 (1967). B. C. Wybourne, Classical Groups for Physicists (Wiley, 1974). V. Bargmann, Z. Physik 99, 576 (1936). D. Park, Z. Physik 159, 155 (1960). P. J. Brussard and H. A. Tolhoek, Physica 23, 955 (1957). R. N. Zare, Angular Momentum (Wiley-Interscience, 1988). D. A. Harmin. In Atomic Excitation and Recombination in External Fields, ed. M. H. Nayfeh and C. W. Clark (Gordon and Breach, 1985). L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 2nd edn (Pergamon Press, 1965).

272

Manipulating Rydberg states

[29] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1965). [30] E. Luc-Koenig and A. Bachelier, J. Phys. B 13, 1743 (1980). [31] D. A. Harmin, Phys. Rev. A 24, 2491 (1981). [32] R. E. Langer, Phys. Rev. 51, 669 (1937). [33] K. Sakimoto, J. Phys. B 19, 3011 (1986). [34] U. Fano, Phys. Rev. A 24, 619 (1981).

Appendix A MQDT normalization

Following the discussion in Chapter 2, the total wavefunction is expressed in the form  (j ) r = i (ξ ) ψi (r)Zi , (A.1) ij

where r is the Rydberg radial variable and ξ includes all other degrees of freedom. Although Coulomb forms do not apply within the core, the radial factors ψi (r) are assumed to be regular at the origin and to behave in the Coulomb region as (j )

ψi (r) ∼ fi (r)δij − gi (r)Kij ,

(A.2)

where K is a symmetric matrix and the solution vector satisfies the uncoupled matrix equation 

 d2 + ε − U (r) ψ(r) = 0, dr 2

(A.3)

in which ψ(r) and U (r) are diagonal matrices and ε = (E − I )/Ry . It is sufficient to consider the two cases in which all channels are either open or closed. The resonant case, involving coupling between the two types, is handled by applying an auxiliary condition (2.44) to relate the open and closed amplitude components Z o and Z c . A.1 Open channels As discussed in Section 2.4, the open-channel basis functions have the asymptotic forms  1 fi (ε, r) ∼ sin [ki r + ϕi ] π ki  1 gi (ε, r) ∼ − cos [ki r + ϕi ] , (A.4) π ki where ε = (E − I )/Ry = Ii + ki2 , in which I is the lowest ionization limit and Ii is the ith channel threshold. We first demonstrate that the regular functions, at energies a and b , are 273

274

MQDT normalization

normalized to the energy delta function [1]  1 ∞ δ(a − b ) = cos [(a − b )t] dt. π 0

(A.5)

The argument rests on observing that the product f (εa , r)f (εb , r) decomposes in the limit a → b into a constant background term plus an oscillatory term that provides a negligible contribution to the overlap integral. Thus for a  b  ∞  ∞ 1 f (εa , r)f (εb , r)dr = cos [(ka − kb ) r] dr. (A.6) 2π k¯ 0 0 The substitutions ka − kb = then lead to1 

∞

f (εa , r)f (εb , r)dr =

0 1

a − b ¯ and r = 2kt 2k¯ 1 π



∞

cos [(a − b )t] dt = δ(a − b ).

(A.7)

(A.8)

0

The argument may be extended to the regular continuum solutions of the general Schr¨odinger equation # $ h¯ 2 d2 + E − V (R) F (E, R) = 0 2μ dR 2 by scaling distances and energies in the forms r = R/a and  = E/E0 . Thus   2 d + k 2 − U (r) f (, r) = 0, 2 dr where f (, r) = F [E0 , ar] and 2μE0 . a 2h¯ 2

k2 = The regular solutions with asymptotic form

 W sin [kν r + φ] kν

fν (, r) ∼

for ν = a, b are found, by the substitution r = (a 2h¯ 2 /μE0 )t, to have overlap  ∞  W ∞ fa (, r)fa (, r)dr = cos [(ka − kb )r] dr 2k¯ 0 0 =

π a 2h¯ 2 W δ (a − b ). 2μE0

It follows that W must be chosen as W =

2μE0 . π a 2h¯ 2

Various cases may be considered. Electron motion, atomic units Electron motion, natural units

E0 = a 2h¯ 2 /me E0 = a 2h¯ 2 /2μe

W = 2π −1 (μe /me ) W = π −1

Nuclear motion, atomic units Nuclear motion, natural units

E0 = a 2h¯ 2 /me E0 = a 2h¯ 2 /2μe

W = 2π −1 (μN /me ) , W = π −1 (μN /μe )

in which me is the electron mass while μe and μN are the reduced masses for electronic and nuclear motion, respectively.

A.2 Closed channels

275

The extension to multiple open channels takes into account that the column vector Z, with components Zi in (A.1) satisfies the matrix eigenvalue equation2 (K − tan π τ )Z = 0,

(A.9)

The N o solutions, which are distinguished by a label ρ, may be used to eliminate the Kij from (A.2). Thus, by combining (A.2) and (A.9), (ρ)

ψi (ε, r)Ziρ = χiρ (ε, r)Tiρ   χiρ (ε, r) = fi (ε, r) cos π τρ (ε) − gi (ε, r) sin π τρ (ε)    1 ∼ sin ki r + ϕi + π τρ (ε) , π ki

(A.10)

where Tiρ = sec π τρ (ε) Ziρ . Notice that the superscript (j ) in (A.1) has been replaced by (ρ) because the eigenphase τ%ρ (ε)&is common to all channels. Each solution is characterized by an orthogonal vector set Tρ , which rotates continuously as the energy changes. The aim is to normalize the magnitudes Tρ of the column vectors to preserve the energy normalization within each separate continuous ρ-branch of the solution. In view of the orthogonality of the channel functions i (ξ ) in (A.1) this requires that  ∞  Tiρ2 χiρ (εb , r)χiρ (εa , r) dr = δ(εa − εb ). (A.11) 0

i

The symbol χiν in what follows is a shorthand for χiρ (εν , r) with ν = a, b. We shall also 2 use the notation νi = kνi for the continuum energy in channel i, bearing in mind that (ai − bi ) is equal to the total energy difference (εa − εb ) because εν = Ii + νi . It remains, on replacing f (εν , r) by χiρ (εν , r) in (A.6)–(A.8), to recognize that  ∞ χi (εa , r)χi (εb , r)dr = δ(ai − bi ) = δ(εa − εb ). (A.12) 0

Thus, on combining (A.11) and (A.12), 

Tiρ2 = 1.

(A.13)

i

Bearing in mind that the column vector Tρ belongs to an orthogonal set, this establishes the orthogonality of the matrix T . In other words T T T = T T T = I , or N  o

i=1

N  o

Tiρ Tiσ = δρσ

and

Tiρ Tjρ = δij .

(A.14)

ρ=1

A.2 Closed channels As a preliminary to the closed-channel normalization procedure, we consider the overlap between a single regular basis function fn (r) with eigenvalue εn = (En − I ) /Ry and a 2

It is assumed that K takes the folded form (2.45) in the resonant case.

276

MQDT normalization

similar regular solution fp (r) at a neighbouring energy εp , which is not an eigenvalue. It is easy to verify from (A.3) that  R  

     R   fp fn − fn fp

dr = W fn , fp R − W fn , fp 0 = εp − εn fn fp dr, (A.15) 0

0

where R lies in the asymptotic region.  Now both fn (r) and fp (r) vanish as r → 0 and their derivatives are finite. Hence W fn , fp 0 = 0. In addition fn (R) contains only an exponentially decreasing term cos βn Fn(−) (R). Thus, with the help of the upper line of (2.13)     W [fn , f ]R = cos βn sin βp W Fn(−) , Fp(+) − cos βn cos βp W Fn(−) , Fp(−) , (A.16) where the notation F (±) indicates an asymptotic exponentially increasing or decreasing term. Also βn = β(εn ). The first term on the right vanishes linearly with (εp − εn ), by virtue of the factor sin βp  (dβn /dε)(εp − εn ), while the Wronskian in the second vanishes exponentially. It is easy to verify by taking the limits R → ∞ and εp → εn , with cos2 βn = 1, that



 ∞ 1 dβn dνn fn2 (r)dr = = , (A.17) π dε dε 0 where νn = β(εn )/π is the number function that interpolates through integers at the eigenvalues. The unit-normalized eigenfunction is therefore obtained by multiplying fn (r) by the square root of the local bound state level spacing – expressed in the units employed for the Wronskian normalization (see footnote to (A.8)) – taken here as Rydbergs. Normalization of the vector Z in (A.1) proceeds by a matrix generalization of (A.15)– (A.17). Note first that the total wavefunction  in (A.1) may be expressed in matrix notation as r = χ, with χ = ψZ, where the vector Z is determined by the MQDT quantization condition [K + tan β(ε)] Z = 0.

(A.18)

Thus, if the internal wavefunctions i are normalized to unity, the normalization of χi requires that  ∞  ∞ χ T (r)χ (r)dr = χi2 (r)dr = 1. (A.19) 0

i

0

Two forms for χ are considered. The first is the solution χn at an eigenvalue εn , with properly decreasing components in all channels. Thus d2 χn − U χn = −εn χn , dr 2

(A.20)

χn = ψn Z ∼ Fn(−) [cos βn I − sin βn K1 ] Z.

(A.21)

with the asymptotic form

The second regular solution, χp , at a different energy εp , is chosen to behave asymptotically as   %  & χp = ψZ ∼ F (+) sin βp I + cos βp K + F (−) cos βp I − sin βp K Z, (A.22)

A.2 Closed channels

277

with the same contraction vector Z as χn . In addition, (A.15) generalizes to the matrix form     R T    χp χn dr, (A.23) W χn , χp R − W χn , χp 0 = εp − εn 0

where

  W χn , χp = χpT χn − χnT χp ,

(A.24)

in which χ T is the transform of the column vector χ . As in the single-channel case, the components of the regular solutions, χn and χp , vanish at the origin, with finite derivatives; hence W [χn , χ ]0 = 0. In addition the generalization of (A.16) becomes       W χn , χp R = Z T I cos βn − KnT sin βn W Fn(−) , Fp(+) R   × sin βp I + cos βp K Z, (A.25) after recognizing that W [Fn(−) , Fp(−) ]R → 0 as R → ∞. The first-order contribution to an energy expansion for the final term gives ( )   sin βn I + cos βn Kn +   sin βpp I + cos βp K Z  Z, (cos βn I − sin βn Kn ) βn + cos βn Kn (ε − εn ) (A.26) in which the first term on the right vanishes, by virtue of (A.18). Moreover, since the trigonometric matrices are diagonal, (A.18) also shows that Kn Z = −tan βn Z. Consequently   (A.27) [cos βn − sin βn Kn ] Z = cos βn + sin2 βn sec βn Z = sec βn Z. Finally W [Fn(−) , Fn(+) ]R = π −1 I . Taken together, these equations yield the normalization condition  ∞   π −1 ZnT sec βn sec βn βn + cos βn K Zn = χnT χn dr = 1. (A.28) 0

When expressed in terms of components, the elements of Z must be scaled such that 1  2 dβi 2 1  dKij sec βi Zi Z + Zj = 1, π i dε i π ij dε

(A.29)

where the subscript n, which was used to denote the solution at the eigenvalue, has been dropped. The factors π −1 dβi /dε in the first term coincide with the single closed-channel integrals in (A.17), and the factors sec2 βi ensure that the weighting, Zi2 , given to channel i decreases according to the separation between the eigenvalue εn and the zeroth order energy level εn0 , at which sec2 β = 1.

Appendix B Alternative MQDT representations

The standard MQDT representation in Section 2.4 is most convenient for exposition purposes, but a number of alternative representations are available in the literature [2], either for practical or historical reasons. The relations between them are discussed below. It is assumed in all cases that the K-matrix in the uncoupled |i picture is related to the quantum defects for coupled states |α in the form Kij =



i |α tan π μα α |j  .

(B.1)

α

B.1 Standard representation The outer part of the wavefunction is written as rout =



  |i fi (r)δij − gi (r)Kij Zj .

(B.2)

ij

In the general case, with N o open and N c closed channels, there are N o possible sets of coefficients Zjρ , where ρ = 1, 2, . . . , N o . They satisfy the matrix equations

K oo − tan π τρ (E)I K co

K oc K cc + tan β(E)



Zρo Zρc

= 0,

(B.3)

which may be folded, by the substitution c =− Ziρ

  −1 co o K Zjρ , K cc + tan β(E) ij

j

(B.4)

into the symmetric eigenvalue form 

 −1 co K oo − K oc K cc + tan β(E) K − tan π τ (E)I Zρo = 0,

o subject to the normalization Ziρ = cos π τ (E)Tiρ , where

278

 i

Tiρ2 = 1.

(B.5)

B.2 Sine–cos representation

279

In the special case, with only closed channels, the discrete energy eigenvalues are given by [K + tan β(E)] Z = 0, 1  2 dβi 2 1  dkij sec βi Zi Z + Zj M = 1. π i dE i π ij dE

(B.6)

B.2 Sine–cos representation The following sine and cosine form [2] is chosen to avoid near divergences of tan π μ. It is related to the standard representation by the definitions1   Sij = i |α sin π μα α |j , Cij = i |α cos π μα α |j  , K = SC−1 , α

α

Bρ = C−1 Zρ.

(B.7)

The outer wavefunction is re-scaled to the form    |i fi (r)Cij − gi (r)Sij Bj , rout =

(B.8)

ij

where the coefficients Bjρ , with ρ = 1, 2, . . . , N o , satisfy the alternative MQDT system Bρ = tan π τρ Bρ , ( sin βi Cij + cos βi Sij , i ⊂ Q (closed) , ij = Sij , i ⊂ P (open) ( 0, i ⊂ Q (closed) ,

ij = Cij , i ⊂ P (open)

(B.9)

where  and are typically non-symmetric, with dimensions N × N and N o × N, respectively. This system, which has rank N o , fixed by the dimensions of , can also be folded, by the substitution B c = − [ cc ]−1  co B o into the form    −1 co o  −1 co o  oo −  oc  cc  B = tan π τρ oo − oc  cc  B . (B.10) The solutions are normalized by combining 6    j cos π τρ Sij − sin π τρ Cij Bjρ = 0 i⊂P  j Cij Bjρ = Ziρ = cos π τρ Tiρ to give

1

   j cos π τρ Cij + sin π τρ Sij Bjρ = Tiρ    j sin βi Cij + cos βi Sij Bjρ = 0

i ⊂ P. i⊂Q

In particular the vibrational sin and cosine matrix elements ! ! ' ' Sv+ v+ = v + sin π μ(R) v + ; Cv+ v+ = v + cos π μ(R) v + remain finite when μ(R) crosses π/2.

(B.11)

(B.12)

280

Alternative MQDT representations

Note  2that the sums over j in (B.11) and (B.12) are taken over all channels, and i Tiρ = 1. In the special case with only closed channels the discrete energy eigenvalues are given by [sin β(E)C + cos β(E)S] B = 0,

(B.13)

and the solutions are normalized by the substitution Z = CB, with Zi subject to (B.6). B.3 Mixed representation The following mixed form, which is frequently applied in the early literature [3], works in terms of the inner amplitudes Aα , rather than Zi , which is sometimes convenient for spectroscopic applications. The connection with the standard representation involves the definitions  Ziρ = i |α cos π μα Aαρ . α

Siα = i |α sin π μα ,

Ciα = i |α cos π μα.

(B.14)

The re-scaled outer wavefunction then becomes  |i i |α [fi (r) cos π μα − gi (r) sin π μα ] Aαρ rout = iα

=



|i [fi (r)Ciα − gi (r)Siα ] Aαρ ,

(B.15)

iα

with the coefficients Aαρ , with ρ = 1, 2, . . . , N o , given by [4] ˜ ρ, ˜ ρ = tan π τρ A A ( sin βi Ciα + cos βi Siα , ˜ iα = Siα , ( ˜ iα = 0, i ⊂ Q .

Ciα , i ⊂ P

i⊂Q , i⊂P

(B.16)

˜ (N o × N o ) is symmetric. Again the first equation has rank N o , but neither ˜ (N × N) or By analogy with (B.12), the solutions are normalized by    i⊂P α cos π τρ Ciα + sin π τρ Siα Aαρ = Tiρ  , (B.17) C + cos β S = 0 i ⊂Q β A [sin ] i iα i iα αρ α  where i Tiρ2 = 1. In the special case with only closed channels the discrete energy eigenvalues are given by  α

[sin βi (E)Ciα + cos βi (E)Siα ] Aα = 0,

(B.18)

B.3 Mixed representation

281

subject to the normalization 1  dβi π i dE

 

2 [cos βi Ciα − sin βi Siα ] Aα

= 1.

(B.19)

α

Notice that this may be reduced to (B.6), by multiplying (B.18) by sin βi and using the first of (B.14).

Appendix C Rotational frame transformations

The analytical forms for a variety of rotational frame transformations are given here. In view of the diversity of angular momentum coupling schemes, attention is first restricted to diatomic molecules, within the framework of Hund’s coupling cases [5], which differ according to the relative importance of three factors – the electronic energy splitting between different components, the strength of spin–orbit coupling, and the rotational energy-level separations. The relative values of these three quantities allow six possibilities, each of which has a characteristic form for its parity-adapted wavefunction, although Hund himself only covered cases (a)–(d). This discussion is restricted to situations in which the Rydberg electron in a neutral molecule, which conforms to case (a), (b) or (c), is uncoupled from the molecular axis, to leave the positive ion in the same case as the parent molecule. Such excitations correspond to transformations of the type (a)→(e), (b)→(d) and (c)→(e ). The first of these has been most fully described by Jungen and Raseev [6]. The second is discussed in its simplest form in Chapter 4.2, along lines pioneered by Fano [7]. A fuller account, applicable to species with open shell cores, is given below. The final (c)→(e ) case, which has as yet found no application in the literature, is mentioned for completeness, but not treated in detail. The final section includes results for the rotational frame transformation for asymmetric tops, in the absence of spin, which goes beyond earlier work [8], by employing permutation inversion symmetry [9]. C.1 Hund’s cases for diatomic molecules The frame transformations of interest are algebraic expressions for the projections of the coupled rotational electronic states, |α = |a or |b, onto the uncoupled ones, |i = |e or |d. Before turning to the angular momentum algebra, it may be helpful to review the physical origin of Hund’s cases [5, 10, 11, 12, 13], which differ according to the relative magnitudes of the leading contributions to the rotational electronic Hamiltonian, Hˆ = Hˆ el + Hˆ so + Hˆ rot , where Hˆ el =



(C.1)

|L  E L |

ˆ · Sˆ Hˆ so = λL ˆ +2 B. Hˆ rot = N 282

(C.2)

C.1 Hund’s cases for diatomic molecules

283

ΔEel

(a)

(b) (d)

(c) Δ Eso

(e′)

(e) Δ Erot

Figure C.1 Diagrammatic representation of Hund’s coupling cases according to the relative magnitudes of Erot , Eel and Eso . Points far from the internal boundaries correspond to the relevant pure coupling case. Cases (a)–(c) in the upper half of the diagram, corresponding to Erot < Eel , have defined body-fixed (or ‘coupled’) angular momentum projections, while the inequality is reversed in cases (d), (e) and (e ), in which the excited electron has no defined body-fixed components. Passing from the upper to the lower half of the diagram therefore involves ‘decoupling’ the Rydberg angular momentum from the molecular axis. Figure C.1 shows a diagrammatic division into the six possible cases, (a)–(e ), allowed by the relative strengths of these three terms. The relevant angular momenta include L S Ja J N N+

electronic orbital angular momentum, electronic spin angular momentum, total electronic angular momentum (L + S), total angular momentum, total angular momentum excluding spin (J − S), nuclear rotational angular momentum (N − L),

as well as the corresponding angular momenta of the Rydberg electron and those of the positive ion, which are denoted, respectively, as ( , s, j, etc.) and (L+ , S+ , N+ , etc.). Note that the symbol N+ is employed for the nuclear rotational term, in place of the normal symbol R [10, 11, 12]. Seen in relation to the divisions in Fig. C.1, Hund’s strict coupling cases correspond to points far from the internal boundaries. The term Hel dominates in cases (a) and (b) in the sense that the energy splitting between different |L  components is large compared with both the spin–orbit coupling matrix elements and the rotational energy separations, Erot . The resulting existence of well-defined components is expressed colloquially as ‘coupling to the diatomic axis’. These two cases differ according to the relative magnitudes of Eso and Erot . Thus Eel  Eso  Erot in case (a), which means that L and S are separately coupled to the axis, in the latter case by the second-order influence of Hso . By contrast Eel  Erot  Eso in case (b), with the result that L remains coupled to the axis, but the coupling of S to the axis is quenched by the relatively rapid rotation. By a similar argument, Eso  Eel  Erot in case (c), with the result that L and S first form a resultant Ja which is then coupled to the axis. The three remaining cases arise when Erot  Eel , in the Rydberg context because Eel arises from interaction with the positive ion core, which decreases with the principal quantum number as n−3 , while Erot is independent of the level of excitation. The excitation schemes of interest, (a) → (e),

(b) → (d),

retain the relative magnitudes of Eel and Eso .

and (c) → (e)

284

Rotational frame transformations

Table C.1 Properties of various angular momentum states under the space-fixed inversion operation E ∗ . Capitals and lower-case symbols refer to total states and single-electron functions, respectively. The notation  − (1/2) means that the fraction is subtracted for half-integer  values. The index q = 0 except for states with orbital symmetry  − , for which q = 1. |basis

E ∗ |basis

|basis

|J M |  |S |SMS 

(−1)J − |J − M (−1)q+ | −  (−1)S− |S −  |SMS 

|n |λ |m  |j (s)ω |j (s)mj 

E ∗ |basis (−1)−(1/2) |n −  (−1)λ | − λ (−1) |m  (−1)s−ω |j (s) − ω (−1) |j mj 

C.2 Parity considerations The behaviour of the various contributions to the wavefunction under the space-fixed inversion operation, E ∗ , are well-known [12, 14, 15]. Functions of the Euler angles (φ, θ , χ ) and of the body-fixed coordinates (xi , yi , zi ) transform respectively as E ∗ f (φ, θ, χ ) = f (π + φ, π − θ , π − χ)

(C.3)

E ∗ g (xi , yi , zi ) = g (xi , −yi , zi ) ,

(C.4)

and

from which the entries in Table C.1 may be deduced. Note that the parities of |λ and |m  ensure consistency with the identity |m  =

 ∗  |λ Dm λ

(C.5)

λ

  ∗ transforms like |J M. In addition the single electron state |j (s)ω because Dm λ arising from coupling |λ and |sσ  has parity (−1)s−ω , because ω = λ + σ is preserved by the coupling. Similarly |j (s)mj  has parity (−1) because the spin components |sms  are invariant under E ∗ . Finally, in situations in which the Rydberg electron with orbital projection λ combines with a core with + = −λ = 0, one may construct the parity-adapted combinations  1  + | = 0, q = √ | + | − +  + (−1)q−q | − + | +  . 2 Bearing in mind that E ∗ | +  = (−1)q

+

− +

(C.6)

| − + , this means that

E ∗ | = 0, q = (−1)q | = 0, q , with q = 0 or 1 for  + or  − states of the neutral molecule.

(C.7)

C.3 Basis functions

285

Table C.2 Basis states for Hund’s cases (a)–(e). Case a =0 a =0 b c d e

|x

Cx−2

| + |λ|S|J M 2(1 + δ + 0 δ 0 δ 0 ) | = 0, q|S|J M 2(1 + δ 0 ) | + |λ|N MN |SMS  2(1 + δ + 0 δ 0 ) |Ja+ + 2(1 + δ + 0 δ 0 ) a |j ω|J M + + + + | |N MN |m |SMS  2(1 + δ + 0 ) + + + + + + | |S  |J  M |j mj  2(1+δ + 0 δ + 0 )

τx p − q+ + J − S p−q +J −S p − q+ + N + p − q + J − Ja+ − 1/2 p − q+ + N + +  p − q+ + J + − S+ + 

C.3 Basis functions The following expressions give the forms for the various basis functions arising from the addition of a Rydberg electron to a positive ion. The quantum numbers J , M and , the overall parity p and the positive ion orbital parity q+ are taken as fixed under the frame transformation. We are concerned with situations in which a Rydberg electron is attached to a positive ion, according to one or other of Hund’s coupling cases, although the reader should interpret the symbol |λ as ‘the |λ component of the relevant orbital’. The compositions of the parity-adapted basis functions are summarized in Table C.2, in a short hand notation   |x, p = Cx |x + (−1)τ x |−x , (C.8) for case (x), in which the composition of |x is listed in the second column and the symbol |−x means that the signs of the body-fixed (greek) angular momentum components are reversed. The index τ x is designed to ensure that E ∗ |x, p = (−1)p |x, p, and the coefficient Cx ensures normalization to unity. Two forms are given for case (a) according to whether or not = 0.1 Thus the states with = 0 take the form

! |a, p = Ca n + |λ |S |J M  ! + + (−1)p−q +J −S n − + | − λ |S −  |J − M , (C.9) which may be verified by use of Table C.1 to satisfy E ∗ |a, p = (−1)p |a, p. Note that the sign of the integer S −  has been reversed in the phase term. The states with integer values of τ a collapse to a single normalized term if + =  =  = 0, provided that p − q + + J − S is even, because λ is also constrained to zero by the identity  = + + λ + . Similarly, when = 0, we have 1 | = 0, q 2(1 + δ 0 )   × |S |J M + (−1)p−q+J −S |S −  |J − M ,

|a, p = √

1

Jungen and Raseev [6] give a composite expression that includes both possbilities.

(C.10)

286

Rotational frame transformations

Table C.3 Quantum numbers appropriate to the transformations (a)→(c), (b)→(d). Note that, for simplicity, spin is ignored in the second case. By convention, states with even or odd values of J (−1/2) + p are labelled e or f , where the factor (−1/2) applies when J is a half odd integer. Similarly the core levels are marked e+ or f + for even or odd values of J + (−1/2) + p + + . Trans (a) → (c) (b) → (d)

Good

Fixed

|α

J, M, p, s N, MN , p

+ , S + , q + , 

+ , q + , , S, MS

|S, ,  | 

|i |j, J + , +  |N + 

where | = 0, q is given by (C.6). There is no need for a similar special specification for case (b) when = 0, because the form given by (C.8) in Table C.2 goes over to |b, p = | = 0, q |N0MN , with q = p + N . Finally, it is important to realize that the total angular momentum is as yet undefined in the uncoupled states |d, p and |e, p given by Table C.2. The proper case (d) total angular momentum state is given by  |d, pN = N + m MN+ |N MN  |d, p . (C.11) m M +

C.4 Diatomic frame transformations Before embarking on details of the individual transformations it is useful to categorize the angular momenta involved. The physical process is the notional decoupling of a Rydberg electron with angular momentum |λ, from the molecular axis, at fixed total angular momentum and parity, without changing the electronic state of the positive ion. The word notional is chosen to underline the mixed  character of a typical molecular orbital, despite which  is taken as ‘fixed’ under the frame transformation – leaving the -mixing to be handled by the electronic K-matrix (see Section 4.1). In other words there are three types of angular momentum variable: ‘good’ quantum numbers such as total angular momentum and parity; ‘fixed’ ones such as  and the positive ion labels; and ‘variable’ ones that label the elements of the frame transformation matrix. Those appropriate to the initial coupled states |α specify the possible electronic states arising from the spin and orbital angular momenta of the ion and electron. The corresponding variables in the uncoupled states |i include the positive ion total angular momenta, consistent with J = J+ + , the total Rydberg angular momentum, and the body-fixed orientation of J. The members in each category, appropriate to case (a)→(e) and case (b)→(d) excitation, are listed in Table C.3. It is simplest to start with the case(b)→case (d) transformation.

C.4.1 Case (b) to case (d) frame transformation The derivation of the case (b) to case (d) transformation follows the procedure outlined in Section 4.2, except that we now make an explicit distinction between the Rydberg electron and the positive ion, with provision for + = 0. A similar argument also applies to symmetric top molecules, apart from the conventional substitution → K for the angular momentum components with respect to the symmetric top axis. The first step

C.4 Diatomic frame transformations

287

is to evaluate the overlap integral d |b between the primitive (non-symmetrized) basis functions in Table C.2. Following the method of Section 4.2, total angular momentum states are represented by Wigner rotation matrices and (C.5) is used to relate the uncoupled case (d) Rydberg states, |m , to the coupled case (b) states |λ. Thus, using the orthogonality of |λ and |SMS , the primitive integral d |b reduces to [11, 16],   N ∗ 3  N+ d |b = A Dm (R)DM dR + + (R) DM (R)

λ N = A(−1)−MN +



N

+

 N N 3 Dm (R)DM + + (R)D−M − (R)d R

λ N

= 8π 2 A(−1)−MN +



N

 m

N+ MN+

N −MN



 λ

N+

+

N , −

(C.12)

√ where 8π 2 A = (2N + 1)(2N + + 1). Furthermore, the first Wigner symbol is related to the Clebsch–Gordan coefficient

√  N+ N + + −N + +MN N m MN |N MN  = (−1) 2N + 1 , (C.13) m MN+ −MN which is shown by (C.11) to determine the proper |N MN  component of the primitive state |d . Hence +

√ N  N , (C.14) d |bN = (−1)N− 2N + + 1

+ λ − where the order of the first two columns of the 3j symbol has been reversed and  has been recognized as an integer. The overlap d, p |b, pN between the parity-adapted case (d) and case (b) states at fixed N gives rise to a similar term −d |−bN , in which the signs in the second row of (C.14) are reversed – a reversal that may be restored by taking account of the index sum τ b + τ d in Table C.2. Finally, it should be noted that the state |d, p for + = 0 vanishes when τ d = p − q + + N + +  is odd and contains two identical terms when τ d is even. Thus the frame-transformation matrix elements at constant N,  and + and fixed parity p − q + may be expressed as

(−1)N− [1 + (−1)τ d δ + 0 ] N +  N + + N + | N pq = √ . (C.15) + λ − (1 + δ + 0 δ 0 )(1 + δ + 0 ) Essentially the same form, with + and replaced by K + and K applies for symmetric top molecules [17]. The analogous form for asymmetric tops is derived in Section C.5 below. It is useful for practical applications to establish the limits on N + and in the two possible parity blocks. To avoid double-counting the number of basis states, it is convenient to impose the conditions that +  0 with  0 for + = 0. The situation with + > 0 allows equal numbers of |N +  or |  states in each parity block, but the number itself is complicated by the restriction N +  + . An enumeration of possible cases, subject to N  | − + |, allows two possible situations, according to the relative values of N and

+ + . Thus

+ −    max ; Nmin  N +  N + ,

288

Rotational frame transformations

where max = min( + + , N) and Nmin = max(N − , + ). The inequalities imply unit steps in or N + over the indicated ranges. Different considerations apply for + = 0. For example the |d states in Table C.2 vanish unless p − q + − N + +  is even, which means that even and odd values of N + belong to different parity blocks. In addition, the Wigner coefficient in (C.15) restricts to the values 0   min(, N), with the added requirement that = λ = 0 is allowed by the Wigner coefficient only for even values of N + +  + N. One finds, by considering the possible cases that the elements of the two parity blocks may be labelled according to the scheme. N + = |N − |, |N − | + 2, . . . , (N + ),

= 0, 1, . . . , min(, N ),

for even values of p − r + + N, while those of the opposite parity block have labels N + = |N − | + 1, |N − | + 3, . . . , (N + ) − 1.

= 1, 2, . . . , min(, N),

The blocks with even and odd values of p + N are assigned e and f symmetries, respectively.

C.4.2 Case (a) to case (e) frame transformation The key to the (a)→(e) frame transformation lies in the identities [11, 16]  ∗ j |j mj  = Dm (R) |j ω jω ω

=



(C.16)

∗ j Dm (R) |lλ |sσ  lsλσ |j ω, jω

ωλσ

together with a spin recoupling transformation, s + S+ = S, in the body frame. The analogues of (C.12) and (C.13) then lead to [6] e |a = (−1)S

+

−+ ++ +J +

[ + + | ] ,

(C.17)

where [ + + | ] is a shorthand notation for the following triple product of 3j symbols, 1/2   [ + + | ] = (2S + 1) (2j + 1) 2J + + 1

s S+ S ×  − + − + + + − + − +

 s j ×

− +  − + − + + − + + +

J j J × , +  − + −

(C.18)

in which the substitutions ω =  − + , λ = − + and σ = ω − λ have been made. The physical interpretation of the successive Wigner coefficients is that the electron spin is first uncoupled from S+ and then coupled to to form j. Finally, j is uncoupled from the molecular axis in a manner analogous to the uncoupling of in (C.14).

C.4 Diatomic frame transformations

289

Table C.4 Allowed values of the case (a) and case (e) basis labels. The notation [n · · · m] implies n, n + 1, n + 2, . . . , m. In addition, S = S ± 1/2 in case (a) and j =  ± 1/2 in case (e). Case (a)

Case (e)

+



Comment

>0 0 0 0

[ + −  · · · max ] [1 · · · ] 0 0

+ [−S · · · S]

+ [−S · · · S] [1 · · · S] 0

max = min( + + , N) − − p − q + + J − S even

+

+

J+

Comment

>0 0 0

 

+ + −S + · · · S +   1 · · · S+ 0

[Jmin · · · J + j ] [Jmin · · · J + j ] [Jmin · · · J + j ]

+ Jmin = max(J − j, + ) − p − q + + J + − S + +  even

The forms of the parity-adapted basis functions in Table C.2 for = 0 therefore lead to the frame transformation matrix elements + +

+ +

+

+

+ J + j |S (J Mp, S  q ,ls) = N(−1)S − ++ +J +   + + + × [ + + | ] + δ + 0 (−1)p−q −S −J + [− + − + | ]

(C.19)

where N −2 = (1 + δ + 0 δ 0 δ 0 ) (1+δ + 0 δ + 0 ). The special term for + = 0 appears because the unsymmetrized state |a has non-zero overlap with both | ± e. The sign change of J + in the final term, compared with that in τ d in Table C.2, arises from an additional phase term 2+ associated with the sign change of + in the bracket. Moreover, J + + 2+ has the same parity as −J + because 2(J + + + ) is even. The corresponding form for = 0 is given by % + + + + + + + J + j |S (J Mp, S  q ,ls) = M(−1)S − ++ +J + [ + + | ] (C.20) & + (1 − δ 0 )(−1)p−q−S−J [ + + | − ] , where M −2 = (1 + δ + 0 )(2 − δ 0 ). Jungen and Raseev, from whom this discussion is taken, combine (C.19) and (C.20) into a single equation [6]. It is useful to recognize the constraints on the various quantum numbers. To avoid overcounting, the convention adopted for the case (a) basis is that +  0, with  0 for

+ = 0 and   0 for + = = 0 [6]. Symmetrized states allowed by this convention appear in both parity blocks except that p − q + + J − S must be even when  = =

+ = 0. Note also that  =  − , which means that the allowed values of  are set by the magnitude of S. Thus  = + [−S, S], where the square bracket implies integer steps between ±S. Allowed combinations of the various quantum numbers, with S = S + ± 1/2 are given in the upper part of Table C.4. The convention for the case (e) basis is that +  0, with +  0 for + = 0. Again, the resulting states occur in both parity blocks except that p − q + + J + − S + +  must be even for + = + = 0. The allowed combinations, with j =  ± 1/2, consistent with this convention are given in the lower half of the table.

290

Rotational frame transformations b(x)

O

a(z)

+c(y)

H1

H2

Figure C.2 The Ir axis system for H2 O. The convention, with regard to e/f symmetry, for integer J values, is that p + J is even or odd for e and f levels, respectively. The corresponding rule when J is a half integer is that p + J − 1/2 is even for the e levels and odd for the f levels. C.5 Asymmetric tops The frame transformation for case (b) to case (d) excitation at fixed orbital angular momentum , in an asymmetric top was given by Child and Jungen [8]. The treatment below extends the discussion by using permutation inversion symmetry to generate case (b) and case (d) bases with arbitrary  consistent with a given irreducible representation [9]. The case of H2 O is employed as an illustration.

C.5.1 Symmetry considerations The molecular symmetry group C2v (M) contains four elements, each of which appears as a product of three terms acting on the vibronic, rotational and nuclear spin parts of the wavefunction [9] E=I (12) = C2b Rbπ p12 E ∗ = σ ab Rcπ p0 (12)∗ = σ bc Raπ p12 . Their effects on the molecular wavefunction depend on the chosen axis system, which will be taken in the Ir convention in Fig. C.2, such that (x, y, z) lie along the (b, c, a) inertial axes. The resulting transformation properties of the basis states |N KM, |λ, and |m are given in Table C.5. The entries for |N KM (and similarly for |N + K + M + ) are ∗ N derived from the transformation properties of DMK (φ, θ , χ ) [11, 9]. The behaviour of |λ follows from the identities [16] Yλ (π − θ , χ) = (−1)−λ Yλ (π − θ , χ) Yλ (θ, −χ ) = (−1)λ Y−λ (θ , χ ) .

(C.21)

Thus for example (x, y, z) → (x, −y, −z) under C2x , which means that (θ , χ) → (π − θ , −χ ), from which C2x |λ = (−1) | − λ. Finally, the space fixed electronic

C.5 Asymmetric tops

291

Table C.5 Transformation properties of the rotational and electronic basis states under the operations of the C2v (M) molecular symmetry group in the Ir representation. E E R0 |N KM |λ |m

(12) C2b Rxπ

E∗ σ ab Ryπ

(−1)N |N − KM (−1) | − λ |m

(−1)N−K |N − KM (−1)−λ | − λ (−1) |m

(12)∗ σ bc Rzπ (−1)K |N KM (−1)−λ |λ (−1) |m

state |m is invariant to the body-centred operation 12, and has parity (−1) under the space-fixed inversion E ∗ . Properly symmetrized case (b) and case (d) states may be generated by use of the projec∗ tion operator [9]. Those with characters (−1)p and (−1)p under (12) and E ∗ , respectively, are given by  ∗ ! 1 + (−1)+K−λ+p+p ∗ |b = λK; pp = √ 2 2(1 + δ λ0 δ K0 )   × |λ |N KM + (−1)N++p | − λ |N − KM  + ∗  ! 1 + (−1)+K +p+p + + ∗ |d = N K ; pp = √ 2 2(1 + δ K + 0 )  ! ! + × N + K + M + + (−1)N +p N + − K + M + |m . (C.22) It follows that p + p ∗ +  + K − λ = even

in case (b)

p + p∗ +  + K + = even

in case (d).

(C.23)

Hence K − λ and K + must be either both even or both odd, within a given representation. Since K and λ are integers, there is an equivalent type |b state with K replaced by −K, and it is convenient to form the following linear combinations appropriate to separately symmetrized electronic and rotational states    |λ + (−1)pe + | − λ |N KM + (−1)pr +N |N − KM |bs  = , (C.24) √ 2 (1 + δ λ0 )(1 + δ K0 ) subject to the constraint that pe + pr = p. Notice that the case (d) form in (C.22) and the above expression for |bs  differ from equations (12) and (15) of Child and Jungen [8]. The present form is more convenient for the simultaneous generation of symmetrized bases, belonging to a given (p, p∗ ) representation, in situations involving multiple  values. It is easy to derive the separate rotational and electronic constraints pe + pe∗ +  − λ = even;

and

pr + pr∗ + K = even.

(C.25)

292

Rotational frame transformations

Table C.6 Properties of the rotational and electronic basis functions for different (pr , pr∗ ) and (pe , pe∗ ) combinations. The subscripted signs on the electronic orbitals are the signs of (−1)pe + . The character with respect to nuclear exchange is given by (−1)pr +pe +pc , where pc is the pe value corresponding to c . pr

pr∗

Ka

Kc

pe

pe∗

−λ

s

p

d

f

c

0 0 1 1

0 1 1 0

Even Odd Even Odd

Even Odd Odd Even

0 0 1 1

0 1 1 0

Even Odd Even Odd

sσ − − −

pπ − − pπ + pσ

dσ ,dδ + dπ + dδ − dπ −

fπ − ,fφ − fδ − fπ + ,fφ + fσ ,fδ +

A1 A2 B1 B2

In addition, since (12)∗ and E ∗ involve rotation about the a and c axes, respectively, (pr + pr∗ + Ka ) and (pr∗ + Kc ) are both even [9], which means that pr + Ka + Kc = even

in case (b)

p + Ka+ + Kc+ = even

in case (d),

(C.26)

where the case (d) equality follows by comparison between the rotational factors in (C.22) and (C.24). These properties are summarized in Table C.6, in which the point group representations c , for the ionic core, have the characters implied by the entries under pe and pe∗ . The representations spanned by Rydberg orbitals |λ±  for  = 0−3 are also given, using the notation ± for the sign of (−1)pe + in (C.24). The table is also useful for determining the nuclear spin symmetry, because the character under the nuclear exchange operator is (−1)p , where p = pr + pe + pc , in which pc is equal to the pe entry appropriate to c . Equations (C.23) and (C.26) provide useful selection rules for both the frame transformation and allowed optical transitions. Since the frame transformation simply involves a projection from one representation to the other at fixed orbital angular momentum , both p and p ∗ are conserved. Hence, by comparison between the two lines of (C.23), K − λ + K + = even.

(C.27)

The case of optical excitation is more interesting because the radiation source is invariant under a (12) permutation, which means that p is conserved. On the other hand, the parity index p∗ under the inversion operation E ∗ changes according to the number of absorbed photons, which is equal to the change in the orbital quantum number, . Thus the transition from a case (b) and case (d) states |λ

K

; pe

pr

 and |N + K + ; p is subject to the selection rule Ka

− λ

+ Ka+ = even,

(C.28)

after identifying the K quantum numbers with the a-axis projections Ka . Taken in conjunction with the difference between the two lines in (C.26) this implies the additional condition pe

+ Kc

− λ

+ Kc+ = even.

(C.29)

C.5 Asymmetric tops

293

C.5.2 The case (b) to case (d) frame transformation The full frame transformation for an asymmetric top must allow for angular momentum mixing in the ionic core and the asymmetric top character of the rotational functions, as well as the angular momentum transformation associated with the Rydberg excitation. It therefore has the general structure  i |α = i |d d |bs  bs |α, (C.30) dbs

where the elements i |d are derived from the transformation matrix that diagonalizes the positive ion asymmetric top Hamiltonian in the |d basis. Similarly, the elements bs |α include electronic factors derived from the matrix that diagonalizes the electronic K-matrix and rotational factors obtained by diagonalizing the neutral molecule rotational Hamiltonian. The angular momentum transformation d |bs , which is block diagonal in  for given p and p ∗ , is obtained by the methods of the previous sections. d |bs N = N + K + |λKNMpp 

∗



(C.31)   1 + (−1)pe +pr +p (2N + + 1) = (−1)J −M 2(1 + δ K + 0 )(1 + δ λ0 )(1 + δ K0 ) 2 ( + +

  N  N N pe + N × + (−1) K + λ −K K + −λ −K  + +

)  N  N N N p+N + pe + + (−1) + (−1) , −K + λ −K −K + −λ −K

which differs from equation (17) of Child and Jungen [8] by the appearance of pe +  in place of pe and of p + N + in place of p + N + + . Naturally, these differences in labelling have no effect on the underlying selection rules. Thus the transformation element vanishes unless the case (d) permutation symmetry index p coincides with pe + pr . In addition, the Wigner coefficients require that K + = |K ± λ|, which confirms that K + and K − λ must be either both positive or both negative, in accordance with the characters under E ∗ in Table C.5. The convenience of the present forms lies in the systematic construction of basis states of a given permutation inversion symmetry for arbitrary values of , because the factors + (−1)pe + and (−1)p+N correspond to those in the properly symmetrized states given by (C.24) and (C.31). The case (b) basis states, for given N, p and p∗ , may be labelled by , λ, K and pr , with pe = p − pr . The projections λ and K must lie in the ranges 0  λ   and 0  K  N subject to the restrictions that  + λ + K + p + p∗ = even, with the additional conditions K = λ = 0 ⇒ N +  + p = even K = 0, λ = 0 ⇒ pr + N = even K = 0, λ = 0 ⇒ pe +  = even ⇒ pr +  + p = even.

294

Rotational frame transformations

The corresponding symmetrized case (d) basis states are labelled by , N + and K + , where |N − |  N +  N +  and 0  K +  N + . Finally, even and odd K + values are given by the rules p + p∗ +  = even ⇒ ( + K + ) = even p + p∗ +  = odd ⇒ ( + K + ) = odd. subject to the special condition K + = 0 ⇒ p + N + = even.

Appendix D Optical transition and photo-ionization amplitudes

This appendix is concerned with the evaluation of the reduced matrix elements [ T k 

], which relate the optical absorption or photo-ionization intensities to the body-fixed compositions of the initial and final wavefunctions. They involve a combination of angular momentum matrix elements multiplied by coefficients Zi or Aα in the equivalent MQDT forms for the wavefunctions    r = i fi (r)δij − gi (r)Kij Zj ij

=



α [fα (r) cos π μα − gα (r) sin π μα ] Aα .

(D.1)

α

The relevant equations for these coefficients are given in Sections B.1 and B.3, respectively. D.1 Discrete absorption amplitudes In the case of discrete absorption, the transition amplitudes are determined by the inner parts of the wavefunction. Hence the matrix elements are most conveniently evaluated in the coupled |α representation,  - k -

  ∗  - k -

  - T - = Aα α - T -α Aα

, (D.2) α α

as described by Hund’s case (a), (b) or (c) in a diatomic molecule. Square brackets are employed here for consistency with the subsequent theory of photo-ionization. The states in question may be denoted as |γ J M = |ηJ M for a linear molecule (or |ηJ KM for a symmetric top). Moreover in cases where the electronic angular momentum is well-defined, the vibrational electronic factor may be expressed as |η = |vλ. The required reduced matrix elements are defined by the Wigner–Eckart theorem

' (k)





!  k J

 J J −M γ J M Tq γ J M = (−1) γ J - T (k) -γ

J

. (D.3) −M q M

  In addition, explicit expressions for η J  - T (k) -η

J



may be obtained by transforming the space-fixed spherical tensor operator Tq(k) to body-fixed axes [11],  ∗ k Tq(k) = Dqν (R) Tν(k) . (D.4) ν

295

296

Optical transition and photo-ionization amplitudes

Hence the transition amplitude between the unsymmetrized states is given by  ' (k)





! !  k ∗  J

∗ 3 ' J DM γ J M Tq γ J M = A (R) Dqν (r) DM d R × η Tν(k) η



(R) ν

(D.5) √

2

where A = [J ][J ]/8π , in which [J ] is a shorthand notation for 2J + 1. It follows by combining (D.3) with the properties of the rotation matrices that [11]

 J  - (k) -







 k J

' (k)

! η J  - T -η J  = (−1)J − [J ][J

] η Tν η . − ν 

ν

(D.6) Furthermore, if the orbital angular momentum of the excited electron is well-defined

-  ' (k)

!

 k 

 (D.7)  - T (k) -

d

λλ

, η Tν η = (−1) −λ v |v



−λ ν λ -   where  - T (k) -

, with round brackets, is a single-electron reduced angular matrix

element, d

λλ

is the radial matrix element1 and v |v

 is the vibrational Franck–Condon factor. The extension to the parity-adapted case (a) states in Table C.2 yields # $



 - (k) -



 1 + (−1)p +p + + a ,p T a ,p = √ 2 (1 + δK 0 δ + 0 )(1 + δK

0 δ + 0 )  -  

× (−1) +J −K −λ [J ][J

]v |v

  - T (k) -

d

λλ



 J



 k J k  × (D.8) −K ν K

−λ ν λ

, ν

Notice that the term in the numerator of the first term is obtained by combining the parity indices τa and τa

in Table C.2, with the properties of the 3j symbols under sign reversal of the elements in the second row, bearing in mind that S = S

. It carries the selection rule p − p

= 

− 

because p and  are integers. The corresponding expression for transitions between case (b) states may be obtained by substituting |N KN  for |J K in (D.6)–(D.8). Moreover, the two expressions may be related by the following identity, implied by equation (5.72) of Zare [11],  - (k) -







 γ J (N S )- T -γ J (N S ) 



= (−1)N +S +J +k [J ][J

] ( )

 J S γ N - T (k) -γ

N

δS S

. × N (D.9)



J N k 1

For example in the case of a single photon [11]  - (1) -



   - rˆ - = (−1) [ , 

] 0

1 

0 0



.

The influence of the quantum defects on the radial matrix elements d

λ

λ for Rydberg–Rydberg transitions is discussed in Section D.3.

D.2 Photo-ionization amplitudes Thus,

  - (k) -





a , p - T -a , p = (−1)N +S+J +k (2J + 1)(2J

+ 1) ( )

  N J S × J

N

k b , p - T (k) -b

, p

,

297

(D.10)

where S = S = S

. Finally, one should note that the corresponding asymmetric top matrix elements, for transitions between case (b) and case (d) states, are given by  - (k) -





 d p - T -b pe pr ) (





N + × 1 + (−1) +

+np  = C(−1) −λ N N

k (



 

N + N

+ (−1)N + +p 

N + N

× λ

K + −K

λ

−K + −K





)

+

N+ N

  N N pe

+

N + +p , + (−1) −λ

K + −K

+ (−1) −λ

−K + −K

(D.11) where

 C=

 - (k) -

 [J

][J ][N + ]  - T - . 2(1 + δλ

0 )(1 + δK

0 )(1 + δK + 0 )

(D.12)

The 3j symbols carry the selection rule |J

− 

|  N +  J

+ 

, while the body-fixed projections, K

and K + , which correspond in asymmetric top notation to Ka

and Ka+ , are restricted by Ka+ = |Ka

± λ

| because the symmetrization in (C.22) restricts K + to positive values. In addition, (C.23) gives the parity-selection rule pe

+ Kc

− λ

+ Kc+ = even. D.2 Photo-ionization amplitudes

D.2.1 Multichannel boundary conditions The theory of photo-ionization is more complicated, because the electron is excited to an outgoing wave. Moreover, the elementary theory in Section 7.1, must be generalized to allow for multiple target states of the positive ion – or in quantum-defect language multiple open channels |j , with channel functions j . The aim is to construct combinations of the channel functions and electron waves with momenta pj = k¯h, which satisfy the incoming boundary conditions,

 fij (θk.r ) −ikj r (i−) −3/2 ikj .r (D.13) j k ∼ (2π) e e δij + j , r i appropriate for ionization in a particular channel |i. To satisfy these conditions in quantum defect notation, recall that there are N o open channels, and hence N o eigenchannel solutions of (2.42), with total wavefunctions of the form r (ρ) =

N  j =1

j ψj(ρ) ,

ρ = 1 . . . N o,

(D.14)

298

Optical transition and photo-ionization amplitudes

where the sum is taken over all channels. In addition, the open-channel components satisfy the boundary condition.2   (ρ) ψj ∼ (π kj )−1/2 Tjρ sin kj r + δj + π τρ ∼

  1  Tjρ eiπτρ eikj r+iδj − e−ikj r−iδj −2iπτρ , 2i π kj

(D.15)

in which Tjρ are elements of the N o × N o orthogonal matrix, related to the eigenchannel coefficients Zjρ of (2.42), in the form Zjρ = cos π τρ Tjρ . The eigenphases π τρ , which arise from short range interaction with the core, are common to all channels, while δj = ηj − j π/2 is a long-range phase, specific to channel j , which includes the Coulomb phase shift   (D.16) ηj = arg (j + 1 − i/kj . After taking particular account of these phase factors and the orthogonality of the T matrix, the above boundary condition may be satisfied by the combination  (ρ) ψj(i−) = ψj e−iπτρ TρiT e−iδi ρ

∼

 1 eiki r δij − e−iki r Sij(−) , √ 2i π ki

where Sij(−) = e−i(δi +δi )



Tiρ e−2iπτρ TρjT ,

(D.17)

(D.18)

ρ

are elements of the photo-ionization scattering matrix. It follows from (D.1), (D.14) and (D.16) that the properly defined wavefunction in channel |i takes the form r (i−) =

N 

j ψj(i−)

(D.19)

j =1 N No  1  1 iki r ∼ j √ e δij − e−iki r Sij(−) . 2i j =1 &=1 π ki

Hence by comparison with (D.1) and (D.2) the required reduced matrix element may be expressed as  (i−) - k -

  (−)  - k -

  - T - = Sii i - T -α Aα

, (D.20) i α

where the state |j  in (D.13)–(D.18) has been replaced by |i . The reduced total transition matrix element therefore includes terms Aα

from the composition of the initial state, from final state interactions between the [i T k α

] from the photon excitation and Sii(−)

photo-excited state |i  and the target state |i. 2

The radial variation of the logarithmic term in (2.5) is negligible compared with kr as r → ∞.

D.2 Photo-ionization amplitudes

299

An alternative expression in terms of the eigenchannel solutions r (ρ) may be obtained by combining (D.14), (D.17) and (D.19). Thus  r (i−) = r (ρ) e−iπτρ TρiT e−iδi , (D.21) ρ

so that (D.20) goes over to  (i−) - k -

   - T - = [ρ T k α

]e−iπτρ T T e−iδi Aα

ρi

(D.22)

ρα

D.2.2 Photo-ionization matrix elements The discussion is restricted for simplicity to diatomic molecules, for which the forms of the ' ! photo-ionization matrix elements j Tqk α

have been discussed by Buckingham et al. [18] and Xie and Zare [19]. The former is more convenient for comparison with the formal photo-ionization theory of Fano and Dill [20, 21]. As an introduction, we consider the ionization of a diatomic molecule in Hund’s case (a) to form an ion also in case (a). The relevant entries in Table C.2 are modified in the forms ! ! ! α

= a

, p

= J



M

η



S



, p

(D.23) ! ! !

+

+ + + + + + +

i = a ; m sms ; p = J  M η S  ; m sms ; p , where |η



 and |η+ +  are vibrational electronic states of the neutral molecule and the positive ion. For simplicity, we shall work initially with the unsymmetrized states |a

 and |a +  and

= 0. Note that p = p+ +  in the second line, is the parity of the total ionized system. After using (D.4) and its analogues to transform |m  and |sms  to the body frame, the matrix element between unsymmetrized states may be written as a combination of products of orientational and electronic integrals [19], ' + !  a ; m sms Tq(k) a

= I1 (λ, σ, ν)I2 (λ, σ, ν). (D.24) λσ q

Thus I1 (λ, σ, ν) =

√

[J + ] [J

] 8π 2



∗ J+  s k J

DM d3 R, + + (R)Dm λ (R)Dm σ (R) Dqν (R)DM



(R)  s (D.25)

and ! ' I2 (λ, σ, ν) = S +  + η+ + λsσ Tνk S



η



,

(D.26)

where [J ] = 2J + 1 and |η  designates a vibrational electronic state. The integrand of the orientational integral I1 (λ, σ, ν) is reduced to a familiar product of three rotation matrices [11] by using pairwise Clebsch–Gordan transformations, each of which corresponds to a particular angular momentum coupling identity. Differences in the order of these latter transformations yield a variety of equivalent formulae, appropriate to alternative angular momentum identities. Thus Xie and Zare [19] emphasize the connection

300

Optical transition and photo-ionization amplitudes

with photo-excitation to a discrete state, by employing transformations appropriate to the total angular momentum identities j= +s J = J+ + j = k + J

.

(D.27)

However, the discussion in Chapter 7 shows that the photo-ionization cross-sections are more conveniently expressed in terms of angular momentum transfer vectors [18, 20, 21]3 t = − k Jt = J − J

= k − j = − t − s,

(D.28)

+

of which Jt determines the rotational branch structure of the photo-electron spectrum, while t is the orbital angular momentum transferred by the radiation. The following account uses this transfer approach. To indicate the method, the first line of (D.28) implies the contraction [11]  k  k Dm (R)[Dqν (R)]∗ = (−1)λ−m [D−m (R)Dqν (R)]∗ λ  −λ

  k t  λ−m = (−1) [t ] −λ ν λt −m t λt mt

k p

t mt

t (R), Dm t λt

(D.29) which is common to all angular momentum coupling schemes. Here the symbol [t ] = 2t + 1 and formal sums over λt and mt are included for later convenience although the non-zero terms are restricted to λt = λ − ν and mt = m − q. The first equality in the second line of (D.28) has a similar contraction, and the final integral is taken over the t Jt s product Dm (R)Dm DM (R), to yield sσ t λt t t    + + I1 (λ, σ, ν) = (−1) −M +λ−m [J + ] [J

][t ][Jt ] × ×

t λt mt Jt t Mt

 −λ  −m

k ν

t λt k q



t mt

t λt



s σ t mt

Jt t s ms



Jt Mt

J+ −+



J



Jt t

J+ −M +

J

M



Jt Mt

,

(D.30)

which coincides, apart from differences in notation, with equation (33) of Buckingham et al. [18]. The present use of the symbol Jt in place of χ is helpful in pointing to the connection with the transfer theory of Fano and Dill [20, 21]. The second integral takes the form ! ' I2 (λ, σ, ν) = S +  + η+ + λsσ Tνk S



η



(D.31) ' + + ! = S  sσ S



 λ| Tνk η



+

! ' + + +

 S s S

η λ Tνk η



, = (−1)S −s+ [S

] +

 σ − where |η



 denotes the initial vibrational electronic state. 3

Fano and Dill [20, 21] actually exclude the spin, which requires only one transfer vector because the second line of (D.28) means that Jt and t then coincide.

D.2 Photo-ionization amplitudes 301 ' + ! The full matrix element a ; m sms ; p Tp(k) a

p

between parity-adapted states is obtained by combining (D.24), (D.30) and (D.31). The forms implied by the parity construction in (C.8) and Table C.2 give rise to a combination of two terms, which combine via the symmetry properties of the Wigner coefficients, to yield a factor # $

1 + (−1)p +p ++t P= , (D.32) 2 which carries the parity condition that p + p

+  + t must be even. Similar arguments apply for other angular momentum schemes. The results for selected cases are given below, of which the first four were derived in a different notation by Buckingham et al. [18]. The photo-ionization selection rules arising from them are discussed in Section 7.2.

|a

p

 → |a + ; m sms ; p  transitions The resulting full matrix element implied by (D.24), (D.30) and (D.31) is given by ' + + + + + + + ! J M  S  η ; m sms Tq(k) J

M



S



η



; p

   - ! ' +

[Jt ] [J + ] [J

] [S

](−1)γ (a ,a ) Paa (, t ) η+ - T (k) (t ) -η

= Jt Mt t σ t





+

s Jt J

Jt  k t t J × −m q mt m t m s Mt −M + M

Mt +



+

S s S

J

Jt t s Jt J × ,  + σ −

λt σ t −+ 

t

(D.33)

where γ (a + , a

) =  + S + − s + 

+ + − M + − m . In addition,

- !  ! ' + - (k)  k t ' + + [t ](−1)−λ η - T (t ) -η

= η λ Tνk η



, −λ ν λt

(D.34)

λν

and Paa (, t ) is the parity factor #

$

1 + (−1)p +p ++t . √ (1 + δ + 0 δ + 0 δ+ 0 )(1 + δ

0 δ

0 δ

0 )

Paa (, t ) =

(D.35)

|b

p

 → |b+ ; m sms ; p  transitions The argument is simplified by conservation of the space quantized spin. Thus ' + + + + ! N MN η ; m sms ; p Tq(k) N

MN

η



; p

  - ! ' +

(−1)γ (b ,b ) Pbb (, t ) η+ - T (k) (t ) -η

= [N + ] [N

]  S+ × Ms+

t

s ms

t

×

N+ − + N

t λt

N



S

−MS

,



 −m

k q

t mt



N+ −MN+

t mt

N

MN

(D.36)



302

Optical transition and photo-ionization amplitudes

where γ (b+ , b

) =  + + − M + − m , and $ #

1 + (−1)p +p ++t Pbb (, t ) = √ (1 + δ + 0 )(1 + δ

0 )

(D.37)

|a

p

 → |b+ ; m sms ; p  transitions Since the electron and positive ions are space quantized, it is convenient to apply the transformation ! 



! S

S



= S MS DM



, (D.38) Ms

S

which yields ' + + + + + + ! N MN η S MS ; m sms ; p Tq(k) J

M



S



η



; p

   - ! ' +

[Jt ] [J + ] [J

] [S

](−1)γ (b ,a ) Pab (, t ) η+ - T (k) (t ) -η

= Jt Mt t σ t





+

N J

Jt S

Jt  k t t × −m q mt mt MS

Mt −MN+ M

Mt +



+

S s S

J

Jt t S

Jt N × , λt 

t Ms+ ms −MS

− + 

t where γ (b+ , a

) =  − s + S + + MS

+ + − M + − m . Finally, # $

1 + (−1)p +p ++t Pba (, t ) = √ . (1 + δ + 0 )(1 + δ

0 δ

0 δ

0 )

+

+

(D.39)

(D.40)

|b

p

 → |a + ; m sms ; p  transitions

In this case |S  is transformed by the analogue of (D.38) to obtain ' + + + + + + + ! J M  S  η ; m sms ; p Tq(k) N

MN

S

MS

η



   - ! ' +

[Jt ] [J + ] [J

] [S

](−1)γ (a ,b ) Pab (, t ) η+ - T (k) (t ) -η

= Jt Mt t σ t



× ×

 −m

k q

S+ Ms+

s ms

+ S+ Jt t J mt −MS + Mt −M +



+ S

S+ Jt t J −MS

λt − + t −+



m



N

MN

N



where γ (a + , b

) =  − s + S + + MS

+ + − M + − m , and # $

1 + (−1)p +p ++t Pab (, t ) = √ . (1 + δ + 0 δ + 0 δ+ 0 )(1 + δ

0 )

Jt Mt

Jt , t

(D.41)

(D.42)

D.3 Dipole radial matrix elements and Cooper minima

303

|c

p

 → |c+ ; j mj ; p  transitions The final example assumes strong spin–orbit coupling in both the neutral and the ion, for which it is natural to consider transitions to an excited |j (s)mj  wave. The relevant matrix element takes the form ! ' + + + + (D.43) J M η  ; j (s)m; p Tq(k) J

M

η



; p

  ! '



[jt ](−1)j −m+M − Pcc (, jt ) η+ j - T (k) (jt ) -η

= [J + ] [J

] Jt



j × −m

k q

jt −mt



J+ M+

where - !  ' + - (k) j η j - T (jt ) -η

= [jt ](−1)j −ω −ω ων

and

# Pcc (, jt ) =

J

−M

jt −mt

k ν

jt ωt





J+ +

jt −ωt

J

−



! ' + + η  j ω Tνk η



,

$

1 + (−1)p +p ++jt −s . √ (1 + δ+ 0 )(1 + δ

0 )

The matrix element ' + + + ! J η  ; m sms ; p Tq(k) J

η



; p

 ! ' j m |m sms  J + η+ + ; j (s)mj ; p Tq(k) J

η



; p

=

,

(D.44)

(D.45)

(D.46)

j

goes over to a form with the same space-fixed terms as (D.33), under a suitable 6j transformation [11]. D.3 Dipole radial matrix elements and Cooper minima The factors affecting radial matrix elements differ according to whether the transitions occur from tightly bound ground state orbitals within the core or from more loosely bound Rydberg orbitals. Transitions from molecular ground states normally occur from orbitals within the core. In the former case, the bound to bound matrix elements are expected to follow the usual ν −3/2 scaling law. In addition, the wavelength of the threshold continuum orbital is large compared with the range of the initial one, which means that the sign of threshold matrix element is dictated by that of the outermost lobe of the inner orbital. The increasing wavelength of the continuum wave at higher energies causes the matrix elements for nodeless, 1s, 2p, 3d, etc., orbitals to decrease monotonically. However, the nodal structure of the higher orbitals allows interference between the contributions from successive lobes, such that the matrix element passes through zero, leading to a ‘Cooper minimum’ in the relevant contribution to the cross-section [22]. Different considerations apply, for example, to the second step of a resonant two-photon transition, because the dominant contribution to the matrix element comes from well outside the core region, as may be confirmed by analysis of the integrand of the matrix elements  ∞ n



dn  = Rn  (r)rRn



(r)dr, (D.47) 0

304

Optical transition and photo-ionization amplitudes

between hydrogenic wavefunctions. The known analytical expressions for such elements with a given positive value of n − n

are also found to be systematically larger when  = 

+ 1 than when  = 

− 1 – a difference that is attributable 2 the more  3p to3pthe oscillatory radial function with the lower 

value. For example, d4d /d4s  9.5 and  3d 3d 2 d4f /d4p  61.5 [23]. Bates and Damgaard (BD), who pointed to the relative unimportance of contributions from the core region, extended the discussion to arbitrary Rydberg wavefunctions by giving tables derived from the outer parts of the asymptotically decreasing Coulomb functions, which allow an assessment of the influence of the quantum defects [24]. The following discussion follows BD in employing this ‘coulomb approximation’ for bound–bound matrix elements and extending related arguments to continuum ones. With regard to the former, BD employed analytical integrals over asymptotic approximations to the appropriate confluent hypergeometric functions, which were truncated to avoid divergence due to the r − behaviour of the irregular components as r → ∞. Normalization factors were determined by analytical continuation of the form appropriate to integer ν values. The alternative followed below, which is equally applicable to bound and continuum functions, is to employ numerical integration over numerically determined Coulomb functions, with the integration range [a, ∞] truncated at the inner turning point of the Coulomb motion appropriate to the larger of the two angular momenta  and 

. The relevant bound functions were numerically integrated inwards from a large radius and normalized to unity over the range [a, ∞]. The corresponding continuum functions4 are given, according to (2.27), by R (r) = cos π μ f (, r) − sin π μ g (, r),

(D.48)

where  = (E − I )/Ry . Figures D.1 (a) and (b) show the resulting variation of the bound–bound 3p→ (νs, νd) and 3d→(νp, νf) radial matrix elements as a function of ν = n − μ, using the phase convention that the outermost lobes of Rν (r) are positive. The oscillations show the influence of the quantum defects on the matrix elements. Given that the outermost lobe of Rν (r) is positive, the inner parts of the wavefunction at integer ν = n values must oscillate in sign, which means that the integral must change sign at some intermediate value of ν = n − μ – an argument that is equally applicable to tightly or weakly bound initial orbitals. By the same reasoning, differences between the quantum defects for different λ projections imply differences between the radial matrix elements, though these will be relatively small for non-penetrating d, f, etc., orbitals. 4

The basis functions (f , g ) were typically evaluated, as (s , −c ) by the Seaton routines [25]. Instabilities at long range and high continuum energies were avoided by matching to the Langer-corrected JWKB forms [26] ) ( 1 f (, r) sin [φ (, r) + π/4] ≈ √ , g (, r) π k (, r) cos [φ (, r) + π/4] where

 φ (, r) =

r

k (, r)dr, a

in which k2 (, r) =  +

( + 1/2)2 2 − r r2

and a is the turning point. Analytical forms for φ (, r) are readily derived.

Integral

D.3 Dipole radial matrix elements and Cooper minima 8 6 4 2 0 −2 −4 4

305

(a) 3p − νs

5

3p − νd

6

7

8

9

10

10 Integral

(b) 5

3d − νp

3d − νf

0 4

5

6

7 ν

8

9

10

Figure D.1 Variation of bound–bound radial matrix elements with ν = n − μ. (a) 3p→ νd (solid) and 3p→ νd (dashed); and (b) 3d→ νf (solid) and 3d→ νp (dashed). Both upper and lower states were normalized to unity. Figure D.1 is actually drawn for integer effective quantum numbers ν

= 3, but similar shifted oscillatory traces are found for ν

= n

− μ

. Moreover, Bates and Damgaard accommodated this shift by adopting the notation [24] ⎡  ⎤  ∞ 3ν ν2 − 2 ⎦ I(ν −1 , ν , ), Rν−1 −1 (r)rRν  (r)dr = ⎣ (D.49) 2 0 in which I(ν −1 , ν , ) is strongly dependent on the difference ν − ν−1 but relatively insensitive to ν . The implication for Fig. D.1 is that the the radial integrals for ν

= 3 − μ

and  = 

+ 1 may be obtained by displacing the oscillatory trace by the quantum defect μ

. The corresponding shift for  = 

− 1 requires an additional scaling in magnitude according to the change in first factor on the right of (D.49). Careful inspection of the traces shows that the separations between successive nodes in the various traces are very close to unity, which could lead to accidental intensity cancellation for an entire ν



λ

→ ν  λ series, if the excited quantum defect were strictly constant. However, the more common weak energy dependence would account for a bound state manifestation of the ‘Cooper minimum’, at which the spectroscopic intensity dips to near zero between successive members of a series. One may also note that the envelopes of the oscillatory traces are larger for the higher value of  = 

± 1, although the displacement between the nodes of the solid and dashed traces means that there are ranges of ν over which this order is reversed. Finally the amplitudes of the envelope functions vary approximately as ν −5/2 , which is significantly stronger than the ν −3/2 variation of quantities accumulated inside the core region. Turning to the bound–continuum matrix elements, Fig. D.2 shows that reorganization of the inner nodes of the continuum function, resulting from changes in the quantum defect, again cause sign changes in the matrix element, arising from the trigonometric terms in (D.48), when μ increases by one unit. The lines in these plots were computed for regular 3p and 3d bound states, with μ

= 0, while the dots were computed for regular continuum

306

Optical transition and photo-ionization amplitudes 20 3p − εd

5 0 −5

(a)

3p − εs

−0.4 −0.2 0

Δμ

0.5

Integral

Integral

10

ε

0.3 0.2 0.1 0

3p − εs

−10

−0.4 −0.2 0

0.2 0.4

−0.4 −0.2 0

0.4

ε

0.3 0.2 0.1

(c)

3d − ε p

(b) 0.2 0.4

0.5

3p− εd

0.4

0

−20

0.2 0.4

3d − εf

10

0

3d − εf 3d − εp

(d) −0.4 −0.2 0

0.2 0.4

Δμ Figure D.2 (a) and (b) Variation of the threshold ( = 0) bound–continuum 3p→ν and 3d→ν radial matrix elements vs the quantum defect difference μ = μ − μ

. Panels (c) and (d) show the loci of zeros of the integral in the (μ, ) plane. In all cases, lines and points apply to μ

= 0 and μ = 0, respectively. functions, with μ = 0 and variable μ

. The agreement between the two types of trace strongly suggests that the bound continuum matrix elements are also mainly dependent on the quantum defect difference, μ, although it is difficult to handle the general case, because the integrals become sensitive to the precise choice of the lower integration limit, particularly for ν

d→ f transitions. The second two panels, in Figs D.2 (c) and (d), show how the energies of possible Cooper minima, at which the integral vanishes, vary with the quantum difference μ – either at μ

= 0 (lines) or at μ = 0 (points). The close agreement between the points and lines suggests that the positions of Cooper minima are again principally dictated by the quantum defect difference μ.

Appendix E Generalized MQDT representation

The quantum defect formalism may be extended from the Coulomb equation to the general Schr¨odinger equation   h¯ 2 d2 J (J + 1)¯h2 − + V (R) + (E.1) E (R) = EE (R), 2μ dR 2 2μR 2 in which V (R) takes the single minimum potentials of the form in Fig. E.1. Two independent solutions F (E, R) and G(E, R) are chosen, of which the first is taken to be regular at the origin, while the second is related to it in the phase–amplitude form [27, 28] " 2μ F (E, R) = α(E, R) sin φ(E, R) πh¯ 2 " 2μ α(E, R) cos φ(E, R), (E.2) G(E, R) = − π¯h2 where the amplitude and phase are connected by φ (E, R) = α(E, R)−2 . The following Wronskian relation is easily verified: W [F, G] = F (E, R)G (E, R) − G(E, R)F (E, R) =

2μ . πh¯ 2

(E.3)

Open-channel solutions At energies above the dissociation limit, these solutions are uniquely determined by the asymptotic conditions "   2μ F (E, R) ∼ sin kR − J π/2 + ηJ 2 πh¯ k "   2μ (E.4) cos kR − J π/2 + ηJ , G(E, R) ∼ − 2 π¯h k with k 2 = 2μE/¯h2 , and the square-root factors ensure the energy normalization  ∞ f (E, R)f (E R)dR = δ(E − E ).

(E.5)

0

307

308

Generalized MQDT representation

V(R)

D

E

R

Figure E.1 The potential function

Normal closed-channel solutions The proper asymptotic behaviour at energies between the potential minimum and the dissociation limit is ensured by analogy with (2.12). Thus F (E, R) = sin β (gen) (E)X+ (E, R) − cos β (gen) (E)X− (E, R) G(E, R) = − cos β (gen) (E)X+ (E, R) − sin β (gen) (E)X− (E, R),

(E.6)

where X + (E, R) and X− (E, R), respectively, increase and decrease exponentially as R → ∞, and β (gen) (E) is defined such that sin β (gen) (E) vanishes at the eigenvalues of (E.1). For example, within the JWKB approximation [26], β

(gen)

1 (E) = h¯



b

   2μ E − V (R) − J (J + 1)¯h2 /2μR 2 dR − π /2.

(E.7)

a

However, the determination of an exact form for β (gen) (E) is complicated by a wellknown ambiguity in its behaviour between the eigenvalues, which arises from an ambiguity in the basis functions in (E.2) [29, 30, 31, 32, 33]. Even the regular function f (E, R) is only defined to within an arbitrary scaling constant. Moreover, even with a given scale factor, the phase–amplitude construction in (E.2) is not unique. Thus different choices for α(E, R) and φ(E, R) in (E.2) lead to different oscillatory variations in β (gen) (E) between the eigenvalues, and the aim is to minimize these oscillations as far as possible. To see the nature of the ambiguity, Jungen and Texier [33] show that   α 2 (E, R) Yf (E, R) − YX (E, R)   tan β (gen) (E) = , (E.8) 1 + α 4 (E, R) Yf (E, R) − Yα (E, R) [YX (E, R) − Yα (E, R)] where Yf =

1 F



dF dR

,

YX =

1 X−



dX− dR

,

Yα =

1 α



dα dR

,

(E.9)

Generalized MQDT representation

309

of which the first two are always available with arbitrary numerical accuracy, by numerical integration from the origin and from R → ∞, respectively. This shows that tan β (gen) (E) always vanishes at the eigenvalues, Ev , because Yf (Ev , R) and YX (Ev , R) are equal at all R, and hence that β (gen) (Ev ) = vπ . The values of α(E, R0 ) and Yα (E, R0 ), or α (E, R0 ), at a suitable reference point, R0 , remain to be chosen such that β (gen) (E) varies as smoothly as possible. The recommended choice [33] is that α(E, R0 ) and α (E, R0 ) should be taken consistent with a third-order corrected JWKB wavefunction, at the point R0 at which the third-order correction takes its smallest value. The forms of α(E, R) and φ(E, R), and hence of the basis functions F (E, R) and G (E, R), at other R values, may be obtained by solving the following Milne equivalent of the Schr¨odinger equation [27, 28]: 

 d2 1 2 + k (R) α = 3, dR 2 α

dφ 1 = 2, dR α

(E.10)

where k 2 (R) =

  2μ J (J + 1) E − V (R) − . R2 h¯ 2

(E.11)

Notice that propagation of the non-linear α equation requires knowledge of both α (E, R) and α (E, R) at the initial point R0 , whereas the shape of the wavefunction determined by (E.1) is fixed by knowledge of the ratio Yα (E, R) = [α (E, R)/α(E, R)]. The initial condition φ (E, R) at R = 0 is required to ensure the regularity of f (E, R).

Strongly closed channels This scheme may be extended to strongly closed channels, with energies close to or below the potential minimum. However, the choice of the initial propagation point, R0 , and the forms of α (, R0 ) and α (, R0 ) must be modified, because the third-order JWKB criterion is no longer appropriate. After considering various possibilities, Jungen and Texier [33] handled the energy range below the minimum by taking R0 at the minimum point and setting α (, R0 ) = κ −1/2 exp [κR0 ] ,

[α /α] = −[κ /2κ],

(E.12)

where κ 2 (R) is the negative of k 2 (R) given by (E.11). A cubic interpolation for α (E, R0 ) with respect to E was then employed to bridge an energy range roughly equal to twice the zero-point energy, around Vmin . When applied to the pure Coulomb problem, the resulting accumulated phase functions β (gen) (E) were found to follow the Seaton forms β (, ) = π [ν() − ], in (2.13), for the normal closed channels, and then to bend away towards β = 0 as ν → 0.

Appendix F Notation

F.1 Angular momenta Total angular momentum states are denoted |J M, where J specifies the magnitude and the roman and greek symbols, M and , are the space-fixed and body-fixed projections, respectively. Upper and lower states are indicated by primes, |J M   and double primes, |J

M



, respectively. Those of the positive ion are given as |J + M + + . The same conventions apply to the following possible components of |J M |LML  electronic orbital angular momentum |SMS  electronic spin angular momentum |Ja Ma a  total electronic angular momentum (L + S) |J M total angular momentum |N MN  ! total angular momentum excluding spin (J − S) N + MN+ + nuclear rotational angular momentum (N − L) |m λ single-electron orbital angular momentum |sms σ  single-electron spin angular momentum |j mω total single-electron angular momentum In addition, space-fixed and body-fixed spherical tensor operators are denoted as Tq(k) and Tν(k) , respectively. F.2 Reduced matrix elements Three types of reduced matrix elements, which can occur in the theory of photo-ionization, are distinguished by the use of square, pointed and round, brackets respectively.

Square brackets The simplest form of the first covers the space-fixed spherical tensor matrix elements between discrete total angular momentum states, as given by (6.33),

' (k)





!  k J

 J J −M γ J M Tq γ J M = (−1) γ J - T (k) -γ

J

. −M q M

The analogous form for photo-ionization applies to the matrix element for transitions to the coupled total angular momentum state |γ + J (J + , j ) associated with positive ion angular momentum J + and a spin-coupled electron wave with angular momentum j . 310

F.2 Reduced matrix elements

311

Following (7.17), ' + + ! γ J (J , j ) Tq(k) γ

J

M



 k J

 + + J J −M = (−1) γ J - T (k) (J , j ) -γ

J

, −M q M

the related linear combination in (7.20),   + + - (k) γ J - T (t , Jt ) -γ

J

  

+ = [j ][J ](−1)J −Jt −t −J − γ + J + - T (k) (J , j ) -γ

J

J j

(

 × Jt

s k

j t

) (

J j

k J+

) J , Jt

applies in the transfer representation, with defined angular momenta Jt = J+ − J

and t = − k.

Pointed brackets Detailed expressions for the photoionization elements above differ according to the angular momentum coupling schemes implied by the labels γ + and γ

but (7.31) and (7.33) show that it proves possible to factor out this dependence in such a way that  - !2 2 ' γ + J + - T (k) (J , j ) -γ

J

= Pγ + γ

(, t ) η+  - T (k) (t ) -η

Qγ + ,γ

(t ), Jt

where Pγ + γ

(, t ) is a parity factor and the pointed bracket is the reduced electronic transition matrix element in (7.22)

-

!  ' + - (k) !  k t ' + + −λ η  T (t ) η = [t ](−1) η λ Tνk η



. −λ ν λt λν

Expressions for Qγ + ,γ

(t ) for different coupling schemes are given in Table 7.1. A related pointed bracket

-

!  ' + - (k) ! j k jt ' + + j −ω η  T (jt ) η = [jt ](−1) η j ω Tνk η



−ω ν ωt ων

is applicable to situations involving strong spin–orbit coupling.

Round bracket Round brackets are employed for single-electron matrix elements,

- 



 (k)





k 

   - T (k) -

d λλ ,  λ Tν  λ = (−1) −λ

−λ ν λ



where d λλ is the radial matrix element.

312

Notation F.3 Other special brackets [J ] = 2J + 1 and [L, S] = (2L + 1)(2S + 1), etc.,

and [J1 · · · J2 ] = J1 , J1 + 1, J1 + 2, . . . , J2 . References [1] P. A. M. Dirac, Principles of Quantum Mechanics, 4th edn (Oxford University Press, 1958). [2] C. H. Greene and C. Jungen, Adv. At. Mol. Phys. 21, 51 (1985). [3] K. T. Lu, Phys. Rev. A 4, 579 (1971). [4] C. Jungen and M. Raoult, Faraday Discus. Chem. Soc. 71, 253 (1981). [5] F. Hund, Handb. der Physik, Band I 24, 561 (1933). [6] C. Jungen and G. Raseev, Phys. Rev. A 57, 2407 (1998). [7] U. Fano, Phys. Rev. A 7, 353 (1970). [8] M. S. Child and C. Jungen, J. Chem. Phys. 93, 7756 (1990). [9] P. R. Bunker and P. Jensen, Molecular Symmetry and Spectroscopy, 2nd edn (NRC Research Press, 1998). [10] G. Herzberg, Spectra of Diatomic Molecules, 2nd edn (van Nostrand, 1950). [11] R. N. Zare, Angular Momentum (Wiley-Interscience, 1988). [12] J. M. Brown and A. Carrington, Rotational Spectroscopy of Diatomic Molecules (Cambridge Molecular Science, 2003). [13] E. E. Nikitin and R. N. Zare, Mol. Phys. 82, 85 (1994). [14] J. T. Hougen, The Calculation of Rotational Energy Levels and Rotational Intensities in Diatomic Molecules (N. B. S. Monograph 115, 1970). [15] M. Larsson, Physica Scripta 23, 835 (1981). [16] D. M. Brink and G. R. Satchler, Angular Momentum, 2nd edn (Oxford University Press, 1968). [17] S. Pan and K. T. Lu, Phys. Rev. A 37, 299 (1988). [18] A. D. Buckingham, B. J. Orr and J. M. Sichel, Phil. Trans. Roy. Soc. London, A 268, 147 (1970). [19] J. Xie and R. N. Zare, J. Chem. Phys. 93, 3033 (1990). [20] U. Fano and D. Dill, Phys. Rev. A 6, 185 (1972). [21] D. Dill and U. Fano, Phys. Rev. Lett. 29, 1203 (1972). [22] J. W. Cooper, Phys. Rev. 128, 681 (1962). [23] E. U. Condon and G. H. Shortley, The Theory of Atomic Spectra (Cambridge University Press, 1957). [24] D. R. Bates and A. Damgaard, Phil. Trans. Roy. Soc. London, 242, 101 (1949). [25] M. J. Seaton, Comp. Phys. Commun. 25, 87 (1982). [26] M. S. Child, Semiclassical Mechanics with Molecular Applications (Oxford University Press, 1991). [27] W. E. Milne, Phys. Rev. 35, 863 (1930). [28] W. E. Milne, Am. Math. Mon. 40, 863 (1933).

References [29] [30] [31] [32] [33]

B. Yoo and C. H. Greene, Phys. Rev. A 34, 1635 (1986). H. J. Korsch, Phys. Lett. A 109, 313 (1985). H. J. Korsch and H. Laurent, J. Phys. B 14, 4213 (1981). I. Fourr´e and M. Raoult, J. Chem. Phys. 101, 8709 (1994). C. Jungen and F. Texier, J. Phys. B 33, 2495 (2000).

313

Index

ab initio 1, 3, 8–9, 210, 225, 236, 266 potential 1, 4, 47 quantum defect 23 R-matrix (see R-matrix) wavefunction 52–54 absorbing potential 1, 152 absorption (see photo-excitation) absorptivity 162 accumulated phase 22, 25, 30, 84, 86, 107, 264, 309 alignment of angular momentum 11, 12, 168, 221 coefficient 169, 222, 229 angular momentum commutation relations 256 control of 251 decoupling 158, 177–183, 283, 288 L (or LN ) projection 98, 103 mixing 81, 197, 205, 286 multipole 222, 227, 229, 230, 231 notation 283, 310 orientation and alignment 11, 12, 167, 192, 226–230, 232 recoupling 180, 196, 203, 227, 234, 288 transfer 12, 191, 196, 197, 203, 227, 300 ArH 79, 81, 95 asymmetric tops 174–177 asymmetry index 175 frame transformation 14, 290–294 transition amplitudes 157, 175, 297 asymmetry parameter Beutler–Fano 35, 135 photoionization 12, 191, 206–212, 208, 224, 230 asymptotic approximation (see also JWKB approximation) coulomb wavefunction 19 Wigner 3j coefficient 168, 258, 270 Wigner 6j coefficient 102, 221 auto-correlation function 246, 248

314

auto-ionization 2, 3, 7, 109–114, 118, 122, 125, 184, 185, 247 branching ratio 31, 109 linewidth 34, 111, 119, 154 time-dependent 252 axial recoil approximation 205, 217 axis convention (see rotational axis convention) band profile (see rotational band profile) Berry phase 116, 189 Bessel function 62 Beutler–Fano lineshape 34–36, 121–122, 134–135, 139, 253 Bloch operator 54, 59, 66 BOS theory 14, 191, 200, 300 Born-Openheimer approximation 1, 4–5, 7, 45 state 5, 90 wavefunction 105, 142 boundary condition 8, 28 closed channel 21–22, 268 complex 6–7, 11, 151 open channel 19, 29, 298 log-derivative 55, 66, 83, 141 photoionization 12, 14, 191, 192–193 branching ratio 31, 75, 109, 110, 147 Breit-Wigner 32, 134 Buckingham–Orr–Sichel theory (see BOS theory) Buttle correction 9, 60–62, 65, 141 CaF 79, 81, 82–85, 95 Casimir operator 257 CH4 188 channel 6–7 closed 6, 21–22, 29, 37–43, 72, 143, 275–277, 308 dissociation 7, 10, 130, 138 ionization 7, 10, 130 open 6, 19, 29, 30, 72, 110, 142, 252, 273–275, 298 strongly closed 6, 22–25, 57

Index channel interactions 9 Jahn–Teller 115–123 rotational 97–105, 120 vibrational 10, 48, 105–114 vibronic 114–123 channel potential 19, 6–7 chiral molecule 207 classical ensemble 243 classical recurrence time (see Kepler time) Clebsch–Gordan series 203, 218, 234 coefficients 268, 287, 299 CO molecule 217 coherence 222 coherent control 13, 239, 249–251 competitive fragmentation 115, 125–155, 135, 139, 142, 146 configuration interaction 45, 114–115 confluent hypergeometric function 86, 259, 304 continuum orbitals 52, 53, 66, 67, 143 Cooper minimum 208, 233, 303–306 core dipolar 79, 81, 95 polarization 58 radius 4, 79, 81, 95 transit time 4, 91, 94 Coriolis coupling 7, 79, 81, 102, 140, 146, 177, 180 correlation diagram 179 Coulomb approximation 266, 304 Coulomb field 3, 8 Coulomb functions 3, 8, 17–22 asymptotic forms 19, 20–21, 29 dipole modified 79, 82, 85–87 nomenclature 19, 24 regular and irregular 18, 22, 55, 273 strongly closed 22–25 Coulomb phase shift 19, 298 Cs atom 232 curve-crossing 10, 47, 74, 125 degeneracy factor 131 delta function 274 density matrix 12, 157, 170, 192, 219–223, 227, 231 diabatic potentia 50 diffuse orbitals 3, 45 digamma function 23 dipole coupling 9, 78–87 region 79, 81 dipole representation 79, 82 dipole moment 78, 260 dipole transition moment 159 direct product 163

315

dissociative recombination 10, 125, 130–155 cross-section 131, 134, 154 direct and indirect mechanisms 131–135, 149 H+ 2 139, 146–148 H+ 3 148–155 NO+ 135, 139 Edlen plot 27, 56 e/f states 99, 103, 112, 288, 290 eigenchannel method 66, 72 eigenphase 29, 31, 75, 76, 142, 143, 252, 275 elliptic (spheroidal) coordinates 73, 82 elliptic integrals 265 Euler angles 215 false eigenvalues 23, 49 field ionization 239, 247, 255, 262, 267 linewidth 264 fine structure constant 159 fluorescence 146, 231 frame transformation 5, 9–10, 90–123, 268, 269 electronic 13, 41 physical basis 91–95 rotational 9, 14, 32, 97–105, 108, 109, 120, 281, 282–294 vibrational 105, 140 Franck–Condon factor 177, 296 gamma function 20 Gaussian orbitals 3, 53 truncated 66 Gaussian pulse 240, 253 geometric phase (see Berry phase) geometrical factor 204–205 Green’s function 128 H atom 22, 239, 246, 255 circular states 258, 261 parabolic states 13, 255, 257, 258–260 Stark effect 260–265 H2 molecule 2, 99, 104, 106, 126, 172, 173 auto-ionization 31–34, 109–114, 111, 253 bound states 107–109 dissociative recombination 139, 140, 146–148 double minimum states 46, 50, 115 photo-ionization 208–210 potential curves 46–47, 73 perturbations 32, 39, 48 quantum defect functions 47–48 H3 radical 34, 121, 148–155, 167, 178 HI molecule 42, 206, 236 H2 O molecule 51, 176, 183, 184 Ham correction 8, 49, 115 Hamiltonian matrix 1, 37, 54, 84, 116 Hartree–Fock 45 Hund’s cases 14, 79, 198, 200, 208, 224, 248, 282–283, 285

316

Index

Hund’s cases (cont.) case(a) 41, 196, 285, 288, 299, 302 case(b) 32, 97–105, 178, 198, 208, 223, 224, 286, 290, 292–306 case(c) 32, 200 case(d) 97–105, 180, 248, 286, 290, 292–306 case(e) 41, 167, 288 hyperspherical coordinates 149 information content 213–215, 223, 225, 229 Inglis–Teller limit 261, 266 inversion operation 284 ion-pair states 46 Jahn–Teller coupling 11, 115–123, 148–150, 153, 188 stabilzation parameter 116, 153 JWKB approximation 14, 93, 127, 263, 264, 265, 268, 269, 308 K atom 27 K-matrix 6, 8, 10, 28, 30, 45, 86, 105, 127, 128, 134, 145 electronic 92, 96, 114, 116, 121 physical (folded) 30, 69, 75, 131 Kepler time 241, 243, 248, 253

doubling 9, 97, 100–101, 109  uncoupling 9, 97, 101–103 Laguerre polynomial 259 Landau frequency 239 Landau–Zener aproximation 267 Langer correction 263, 304 laser induced fluorescence 232 Legendre polynomial 205, 217, 218, 258 line stregth 231 log-derivative 53, 62 Lorentzian 33 Lu–Fano plot 40–41 magnetic field 239 Michelson interferometer 247, 249 Milne equations 14, 309 molecular symmetry group C2v (M) 290 D3h (M) 121 Td (M) 187 MQDT 2, 8, 37, 140 alternative representations 14, 278–281 generalized 13, 14, 127, 307–309 quantization condition 17, 25–27, 49, 104, 115, 270, 276 Stark effect 255, 266, 267–271 time dependent 252–255 wavefunction 25, 92, 136–137, 142, 222, 273, 278, 280 working equations 30, 110, 129, 131, 143, 252

multichannel quantum defect theory (see MQDT) multi-photon 157 ionization 210–212 spectroscopy 11 N2 molecule 183, 186 Na2 molecule 103, 180, 186 NH3 molecule 174, 177 NH4 radical 178 NO molecule 56–58, 75, 76–77, 127, 130, 172, 181, 217, 223–226, 248, 250–251 normal coordinates 149 normalization 13, 31, 167, 273–277 to energy delta function 13, 29, 127, 193–194, 274, 278, 307 to unity 20, 26, 37, 63, 276, 277 notation 7, 14, 162, 164, 196, 202, 295, 310–312 nuclear spin-statistics 120, 292 nuclear hyperfine coupling 91, 292 O2 molecule 183, 186, 217 orientation of angular momentum 11, 12, 168, 220 coefficient 169, 222, 229 orthogonality 245, 275 3j coefficient 164, 234 6j coefficient 231 frame trasformation 92, 97, 133, 270 open channel solutions 13, 55, 69, 143, 252, 275, 298 overlap integral 126, 135 P branch 167, 170, 179, 222, 232 parabolic coordinates 13, 258 partial waves 12, 193, 207, 216, 225, 230 parity 99, 165, 179, 180, 182, 234, 284–286 parity-adapted wavefunctions 8, 14, 284, 285–286 parity factor 197, 234, 301 parity favoured and unfavoured 205–206, 210 Pauli spin matrix 117, 233 permuation-inversion symmetry 290 perturbations 2, 7, 32, 186 matrix element 77, 106, 114, 118 rotational 39–41 vibrational 85, 106, 113 PGOPHER package 173 phase-amplitude representation 14, 20, 307 phase shift 26 photo-electron angular distribution 12, 191, 202–212, 219, 225 fixed molecule 12, 192, 215–219 spin polarized 233–235 photo-excitation cross-section 160, 162 multiphoton band structure 11, 157, 171–177, 183 resonant 162–163, 170–171 transition amplitudes 178–180, 295–297 photo-ionization 12–13 boundary conditions 12, 14, 191, 192–193, 297

Index decoupling factor Q(t ) 200, 201, 204, 208 differential cross-section 202–212, 219, 225 integrated cross-section 194, 199–202 matrix elements 14, 194, 195–199, 297–303 phase 193, 208, 216, 223, 225, 230 selection rules 12 photon field 163, 205, 206, 211, 213, 229 photon flux 158, 160, 162 plane wave 191, 192 polarizability tensor 11, 161 symmetric and anti-symmetric parts 162, 166 polarization 157, 167, 204, 220 axis 219, 223, 229, 233 circular 158, 164, 166, 206, 207, 232, 234 crossed 211, 214, 229, 230 index 220, 230 linear 158, 164 vector 159, 161 predissociation 3, 10–11, 51, 125, 129–130, 142, 162, 176, 247, 255 homogeneous and heterogeneous 7, 146 linewidth 130, 146 probe laser 219, 231, 246, 249 propensity rule 180 pseudo-potential 82 pulse sequence 240, 249, 250, 251 Q branch 168, 170, 179, 184, 208, 222, 232 quadrupole moment 78, 103, 248 quantization 17, 25–27, 49, 104, 115, 276 axis 212, 215, 222, 233 quantum beats 229, 255 quantum chemistry 45–52 quantum defect 2, 3, 5, 20, 26, 56, 80, 265 η (Ham) defects 26, 49, 115 μ ˜ (modified) defects 26 quantum defect function 3, 8, 46, 47–48, 51, 57, 105, 108, 115, 140 quantum number fixed 286 function 22 good 286 magnetic 195, 199, 219, 233 orbital 79, 205, 208, 283, 290, 310 principal 1, 22, 45, 79, 260, 266 R branch 170, 179, 222, 232 R-matrix 4, 9, 11, 27–28, 45, 54–56, 86, 139, 140–148 convergence of 65, 70–73 eigenchannel 66, 72 variational 54, 62–66, 70, 83, 141–146, 147 Wigner–Eisbud 9, 53, 58–60, 65, 141 Rabi oscillations 251 radial matrix element 172, 178, 197, 225, 232, 296, 303–306 Ramsey fringes 249

317

rank (of matrix) 69, 75, 279, 280 recurrence time (see Kepler time) reduced mass 18, 105, 107 reduced matrix element 208, 228, 295, 298 J representation 164, 172, 229, 230, 310  representation 172, 178, 181, 197, 311 photo-ionization 234, 311 relaxation rate constant 162 REMPI 12, 157, 219–226 Renner–Teller coupling 91 resonance (see auto-ionization, predissociation) resonant multiphoton ionization (see REMPI) revival time 242 rotational axis convention 176, 290 band structure 11, 157, 171–177, 183 Runge–Lenz vector 255, 266 Rydberg constant 2, 3 mass correction 107 Rydberg formula 2, 3, 22, 26, 39, 42, 46, 105, 144 Rydberg molecule 178–183 Rydberg population 250 Rydberg orbitals 4, 17, 43, 45, 51, 54, 58, 177, 292, 303 Rydbergization 48, 58 Rydberg/valence interactions 74–77, 127, 128, 129, 131, 136, 138 S-matrix 6, 10, 131, 133, 137, 138, 145, 152, 198, 298 scaling law 5, 22, 105, 120, 130, 153, 239, 240, 303 scattering 3–7, 27, 93, 131, 152, 191, 192–193, 200, 204 selection rule 41, 167, 175, 181, 183, 186, 188, 200–202, 207, 216, 292 symmetry 173, 175, 187–189 parity 200, 202, 215 Siegert states 151 separation constant 263 SO(4) algebra 256 spatial selectivity 167–170 spherical harmonic 18, 202 modified 203, 234 spherical tensor representation 11, 12, 157, 163–167, 191, 227 coefficient 163, 210, 211, 213 component 163, 173, 194, 205, 229 spin decoupling 41–43 spin–orbit coupling 43, 91, 205, 208, 233, 282, 303 spin polarization 13, 192, 232–237 Stark effect 13, 183, 255–271 H atom 260–265 general 265–267

318 storage ring 149, 153 stroboscopic fringes 10, 97, 103–105, 239 revivals 247–249, 250 suddenness parameter 95 superelastic energy transfer (SEC) 131, 139 super symmetry 13, 255–258 symmetric top 157, 172–174, 286 symmetry considerations 290–292 target function 53, 121 transition amplitude 178–180, 215 two-step fragmentation theory 125, 135–140 unitarity 30, 152 unitary defect 152 units 274 atomic 18 natural 18 physical 274 vector potential 158

Index wavefunction 192 ab initio 52–54 coulomb 3, 8, 17–22 MQDT representation 25, 222, 273, 297 wavepackets 13, 239–255 classical 241 partial revival 243, 245 recurrence and revival 239, 240–247 spreading 242 Wigner–Eckart theorem 164, 295 Wigner coefficients 3j 99, 181, 184, 197, 207, 210, 222, 227, 234, 258, 287, 288, 296, 297 6j 102, 221, 229, 231 Wigner rotation matrix 98, 168, 215, 222, 258, 287 window resonance 36, 135, 139 wronskian 20, 22, 23, 127, 259, 274, 307 Xe atom 206 zero kinetic energy spectroscopy (see ZEKE/PFI) ZEKE/PFI 11, 13, 130, 158, 183–189, 239, 255, 267