Advances in Atomic and Molecular Physics, Volume 7

Advances in ATOMIC A N D MOLECULAR PHYSICS VOLUME 7

CONTRIBUTORS TO THIS VOLUME C. AUDOIN J. C. BROWNE MAURICE COHEN GY. CSANAK J. GERRATT A. J. GREENFIELD P. GRIVET B. R. JUDD THOMAS F. O’MALLEY RUBEN PAUNCZ J. P. SCHERMANN

H. S. TAYLOR HAREL WEINSTEIN NATHAN WISER ROBERT YARIS

ADVANCES IN

ATOMIC AND MOLECULAR PHYSICS Edited by

D. R. Bates DEPARTMENT OF APPLIED MATHEMATICS A N D THEORETICAL PHYSICS THE QUEEN’S UNIVERSITY OF BELFAST

BELFAST, NORTHERN IRELAND

Immanuel Esterman DEPARTMENT OF PHYSICS THE TECHNION ISRAEL INSTITUTE OF TECHNOLOGY HAIFA, ISRAEL

VOLUME 7

@

1971

ACADEMIC PRESS New York

London

COPYRIGHT 0 1971, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM, BY PHOTOSTAT, MICROFILM, RETRIEVAL SYSTEM, OR ANY OTHER MEANS, WITHOUT WRIlTEN PERMISSION FROM THE PUBLISHERS.

ACADEMIC PRESS,

INC. 111 Fifth Avenue, New York, New York 10003

United Kingdom Edition published by ACADEMIC P R E S S , INC. (LONDON) LTD.

24/28 Oval Road, London NWl I DD

LIBRARY OF CONGRESS CATALOG CARDNUMBER:65-18423

PRINTED IN THE UNITED STATES OF AMERICA

Contents ix

LISTOF CONTRIBUTORS CONTENTS OF PREVIOUS VOLUMES

xi

Physics of the Hydrogen Maser C. Audoin, J. P . Schermann, and P . Grivet I. Introduction 11. Hydrogen Maser Techniques 111. Dynamical Behavior of the Maser IV. Hyperfine Spectroscopy V. Interaction of Atomic Hydrogen with Radio Frequency Fields References

2 5 8 29 33 42

Molecular Wave Functions: Calculation and Use in Atomic and Molecular Processes J . C. Browne I. Introduction and General Principles 11. Computations of Wave Functions, Potential Surfaces, and Coupling Matrix Elements 111. Some Results and Expectations for the Future IV. Atom-Atom Scattering V. Radiative Processes References

47 53 78 79 83 87

Localized Molecular Orbitals Hare1 Weinstein, Ruben Pauncz, and Maurice Cohen I. Introduction 11. Density Matrix Formalism 111. The Edmiston-Ruedenberg Localization Method IV. The Method of Boys and Foster V. Direct Localization Methods VI. Internal and External Localization Criteria VII. The Method of Magnasco and Perico VIII. The Method of Peters IX. Molecular Orbitals Determined from Localization Models X. Localized Orbitals in Expansion Methods XI. Concluding Remarks References V

97 99 102 109 116 121 122 126 128 134 138 138

vi

CONTENTS

General Theory of Spin-Coupled Wave Functions for Atoms and Molecules J . Gerratt I. Introduction 11. Properties of the Exact Electronic Eigenfunction

111. IV. V. VI. VII. VIII. IX. X.

Construction of the Spin Functions The Spin-Coupled Wave Functions Calculation of Matrix Elements of the Hamiltonian The Orbital Equations Symmetry Properties of the Spin-Coupled Wave Functions The Hund's Rule Coupling The General Recoupling Problem and Bonding in Molecules Conclusions Note Added in Proof Appendix A. Proof of the Relations (59)-(62) Appendix B. Matrix Elements of Spin-Dependent Operators Appendix C. Proof That the Orbital Equations Are Invariant under 2 Appendix D. Proof That the Operators F'"' and F'@' Are Invariant under Unitary Transformations of the 4" and #,, Sets of Orbitals References

Diabatic States of Molecules-Quasistationary

141 144 147 152 154 163 168 180 194 206 207 207 21 1 213 215 219

Electronic States

Thomas F. O'Malley I. Introduction 11. Mathematical Preliminaries 111. Molecular Ground States-The

IV. V. VI. VII. VIII.

One-State Problem-The Stationary Adiabatic Representation The Na CI Two-State Problem-Covalent and Ionic States Charge Exchange in Helium-Single Configuration Diabatic States Dissociative Recombination and Attachment-The Quasistationary State Represen tat ion Slow Heavy-Particle Collision Theory-Extension of the Quasistationary Representation to Rydberg States Summary and Conclusion References

+

223 225 228 230 232 236 243 245 248

Selection Rules within Atomic Shells B. R. Judd I. Introduction Groups Irreducible Representations Generalized Triangular Conditions Generators Conflicting Symmetries

11. I I I. IV. V. VI.

252 252 258 270 273 276

CONTENTS

VII. Oriented Spins VIII. Special Cases IX. Conclusion References

vii 280 282 284 285

Green’s Function Technique in Atomic and Molecular Physics Gy. Csanak, H . S. Taylor, and Robert Yaris I. Introduction 11. Many-Particle Green’s Function and Physical Quantities

288 290

111. Coupled System of Equations for Green’s Functions (The Method of

Functional Differentiation; The Dyson Equation; The Bethe-Salpeter Equation) IV. Scattering V. Nonperturbative Approximation Method VI. Perturbation Methods Appendix A Appendix B Appendix C References

305 321 330 339 354 358 359 360

A Review of Pseudo-Potentials with Emphasis on Their Application to Liquid Metals Nathan Wiser and A . J . Greenfield I. Introduction 11. Underlying Ideas of the Pseudopotential 111. Simplification of the Form Factor IV. Formulations of v ( q ) Useful for Liquid Metals

V. Screening VI. Comments about the Various Pseudopotentials VII. Conclusions References

AUTHOR INDEX

SUBJECT INDEX

363 365 370 374 381 384 385 386 389 399

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List of Contributors Numbers in parentheses indicate the pages on which the authors’ contributions begin.

C. AUDOIN, Section d’Orsay du Laboratoire de 1’Horloge Atomique du CNRS, BBtiment 220, Faculti: des Sciences, Orsay, France (1)

J. C. BROWNE, Departments of Physics and Computer Science, University of Texas, Austin, Texas (47) MAURICE COHEN, Department of Physical Chemistry, Hebrew University, Jerusalem, Israel (97) GY. CSANAK, Department of Chemistry, University of Southern California, Los Angeles, California (287) J. GERRATT, Department of Theoretical Chemistry, University of Bristol, Bristol, England (141) A. J. GREENFIELD, Department of Physics, Bar-Ilan University, RamatGan, Israel (363) P. GRIVET, Institut d’Electronique Fondamentale, Laboratoire Associ6 au CNRS, BBtiment 220, Faculti: des Sciences, Orsay, France (1) B. R. JUDD, Department of Physics, The Jzhns Hopkins University, Baltimore, Maryland (251) THOMAS F. O’MALLEY, Physics Department, University of Connecticut, Storrs, Connecticut (223) RUBEN PAUNCZ, Department of Chemistry, Technion-Israel Technology, Haifa, Israel (97)

Institute of

J. P. SCHERMANN,* Section d’Orsay du Laboratoire de 1’Horloge Atomique du CNRS, Bgtiment 220, Facultk des Sciences, Orsay, France (1) H. S. TAYLOR, Department of Chemistry, University of Southern California, Los Angeles, California (287) HAREL WEINSTEIN, Department of Chemistry, Technion-Israel Institute of Technology, Haifa, Israel (97) NATHAN WISER, Department of Physics, Bar-Ilan University, Ramat-Gan, Israel (363) ROBERT YARIS, Department of Chemistry, Washington University, St. Louis, Missouri (287)

* Present address:

NASA-Goddard Space Flight Center, Greenbelt, Maryland

ix

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Contents of Previous Volumes Volume 1 Molecular Orbital Theory of the Spin Properties of Conjugated Molecules, G. G. HaN and A. T. Amos Electron Affinities of Atoms and Molecules, B. L. Moiseiwitsch Atomic Rearrangement Collisions, B. H. Bransden The Production o f Rotational and Vibrational Transitions in Encounters between Molecules, K. Takayanagi The Study of Intermolecular Potentials with Molecular Beams at Thermal Energies, H . Pauly and J. P. Toennies High Intensity and High Energy Molecular Beams, J. B. Anderson, R. P. Andres, and J. B. Fenn AUTHOR INDEX-SUBJECT INDEX Volume 2 The Calculation of van der Waals Interactions, A. Dalgarno and W. D. Davison Thermal Diffusion in Gases, E. A. Mason, R. J. Munn, and Francis J. Smith Spectroscopy in the Vacuum Ultraviolet, W . R. S. Carton The Measurement o f the Photoionization Cross Sections of the Atomic Gases, James A. R. Samson The Theory of Electron-Atom Collisions, R. Peterkop and V. Veldre Experimental Studies of Excitation in Collisions between Atomic and Ionic Systems, F. J. de Heer Mass Spectrometry of Free Radicals, S. N. Foner AUTHOR INDEX-SUBJECT INDEX Volume 3 The Quanta1 Calculation of Photoionization Cross Sections, A. L. Stewart Radiofrequency Spectroscopy of Stored Ions. I: Storage, H. G . Dehmelt Optical Pumping Methods in Atomic Spectroscopy, B. Budick Energy Transfer in Organic Molecular Crystals: A Survey of Experiments, H . C. Wow Atomic and Molecular Scattering from Solid Surfaces, Robert E. Stickney Quantum Mechanics in Gas Crystal-Surface van der Waals Scattering, E. Chanoch Beder Reactive Collisions between Gas and Surface Atoms, Henry Wise and Bernard J . Wood AUTHOR INDEX-SUBJECT INDEX xi

xii

CONTENTS OF PREVIOUS VOLUMES

Volume 4

H. S. W. Massey-A

Sixtieth Birthday Tribute, E. H. S. Burhop

Electronic Eigenenergies of the Hydrogen Molecular Ion, D. R. Bates and R. H. G. Reid Applications of Quantum Theory to the Viscosity of Dilute Gases, R . A. Buckingham and E. Gal Positrons and Positronium in Gases, P. A. Fvaser Classical Theory o f Atomic Scattering, A. Burgess and I. C. Percival Born Expansions, A. R. Holt and B. L. Moiseiwitsch Resonances in Electron Scattering by Atoms and Molecules, P. G. Burke Relativistic Inner Shell Ionization, C. B. 0. Mohr Recent Measurements on Charge Transfer, J. B. Hasted Measurements of Electron Excitation Functions, D. W. 0. Heddle and R. G. W . Keesing Some New Experimental Methods in Collision Physics, R. F. Stebbings Atomic Collision Processes in Gaseous Nebulae, M. J. Seaton Collisions in the Ionosophere, A. Dalgarno The Direct Study of Ionization in Space, R . L. F. Boyd AUTHOR INDEX-SUBJECT INDEX

Volume 5 Flowing Afterglow Measurements of Ion-Neutral Reactions, E. E. Ferguson, F. C. Fehsenfeld, and A . L. Schmeltekopf Experiments with Merging Beams, Roy H. Neynaber Radiofrequency Spectroscopy of Stored Ions I1 : Spectroscopy, H. G. Dehmelt The Spectra of Molecular Solids, 0. Schnepp The Meaning of Collision Broadening of Spectral Lines: The Classical-Oscillator Analog, A. Ben-Reuven The Calculation of Atomic Transition Probabilities, R. J . S. Crossley Tables of One- and Two-Particle Coefficients of Fractional Parentage for Configurations sAsfUpq, C. D. H . Chisholm, A . Dalgarno, and F. R. Innes Relativistic Z-Dependent Corrections to Atomic Energy Levels, Holly Thomis Doyle AUTHOR INDEX-SUBJECT INDEX

Volume 6 Dissociative Recombination, J. N. Bardsley and M . A. Biondi Analysis of the Velocity Field in Plasma from the Doppler Broadening of Spectral Emission Lines, A . S. Kaufman The Rotational Excitation of Molecules by Slow Electrons, Kazuo Takayanagi and Yukikazu Itikawa

CONTENTS OF PREVIOUS VOLUMES

...

XI11

The Diffusion of Atoms and Molecules, E. A . Mason and T. R . Marrero Theory and Application of Sturniian Functions, Manuel Rotenberg Use of Classical Mechanics in the Treatment of Collisions between Massive Systems, D. R . Bates and A . E. Kingston AUTHOR INDEX-SUBJECT INDEX

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Advances in ATOMIC A N D MOLECULAR PHYSICS VOLUME 7

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PHYSICS OF THE HYDROGEN I I MASER C . AUDOIN and J . P . SCHERMANN* Section d'Orsay du Laboratoire de I'Horloge Atomique du CNRS Britiment 220, Faculte' des Sciences 91 Orsay, France

and

P . GRIVET Institut d'Electronique Fondamentale, Laboratoire Associe' nu CNRS Britiment 220, Faculte' des Sciences 91 Orsay, France

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ............................ A. History. .. . . . . . . . .. ... .. B. The Maser as a F r C. The Maser as a Researc .................................. 11. Hydrogen Maser Techniques . . . . . . . . A. Atomic Hydrogen Source ............................. B. Magnetic State Selector ........................ C. Storage Bulb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Microwave Cavity . . . . . . . . . . . . . . . . E. Magnetic Shields.. . . . . . . . . . . . . . . . . F. Frequency Stability and Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .................... 111. Dynamical Behavior of the Maser A. Two-Level Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Electromagnetic Field in the Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . ................ C. Amplitude and Phase Equations D. Steady State and .............. E. Molecular Ringing . . . . . . . . . . . . . . . . F. Maser under Continuous Excitation .............. 1V. Hyperfine Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Ground State Hyperfine Splitting of Atomic Hydrogen, Deuterium, I

. . . . .. . . . . .. .. .. ... . . B. Quadrupole Coupling Constant of the I4N C. Absolute Measurement of the Lande g-Factor of the Proton. . . . . D. Stark Shift of the Hydrogen Hyperfine Separation . . . . . . . . . . . . . . . . E. Spin-Exchange Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V. Interaction of Atomic Hydrogen with Radio Frequency Fields . . . . . . . . A. Quantum Description of the Interaction between Atomic Hydrogen and Radio Frequency Fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Double Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Modification of the Zeeman Hyperfine Spectrum of Atomic Hydrogen Interacting with a Nonresonant Radio Frequency Field . . . . . . . . . . , . References .. ..............

* Present address : NASA-Goddard

2 2 3 4 5 5 6 6 7 7 7 8 9 13 14 15 20 25 29 29 31 31 32 32 33

33 34

41 42

Space Flight Center, Greenbelt, Maryland.

1

2

C. Audoin, J. P. Schermann, and P. Grivet

I. Introduction A. HISTORY Townes and his colleagues, Shimoda and Gordon (Gordon et al., 1954; Shimoda el al., 1956), in the United States and Basov and Prokhorov (1954) in the U.S.S.R. created the two-level maser as an important tool for physicists and metrologists in 1954, with their description of a successful maser working on a electric dipole (ED) transition in the NH, molecular inversion spectrum. It soon appeared that the accuracy of this new molecular frequency standard suffered from the complicated structure of the inversion line, and that its frequency stability, although very good, was limited by the short interaction time of the NH, molecules with the high frequency field. The time of flight of the molecules through the resonator, 100 psec in order of magnitude, is the useful lifetime, and by the uncertainty principle, the width of the line is rather large, a number of kilohertz. As far as the short term frequency stability is concerned, this drawback is partly compensated by the rather high power of the ammonia maser, about lo-'' W, as shown by the general formula due to Berstein (1950) and Blaqui6re (1953) for a classical oscillator and to Townes (Shimoda et al., 1956) for quantum oscillators; (Af2)'I2/f0

=(kT/Pz)'"/2Q,

where (Af2)'12ifo is the fractional frequency fluctuation, Q, is the atomic line Q-factor, P is the emitted power, and z is the averaging time. For overcoming this drawback, Ramsey proposed the hydrogen maser and demonstrated (Goldenberg et al., 1960) its success in 1960 at Harvard University. At that time, the idea was to use a simpler atomic spectrum and to achieve a new optimization based on a different tradeoff between power and linewidth, for the benefit of its overall properties as a frequency standard. To achieve this aim, Ramsey proposed to store active hydrogen atoms in a bulb inside the interaction cavity (Fig. I). Long interaction time, between 0.1 and 1 sec, and therefore storage of particles in the field are needed to obtain self-oscillation with a magnetic dipole ( M D ) transition such as the A F = I , AmF = 0 transition of the hyperfine structure of hydrogen atoms. Indeed, the probability of this M D transition is smaller than the probability of a ED transition by about four orders of magnitude. The feasibility of storing polarized hydrogen atoms in a bulb, without buffer gas, had been suggested from results obtained in atomic beam resonance experiments with stored beams of cesium (Goldenberg et a/., 1961). Owing to the low polarizability of hydrogen atoms, they can be stored up to 1 sec, sustaining about lo5 collisions with suitably coated bulb walls, with only slight perturbation of the

3

PHYSICS OF THE HYDROGEN MASER

Microwave cavity

-loop

I Source

Hexapolar magnet

/

Storage bulb

FIG.1. Schematic diagram of the hydrogen maser.

energy levels. Furthermore, storage of atoms eliminates the first-order Doppler effect. The linewidth is then as low as I Hz. The basic theory of the hydrogen maser has been very ably described by Ramsey and his collaborators in two fundamental articles (Kleppner et al., 1962, 1965). The first article is more theoretical than the second which deals with the experimental aspects of the work. B. THEMASERAS

A

FREQUENCY STANDARD

I . Frequency Stability

In practice, the short term frequency stability of the H maser is not as good as given by the above formula (see Fig. 4) because the power delivered by atoms to the H maser oscillation is small-about W. It is rather determined by the noise added to the maser signal in the external electronic circuits (Vessot et al., 1968). Nevertheless, at the present time, the hydrogen maser is the oscillator which exhibits the best frequency stability, up to several parts in for averaging times larger than 10 sec. This results from, among other factors, the very high Q-factor of the atomic line which minimizes the cavity pulling effect [Eq. (41)]. Under actual conditions, collisions of hydrogen atoms with the bulb walls, or with other hydrogen atoms determine the linewidth. Several modifications of the initial hydrogen maser design have proved to be efficient in reducing these sources of line broadening.

4

C. Audoin, J. P . Schermann, and P . Grivet

Average effects of wall collisions are reduced by increasing the mean bulb diameter. The wall contribution to relaxation times have been lowered by a factor of 1.7 by using a long cavity operating near the cutoff condition of wave propagation (Peters et al., 1968), and by a factor of 10 in the “big bulb” maser of Ramsey (Uzgiris and Ramsey, 1970). In mutual collisions of hydrogen atoms spin exchange may occur with a loss of coherence of the radiating atoms resulting in a line broadening. In this respect, the classical hexapole magnetic state selector is unsatisfactory because it allows atoms in both F = 1, mF = 0, and F = 1, mF = 1 states to enter the bulb. One can prevent atoms in the undesirable F = 1, m F = 1 state to reach the bulb by means of an improved state selector described in Section II,B.

2. Accuracy The confined hydrogen atoms are of course not absolutely free of their environment. During the very short duration of the wall collisions, the atom is perturbed and one must take into account the resulting shift of the hyperfine frequency separation. In the mean, this correction is small (lo-”) because the time of flight inside the bulb volume is much larger than the collision time. Nevertheless it is the main factor impairing the accuracy of the maser as a primary frequency standard. Its measurement is performed by noticing that the shift is proportional to the mean collision frequency, and thus to theinverse of the mean bulb diameter. Theprecision of this measurement can be improved by varying the volume of a deformable bulb instead of having to coat two or more different bulbs. The question of coating reproductibility is then avoided (Debely, 1970; Brenner, 1970; Vessot et al., 1970). The size of the shift can be reduced by increasing the bulb diameter (Peters et al., 1968; Uzgiris and Ramsey, 1970). Furthermore, experimental studies have been conducted indicating a zero shift at a certain bulb temperature (Zitzewitz, 1970; Vessot et al., 1970). A more detailed survey of the performances of the hydrogen maser as a frequency standard can be found in the literature (Ramsey, 1965; McCoubrey, 1966; Grivet and Audoin, 1969). C. THEMASERAS

A

RESEARCH TOOL

As summarized in Sections IV and V below, the hydrogen maser is a very powerful spectrometer for studies of frequency deviations or level variations arising in the interaction of hydrogen atoms with fields or with atomic particles introduced into the bulb. Its interest is due to the simple structure of hydrogen atoms and to the possibility of observing narrow resonance lines with stored atoms.

PHYSICS OF THE HYDROGEN MASER

5

In the following sections, we will survey some of the results obtained in recent years, giving more details on maser dynamics and spectroscopic measurements as studied at Orsay.

11. Hydrogen Maser Techniques Figure 2 shows the different parts and the proportions of a recent laboratory design. More compact designs for field use are described by Peters et al. (1968) and Vessot et al. (1968).

A. ATOMIC HYDROGEN SOURCE Molecular hydrogen, fed through a silver-palladium leak, is dissociated into atoms in a rf discharge at pressures between lo-' Torr and 5 x lo-' Torr. The rf power, delivered by an oscillator at a frequency up to 300 M H z , is typically 10 W. In order to reduce the spreading of the atomic beam, one or several capillaries are used giving a directivity of a few degrees. The mean velocity in the beam, which corresponds to the source temperature, is about 2.8 x lo3 mjsec. The pressure in the source is regulated by means of a servosystem acting upon the hydrogen-leak rate through its temperature. ___-__

state selector

discharge1 tube

/

palladium leak

',

IHydrogen tank

FIG.2. Example of design.

C. Audoin, J. P . Schermann, and P . Grivet

6

The A F P exchanges the or

the F,l,m,l

populations

and F - 1 , m,-1

states.

F- 0 Hexapolar magnets

_____----------_______

Storoge

F-1, m=?l F=O,m=O Source

A F P region

FIG.3. Improved state selection with adiabatic fast passage.

B. MAGNETIC STATESELECTOR Before entering the microwave active region, the atoms must be separated according to their energy states. This is generally realized by hexapole magnets which produce a high-beam intensity for a given source emittance because of their focusing properties. Such magnets behave as convergent lenses for atoms in energy states I 1, 1 ) and I 1, 0), and as divergent lenses for atoms in energy states 11, - 1) and 10,O) (Audoin et at., 1969a). With a single hexapole magnet, the atomic fluxes of atoms in the states I I, 1) and I 1 , O ) are nearly the same. But only the atoms in the state I 1 , O ) are involved in the maser transition AmF = 0. The undesirable atoms in the state 11, 1) merely increase the atomic density in the storage bulb and thus the atomic linewidth through spin exchange collisions. The elimination of the atoms in state I 1, 1 ) can be achieved by means of a double focuser device (Schermann, 1966) with two hexapole magnets separated by an adiabatic fast passage (AFP) region (Fig. 3). The AFP, produced by a nearly static magnetic field varying slowly around 8 G and by a 12 MHz rf field, exchanges the populations of states 1 1, 1 ) and I 1, - 1 ) (Section V,B,2), and one of the two hexapole magnets acts, therefore, as a divergent lens for mF = 1. It has been demonstrated that 85 % of the undesirable atoms in state I I , 1 ) can be eliminated while the atoms in state I 1,O) remain unaffected (Audoin et al., 1969b). C. STORAGE BULB The state-selected atoms are introduced into a storage bulb and are stored there for about 1 sec in the central region of a resonant microwave cavity. The flux of atoms in state I 1 , O ) is about 1013 atoms/sec. Inside the bulb,

PHYSICS OF THE HYDROGEN MASER

7

the vacuum remains very high and the mean free path is very much larger than the bulb diameter, which is between 10 and 20 cm. The atoms suffer about lo5 collisions with the wall which must thus have an inert non-relaxing surface. The best results have been obtained by coating the bulb walls with Teflon homopolymer (Du Pont TFE 42) and copolymer (FEP 120) giving relaxation times of a few seconds and small frequency shifts (about lo-” in relative value for a 16 cm bulb diameter) measurable within a precision of a few percent (Menoud and Racine, 1969, Vanier and Vessot, 1970; Zitzewitz et al., 1970; Hellwig et al., 1970).

D. MICROWAVE CAVITY The microwave cavity operates on mode TEoll. It is tuned to the hyperfine transition frequency. The diameter and the length are about 27 cm and the loaded Q factor is about 40,000. The energy losses originating in the cavity walls and in relaxation processes are small enough to obtain a self-oscillation at 1420 MHz when the flux of atoms in the higher energy level is greater than approximately 10’’ atoms/sec. In order to avoid pulling effects due to cavity mistuning (see Section 111, D,l,b), the resonant frequency of the cavity and, thus, its temperature must be held as constant as possible. Two possibilities have been studied: (1) a temperature controlled cavity made of cervit with a large thermal time constant (Vessot e l al., 1968) and (2) a metallic resonant cavity, the temperature of which is directly controlled with a small time-constant (Peters et al., 1968). A part of the radiated energy is coupled out by a loop giving a typical output power of W. To obtain a useful reference frequency of sufficiently high power level, the maser signal is commonly used to control a 5 MHz quartz oscillator with a phase-locking servo-system.

E. MAGNETIC SHIELDS Magnetic shields reduce the earth’s magnetic field and its fluctuations in the microwave interaction region. Several layers of mu-metal are most commonly used, providing residual fields as low as a few microgauss and fluctuations of a few tenth of a microgauss. A solednoid, coaxial with the resonant cavity produces a static magnetic field, between 0.1 and 1 mG. Inside the magnetic shields, the materials must be nonmagnetic. Aluminium, copper, quartz, etc., are commonly used.

F. FREQUENCY STABILITY AND ACCURACY

-

The very high Q-factor of the resonance line (Q lo9) due to the long interaction time between the atoms and the electromagnetic field (-J 1 sec)

C. Audoin, J . P. Schermann, and P. Grivet

8

"1'

Frequency stability

time

FIG.4. Frequency stability of the hydrogen maser: solid line, measurement; dotted lines, theoretical limitations.

and the insensitivity to external factors gives the hydrogen maser oscillation a very high frequency stability. A fractional frequency stability of several parts in 10'' have been obtained by Vessot et al. (1968) as shown in Fig. 4. The accuracy of the frequency standard is in the order of lo-". The main contribution to this figure comes from the uncertainty in the measurement of the wall shift. Hellwig et al. (1970) and others cited therein have measured the atomic hydrogen hyperfine transition frequency in terms of the Cs 133 hyperfine transition frequency. The result is

v H = 1, 420,405, 751.768

0.003 Hz.

This is the most accurately known physical constant. Applications of the hydrogen maser as a frequency and time standard are in the field of time-keeping, very long baseline radio interferometry, and spectroscopy.

111. Dynamical Behavior of the Maser In this section, the maser description is classical in character. A fully quantum mechanical model has been worked out by Shirley (1968) and developed to obtain as special cases the hydrogen maser dynamical equations and also the laser rate equations of Statz and de Mars (1960).

PHYSICS OF THE HYDROGEN MASER

9

A. TWO-LEVEL HYPOTHESIS 1. Fictitious Spin

As we will justify later, we assume that only the two mF = 0 hyperfine levels are to be considered. The F = 1, mF = 0 and the F = 0, m F = 0 states will be labeled as I 1 ) and I2), respectively. The density operator p of a hydrogen atom, in IF, m F ) representation reduces then to

The static magnetic field being parallel to the 2 axis, the magnetic dipole moment operator of the atoms is PZ

= PB(g1 I2

+ g J JZ)?

(2)

where I , and J , are, respectively, the 2 components of the nuclear and the electronic angular momenta, pB is the value of the Bohr magneton, and gr and gJ are the nuclear and the electronic Land6 factors. This operator has nondiagonal elements only, given by 1P2l2>

=PB(gJ

-gI)/2

=PB*

(3)

The expectation value of the atomic magnetic dipole moment is

When a time dependent magnetic field H ’ ( t ) = H,‘ cos (ot + cp) parallel to the static field is applied, the perturbation operator is V = - p z H ’ ( t ) , and the perturbed Hamiltonian of an atom can then be written as follows:

The origin of the energy scale is chosen midway between the 11) and 12) levels. wo is the (angular) frequency of the AmF = 0 hyperfine transition. By expanding the 2 x 2 density and Hamiltonian matrices on a complete set of orthogonal operators, namely the identity and the Pauli vectorial operators, it can be seen that the Schrodinger equation of such a two-level

10

C. Audoin, J . P. Schermann, and P. Grivet

system is equivalent to the equation of motion of a fictitious spin one-half M’ acted upon by a high frequency magnetic field H’ (Abragam, 1961a; Fano, 1957; Feynman et al., 1957): (aM’/lJt),,

=

-M’

H’.

(6)

hH’ = Tr (ZO),

(7)

A

The two vectors are defined as follows:

M’ = Tr ( P O ) ;

where o is the Pauli vectorial operator. Note that the magnetic field H’ is expressed in frequency units. The components M,’ and H,’ are related to the expectation values of the atomic magnetic-dipole moment and to the variable field, respectively, by Ml‘

H,’=(V12

= P12

+ PZl

+ V,l)/h=(-pBH,’/h)cos(ot

The component M3’-the

(8)

= p/pB

+q).

(9)

static magnetic field in EPR experiments-equals

coo. In a maser, it is much larger than p B H z ’ / h . The Bloch-Siegert frequency shift, referred to the transition frequency, amounts to about lo-’*. It is

completely negligible, and only one of the two rotating components of M,’ and H,’ needs to be considered. The component M,’ is the population difference: M3’

= P11 - P 2 2

(10)

2. Validity of the Two-Level Hypothesis The validity of the two-level hypothesis follows from the properties of the atomic relaxation processes : spin exchange, wall collisions, inhomogeneous magnetic field mixing. a. Spin Exchange. When two alkali-like atoms collide, the unpaired electron spin coordinates may be exchanged (spin exchange). This effect has been extensively studied (Balling et af., 1964; Bender, 1963; Dehmelt, 1958; Grossetcte, 1964; Purcell and Field, 1956; Vanier, 1968; Wittke and Dicke, 1956). As a result, the rate of change of the population difference ( p , , - p Z 2 ) can be expressed for all possible values of the nuclear spin’,’ by (Balling et aZ., 1964; Grossetzte, 1964): (aM3‘/dt),, = - M3‘/Tl . It is justified to neglect the indistinguishability of identical atoms. - p Z 2 )is approximately zero.

* Due to the condition Amo < kT, the equilibrum value of ( p ,

PHYSICS OF THE HYDROGEN MASER

11

T,, is the longitudinal spin exchange time constant given by

l / T l e= naoijr,

(12)

where n, is the atomic density, ijr is the mean relative atomic velocity, and 0 the spin exchange cross section. The rate of change of the oscillating dipole moment of a hydrogen atom is described in our formulation by (Balling et al., 1964) (aM,’/dt),,

= -M l ’ / T 2 ,

+ il.M,’/4oT1 ,.

(13)

T 2 , is the transversal spin-exchange time constant. For hydrogen atoms, the ratio of the two spin exchange relaxation times is (Wittke and Dicke, 1956) T2elT1e = 2.

(14)

The last term of Eq. (13) yields a small frequency shift (Bender, 1963), given by Eq. (36) where A is the spin exchange frequency shift parameter in Crampton’s notation (Crampton, 1967). One can see from the previous results that the population difference and the magnetic moment of m F = 0 states of hydrogen atoms do not depend upon other state parameters. The last statement does not strictly apply to rubidium atoms in a rubidium maser whose hyperfine structure is more complicated (nuclear spin > 1/2), but it can be used as a good approximation (Vanier, 1968) in practical experimental situations. 6. Wall Relaxation. Experimental investigation (Berg, 1965) of the storage-bulb-wall-relaxation time constants agrees with the two-level hypothesis, where the rate of change of the population difference and the induced magnetic dipole moment are each characterized by a single T I , and a single T2wrelaxation time

The small frequency shift due to wall interactions does not depend upon the atomic density in the gas. It is included (as are the second-order Doppler shift and the second-order magnetic shift) in the value of the uncorrected hyperfine separation frequency wo . Thus it does not appear in Eq. (16). c. Inhomogeneous Static Field Mixing. Due to the random motion of the hydrogen atoms a component of the magnetic-field gradient perpendicular to the static field can induce transitions between the F = 1 sublevels (Kleppner et al., 1962; Vanier and Vessot, 1966). This occurs when the mean frequency of the collision with the walls of the bulb is greater than the frequency separation between the F = 1 sublevels (1.4 kHz per milligauss of static field).

C. Audoin, J. P . Schermann, and P . Grivet

12

-

Coupling of the F = 1, mF = 0 state to the F 1, mF = k 1 states does not strongly disturb the two-level approximation (Audoin, 1967). Furthermore, magnetic field inhomogeneities can be made small enough to neglect mixing effects by properly degaussing the magnetic shields and by using compensation fields. 3. Bulb Storage a . Time Averaging. Atoms can disappear from the bulb by recombination on the wall or by escaping from the bulb. The probability that an atom entering the bulb at time to be present at time t > t o is exp [ - (t - to)/Tb], where Tbis the storage time constant. The atomic density is sustained by the beam flux Z(t,) which brings in atoms mostly in the upper of the two considered levels. At time t the density operator p(t) of atoms is the mean value of the density operator p ( t , to) of atoms entered at time to and still present at time t. p ( t ) is then given by (Lamb, 1960) N(t)p(t) =

f dt,

tO)Z(tO)

-m

exp [-(l - tO)/Tbl d t O .

(17)

N ( t ) is the total number of atoms in the bulb:

The fictitious spin vector of all atoms confined in the bulb is then

M(t) = N ( t ) . Tr [p(t)o]

(19)

b. Motionaf Averaging. The phase of the electromagnetic field applied to atoms is a constant on both sides of the nodal surface of the TE,,, mode cavity field. Its amplitude is a function of the instantaneous atomic position. One can then show (Kleppner et al., 1962; Hartmann, 1969) that the atomic line emerges strongly above a Doppler pedestal as far as the bulb volume does not extend to much beyond the nodal surface. The line appears with its natural width. Only the second-order transverse Doppler effect is significant. The corresponding frequency shift Ao amounts to Awlw,

=

-3kT/2m0 c2,

(20)

where 3kT/2 is the kinetic energy of atoms and m, c2 their rest energy. Furthermore, the magnetic field amplitude is motionally averaged to its mean value calculated over the bulb volume. We then put =
(21)

13

PHYSICS OF THE HYDROGEN MASER

The filling factor y~may be defined as (Kleppner et al., 1962)

4. BIoch Equation f o r Hydrogen Atoms The density operator varies very slightly in the interaction representation, during the very small correlation time of the relaxation processes. These processes are uncorrelated. Equations (6), (I l), (13), (15), and (16) can then be added to give the rate of change of the density matrix p(t, to) of atoms acted upon by all the perturbations. It can then be easily established (Audoin, 1967) from Eq. (17), that the whole set of spins confined in the bulb fulfills a Bloch equation:

The overall relaxation times Tl and T, are given by 1

-

1

1 1 +-+--;

1

Tb

T2

-

1

1 +-+-.

1

T2w

T2e

Tb

k,, k,, and k, are the unit vectors of the fictitious frame. Z(t) is the excess of the flux of atoms in the 1 1 ) state over the flux of atoms in the 12) state. The validity of the Bloch equation (in a suitable representation) for hydrogen atoms in a hydrogen maser arises both from the simple structure of the hydrogen atom ( I = f) and the circumstance that the population inversion is performed outside the microwave cavity and atoms are confined for a long time in this cavity. All the properties of the H maser can then be calculated and quantitatively compared to experimental observations to give information on atomic processes. The situation is not so favorable for other masers. In ammonia and HCN masers, the molecules pass through the cavity. It is then impossible to find an exponential law for the interaction time between the molecules and the electromagnetic field. As a consequence, the frequency of oscillation does depend upon the oscillating level (Shimoda et al., 1956). In the optically pumped 85Rb and 87Rb masers, spin exchange and optical pumping prevent the two-level hypothesis from being rigorously valid. Nevertheless, it can be used as a good approximation for measurements performed in the dark (Vanier, 1968).

B. ELECTROMAGNETIC FIELDIN THE CAVITY The effect of the induced magnetic dipole moment pB M , of all the hydrogen atoms driving the electromagnetic field in the resonant cavity is calculated

14

C. Audoin, J . P . Schermann, and P . Grivet

from Maxwell’s equations. For experimental purposes, we will also consider an excitation of a frequency w produced by an external generator connected to the cavity through a coupling loop. In order to avoid the introduction of the coupling factor, we define this excitation in such a way that the amplitude of the cavity response H , is equal to the amplitude of the excitation field He when the cavity is empty of atoms and when the excitation frequency is the same as the cavity resonant frequency 0,. The magnetic field H I is then a solution of the equation:

+ 2H1/T, + wC2H1= K M , - 2w, He/T,,

(25) where T, is the cavity time constant related to the loaded cavity quality factor by T, = 2Q,/w,. The parameter K depends on the filling factor and the cavity volume V , and is given by 3 1

K

=

4nqpB2/hV ,

(26)

c. AMPLITUDE A N D PHASE EQUATIONS Equations (23) and (25) allow the exact calculation of all the macroscopic properties of the maser, either in free running conditions (He 3 0) or when it is externally driven ( H e # 0). The directly measurable parameters are the frequency, the amplitude, and the phase of the cavity electromagnetic field. A ready way to calculate the last two parameters as a function of time is to look for amplitude and phase equations (Kryloff and Bogoliuboff, 1947) connected to Eqs. (23) and (25). We set He = p sin wt H,

=b

M,

=m

(27)

cos (at + cp)

(28)

sin (ot + I)).

(29)

The involved time constants T,, T 2 ,T, are much larger than the hyperfine transition period. The parameters p , 6 , m, cp and Ic/ are, therefore, slowly variable functions of time. We then obtain (Grasyuk and Orayevskiy, 1964) the amplitude equations:

+

rn T2Gq = M3 b T2 cos 8

M3

+ TlM3 = -T,bm cos 8 + T,I b + T, b = K Q , m cos 8 + p cos cp

(30)

(31) (32)

where

O=cp-*

(33)

PHYSICS OF THE HYDROGEN MASER

15

and the phase equations :

@=(w,-co)-

[ic(

! )

- I--cosrp

(34)

t]

1 P . t - tanO---sincp. T, b

(35)

The spin exchange frequency shift 0,'- oois expressed by COO'

-

00

= -M ,

A / ~ N u T, ~ ,

(36)

In the next three sections, we will successively consider several conditions of maser operation.

D. STEADY STATEAND TRANSIENT BEHAVIOR We here consider the oscillating maser, without externally applied signal

(P = 0). I . Steady State Operation

Of course, the steady state solution of Eqs. (30) to (36) give known results obtained by direct calculations of the density matrix elements. We recall the most important ones. a. Oscillation Level. When the cavity is tuned, and the equilibrium conditions are reached, the normalized magnetic field amplitude b takes the value b, given by 1

+ TlT2b02= KQ,TlT2Z=a,

> 1.

(37)

The oscillation level is not a linear function of the atomic flux I because the relaxation times TI and T2 are decreasing functions of the atomic density on account of spin exchange collisions. The power P delivered by atoms to the cavity electromagnetic field then varies as a quadratic function of the atomic flux according to the following expression (Kleppner et af., 1965). PIP,

=

-2q2(Z/Ir,)2

+ (1

-

cq)(Z/Z,/J - 1.

(38)

The parameter q characterizes the spin exchange reduction of the oscillation level. It is given by D6,h Tb v, 1 ztot (39) (I=-------. 8ZPii2 T qvb Qc 1 The parameters P , , I,,,and T, are normalization factors defined in the given reference. The value of the c factor is close to 3, and I,,, > I .

C. Audoin, J . P. Schermann, and P. Grivet

16

The value of q must be smaller than 0.172 for oscillation to occur. In practice, half this value can be easily obtained; typical values of Zth and P, are 10l2 atoms/sec and 5 x W, respectively. b. Oscillation Frequency. The value of the oscillation frequency is determined by self-consistency of the phase shift between the fictitious spin and the rotating field components, calculated both from the atomic and the cavity equations (34) and (35). We then get

-w

w 0

TC T2

= - ( W - w,)

3A 1 + -4l M---. N dT1,’

(T, < Tz).

(40)

This expression can be modified by noticing that M 3 T 2 is a constant to second order of cavity mistuning in a steady state free running maser of given configuration [Eqs. (30) and (32)]. Then, 1/T2factors out in the right-hand side of Eq. (40) to give 00

- o = (T,/Tz)(wo - 0,

+ h,) (T, G T2)

(41)

When the cavity is tuned t o w, = w , + 60, the frequency of the maser oscillation equals coo, the transition frequency unshifted by spin exchange (Crampton, 1967; Vanier et al., 1964). Furthermore, the oscillation frequency does not change when the T2 relaxation time is varied by modulating the atomic beam flux. This provides the commonly used cavity tuning check. The measurement of the maser frequency as a function of the cavity detuning allows the determination of the T, relaxation time under continuous oscillation, i.e., with a bulb atomic density of about lo9 atoms/cm3 (Fig. 5). 2. Dynamical Behavior of’ the Oscillating Maser

a. Amplitude Dynamics. When the cavity is properly tuned, cos 0 = I , and the amplitude variations are calculable from Eqs. (30) to (32) only. In a maser, the cavity time constant is much smaller than the atomic relaxation times, i.e., the cavity field follows, with a negligible delay, the variations of its excitation. Then, for practical experimental situations, the following simplified differential equation describes the level variations (Audoin, 1967)

d b

)(;

1 6

+

+ b2 - bo2= 0.

(43)

It is clear that these variations are defined by the values of the Tl relaxation time and the steady state oscillation level b,. This must be related to the

17

PHYSICS OF THE HYDROGEN MASER

1.0

0.5

Af,,,

0

10

( kHz)

20

>

FIG. 5. Variation of the oscillation frequency vs. cavity detuning, for several atomic fluxes. Slopes of the straight lines are proportional to 1/T2.

fact that the oscillation level is bound by the saturation of the atomic line and that the population difference varies with the T , time constant. 6. Amplitude Transient near the Equilibrium Level. For small variations around the steady state level, the last equation is linearized by setting b = b, + E , E 4 6,. The return to the steady state is then described by T l i + i + 2T1bO2& = 0.

(44)

When the amplitude of the oscillation is low (8Ti2b02< I), it can easily be shown that the amplitude transient is damped exponentially and the amplitude variations are slower as the level is reduced. This conclusion is readily verified by operating the maser near the threshold conditions. For high oscillation levels (8Tl2bO2> I ) , the amplitude transient is damped oscillatory (Grasyuk and Orayevskiy, 1964). E =

C exp ( - t/2T1)cos (w,t

+ 4);

w I 2 + 1/4T12= 2bO2,

(45)

where C and 4 are constant parameters. This last behavior has been reported for the proton maser (Combrisson, 1960), the ammonia maser (Laine and Bardo, 1969), and the hydrogen maser (Audoin, 1966; Nikitin and Strakhovskiy, 1966). c. Measurement of the T , Relaxation Time. Equation (45) shows that recording small amplitude variations provides the measurement of the T , relaxation time under oscillation conditions (high atomic flux). The normalized

18

C. Audoin, J. P. Schermann, and P. Grivet

@

Isolators

Parametric amplifier

F

-@ l

-

F

k

S

y

hrturbation n c h r

o

FIG.6. Experimental setup for measurement of T,

cavity electromagnetic field b, can also be directly determined, without knowledge of the maser configuration and coupling loop factor. A detailed experimental study confirms these conclusions (Audoin et al., 1968). Although the damped oscillatory behavior can be observed under ordinary oscillation condition ((2, 4 x lo"), the determination of TI by observing an amplitude transient is easier to carry out when the oscillation level is higher. Indeed, the envelope is more precisely drawn when the recorded signal presents a large number of extrema, i.e., when the pseudopulsation is large and the cavity Q, factor high. The apparent Q, factor can be increased by connecting a feedback loop to the maser cavity. It consists of

-

lb

1.05 b,

t

0

0.4

(S) 3

0.8

1.2

FIG.7. Example of recorded amplitude transient.

PHYSICS OF THE HYDROGEN MASER

19

a variable gain amplifier together with a phase shifter (Audoin, 1966). The experimental setup is shown in Fig. 6. The amplitude transient is induced by switching off an inhomogeneous static magnetic field previously applied to the atoms. Figure 7 shows a record of the small amplitude variations. Amplitude transient observations (and other measurement methods) also enable the determination of the Z/Zr,, ratio and the q parameter (Audoin et al., 1967). Figure 8 shows the variation of TI and T, vs. Z/Zth. Calculation of Z,, from the maser configuration gives the storage bulb atomic density and allows the determination of the H-H spin-exchange cross section (Section IV,E,2).

FIG.8. Variation of atomic time constants vs. atomic flux.

d. Oscillation Buildup. Figure 9 shows that a computer solution of Eq. (43) describes very well the oscillation buildup. The maser oscillation starts after switching off an inhomogeneous static field. It can also be shown that the amplitude of the oscillation varies as exp (Tlbo2t)as long as it is very low. e. Phase Transient. For the determination of the phase behavior (Audoin et al., 1968) it is necessary to take into account the small spin-exchange frequency shift. Equation (36) shows that the transition frequencyw,’ depends on time as does M 3 . Using Eqs. (30) and (32) and the condition Tc G T, , this time variation can be written as follows for small cavity mistunings o0’(t) - 6,’= (6,’ - w,)T,

bjb,

(46)

where W,,is the steady state value of wo’(t). Elimination of 0 between phase equations gives, after some widely justified approximations

TC c p “

+ 9 = - TZ(60’ - w)(b/b)+ [COO‘(?)- Go’].

(47)

20

C. Audoin, J . P. Schermann, and P . Grivet

1.0 0 05 FIG.9. Oscillation buildup : solid line, theoretical curve; circles, experimental points. (Experimental and calculated curves have been horizontally translated to fit at the first maximum.)

Taking into account Eq. (46), the differential equation for phase variations becomes Tc@+ ci, = T2(w- w0)(b/b).

(48)

The bandwidth of the phase measurement setup is always much narrower than the cavity bandwith. The first term of the left-hand side of Eq. (48) can then be neglected. Thus the phase of the oscillation varies during an amplitude transient as follows:

440 - 'Po

= T2(0 - 0 0 ) In

"0lb01,

(49)

where 'po is the value of the phase at the end of the transient. This equation shows that the phase is a constant during an amplitude transient when the maser oscillates with the frequency w = coo, the spin exchange unshifted transition (Fig. lo). A check of the cavity tuning based on the observation of phase transients must then be equivalent to the usual one. An experimental verification of this point has been carried out (Fig. 11). A good quartz crystal oscillator provided the phase reference. Its phase instabilities limited the precision of the measurement.

E. MOLECULAR RINGING The maser's active medium is turned into a radiative state by applying an excitation with a frequency very close to wo during a time interval T (Kleppner et al., 1962). Afterwards, stimulated emission of radiation is observed during a time equal to a few times T 2 ,while atoms are going back to their equilibrium state.

PHYSICS OF THE HYDROGEN MASER Phase

21

transient

w
transient

FIG.10. Phase transient during an amplitude transient. For purpose of illustration, the cavity Q-factor has been raised to 135,000.

I . Preparation of the Radiative State We assume that the atomic beam flux I has been constant for a long time compared to the T, relaxation time, and that the maser does not oscillate. The value of the population difference is then IT, and the magnetic dipole moment of the atoms is zero. a . Constant Atomic Beam Flux. At time t = 0, the resonant magnetic field is applied for a pulse of duration z, while the atomic flux is held constant. The variation of M , and m can be calculated from Eqs. (30) and (31) with ‘f-&

(rod)

L-

-2

FIG. 11. Transient phase variations as a function of maser detuning. yi and bi are the initial values of the corresponding parameters.

22

C. Audoin, J . P. Scliermann, and P. Grivet

t3

t3

(a )

(b)

FIG.12. Preparation of the radiative state described in the rotating frame: (a) the initial position of the spin; (b) its final position. w = w, = w o andp = p o wherep, is the normalized amplitude of the excitation

pulse. This variation can also be easily understood in a very classical manner (Slichter, 1963) by using the fictitious spin representation. In the rotating frame, the spin vector turns around the excitation vector (Fig. 12). The rotation angle is p = p O z .The spin vector length is a constant as long as the condition T 4 T l , T , is experimentally fulfilled. Then at the end of the pulse, the population difference is M , = IT, cos p and the magnetic dipole moment amplitude is m = IT, sin 0.The last parameter becomes a maximum for p =42. b. Atomic Bean? Switched Off. At time t = 0, the atomic beam is switched off. At time t = T’ later, the population difference decreases to IT, exp ( - T ’ / T ~ ) . At the end of the pulse applied at t = T ’ , the population difference and the magnetic dipole moment amplitude are then M,

= (cos p)IT, exp ( -

T’/T~); m

=

(sin @ITl exp ( - T ’ / T ~ ) .(50)

Of course, the sustained-flux condition can be obtained by setting in the last two relations.

T’ = 0

2. Atomic Radiation after the Pulse

a. Induced-Cavity-Field Equation. The magnetic dipole moment of atoms rotating in the fixed frame induces an electromagnetic field in the cavity. Its frequency is nearly unaffected by cavity pulling (Arditi, 1964) The field amplitude is then calculable from Eqs. (30) and (32), with cos 0 = 0 (Strakhovskiy and Uspenskiy, 1966). We obtain bjb = ( L Y , ~ I)/7‘, (51) j , = -y/T1 - T, b2/ao + cCo‘/TlcC0.

PHYSICS OF THE HYDROGEN MASER

23

where y = M,/IT,

and

T, < T I , T, .

The parameter u0 is defined by uo = K Q , TIT2I ; it is greater than one when the maser is above the threshold condition of oscillation [see Eq. (37)] and smaller than one in the inverse case. ao’ characterizes the beam flux for t > z. We have ao’ = uo when the atomic beam is held constant, and ao’ = 0 when it is switched off. The initial conditions are determined by the atomic state at the end of the pulse. We get T , b(z + T’) = a. sin p exp ( - ? / T I ) y(z

+ 7’) = cos p exp (-z’/T,)

(52)

These equations describe as well the field variation in the rubidium masers after switching off the pumping light (uo’ = 0). 6. Maser below Threshold Condition. Measurement of T, and T I . From the first of the two Eqs. (51) one can see that the amplitude of the field radiated by atoms after the pulse decreases as exp ( - t / T 2 ) when the quantity u o y is much smaller than one. In that case, the maser is far below the threshold condition uo = 1, i.e., the atomic flux is very small. When the oscillation condition is approached, the variation of the population difference can no longer be neglected. The field decrease b(t) is no longer exponential. The gap between the maser signal and the exponential curve of the same initial value and time constant T , can be defined by E(t) = ( - T2/0 In [b(t)lb(O) exp (- t/T2)1

(53)

where the time origin is taken at the end of the pulse. This quantity is unity when the atomic flux is very low. The proper flux condition is obtained when E remains close enough to unity to insure an accurate measurement of T, (within 2 % for example). But it is also desirable to operate the maser with a flux high enough to have a good signal-to-noise ratio (for uo = 0.01, the emitted power is about W only). In order to determine that condition, a solution of the field variations has been computed for several values of a. for z’ = 0, T2/Tl= 413, and ao’ equal either to a. or to zero. Figure 13 shows the values of E ( t ) computed for uo = 0.25, 0.5, and 0.75 and 0 < t/T2 = 3. The best compromise corresponds to a. = 0.25, the beam flux being interrupted while the n/2 pulse is fed into the cavity. The variation of T 2 due to spin exchange through the atomic density decrease is then about 2 %. It is compatible with the required precision of measurement. This method then provides the measurement of the zero flux T, relaxation time. It is also possible to measure, under the same condition, the T I relaxation

C. Audoin, J. P . Schermann, and P. Grivet

24

-

I/----1.0.

a,, = 0.75 a, = 0.5 a,., 0.25

a: =o

a,,o.25

0.sa,

= 0.5

a:=a,

0.6a,,

0.75

0.q

c

t/T?

1.0

20

3.0

B

FIG.13. Computed gap between actual and exponential decrease of the cavity field after the 7~/2pulse.

time because the initial amplitude of the maser signal varies as exp ( - T ’ / T ~ ) when a time delay 7’is provided between the beam interruption and the pulse (Vanier et al., 1964). This simultaneous measurement method of TI and T2 is also widely used in experimental studies of rubidium masers(Arditi and Carver, 1964; Vanier, 1968). c. Maser above Threshold Condition. Going back to Eqs. (51) and (52) with 7’ = 0, one can see that the initial value of b is made positive when the conditions a. > I/cos fl and 0 < fl < 7c/2 are fulfilled together. The first of them implies that the maser is able to oscillate when applying the pulse. Moreover, the stimulated emission of radiation necessarily cancels after the pumping process has been suppressed (ao = 0). The maser signal may then present a maximum after the pulse. This behavior has been foreseen by Bloom (1956) in the particular case TJT, = I where an analytical solution of Eqs. (51) can be found. It has been exploited by Vanier (1967) for the measurement of the spin exchange cross section of 87Rb atoms. In the hydrogen maser, the T2/Tlratio is necessarily greater than one and smaller than two. The lower limit is approached when the bulb escape time constant is very short, and the upper limit when spin exchange prevails as the relaxation mechanism, i.e., at high flux. Under these conditions computer solutions of Eqs. (51) show that the ordinate of the maximum of stimulated emission is a decreasing function of T2/Tl (Jousse, 1967) as it appears in Fig. 14.

PHYSICS OF THE HYDROGEN MASER

25

1.01

FIG.14. Maximum of stimulated emission of radiation after the pulse.

F. MASERUNDER

CONTINUOUS

EXCITATION

We here assume that an excitation with the frequency w is continuously applied to the cavity through a coupling loop. The maser’s behavior is then described by Eqs. (30) to (35) withp # 0. We will neglect here the spin exchange frequency shift, and will consider that the cavity is tuned to o,= o,, . In this section, the steady-state value of b is called b,. The electromagnetic field inside the cavity is observed with a second coupling loop. The signal so detected is, with the same factor, proportional to 6, or p corresponding to excitation with and without atoms, respectively.

I. The Maser as an Amplifier When the maser is operated below the threshold of oscillation it behaves as an amplifier. We define its gain by G = b,/p as given by the expression G = (I

+ So)/[(I + Soy +

%,,I

So

=

T I T ,b:

0;

COS’

COSI

+ So) cos2 011’2

(54)

+ Tz2(o0- o ) ~ ] - ’

(55)

cos2 0 - 2a,( 1

with

0 = [I

The parameter So is the saturation factor of the atomic transition. It must be negligible for the amplifier to be linear. This condition is So < ( 1 - a0) The maximum value of the gain is then

so= 0) = 1/(I

G(w = o,, ,

- a,,).

(56)

The measurement of this gain provides the determination of the parameter a,, , as illustrated in Fig. 15. This parameter is related to the difference

26

C. Audoin, J. P. Schermann, and P. Grivet

1

N

I

N

I

I

L

FIG. 15. Record of the atomic line and principle of the measurement of

cto.

of population between levels 1 I ) and 12). When the level 11) is split by the Autler-Townes effect (Section V,B,3) the maser, used as an amplifier, allows the measurement of the relative difference of population between each of the new levels and the 12) level. Figure 16 shows the experimental verification, for w = coo, of the relation (54). The measurement of T I and T2 is performed as indicated in Sections III,D,2,c and b, respectively. That of bS2is done after the receiver has been calibrated by recording an amplitude transient of the free oscillating maser.

2. Synchronization of the Maser Oscillator a. Level of the Synchronized Maser. When a signal is applied to the maser, the frequency w of which is close enough to the undisturbed oscillator frequency, it imposes its frequency on the oscillator: the maser is synchronized to the external signal. Relations (30) to (35) are valid again to describe the level of the synchronized maser. Now, a. is greater than unity, and is defined by Eq. (37). Figure 17 shows the variations of b,/p as a function ofp, the excitation level, for given undisturbed maser operating conditions, and w = wo . Variations of b, vs. o - coo are represented in Fig. IS. b. Synchronization Bandwidth. Search for the stability conditions of the synchronized solution of Eq. (30) to (35)-by applying the Routh-Hurwitz

27

PHYSICS OF THE HYDROGEN MASER

2.2: Gain

2

1

3 4 Saturation factor 5,

FIG.16. Saturation of the gain, below oscillation threshold: solid line, theoretical curve; circles, experimental points.

t

/-T,= 0.092

s

T2= 0.118 s b,=l0.6 s*

a25 0 -

I

3

10

30

100

FIG.17. Variation of the gain of the synchronized maser: solid line, theoretical curve; circles, experimental points.

28

C. Audoin, J. P . Schermann, and P . Grivet

101

i

I

i

T,= 0.0975 s T p 0.122 s b,= 9.9 s-’

51

.

I

{

a5

-

f ,f

I L

1.0

015

-1.0

(Hz) e

FIG. 18. Variation of the synchronized level in the synchronization bandwidth : solid line, theoretical curves; circles, experimental points.

criterion, for example-is a rather cumbersome problem. We give here a simpler calculation. Its validity is confirmed by an experimental investigation (Audoin and Viennet, 1969) together with a more complete analysis. When the steady state synchronized level is reached, the phase equations combine to give

(57) The value of sin q being bound by + I , the synchronization bandwidth turns out to be 0 3

-0

2

= 2PIT2 b,(w2),

(58)

where 6, is the common value of the synchronized level for w = w2 or w = 03, at the limits of the synchronization bandwidth. Experiment shows that the formula is valid within p/b, 0.3. Its experimental verification is represented in Fig. 19. In Table I it can be seen that the values of T2 measured by the cavity pulling method, and the synchronization method are in very good agreement. It is thus possible to determine the relaxation time T 2 , without detuning the cavity, by measuring the synchronization bandwith.

29

PHYSICS OF THE HYDROGEN MASER

(Hz)LSynchronization i bandwidth

FIG.19. Determination of T2 by measuring the synchronization bandwidth : experimental curves. Values of T 2 :(a) 0.112 sec; (b) 0.130 sec; (c) 0.146 sec. TABLE I COMPARISON OF Two METHODS OF MEASURING Tz Measurement method Cavity pulling Synchronization

Measured value of Tz (msec) 112 112

131 130

143

138

154 146

IV. Hyperfine Spectroscopy A. G R O ~ ~STATE N D HYPERFINE SPLITTING OF ATOMIC HYDROGEN, DEUTERIUM, A N D TRITIUM Hydrogen, deuterium, and tritium are the elements that possess the simplest atomic structures. One can calculate the hyperfine transition separation (hfs) from quantum electrodynamics and, thus, obtain the fundamental constants from spectroscopic measurements (Taylor et af.,1969). Comparison between the measured (Table 11) and the calculated hydrogen hfs provides a determination of the fine structure constant. The result is (Taylor et a/., 1969) tY1 =

137.03591(35).

(59)

30

C. Audoin, J . P . Schermann, and P . Grivet TABLE I1 HYPERFINE FREQUENCY SEPARATION OF HYDROGEN, DEUTERIUM, AND TRITIUM"'b

Hydrogen Deuterium Tritium a

Nuclear spin

Hyperfine transition frequency

112 1 112

1,420,405,751.768 i0.003 Hz 327,384,352.51 0.05 Hz 1,516,701,470.7919& 0.0071 Hz

From Ramsey (1970).

* Referred to cesium hfs (9, 192, 63 I ,

770 Hz).

Precision of the measurement of hfs of deuterium and tritium has been greatly improved by the maser technique, by a factor of 100 for deuterium and 4000 for tritium. The hyperfine transition frequencies of hydrogen and tritium differ only by about 130 MHz. A suitable modified maser has been operated with an atomic tritium source (Mathur et af., 1967). The ratio of the hydrogen and tritium hfs found in this experiment is v ( T ) / v ( H )= 1.0677945149734 (+50 x

(60)

Due to the small influence of the triton structure, the measurement of this ratio provides a very good check of the reduced mass corrections on the hfs shift of tritium. The hyperfine transition frequency of deuterium (Table 11) is about 4.35 times lower than for hydrogen. Thus the realization of a deuterium maser presents difficult technical problems, requiring the design of large resonant cavity, vacuum tank, magnetic shields, etc. The measurement of this transition frequency has nevertheless been performed by Crampton et al. (1966) and Larson et al. (1969) employing a technique similar to the spin-exchange optical-pumping experiments of Dehmelt (1958). A hydrogen maser oscillates on the field dependent (AF = 1 , Am, = 1 ) transition. The spin-exchange process couples the polarization of hydrogen and deuterium, so that, when the deuterium polarization is modified by applying a resonant microwave field at one of the deuterium hyperfine transition frequencies, the resonance is monitored by observing the decrease of the oscillation level of the hydrogen maser. In the Larson et al. (1969) experiment, the ratio of the hydrogen and deuterium Lande g-factors has also been measured. The result, g,(H)/q,(D) = 1 + (9.4 1.4) x is in slight disagreement with the theoretical value of

31

PHYSICS OF THE HYDROGEN MASER

Hegstrom (1970) and Grotch (1970) which isgJ(H)/g,(D)

=

1 + 7.25 x

In an optical pumping experiment, Robinson and Hughes (1970) have obtained gJ(H)/gJ(D) = 1 -k (7.2 k 3.0)

X

lo-’.

B. QUADRUPOLE COUPLING CONSTANT OF THE 14N ATOMIC GROUND STATE The effective Hamiltonian of the groundstate of

2

= AIJ - gJpu,H.J- g r p g H I

14N

atoms is

+ i B [ 3 ( 1 ~ J +) ~4I.J - (I)2(J)2].

(61)

The hyperfine coupling constant A and the quadrupole coupling constant B have been determined by a spin-exchange experiment similar to the measurement of the deuterium hyperfine frequency. The following results have been obtained (Crampton et al., 1970):

A

=

B=

10,450,929.06 k 0.19 HZ

(62)

+ 1.32

(63)

0.2 HZ

The experimental value of B disagrees with the predicted value which is

B = -3.4

c. ABSOLUTEMEASUREMENT OF THE

_+

1.6 Hz.

LAND^ g-FACTOR

(64) OF THE PROTON

In a high magnetic field, the electron and the proton of the hydrogen atom are uncoupled. It is thus possible to measure simultaneously in a double resonance experiment the resonance frequencies corresponding to the electron and proton spin-flips in the same magnetic field. With a modified hydrogen maser, operating in a 3500 G magnetic field. Myint et al. (1966) have determined, with great accuracy, the ratio of the Land6 g-factors gs of the electron to g p of the proton. The relative uncertainty of this measurement is 3 x 10- ’. Using the value of gs previously obtained by Wilkinson and Crane (1963) gs = 2 x 1.001159(3), this experiment provides the following value of the proton Lande factor g p = 0.0030420652(9).

(65)

This quantity has been measured before by nuclear magnetic resonance (NMR) of protons in water or mineral oils by Lambe (1959), but this determination required several corrections, such as diamagnetic shielding and chemical shift in the molecules. The hydrogen maser experiment provides the first absolute calibration of NMR spectrometers and verifies Ramsey’s theory (Ramsey, 1950) of diamagnetic shielding with a very good accuracy. A more recent measurement (Kleppner, 1970) gives the value gJg, = 658.2107061(65).

32

C. Audoin, J. P . Schermann, and P . Grivet

D. STARK SHIFT OF THE HYDROGEN HYPERFINE SEPARATION The effect of a static electric field on the hydrogen hfs has been calculated by Schwartz (1959). This theoretical prediction has been experimentally verified by Forston e f al. (1964). The effect is proportional to the square of the applied field, within the range of measurement (0 to 10 esu). The maximum The high observed shift of the maser oscillation frequency was 6 x performance of an hydrogen maser is required to observe such a small effect. The obtained experimental value is

dveXp= -(6.7

0.4) x IO-’E2

Hz/(esu)2.

(66)

It is in agreement with the theoretical value

B v , ~= -7.14 x 10-5E2 Hz/(esu)’.

(67)

E. SPIN-EXCHANGE COLLISIONS 1. Verijication of the Spin-Exchange Theory

During a collision between a hydrogen atom and an atom or a molecule containing an unpaired electron, the relative orientations of the electronic spins can be exchanged. The theory of this effect (Wittke and Dicke, 1956; Balling et al., 1964) shows that the ratio T Z e / T lof e the corresponding relaxation rates is independent of the atomic mass, the nuclear and electronic moments, and the vibrational-rotational structures. The spin-exchange process is coherent: T2e> T I , . These assumptions have been verified by Berg (1965) who used a hydrogen maser operating below the oscillation threshold. Several foreign gases were introduced into the storage bulb. Another measurement has been made by Audoin (1 966) for hydrogen-hydrogen collisions with an oscillating hydrogen maser. The results are shown in Table 111. TABLE 111 MEASURED VALUEOF THE T2e/T,eRATIO Colliding System

H-H

H-D

Tz./ TI (theoretical)

2

1.33

1.33

1.33

T,,/T,. (measured)

1,9 (Berg) 2.10 f 0.15 (Audoin)

1.33 i 0.03

1.33 & 0.03

1.33 & 0.03

H-NO

H-02

33

PHYSICS OF THE HYDROGEN MASER

2. Measurement of the Hydrogen-Hydrogen Spin Exchange Cross Section When the hydrogen maser is oscillating, the spin-exchange mechanism provides a significant contribution to the relaxation times T, and T, . Hellwig (1968) has measured the oscillation threshold of the maser as a function of the atomic flux penetrating the storage bulb, and obtained the following spin-exchange cross section:

u

=

(2.65 & 0.4) x

cm2.

(68)

Another determination has been performed (Audoin, 1968) by direct measurement of the relaxation times TI and T2 (Section 111,D) as a function of the atomic density (Fig. 8). It gives o = (2.57 & 0.5) x

cm'.

(69)

The two results are in good agreement.

V. Interaction of Atomic Hydrogen with Radio Frequency Fields The hydrogen maser is a very convenient spectrometer to investigate the interaction between the atomic system and radiation fields, because atomic hydrogen possesses a very simple structure, and can be obtained in nearly free space conditions. In this section, we consider interactions between a radio frequency field and the three F = 1 sublevels of hydrogen atoms. The effect of the interaction on these levels can be detected by observing transitions between the 1; = 0 level and the 1; = 1 modified levels. 'The theory of this interaction can be more easily understood if one considers the system as a whole consisting of the atoms and the quantized rf field. This formalism, developed mainly by Cohen-Tannoudji and Haroche (1969), will be first briefly recalled and applied to the description of phenomena such as multiplet splittings (Autler-Townes effect), multiple quantum transitions, and modification of the Lande g-factor. DESCRIPTION OF THE INTERACTION BETWEEN ATOMIC A. QUANTUM HYDROGEN AND RADIOFREQUENCY FIELDS The hydrogen atoms are assumed to be subjected to a radio frequency field of amplitude H,, and frequency urf,directed along the x axis (Fig. 20). With the convention h = I , the field Hamiltonian is

Xrf= q f a c a ,

(70)

34

C. Audoin, J. P. Schermann, and P. Griuet

t

“O

e

X

t

State selected atomic beam

FIG.20. Field configuration for double resonance experiments.

where a’ and a are, respectively, the creation and annihilation operators of a photon of energy oIf.The eigenstates of Xrfof energies of no,, are labelled In>. The static magnetic field is directed along the z axis and has a low value (below gauss). The three Zeeman sublevels F = 1 can then be considered as equidistant and, thus, as the three levels of a spin 1, hereafter called S, with a gyromagnetic factor y equal to yelectron/2. The Zeeman Hamiltonian can be written xzeeman

= YSZH O = wz SZ

(71)

with eigenstates I I , 1 ), I 1, O ) , I I , - 1 ) of respective energies w Z , 0, -oz. The interaction Hamiltonian, written classically as yH,, Sx,can also be expressed in the following form (Cohen-Tannoudji and Haroche, 1969): flint = H,, n - 1’2y(u

+ u+)S, ,

(72)

being the mean number of photons of the coherent field. Two extreme cases will be considered, according to the relative magnitudes of p z e e m a n and x i n t e r a c t i o n . iz

B. DOUBLE RESONANCE 1. Triplet Splitting of the AmF = 0 HyperJne Transition. Autler-Townes EfSect

In the following double-resonance experiment, the amplitude of the rf fields has a low value (yH,, 6 oz).

35

PHYSICS OF THE HYDROGEN MASER

D

Wz

Ho /Y

FIG.21. Energy levels of hydrogen atoms and rf field: (a) without interaction; (b) with interaction.

The unperturbed Hamiltonian to be considered is

20= z r f + x z e e r n a n . (73) Its eigenstates are labeled IF, m F ) In) = IF, in,; n ) . Without any interaction, the energy diagram of the whole system, atoms + rf field is shown in Fig. 21a. On this type of diagram the usual magnetic resonance condition orf= ozappears as a crossing of the following levels, for example, lb)= II,O;n)

la)= 11,I;n-1);

and

Ic)=

11, - 1 ; n + l )

(74) of respective energies :

E ,=w,+(n -

l)Urf:

Eb=O,;

E,= - U z + ( n +

l)mrf

(75)

The interaction Pint between the atoms and the rf field removes the degeneracy. To the first order of perturbation, the three eigenvalues of the total Hamiltonian 2 = So+ Xin,become (Fig. 21b)

Eo = Eb;

E

_+

=Eb _+ [ ( o ,-wZ)' ~

+ (YH,~)']'''.

(76)

When the hydrogen maser is used as an amplifier, the hyperfine transition between states 10, 0) and 11, 0) is a singlet. In presence of the rf field, the three Zeeman sublevels F = 1 are coherently mixed and the (AF = 1, AmF = 0) line becomes a triplet, the two side-band components being separated from

36

C. Audoin, J. P. Sckermann, and P. Grivet

FIG. 22. Comparison between Am,=O lines amplitudes for the same value of a,,: (a) without Zeeman transitions; (b) with Zeeman transitions.

+

the central line by the quantity [(arf- mZ)’ (yHrf)2]1”, which differs from mZ. This splitting is shown in Fig. 22. Such a splitting has been reported for the first time by Autler and Townes (1950). The perturbed eigenstates of 2, labeled Iq,) I q 0 ) and Iq-) are admixtures of the unperturbed states I a ) , I b ) , and I c ) : [(l - u ‘ ) / 2 ] ’ / 2

(1

+4/2

U

(1

+ u)/2

-

[(I - u)2/2]’/2

The dimensionless parameter u equals : u = (mrf - mZ>/[(mr,

- ,A2

+ (~Hrf)~l”’.

(78)

Andresen (1968) has investigated the maser oscillation frequency shifts due to the induced Zeeman transitions.

2. Adiabatic Fast Passage In an adiabatic fast passage (AFP) experiment, the atomic beam is introduced into the rf field far from the resonance condition. The static magnetic field varies along the direction of the beam and crosses the resonance value. At the exit of the rf field, the atoms are again in a condition far from resonance. The effect of the AFP on a spin can then be interpreted as follows. Far

PHYSICS OF THE HYDROGEN MASER

37

FIG.23. Effect of the adiabatic rapid passage (ARP) on hydrogen atoms.

from resonance u z k 1 , the energy states I q+), I cpo), I cp-) of the system atom rf field are very near the unperturbed states I a ) , I b ) , and I c ) . The passage is adiabatic if the atoms do not undergo transitions between the energy states I cp+), I cpo), and I cp-). We can see in Fig. 23 that the role of the AFP is to exchange the populations of states I 1 , 1 ) and I 1 , - 1 ) and to leave unaffected the population of the state I 1,O). This fact has been used to improve the magnetic state selection of hydrogen atoms (Section 11,B). Figure 24 shows the efficiency of this selector.

+

3. Coherent Method for the Determination of Atomic Level Populations

In principle the determination of the relative population differences between the hyperfine states can be obtained by comparing the intensities of the induced transitions when the static magnetic field is parallel (for Am, = 0 transition) or perpendicular (for Am, = f 1 transition) to the 1420 MHz high frequency field. 1 T2

3

(s-') Spin exchange

J

/a

2

1 Flux of atoms

0

FIG.24. Effect of the ARP on the spin-exchange contribution to the relaxation time T z : (a) ARP switched off; (b) ARP switched on; (c) All atoms in state F = 1, m F = 1 supposed eliminated.

38

C. Audoin, J . P. Schermann, and P. Grivet

For a correct measurement of the different transition intensities, it is necessary to avoid the broadening of the lines due to the magnetic field inhomogeneities. Thus, very uniform magnetic fields are required and this, in practice, may be very difficult to achieve in the same experiment in two orthogonal directions. In the following method, Schermann eliminates this experimental problem : the populations are determined by the measurement of the intensities of the triplet lines which are observed in the common configuration where the static field is parallel to the high frequency magnetic field (Audoin et al., 1969b). The intensities of the three components of the Am, = 0 line are the products of their probability amplitudes by the differences of populations of the involved states. The probability amplitudes are proportional to the coefficient of the state I 1,O; n ) in the expansion of the perturbed states. The populations 1 q0), 1 cp-) depend upon the pumping process achieved by of states I cp,), the magnetic selector, and upon the time t required to introduce the atomic beam into the rf region. The interpretation of experiments is easier in both cases of adiabatic [o,t))l] or sudden [o,t((l] introduction. In the latter case, which corresponds to the usual experimental conditions, the intensities and the frequencies of the lines are, respectively,

2

2

[p,,,

(1 - u)’ 4

+Pl,O

“”1

1 - u2 -+P1,-1 2 ( l +4 ’. 0 0

(1

+ u)’ 2

1-u’ 2

+p1,0-

+ [(or,

+ Pl, - 1 0 0

-

-WzY

+ (Yffrf)zl”z

(79)

+ (rffrf)21”2

(81)

1-

(1 - u)’ 4 ’ ~

Kerf - 02)’

w h e r e P , , , , P l , o , P l , -1 arethepopulationsofstates 11, l), Il,O), 11, - 1 ) in the atomic beam, before introduction into the rf region. TABLE IV BEAMCOMPOSITION OBTAINED BY Two DIFFERENT MEANSOF MEASURING THE EFFICIENCY OF AN IMPROVEDSTATE SELECTOR Hyperfine level Triplet coherent method Measurement of the spin-exchange T2 relaxation time

11,1)

11, - 1 )

lll,O>

1313%

3&3%

841t5%

13 i- 2%

87+2%

39

PHYSICS OF THE HYDROGEN MASER Relative amplitude of the lines

4

Central line

0

1

a5

FIG. 25. Relative amplitude of the triplet lines for a beam equally populated in states F = 1 , mF = 0, and 1 , and completely depopulated in states F = 1, mF= - 1 and F = 0, mp = 0:solid lines, theoretical curves (sudden condition); circles, experimental points.

The validity of the method has been verified for a known composition of the atomic beam = 0) obtained with a single = Pl,o= 50%;P l , hexapolar magnet in a maser where the distance between the magnet and the bulb is about 85 cm (Fig. 25). The performances of a double-focuser device (Section II,B) have also been measured by means of this method. Figure 26 shows examples of triplet splitting with or without AFP on the beam. The results are compared in Table IV with those given by precise measurement of the contribution of spin exchange to the relaxation time T2 of the hydrogen atoms. If20505769 Hz I

1,42O$Il$752 Hz

1,42Of0$735Hz

I

I

FIG.26. Examples of triplet splitting of the AmF = 0 line: (a) without ARP on the beam; (b) with ARP on the beam.

40

C. Audoin, J . P . Schermann, and P . Grivet

FIG. 27. Comparison of two means of observing the triplet splitting of the F = 0, AmF = 0 line: (a): record of the triplet lines; (b) free precession signal in presence of Zeeman transitions (1 div = 100 msec.)

Observation of the triplet by recording the amplification lines is a steady state method. The splitting can also be observed (Bangham, 1969) on the free precession signal (Section 111,2) as a beat with a frequency equal to the triplet splitting (Fig. 27); this is a transient method. It is known that these two methods are related by Fourier transformation (Abragam, 1961b). 4. Multiple Quantum Transitions

The triplet splitting of the AmF = 0 line occurs, not only in the vicinity of resonance condition wrr = ozbut also, when the frequency wrf of the applied signal is an odd-multiple3 of the Zeeman resonance frequency w z . The resonance condition orr= w z corresponds to a crossing of the three levels / I , 1 ; n - l)ll ,O;n) and [ I , - l ; n + 1). On the other hand, when wz = (2r + ])arr ( r is an integer equal to 0, 1, 2, . . .) the atoms can undergo multi-quantum transitions, for instance, there is a crossing of the three levels 11, l ; n - 3 ) , 11,0;n),and 11, l ; n + 3 ) . In order to satisfy the conservation of angular momentum, an odd number of rf protons T + or T - ( H x perpendicular to Ho)is required.

PHYSICS OF THE HYDROGEN MASER

41

The transient method of observing the triplet splitting is very sensitive. It allowed us to observe the Autler-Townes effect due to 1, 3, and 5 quantum transitions.

C. MODIFICATION OF THE ZEEMAN HYPERFINE SPECTRUM OF ATOMIC HYDROGEN INTERACTING WITH A NONRESONANT RADIOFREQUENCY FIELD We consider the following experiment: hydrogen atoms, oriented by a magnetic state selector, enter the storage bulb of an hydrogen maser in both hyperfine levels I:= 1, m, = 0, and F = 1, mF = 1. The static field Ho is applied perpendicular to the microwave field Ifhyp. Because of the polarization of Hhyp,only the field dependent transitions F = 1, Am, = k 1 can be induced. The maser oscillation which produces Hhypis observed on the AF = 1 , Am, = 1 transition, with a frequency uihyp= w o + wz ,where wo is the zero magnetic field transition frequency and w z the Zeeman resonance frequency (wz = gpBI f o ) . If a linear rf field of amplitude H,, is applied along the direction of the microwave field Hhyp(Fig. 28), with a frequency wrf much greater than w, but much smaller than wo , the maser oscillation frequency is modified. The same experiment has been performed with rubidium atoms oriented by optical pumping (Haroche et al., 1970) and the experimental results fit the same curve (Fig. 29). This strong modification of the Zeeman hyperfine spectrum has been predicted by Cohen-Tannoudji and Haroche (1969). For the relative dis-

4"

I

State selected atomic beam

FIG.28. Field configuration for modification of the LandC factor.

42

C. Audoin, J. P. Schermann, and P. Grivet

I FIG.29. Variation of the Lande factor of ”Rb (circles) and H (triangles) atoms.

position of the fields shown in Fig. 28, the Lande g-factor of the atoms interacting with the rf field is given by

s = gJo(YHrf/wrf)

(82) where g is the Land6 factor of the unperturbed atoms and Jois the zero-order Bessel function. This effect can be related to the Lamb shift and to the anomalous magnetic moment of the electron and the muon which are interpreted, in the quantum electromagnetic theory, by virtual emission and absorption of photons in the vacuum . In the light shift experiments (Cohen-Tannoudji, 1962), or in this experiment, the virtual transitions are induced by the externally applied nonresonant electromagnetic field. ACKNOWLEDGMENTS The authors wish to thank their colleagues M. Desaintfuscien, J. L. D u ch h e, P. Petit, and J. Viennet for many useful discussions, efficient equipment design, and participation in the measurements. They are indebted to D r F. G . Major and Dr W. M. Hughes who read the manuscript and made useful comments and suggestions. The author’s work has been sponsored by Centre National de la Recherche Scientifique and Direction des Recherches et Moyens d’Essais.

REFERENCES Abragam, A. (1961a). In “Principles of Nuclear Magnetism,” p. 36. Oxford Univ. Press (Clarendon), London and New York. Abragam, A. (1961 b). In “ Principles of Nuclear Magnetism,” p. 114. Oxford Univ. Press (Clarendon), London and New York.

PHYSICS OF THE HYDROGEN MASER

43

Andresen, H. G. (1968). Z. Phys. 210, 113. Arditi, M. (1964). Proceedings of the IEEE-NASA Symposium on Short-Term Frequency Stability, Goddard Space Flight Center, Greenbelt, Maryland p. 177. Arditi, M., and Carver, T. R. (1964). Phys. Rev. A 136,643. Audoin, C. (1966). C. R. Acad. Sci. 263, 542. Audoin, C. (1967). Rev. Phys. Appl. 2, 309. Audoin, C. (1968). Phys. Lett. A 28, 372. Audoin, C., Desaintfuscien, M., and Schermann, J. P. (1969a). Nucl. Instrum. Methods 69, 1 . Audoin, C., and Viennet, J.(1969).Actes Colloque lnternationalde Chronometrie, Paris, p . AS. Audoin, C., Desaintfuscien, M., and Schermann, J. P. (1967). C. R. Acad. Sci. 264, 698. Audoin, C., Desaintfuscien, M., and Schermann, J. P. (1968). Proceedings of the 22nd AnnualSymposium on Frequency Control, USAEC, Fort Monmouth, New Jersey p. 493. Audoin, C., Desaintfuscien, M., and Schermann, J. P. (1969a). Nucl. Instrum. Methods 69, 1 . Audoin, C., Desaintfuscien, M., Pikjus, P., and Schermann, J. P. (1969b). IEEEJ. Quantum Electron. 5, 431. Autler, S. H., and Townes, C. H. (1950). Phys. Rev. 78, 340. Balling, L. C., Hanson, R. J., and Pipkin, F. M. (1964). Phys. Rev. A 133,607. Bangham, M . (1969). Private communication. Basov, N. G., and Prokhorov, A. M. (1954). Sou. Phys. -JETP27,437. Bender, P. L. (1963). Phys. Rev. 132, 2154. Berg, H. C. (1965). Phys. Rev. A 137, 1621. Berstein, I. (1950). BUN.Acad. Sci. (URSS), Ser. Phys. 14, 187. Blaquiere, A. (1953). Ann. Radioelec. 8, 36, 153. Bloom, S. (1956). J. Appl. Phys. 27, 785. Brenner, D. (1970). J. Appl. Phys 41, 2942. Cohen-Tannoudji, C. (1962). Ann. Phys. (Paris) 7, 423, 469. Cohen-Tannoudji, C., and Haroche, S. (1969). J. Phys. (Paris) 30, 125, 153. Combrisson, J. (1960). In “Quantum Electronics” (C. H. Townes, ed.), p. 167. Columbia Univ. Press, New York. Crampton, S. B. (1967). Phys. Rev. 158, 57. Crampton, S. B., Robinson, H. G., Kleppner, D., and Ramsey, N. F. (1966). Phys. Rev. 141, 55. Crampton, S. B., Berg, H. C., Robinson, H. G., and Ramsey, N. F. (1970). Phys. Rev. Lett. 24, 195. Debely, P. E. (1970). Proceedings of the 24th Annual Symposium on Frequency Control, USAEC, Fort Monmouth, New Jersey. Rev. Sci. Znstr. 41, 1290. Dehmelt, H. G. (1958). Phys. Rev. 109, 381. Fano, U. (1957). Rev. Mod. Phys. 29, 74. Feynman, R. P., Vernon, F. L., and Hellwarth, R. W. (1957). J. Appl. Phys. 28,49. Forston, E. N., Kleppner, D., and Ramsey, N. F. (1964). Phys. Rev. Lett. 13, 22. Goldenberg, H. M., Kleppner, D., and Ramsey, N. F. (1960). Phys. Rev. Lett. 5, 361. Goldenberg, H. M., Kleppner, D., and Ramsey, N. F. (1961). Phys. Rev. 123, 530. Gordon, J. P., Zeiger, H. J., and Townes, C. H. (1954). Phys. Rev. 95, 282. Grasyuk, A. Z., and Orayevskiy, A. N. (1964). Radio Eng. Electron. Phys. ( U S S R ) p. 424, 443. Grivet, P., and Audoin, C. (1969). XVIth General Assembly of U.R.S.I., Ottawa. In “Progress in Radio Science 1966-1969” (J. A. Lane, J. w. Findlay, and C. E. White, eds.), p.289. International Union of Radio Science, Brussels, Belgium, 1971. Grossetcte, F. (1964). J. Phys. (Paris) 25, 383.

44

C. Audoin, J . P . Schermann, and P . Grivet

Crotch, H. (1970). International Conference on Precision Measurement and Fundamental Constants, Nat. Bur. Stand., Gaithersburg, Maryland. Haroche, S., Cohen-Tannoudji, C., Audoin, C., and Schermann, J. P. (1970). Phys. Rev. Lett. 24, 861. Hartmann, F. (1969). ZEEE J. Quantum Electron. 5, 595. Hegstrom, R. (1970). International Conference on Precision Measurement and Fundamental Constants, Nat. Bur. Stand., Gaithersburg, Maryland. Hellwig, H. (1968). Phys. Rev. 166, 4. Hellwig, H., Vessot, R. F. C., Levine, M., Zitzewitz, H., Allan, D., and Glaze, D. (1970). IEEE Trans. Znstrum. Meas. 19, 200. Jousse, M. (1967). DiplBme d’Etudes Superieures, Paris University, Orsay, France. Kleppner, D. (1970). International Conference on Precision Measurement and Fundamental Constants, Nat. Bur. Stand., Gaithersburg, Maryland. Kleppner, D., Goldenberg, H. M., and Ramsey, N. F. (1962). Phys. Rev. 126, 603. Kleppner, D., Berg, H. C., Crampton, S. B., Ramsey, N. F., Vessot, R. F. C., Peters, H. E., and Vanier, J. (1965). Phys. Rev. A 138, 972. Kryloff, N., and Bogoliuboff, N. (1947). “Introduction to Non-Linear Mechanics.” Princeton Univ. Press, Princeton, New Jersey. Laink, D. C., and Bardo, W. S. (1969). Electron. Lett. 5, 364. Lamb, W. E. (1960). Zn “Lectures in Theoretical Physics” (W. E. Brittin and B. W. Downs, eds.), Vol. 2, p. 435. Wiley (Interscience), New York. Lambe, E. B. (1959). Ph.D. Thesis, Princeton Univ., Princeton, New Jersey. Larson, D. J., Valberg, P. A., and Ramsey, N. F. (1969). Phys. Rev. Lett. 23, 1369. McCoubrey, A. 0. (1966). Proc. ZEEE 54, 116. Mathur, B. S., Crampton, S. B., Kleppner, D., and Ramsey, N. F. (1967). Phys. Rev. 158, 14. Menoud, C., and Racine, J. (1969). Actes Colloque International de Chronometrie, Paris, p. A8. Myint, T., Kleppner, D., Ramsey, N. F., and Robinson, H. G. (1966). Phys. Rev. Lett. 17, 405. Nikitin, A. I., and Strakhovskiy, G. M. (1966). Radio Eng. Electron. Phys. (USSR) 11, 1650. Peters, H . E., McGunigal, T. E., and Johnson, E. H. (1968). Proc. 22nd Annu. Symp. Frequency Control, USAEC, Fort Monmouth, New Jersey, p. 464. Purcell, E. M., and Field, G. B. (1956). Astrophys. J. 124, 542. Ramsey, N. F. (1950). Phys. Rev. 78, 699. Ramsey, N. F. (1965). Metrologia 1, 7. Ramsey, N. F. (1967). Phys. Rev. 158, 14. Ramsey, N. F. (1 970). International Conference on Precision Measurement and Fundamental Constants, Nat. Bur. Stand., Gaithersburg, Maryland. Robinson, H. G . ,and Hughes, H. M. (1970). InternationalConferenceonPrecision Measurement and Fundamental Constants, Nat. Bur. Stand., Gaithersburg, Maryland. Schermann, J. P. (1966). C. R. Acad. Sci. 263,295. Schwartz, C. (1959). Ann. Phys. (New York) 2, 156. Shimoda, K., Wang, T. C., and Townes, C. H. (1956). Phys. Rev. 102, 1308. Shirley, J. H. (1968). Amer. J. Phys. 36, 949. Slichter, C. P. (1963). In “ Principles of Magnetic Resonance,” p. 78. Harper & Row, New York. Statz, H., and de Mars, G. (1960). I n “Quantum Electronics” (C. H. Townes, ed.), p. 530. Columbia Univ. Press, New York.

PHYSICS OF THE HYDROGEN MASER

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Strakhovskiy, G. M., and Uspenskiy, A. V. (1966). Sou. Phys.-JETP 23, 247. Taylor, B. N., Parker, W. H., and Langenberg, D. N. (1969). “The Fundamental Constants and Quantum Electrodynamics.” Academic Press. Uzigiris, E. E., and Ramsey, N. F. (1970). Phys. Rev. A l , 429. Vanier, J. (1967). Phys. Rev. Lett. 18, 333. Vanier, J. (1968). Phys. Rev. 168, 129. Vanier, J., and Vessot, R. F. C. (1966). IEEE J. Quantum Electron. 2, 391. Vanier, J., and Vessot, R. F. C. (1970). Metrologia 6, 52. Vanier, J., Peters, H. E., and Vessot, R. F. C. (1964). IEEE Trans. Instrum. Meas. 13, 185. Vessot, R. F. C., Levine, M., Cutler, L., Baker, M., and Mueller, L. (1968). Proceedings of the 22nd Annual Symposium on Frequency Control, USAEC, Fort Monmouth, New Jersey p. 605. Vessot, R. F. C., Levine, M., Zitzewitz, P. W., Debely, P. E., and Ramsey, N. F. (1970). International Conference on Precision Measurement and Fundamental Constants, Nat. Bur. Stand., Gaithersburg, Maryland. Wilkinson, D. T., and Crane, H. R. (1963). Phys. Rev. 130, 852. Wittke, J. P., and Dicke, R. H. (1956). Phys. Rev. 103, 620. Zitzewitz, P. W. (1970). Ph.D. Thesis, Harvard Univ., Cambridge, Massachusetts. Zitzewitz, P. W., Uzgiris, E. E., and Ramsey, N. F. (1970). Rev. Sci. Instrum. 41, 81.

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MOLECULAR WAVE FUNCTIONS: CALCULATION AND USE IN ATOMIC AND MOLECULAR PROCESSES J . C . BROWNE Departments of Physics and Computer Science, University of Texas Austin, Texas

I. Introduction and General Principles . . . . . . . . . . Matrix Elements . . . A. A Brief Survey of

41 53 53 59 61

C . Integral Evaluation

61

F. The Eigenvalue-Eigenvector Problem. . . . . . G . Calculation of Coupling Matrix Elements ........................ 111. Some Results and E IV. Atom-Atom Scatter A. Elastic Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Inelastic Processes V. Radiative Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Spectroscopic Processes . . . . . . . . . . . . B. Scattering Processes . ............ ................ References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69 73 15 78

19 19 81 83 84 85 87

I. Introduction and General Principles The description and comprehension of many of the most interesting processes in atomic and molecular physics involves a knowledge of the molecular wave functions and potential curves (surfaces) accessible to a pair (set) of interacting atoms. The fomulation of principles will be in terms of diatomic molecules. The generalization to polyatomics is straightforward, but greatly increases notational complexity. The assumption underlying most of our discussion is that the eigenfunctions $e,,k(r,R ) and eigenenergies of the electronic, nonrelativistic,fixed nuclei Hamiltonian, Eq. (3) together with the nuclear motion wave functions $"",, k(R)determined by the use of the electronic 47

J. C. Browne

48

eigenenergies constitute an adequate zero-order basis for representation of a variety of atomic and molecular processes. $total(r,

R, = C $el,

k(r, R)$nuc,k(R).

k

(1)

The set of states, $total(r, R), are in this approximation characterized and differentiated by a set of operators which commute with the He, of Eq. (3). The set of eigenvalues of the operators is denoted by k . The subscript notation “el ” and “ nuc will be suppressed in instances where no confusion should result. The nuclear wave function $, k(R) has the form ”

$nu,,

k

=

- lFk,

u, K ( R ) D k f ( e C P )

(2)

where Fk, ”, K ( R )is the wave function for translational (or vibrational) motion and Dx,,(O, CP) is an eigenfunction of the symmetric top. (Rose, 1957; Herzberg, 1950). R is the relative coordinate for internuclear distance, and 0 and CP are the angles of R in a space-fixed reference frame. The significance of K , A, and M depends on the coupling of the various angular momenta of the molecular states. Hund’s case b coupling (Herzberg, 1950) is appropriate for the low lying states of light diatomic molecules for which most accurate calculations have been made. Then the angular momentum operator K is K = L N, where L is the electronic angular momentum and N is the nuclear rotation angular momentum. S, the electronic spin, remains quantized in the space fixed reference frame, A is the component of K along the direction of R, and M is the component of K along the z axis of the fixed reference frame. or $,,,,k, its Intrinsic nuclear spin has no effect on the computation of effects usually being seen only in weight factors and in selection rules.

+

l N He,=- CVi22 i=1

N

1 1 >Zrci+

i= 1

C=U,

b

x Z r i l + - zazb R

i> j

(3)

Atomic units are used in Eq. (3) and throughout except where noted. The indices i and j refer to electrons and the index c refers to nuclei a and 6. The coordinates are defined in a system which rotates with the molecular axis, and Eq. (3) is in the form appropriate for a diatomic molecule. Equation (3) is derived from the full Breit-Pauli Hamiltonian.

Where ps = - iV,, mu and mb are nuclear masses, c1 is the fine structure constant, and H , contains the first-order relativistic and magnetic corrections

49

MOLECULAR WAVE FUNCTIONS AND PROCESSES

(see, for example, Pack and Hirschfelder, 1968). The steps are transformation to a relative set of coordinates, usually the center-of-mass of the nuclei (CMN) system, transformation to a set of axes which rotates with the internuclear axis; and eliminating terms resulting from action of the operators representing nuclear motion on the electronic wave function, dropping H a , and eliminating electronic operator terms which have coefficients of order l/(rn, + mb).The nuclear wave functions are then the solutions of t(2p)-'(a2/dR2) - Ek,

o,

K Fk,

+ ( 2 p R 2 ) - ' ( K ( K + 1) -A2) + Eel.k(R)IFk,v . K(R) u,

(5)

K(R).

Correction of E e l . kin Eq. (5) by evaluation of the diagonal matrix elements of the neglected terms over the fixed nuclei basis $ e l , k ( r , R ) leads to the adiabatic" approximation. This is the most general practicul means of solution which retains the concept of unique electronic potential curves. The terms arising from the action of the nuclear motion operators on the $ e l , k couple the set of fixed nuclei (or adiabatic) states. They are the transition operators for the inelastic processes of atom-atom scattering in a zero-order basis of molecular states. Pack and Hirschfelder (1970) give a detailed discussion of the nature of the equations and solutions obtained by retaining one or more of the terms eliminated in arriving at the operator of Eq. (3). The scope of this article is primarily the calculation of approximate eigenfunctions and eigenenergies of Eq. (3) and the application of these eigenfunctions to the study of atomic and molecular processes. The calculation of the so-called " diabatic" states (Lichten, 1963, 1965, 1967a; O'Malley, 1967, 1969; F. T. Smith, 1969a) and potential curves for which an unambiguous computational definition has not yet been fully worked out (see F. T. Smith, 1969a) will be discussed briefly. Until the large scale digital computer became fairly widely available during the 1960s, it was almost impossible to calculate a molecular wave function or potential curve to the accuracy needed to carry out quantitative studies or analysis of molecular processes. There were a few notable exceptions, principally in the case of H 2 + (Bates et al., 1953a), HeH2+ (Bates and Carson, 1956), and H, (James and Coolidge, 1933; James et a[., 1936; Present, 1935). Extensive computer programs have beendeveloped at several research laboratories which will, with almost trivial labor on the part of the experienced investigator, obtain approximate molecular wave functions and potential curves. The capabilities of these programs range from extremely accurate energy surfaces for very small (2-, 3-, or 4-electron) systems to the calculations on a semiempirical basis of energy levels and charge densities for very large organic or even biochemical systems. This review is confined to methods of calculation appropriate for systems that are usually regarded as the province of the atomic and molecular physicist rather than the quantum "

50

J . C. Bvowne

chemist, organic chemist, or biochemist. As a further restriction, only applications to state changing processes such as atom-atom scattering (elastic scattering is an exception), and interaction with radiation will be considered. Calculations of molecular geometries, valence binding, and static properties, e.g., dipole moments, electric field gradients, etc., will be covered only incidentally. Finally, only applications where calculated molecular wave functions and/or potential energy curves have been used directly will be considered. The use of molecular theory concepts in the analysis and interpretation of experiments is too large an area to be considered together with the computational work. Direct calculations of molecular processes (low or intermediate energy atom-atom scattering, electron-molecule scattering, photon-molecule processes, etc.) requires the wave functions and potential curves of the basis of zero-order states accessible under the action of the transition operator for the process and the coupling matrix elements of the transition operator over the basis ofmolecular states. All ofthese quantities are usually required over a range of geometries or internuclear separations. The computation of the set of molecular eigenstates necessary for the direct calculation of a rate or cross section for a specific molecular process or set of processes has been attempted in only a few cases (see Sections 1V and V). The calculation of the coupling matrix elements between molecular states (except for electric dipole radiative transitions) has received virtually no attention. Even for the hydrogen molecule, where very accurate calculations (Kolos and Wolniewicz, 1964a,b, 1966a,b, 1969) for a number of states have been carried out, there does not exist a coherent set of calculated molecular eigenstates suitable for the accurate computation of general inelastic or state changing processes. The coupling matrix elements between the available set of wave functions has not been computed except for the dipole radiation transitions (Rothenberg and Davidson, 1967; Wolniewicz, 1969). Theoreticians wishing to study molecular processes have, for the most part, been forced to rely on qualitative arguments or such fragmentary information on molecular wave functions and energies as had been produced in the interest of static energetics. This situation is beginning to change both by the establishment of research groups with joint interests in molecular wave function calculations and molecular processes, and by the joining of forces between those interested in wave function calculation and the process theorists. Additionally, some of the automatic programs for the calculation of approximate molecular wave functions previously mentioned are available as standard package programs that can be used by the process theorist if adequate computer facilities are available. The work of the Quantum Chemistry Program Exchange (QCPE, University of Indiana, Bloomington, Indiana) in disseminating programs relating to molecular calculations has been very valuable and should be recognized. The recent organ-

MOLECULAR WAVE FUNCTJONS A N D PROCESSES

51

ization of a journal (Computer Physics Communications, North-Holland, Amsterdam) devoted to the publication and distribution of refereed computer programs and documentation, should also greatly assist in the dissemination of the necessary computational capabilities. The steps involving the discarding of the effects of nuclear motion operators on the electronic wave functions and leading from the full Hamiltonian, H , to Eq. (3) are strictly valid only for infinitely slow nuclear motion. The zeroorder states are coupled to some degree for all finite nuclear velocities. It has become clear from analysis of scattering data, particularly for the collisions of He“ with He (Lichten, 1963; Marchi and Smith, 1965; Kennedy and Smith, 1969) that the fixed-nuclei (or adiabatic) molecular potential curves and wave functions are not always the most appropriate zero-order basis for representing atom-atom collision processes over the entire range of internuclear separations in the moderate to fairly low energy (0.05-2.0) keV range. Adiabatic potential curves for states with identical sets of eigenvalues for the set of operators which commute with H e , cannot intersect (Von Neumann and Wigner, 1929). I t appears that when a pair of colliding atoms moving on an adiabatic potential curve passes an internuclear separation at which the curve is closely approached by a second curve (or set of curves) of the same symmetry, the elastically scattered or resonantly charge-exchanged atoms (or ions) appear to move according to potentials that differ markedly from the adiabatic continuation of the potentials exterior to the crossing region. Lichten (1963) originated the phrase diabatic” to describe these potential curves and their associated molecular states. It is probable that the flux into inelastic channels also sees potentials fairly markedly different from a continuation of the adiabatic curves to smaller internuclear separations, although the direct evidence for this is sparse. A number of suggestions have been made for diabatically continuing the adiabatic curves through and past the regions of strong interaction. Lichten (1963, 1965, 1967a,b) has constructed diabatic curves through the use of elementary molecular orbital theory, which have good qualitative justification but which basically cannot be refined to account for increasing detailed resolutions of experiments. O’Malley (1967) has suggested defining some selected state as a “quasiadiabatic” state by projecting it out and rendering it orthogonal to all other potentially intersecting states. This proposal lacks an effective means of defining the projection operator. Additionally, generalizing to a set of quasi-adiabatic states as would be required for consideration of inelastic processes would probably be complex. O’Malley (1969) has also formulated and carried out calculations for diabatic curves with a valence-bond equivalent -the Projected Atomic Orbital (PAO) method-of Lichten’s MO formulation. This method of calculation is suggested as a means of obtaining welldefined diabatic states to be used in the earlier (1967) formulations of the “

“

“

”

”

52

J . C. Browne

atom-atom collision problem. The curves O’Malley obtains with the P A 0 method would appear to be as qualitatively valid as those obtained by other methods (Lichten, 1963; Marchi and Smith, 1965) but have not been tested by the computation of cross sections in the cases where detailed comparison to experiment is possible. The P A 0 method as currently formulated, however, is only applicable to a partial set of states, and suffers from some nonuniqueness problems. O’Malley (1969) also suggests that the use of the “ stabilization” method (Lipsky and Russek, 1966; Taylor et ul., 1966) merits further investigation as a means of computation of diabatic states. F. T. Smith (1969a) defines carefully the role played by different portions of the full nonrelativistic Hamiltonian in the formulation of the collision problem. He also gives a careful discussion of the work done toward the definition of diabatic states prior to 1968. He suggests that it may be appropriate to diagonalize different portions of the full Hamiltonian as internuclear separation varies so as to minimize the coupling matrix elements between the set of zero-order basis states (which might then be eigenfunctions of different Hamiltonians for adjacent segments of R ) . For example, when the coupling matrix elements between states of the same value of A become larger, one might diagonalize the adiabatic coupling operators H A , [Eq. (39)] to obtain an appropriate zero-order basis. The coupling operator in the new (diabatic) basis would be H e , of Eq. (3). These new coupling matrix elements would, hopefully, be small. F. T. Smith has significantly advanced the status of the formulation of diabatic states by relating the different possible types of zeroorder basis states to an appropriate Hamiltonian. There may well be difficulties, however, in fitting the different zero-order potential curves and wave functions together. The problem of diagonalizing a basis set over the nuclear momentum Hamiltonian is roughly equivalent to the computation of the coupling matrix elements between adiabatic states. These problems, discussed in Section II,E, do not yet have fully defined solutions. The attention recently focused on attempts to find and develop rigorous methods of calculating diabatic states has tended to obscure the fact that the diabatic states supplement, but do not replace, the role played by adiabatic states in collision problems. The adiabatic potential curves are apparently excellent representations of the potentials seen by colliding atoms in the regions exterior to regions of strong interaction in both elastic and inelastic channels (Kennedy and Smith, 1969). A full solution of the set of coupled equations defined by some zero-order basis (diabatic or adiabatic) represents the only complete solution to the collision problem. It is clear in some cases that including coupling among adiabatic states is an effective means of extending the adiabatic approximation (Bates and Williams, 1964; Knudson and Thorson, 1970) to represent the experimental cross sections. This is probably true whenever the interactions are due to actual crossings of states with different symmetries or

MOLECULAR WAVE FUNCTIONS AND PROCESSES

53

where degeneracies at the united atom limit occur. It is also clear that for regions of R where there is strong interaction among a set of adiabatic states ofthe same symmetry that some set of diabatic states will provide a better zeroorder basis set for representation of the collision problem. An effort has been made to be reasonably complete in the literature relevant to the primary theme of calculations of molecular wave functions as basis sets for analyzing and computing atomic and molecular processes. There is a vast literature peripheral to our concern. Only representative citations to this body of work are made, usually to recent work with good lists of references or to review articles. Where a recent review on a peripheral subject or topic is available, it will be referred to and other references omitted.

11. Computations of Wave Functions, Potential Surfaces, and Coupling Matrix Elements The basic problem is to approximate a set of solutions of the Schrodinger equation defined by the H of Eq. (3) which are also symmetry-adapted to conform to the states of interest. The difficulty in obtaining solutions arises from the presence of the electron repulsion operators r i j l in H . The solutions for the one-electron diatomic molecules can be obtained analytically (Bates and Carson, 1956; Bates et al., 1953a and early references therein contained; Peek, 1965; Wind, 1965). The difficulties raised by the r;’ operator are referred to in the quantum chemistry literature as the “ correlation problem.” See Kelly (1969a), Lowdin (1969), Miller and Ruedenberg (1968a), Nesbet (1969), and Sinanoglu (1969) for recent reviews of some of the principal formalisms for dealing with the correlation problem.” These articles contain extensive further references on the subject. Condon (1968) and Nesbet (1971) summarize briefly the relationships of several theories and cite a useful summary list of references. “

A. A BRIEFSURVEY OF FORMALISM It is the purpose of this section to survey methods of computation of molecular wave functions and potential curves from the viewpoint of their utility (or potential utility) in the calculation and analysis of atomic and molecular processes. The varied (but often closely related) different formalisms for carrying out computations on atoms and/or molecules will not be reviewed in detail. Attention will be paid, however, to the common set of computational problems. The characteristics of a computational formalism are often defined in terms of a basis set definition. Thus, some overlap between this section and the immediately succeeding section is unavoidable.

J . C. Browne

54

It was recognized very early (Eckart, 1930; Hylleraas and Undheim, 1930) that the use of an expansion $i

=

1

cijQj(uj)

j

together with the Rayleigh-Ritz variation principle (RRVP)

was a very powerful tool for obtaining approximate $; and E i . The Qj(uj) are N-particle composite functions of the elementary basis set {4i}. The Qj(uj) are antisymmetric functions of electronic space-spin coordinates and are usually (but not necessarily) eigenfunctions of the operators which commute with H e , . The 4; may be one-particle (orbital) functions, two-particle (geminal or pair) functions, or even multiparticle functions (although this latter alternative has never been used in practical calculations). The use of Eqs. (6) and ( 7 ) as a basis for carrying out calculations is called the ’‘ configuration interaction” (CI) method. It is quite clear the direct use of Eqs. (6) and (7) for systems of more than two electrons is virtually intractable without the aid of powerful computing facilities. I f the Q j ( u j ) do not contain interelectronic distance coordinates directly, the convergence of the expansion is slow. If directly correlated functions are used, the more difficult integrals in Eq. ( 7 ) have dimension 3 N , where N is the number of electrons and are very difficult to evaluate. The self-consistent field (SCF) method (Slater, 1960), which yields a wave function compounded of independent particle orbitals, was computationally much more tractable. The SCF procedure has a number of advantages ; its orbitals are eigenfunctions of a known Haniiltonian, and diagonal expectation values of one-electron operators are estimated correctly to second order. A number of calculations on atoms were carried out in the 1930s and 1940s. Roothaan (1951) (see also Hall, 1951) proposed a method of obtaining approximate SCF orbitals by an expansion in analytic functions (linear combinations of atomic orbitals, LCAO’s) which applied to systems which could be represented by all closed-shell molecular orbital wave functions (LCAO-SCF-MO method). This basic method in several modifications (Nesbet, 1955; Roothaan and Bagus, 1963) was coupled to several of the early molecular integral programs and provided the basis for the bulk of molecular calculations until the middle 1960s (see for literature citations, Krauss, 1967; Nesbet, 1967a). There are a number of deficiencies in this approach that prevented its wide-scale application to the calculation of the basis sets for calculations on atomic and molecular processes. Closed-shell MO wavefunctions cannot represent all of the eigenfunctions for some of the operators that commute (or nearly commute) with Hand thus serve to charac-

55

MOLECULAR WAVE FUNCTIONS A N D PROCESSES

terize and differentiate the molecular states. Closed-shell MO wavefunctions cannot in general correlate with the separated atom states that couple to form a given molecular state. The “ correlation energy ” tends to be a fairly strong function of internuclear separation so that an independent particle molecular wave function has an inherent error in its calculated potential curve. Notwithstanding these defects, this computational procedure has been, and apparently will continue as, a staple of chemical valence theory and static properties calculations (see, for example, Clementi, 1969 and Yoshimine and McLean, 1967). A great deal of effort has gone into the development of formalisms that retain the advantages of the SCF model while overcoming the defects of the closedshell LCAO-SCF-MO theory. These are the projected, open-shell, or multiconfigurational Hartree-Fock (MCSCF). (The literature on this subject is extensive, see Wahl and Das, 1970 for a list of references; also Adams, 1969b for a concise summary of methods; Clementi and Veillard, 1967; Gallup, 1969; Ladner and Goddard, 1969; Lowdin, 1969; McWeeny, 1955; Poshusta and Krambling, 1968.) The casual reader should not allow himself to be overwhelmed by the intimidating complexity of the defining equations of MCSCF theory. These equations are never considered except with the assistance of a powerful computer system. Wahl, Das, and co-workers (see Wahl and Das, 1970) have developed and implemented a computer program for a form of MCSCF formalism that omits the changes in atomic core (e.g., ls2 shell) functions as a function of internuclear separations. This is called the optimized valence configuration (OVC) method. This has been used to compute complete potential curves for several molecular systems. The OVC method has been formulated primarily with chemical bonding and valence effects as primary concerns, but it appears to have potential for applications to calculations of the sets of wave functions needed for consideration of transition rates and cross sections. Goddard and co-workers (Wilson and Goddard, 1969; Palke and Goddard, 1969) have also made extensive applications of his “ G I ” method. Very extensive work has gone into the development of extensions to the independent particle model which have as their aim the computation of correlation energy. One branch of this work arises from the chemical notions of localized bonds and separated pairs. Miller and Ruedenberg (1968a,b,c) give a good general discussion making the connections to the earlier work and reviewing the more recent literature. Silver et al. (1970) extend this work and discuss computational techniques. Ahlrichs and Kutzelnigg (1968) and Jungen and Ahlrichs (1970) give a somewhat different view and offer comparison to other approaches. Another branch arises from the many-body theory techniques (Cizek, 1969; Kelly, 1969a; Nesbet, 1969; Schaefer and Harris, 1968a; Sinanoglu, 1969). Attention is focused in each case on the correlation “

”

J . C. Browne

56

energy of separate pairs of electrons. McWeeny and Steiner (1965) and Sutcliffe and McWeeny (1969) give useful surveys of composite particle formalisms for wave function calculation. Wahl and Das (1970) have succinctly tabulated with appropriate references the computational and conceptual properties of some of the formalisms, particularly the two-particle natural orbital ones, which have had at least reasonably successful application to molecules. Illustrative applications to atoms (Kelly, 1963, 1964, 1966; McKoy and Sinanoglu, 1964; Nesbet, 1967b,c; Scheafer and Harris, 1968a; Sinanoglu and Oskutz, 1969) of the many-body theory formalisms have recovered a large percentage of the correlation energy. Applications of these methods to molecules is very limited. Bender and Davidson (1967, 1969) have applied the Nesbet pair correlation theory to diatomic hydrides. Kelly (1969b) and Schulman and Kaufman (1970) (H2) and Lee et al. (1970) (HF) have made calculations using single-center wave functions as basis sets for Kelly’s many-body-perturbation theory approach with encouraging results. Barr and Davidson (1970) have carefully analyzed some of the ambiguities in the pair correlation approaches. Two techniques that make computationally tractable the use of directly correlated wave functions for molecules with more than two electrons are currently in development and/or use. Conroy (1964a,b,c,d) uses the technique of minimizing the variance of the wave functions over a grid of points in configuration space “

Min [ U 2 ]=

”

[

( H i ) - E$)’ dx// i)2 dx] .

The integrations are done by purely numerical (random variate) methods. The trial wave functions is chosen in a form which smoothes the electronnuclear and electron-electron singularities in the Hamiltonian so that a smoothly distributed numerical integration grid of manageable size can be used. This method has been used to produce the most accurate potential surfaces yet available for H 3 + (Conroy, 1969), H, (Conroy and Bruner, 1967), and H, (Conroy and Malli, 1969). It is not yet certain that application to systems of many electrons will be computationally attractive. Boys and Handy (1969a,b) have developed and implemented a means of calculation based on the method of moments (Hegyi et al., 1969; Szondyand Szondy, 1966)whereby directly correlated wave functions can be obtained for polyelectron molecules. The general form of the equations of the method of moments (Handy and Epstein, 1970) can be written (9)

MOLECULAR WAVE FUNCTIONS AND PROCESSES

57

=xi

The set $ r is disjoint from the set mi, and Q ciQi is an approximate solution to Schrodinger’s equation. Boys and Handy (1969a,b) choose

($, = c-lQ’,Ql = CQ’)

ni,

where C is a correlation function C = f ( r i j , ri , r j ) . @’ is an independent particle determinantal function and the /I,are the parameters upon which the C and Q ’ depend. It is possible to choose C such that the most difficult integrals involve integrations of not more than nine dimensions and usually only six dimensions instead of 3 N , where N is the number of electrons; the electron-electron singularities are removed from the integrals of the matrix elements. Boys and Handy (1969a,b) evaluate the integrals of the matrix elements by direct numerical integration over six dimensions. Boys and Handy (1969c, d) have applied the formalism to the neon atom and to the equilibrium internuclear separation of the LiH molecule with encouraging results. Handy (1969) has obtained excellent results for the Be atom. The primary drawbacks of this method are the absence of an upper bound relation for the energy eigenvalue and the necessity for dealing with unsymmetric matrix problems which sometimes have undesirable stability characteristics. It is clear that the methods of moments offers the possibility of useful alternatives tothe standard variational procedure. Note that the standard variational procedure is recovered if the sets $, and Q iare identical. Rayleigh-Schrodinger perturbation theory has had relatively little impact in the calculation of molecular wave functions. The primary use has been in the calculation of long range interaction potentials (Dalgarno and Davidson, 1966; Kolos, 1967) and more recently asymptotic forms of matrix elements (Brown and Whisnant, 1970). Several authors have carried out perturbation calculations for the energy of H, (DvofaZek and Horak, 1967; Goodisman, 1968; Kirtman and Decious, 1968; Matcha and Brown, 1968) and HeH’ (Bartolotti and Goodisman, 1968). The best calculation for H, , that of Matcha and Brown (1968), is of accuracy comparable to the best variational calculations but would appear to be equally laborious. Goodisman (1969a,b) has obtained suprisingly good results for He, and LiH by evaluation of perturbation energies through second-order starting with a zero-order basis of scaled independent particle molecular orbitals. The labor of these calculations would appear to be comparable to that for variational calculations of roughly comparable accuracy. Successful extension of this approach probably calls for better zero-order starting functions. It is of interest that directly correlated

58

J. C. Browne

first-order wave functions can be obtained with the calculation of only twoelectron integrals. Direct configuration interaction (CI) calculations based on the use of single particle (atomic or molecular) orbitals and some type of generalized valence-bond approach (Browne and Matsen, 1962; Michels and Harris, 1968) has been the only method which has as yet been applied to produce basis sets for direct calculation of rates or cross sections for processes. A primary reason is that direct CI wave functions can represent states of arbitrary symmetry and can easily be adapted to pass to appropriate separated- and united-atom states. It seems well established that the most effective expansion not involving the use of interelectronic coordinates will be based on sets of single particle functions which are optimized (by a linear transformation) for use in configuration interaction wave function terms (see Section 11,B). The determination of the c i j in Eq. (6) leads directly to the familiar secular equation for the eigenvectors and eigenvalues ( H -ES)C=O.

The solution of this matrix equation provides approximate eigenvectors, lc/i, and eigenvalues, Ei, for the lowest n of the symmetry states present in the expansion basis. The E iare rigorous upper bounds (Hylleraas and Undheim, 1930; MacDonald, 1933) to the first nEi, true eigenvalues [to the extent that the H of Eq. (3) is valid]. Dalgarno and Epstein (1969) have recently derived some sum relations satisfied by sets of states tjiwhich can be obtained by the solution of Eq. (10). The E iand t,hi are functions of the ajof the expansion basis. The aj are usually the " orbital exponents " (effective nuclear charges) of the orbital basis functions. The determination of optimal (those which minimize a chosen Ei)values of the a j thus generally requires the computation of the Eifor several values of the aj followed by an interpolation (or extrapolation). Thus, the estimatibn of optimal values for the ajincreases the labor of the calculation by a large multiplicative factor. It appears that the calculation of accurate wave functions in the CI method requires near optimal values of the orbital exponents unless intractably large orbital basis sets are employed. There does not seem to be any general procedure for determining optimal values of the aj.Very often it is desired to optimize only a few of the ajin a given basis. Then it is usually efficient to assume that the optimal values of a given a j are not coupled to the values of the other aj,and iterate independent variations until no further improvement occurs. Usually only one or two iterations are necessary. Recourse can always be taken to the standard methods of minimizing nonlinear functions (Hooke and Jeeves, 1961 ; Fletcher, 1965). Brown (1968) has shown how the Virial theorem can be used to determine nonlinear parameters directly. Fletcher (l970), Olive (1969), and Sabelli and Hinze (1969) give techniques for obtaining optimum values of the ajfor

MOLECULAR WAVE FUNCTIONS AND PROCESSES

59

SCF wave functions. The direct CI method displays conveniently the principal phases of the computation of a molecular wave function: (i) basis set selection, (ii) integral evaluation (iii) wave function construction, (iv) matrix element formation, and (v) secular equation solution. All of the different formalisms involve at least these basic steps in some form. The differences in each formalism usually lies in the degree to which the labor involved in a given step is minimized or in the degree of specification of a given step. B. ELEMENTARY BASISSETS The elementary basis sets {4i} used in molecular calculations include among the orbital functions, Slater-type orbitals (Eyring et al., 1944), several types of exponential quadratic (Gaussian) type orbitals (Boys, 1950, 1960; Browne and Poshusta, 1962; Harris, 1963; Singer, 1960; Wright, 1963), and for diatomics, elliptic orbitals (Harris, 1960). A basis of Slater-type orbitals usually includes some functions centered on each nucleus of the molecules. This leads to integrands defined over several coordinate centers, (11) <4d1)4@) I r;21 I4C(2)4&)). For the functions 4a(1) notational convention is that the number in parentheses is a coordinate index while the roman subscript denotes the nuclear coordinate center from which the function is defined. Basis sets defined with respect to a single center are sometimes used to avoid the difficult multicenter integrals. Convergence of expansions in single center basis sets are so slow, however, as to severely limit the utility of this practice for making accurate calculations (Bishop, 1967; Hayes and Parr, 1967). Single-center wave functions can, however, yield fairly accurate values of the moments of molecular charge distributions. They have been used to obtain potentials for the scattering of electrons by H, (Dalgarno and Henry, 1965). Slater-type orbitals (and elliptics where they can conveniently be used) offer a faster convergence while multicenter integrals over gaussians are generally much easier to evaluate. For this reason, there has been considerable interest in approximating Slater-type orbitals by sums of Gaussians (R. F. Stewart, 1970,reviewsearlier work; Allen, 1962; Huzinaga, 1965; Huzinaga and Sakai, 1969; Whitten and Allen, 1965). This reduces the difficult integrals to sums of simpler integrals. The method of moments suggests the possibility of using different basis sets in the conjugate function (to left of operator in matrix element) from the direct wave function (to right of operator in matrix element) (Handy and Epstein, 1970). It is sometimes advantageous, in order to simplify the evaluation of matrix elements and/or to speed convergence, to define the Oi(a)in terms of a set of single particle functions {ui}obtained from the {+i} by a linear transformation, u = 4T. (12)

J . C. Browne

60

The evaluation of matrix elements, (ailHI Q j ) , is much more rapid when an orthonormal set of orbitals (ui} is used as an elementary basis for the Q i . Arbitrary orthogonalization of the naturally nonorthogonal multicenter molecular basis sets is not necessarily a very satisfactory computational procedure. Configuration interaction tends to be more effective when the elementary basis set has a localized character that allows configurations to be constructed that interact effectively with the localized structures that tend to have a high correlation error (see Allen, 1969, for a list of references on this topic). Convergence of the expansion where the Q i are constructed froni arbitrarily orthogonalized basis sets {ui} tends to be slower than when the original localized 4iare used (Joy et af., 1964). Additionally, the transformation of the 2-electron integrals into the orthonormal basis can be excessively laborious if not done in a near optimal manner. Selective and/or partial orthogonalizations (see for example Harris, 1967a; King et af., 1967) may be computationally effective procedures. The natural orbitals ” (NO’s) (Lowdin, 1955) are defined as the set which diagonalize the first-order generalized density matrix y(X’ 1 X). “

y(X’1X) = N j $ i * (X;X2-.*X,)$ i (X1X2... X,)(dX2dX,...dX,) .

(13)

The NO’s can be shown (Lowdin, 1955) to be the most rapidly converging set of single particle functions obtainable from a given elementary basis by a linear transformation. This is a very significant property in that even fairly modest basis sets can generate thousands of Q j . For example, a typical twentyorbital basis for BH can produce 5000 configuration of ‘C+ symmetry. It is essential to make an optimal or near optimal choice of Q j or convergence becomes intractably slow. Note, however, that y(X’ I X) is a function of the c i j and a j of Eq. (5). Therefore, the determination of y(X’1 X) (and thus the NO’s) requires that the full calculation be done in advance rendering the use of NO’s apparently uninteresting. Kutzelnigg (1963a,b; 1964) and Reid and Ohrn (1963) (see Miller and Ruedenberg, 1968a for further references) have obtained explicit equations for the NO’s of 2-electron systems. It appears, however (Bender and Davidson, 1967; Bender and Davidson, 1969), that it is a computationally satisfactory procedure to make a limited calculation, form the approximate NO’s on the basis of the c i j and a j of the limited calculation repeat the calculation using the NO’s and iterate the last two steps until the NO’s approximately converge. Edmiston and Krauss (1965, 1967) and Alrichs and Kutzelnigg (1968) have calculated approximate NO’s for electron pairs in H,’ and He,’, and LiH and BeH,. Adams (1969a,b) has developed a formalism which generates the approximate NO basis directly as the eigenvectors of a pseudo-SCF procedure. Iteration on the CI calculation is, however, still required. Directly correlated basis sets are now fairly widely used in atomic calculations (Larsson and Burke, 1969; Perkins, 1968, 1969). The

MOLECULAR WAVE FUNCTIONS AND PROCESSES

61

difficulties of integral evaluation has up to now prevented a n y very extensive use (see work of Boys and Conroy referred to in previous section) in molecular wave functions. Boys (1960), Lester and Krauss (1964), and Singer (1960) have developed integral formulas that allow ready use of exponential quadratic correlation functions in a Gaussian basis. A number of other orbital forms have been suggested. Somorjai (1968) has suggested the use of integral transform functions

as elementary basis sets. Yue and Somorjai (1970) have used Gaussian transform orbitals in a calculation on H z f . This is a generalization of the use of functions of the form

4(r) = r-1Le-n' - e-Pr]

(15)

by Parr and co-workers (Parr and Weare, 1966; Weare et al., 1969). Wilson and Silverstone (1970) suggest the use of rational function orbitals

These latter two basis sets have not as yet been used in molecular calculations. A linear transformation from one elementary basis set to another is itself a computationally trivial task. The problems arise in the transformation of the two electron integrals over the elementary basis set. The transformation of Eq. (12) leads to

Direct application of Eq. (7) is intractable for large elementary basis sets (large n). Tang and Edmiston (1970) suggest a method whereby the computational labor goes as n 5 , rather than as n6 for the method of Nesbet (1963). It is in some instances desirable to place this transformation directly in the analysis for integral computation.

C . INTEGRAL EVALUATION The accurate evaluation of multicenter integrals over the electron repulsion operator rij

'

<4a(l)4b(l)

I t.2 I 4 C ( 2 ) 4 d P ) )

(18)

62

J . C. Browne

with realistic basis sets such as Slater-type orbitals has long been regarded as the major stumbling block to carrying out molecular calculations. This is no longer true for diatomic molecules. There are now available a number of package programs which will evaluate %-electron 2-center integrals at a reasonable cost on even moderately sized computer facilities (QCPE program catalog). The computational procedures for the evaluation of 3- and 4-center 2-electron integrals are not nearly so complete or satisfactory, although a great deal of progress has been made in the past 5 or 6 years. Huzinaga (1967) has Fairly recently reviewed the literature on evaluation of molecular integrals. He gave a through coverage to the evaluation of integralsover Gaussianbasis sets. Integrals over Gaussians will not be further discussed. Thethrust of most current work on molecular integrals is to produce mathematical analysis and computational structures suitable for the simultaneous calculation of large numbers of integrals. There is possibly no other research field where the mathematical analysis has been so influenced by the availability and use of computers. The calculation of 2-center 2-electron integrals has been steadily refined. That part of this work which has resulted in essentially complete computational procedures will be reviewed before passing on to the work on 3- and 4center 2-electron integrals. The phrase ‘‘ charge distribution refers to an orbital product with the same coordinate index = d)a*(l)d)b(l).a and b index the nuclear coordinate centers and may be the same or different. Two-center Coulomb integrals ”

(d)a(l)@a(l)1 r;Zl

I d)a(2)@b(2))

( 19b)

can be evaluated in essentially closed form. Silver and Ruedenberg (1968) and Harris (1969) have developed numerically stable computational procedures for coulomb integrals over Slater-type orbitals with arbitrary integer quantum numbers. Previous analytic formulations for Coulomb integrals (see Harris, 1969, for a brief discussion) suffered from numerical instabilities for certain parameter ranges, and it was common to compute Coulomb integrals by the numerically sound but cumbersome procedures needed for exchange integrals. Similar considerations were true for hybrid integrals. Flannery and Levy (1969) give an analysis for Coulomb integrals whichcould bevaluable when valuesof a few integrals are needed at closely spaced values of internuclear separation. Christofferson and Ruedenberg (1968) have developed an analysis for the hybrid integrals, which requires only a single numerical quadrature over functions defined in finite sums. It is interesting to note that these procedures are computationally efficient even though they do not exploit the factorization

63

MOLECULAR WAVE FUNCTIONS A N D PROCESSES

of the integrals into charge distributions (Ruedenberg, 1951) that made possible efficient evaluation of the 2-center exchange integrals. A possible drawback of these methods for future use is that they do not allow linear transformations on the elementary basis set to be carried out within the integral computation process as do the charge distribution based analysis. The elliptic coordinate formulations for the evaluation of exchange integrals <4#)4b(1)l r121 I4a(2)4b(2))

(20)

(Harris, 1960; Miller and Browne, 1962; Ruedenberg, 1951; Ruedenberg, 1964) involving a single iterated numerical quadrature of functions defined inside a single infinite (but rapidly converging) summation still appears to be the most effective means of evaluating these integrals. Mehler and Ruedenberg (1969) quote timing studies for large blocks of exchange integrals of about 3 msec per integral on an IBM 360/65. Harris and Michels (1967) give a very detailed report of a comprehensive analysis (including earlier work, Harris and Michels, 1965, 1966) for all types of integrals that arise from the use of the Hamiltonian Eq. (3) for a Slaterorbital basis. The significant new result in this analysis is in the analysis for 3and 4-center electron repulsion integrals. Harris and Michels develop and generalize the Barnett-Coulson (M. P. Barnett. 1963) single-center expansion procedure into a form efficient for large blocks of integrals. The essential point is the use of a transformation (Harris and Michels, 1965)

m

= J0 ~ X ( X ~ ' + ~ ) j- x' d r l r ~ + 2 f ~ ~ mxr:'2f;4m(r2) (rl)~ dr, 0

0

.

(21)

This transformation factors the integral into integrals over separate charge distributions. The problem of evaluating 3- and 4-center integrals by the single center expansion method is thus reduced to the evaluation of n2 charge distributions (where n is the number of orbitals in the elementary basis) and condensation of the charge distributions into integrals by a numerical integration of the terms of a doubly infinite sum. This result is equivalent to the Ruedenberg (1951) transformation of the 2-center exchange integral in the elliptic coordinate method. Ellis and Ros (1966) used such a transformation in the evaluation of 1-center 2-electron integrals. McLean and Yoshimine (1968) and Wahl and Land (1967, 1969) have developed the elliptic coordinate method originally developed for 2-center exchange integrals into an effective computational procedure for multicenter integrals (see also Guidotti et al.,

64

J . C. Browne

1966; Magnusson and Zauli, 1961; Magnasco and Dellepiane, 1963, 1964). The principal strategy in this approach is the computation and tabulation of numerical values of the potential produced by charge distribution 1 in the space ofcharge distribution 2.

This procedure relies much more heavily on numerical quadratures than the Harris-Michels single-center expansion procedure. At least two and usually three of the integrations over rz are carried out numerically while the computation of the values of Uab(rz)will require from one to three numerical quadratures. The formulation of Wahl and Land and McLean and Yoshimine yields a final integral expression that is a doubly infinite sum of terms, each of which is evaluated by three-dimensional quadratures. Boys and Rajagopal (1965) show how direct 6-dimensional quadrature in spherical polar coordinates can be made feasible for multicenter 2-electron integrals. Rajagopal (1965) gives additional details on this method. Boys and Handy (1969a,b) have employed similar techniques to the evaluation of integrals involving interelectronic distances. Scheafer (1970b) has applied 4-dimensional quadrature to 2-center 2-electron integrals. He points out the significant advantage that integrals over composite functions of elementary basis sets can be conveniently treated as single integrals if purely numerical integration procedures are used. This potentially significant point has also been made by Boys and Rajagopal (1965) and Rajagopal (1965). Goodisman (1967, 1968, 1969a,b) has used direct 6-dimensional quadrature for the evaluation of 2-center 2electron integrals. He has never, however, strived for really high accuracy. He has also used purely numerical methods for all nontrivial 4-dimensional integrations to evaluate 2-center 2-electron integrals including some resulting from the use of a directly correlated basis (Goodisman, 1964, 1965). He studies the effect of systematically increasing the set of quadrature points, and concludes that modest accuracy is easily attained. This work appears to have been overlooked by subsequent workers in integral evaluation. Conroy (1967) has proposed the use of closed Diophantine sets of points for the numerical quadrature of multicenter 2-electron integrals. Lyness and Gabriel (1 969) discuss some of the difficulties involved in evaluating error bounds for quadrature by this method. Ellis (1968) points out that this technique (as is any totally numerical procedure) is readily applicable to the evaluation of matrix elements directly without preliminary reduction to elementary integrals. He also points out some of the difficulties in obtaining appropriate sets of sample points to provide adequate accuracy when the total distribution function has maxima at

MOLECULAR WAVE FUNCTIONS AND PROCESSES

65

widely separated points in space, and provide sufficient accuracy given the presence of the singularities of the Hamiltonian (see also Whittington and Bersohn, 1969). There is considerable work in the mathematics literature on this class of methods (see for example, Zaremba, 1968). Chemists and physicists do not seem to be familiar with this literature. A number of authors (Huzinaga, 1967; Taketa et a/., 1966; Whitten and Allen, 1965) have proposed to evaluate multicenter integrals over Slater-type orbitals by expanding the STO’s in terms of Gaussians. This procedure replaces the difficult integrals by a linear sum of much simpler integrals. This procedure has been widely applied (see Allen, 1969; Krauss, 1967, for literature citations) in moderately accurate calculations on polyatomic molecules. This method suffers from a non-uniqueness in choice of expansions. It appears that fairly lengthy expansions may be necessary to obtain high accuracy. Monkhorst and Harris (1969) have approximated charge distributions rather than orbitals. They minimize

where R is a 2-center charge distribution, and G iare Gaussian orbitals subject to a set of constraints. Such a procedure greatly reduces the number of Gaussian integrals that must be evaluated. The continuous analog of the discrete expansion i n gaussians is the integral transform method of Shavitt and Karplus (Shavitt, 1963; Shavitt and Karplus, 1965). The Gaussian integral transform procedure has the basic disadvantage that the analysis does not decompose the integrals into independent charge distributions. The basic computational labor thus varies as n4, where 17 is the number of orbitals in the elementary basis set although the labor is considerably reduced by overlap in the computation of intermediate auxiliary functions for different orbitals. It is not clear that any of the approaches is markedly superior to all the others. Harris and Michels (1967) and Wahl and Land (1969) report similar timings of less than 100 msec per integral for large blocks of integrals forcomparableaccuracy. McLeanand Yoshiniine(Clementi, 1970) report considerably faster timings on the very powerful IBM 360/91 computer. The effectiveness of all of these methods are as dependent on the computational skill of the implementer as on the analysis utilized. A choice of methods may depend on the character of the computational facilities at hand, e.g., the purely numerical methods tend to require less fast memory but require a very fast processor to be efficient. Some of the trends in developing computer technology such as parallel processors and very fast functional units for composite operations such as direct scalar product formation favor the use of numerical methods. A breakthrough in the cost of fast memory would favor the use of more analytically based methods.

66

J . C. Browne

Boys (1969) has shown that the use of a consistent numerical integration scheme for all integrals of the matrix elements in a secular equation gives a leading term in the error analysis of the quadrature of M,M, where M, is the truncation error due to use of quadrature, and M is the error in the calculated energy. This is a potentially significant result. Gaussian expansion methods may be effective for fast relatively approximate integral calculation. Silverstone and coworkers (Silverstone, 1967, 1968a,b; Silverstone and Kay, 1968, 1969a,b) have carried out an extensive set of investigations via the Fourier transform and Taylor-series (Roberts, 1967a) methods aimed at producing analytical expressions for the general 3- and 4-center electron repulsion integrals. These expressions in their current form involve taking multiple derivatives of extremely complex products and sums of auxiliary functions. General purpose integral programs have not yet evolved from this analysis. Useful expressions for asymptotic forms of the integrals have been obtained (Silverstone and Kay, 1970). The value of such analytic procedures may be expected to increase asthe “ state-of-the-art ” incarryingout analytical mathematics by computer improves (Engeli, 1968). There has been a revival of interest in the use of bipolar coordinates in the analysis of multicenter integrals (Ruedenberg, 1967; Salmon, Birss and Ruedenberg, 1968, this reference has citations to earlier work; Christoffersen and Ruedenberg, 1968; Silverstone and Kay, 1969c; Steinborn, 1969; Taylor, 1969). The utility of much of this analysis for computer implementation is not yet clear. Roberts (1966) has expressed the general multicenter integral as an integral related to 2-center Coulomb integrals. This result was rederived by Sack (1967). Roberts (1969) has also developed an interesting analysis for 3-center 2-electron integrals based on the use of Sack’s (1964) modified LaPlace expansion of .:Y; Sharma (1968) has developed a form for the radial expansion coefficients for the expansion of an orbital about a different coordinate center, which may be useful in the evaluation of integrals over numerical orbitals and which has some interest for development of the singlecenter expansion method for multicenter integrals. Peterson and McKoy (1967) have studied the use of nonlinear transformations on the partial sums of the infinite series of the single-center expansion method and find considerable acceleration of convergence. We have omitted mention of several analytic studies where the work does not appear to be of direct or immediate computational significance or where the results obtained are duplicated in more general treatments. We also omit discussion of the evaluation of the integrals over the I-electron operators of Eq. (3) since they are essentially trivial on modern computing equipment. There are, however, suggestions in current lines of research on multicenter integrals that resolution into 2-center 1- and 2-electron integrals may prove to be an efficient means of calculation for the 3- and 4-center integrals. If this proves true, then optimal analysis of the

MOLECULAR WAVE FUNCTIONS AND PROCESSES

67

simpler integrals will be of great value. It should be possible to prove optimality of solution for these simple integrals. The calculation of 3- and 4-center, 2-electron integrals would seem to be in a state similar to that of 2-center integrals several years ago. It seems reasonable to expect a similar refinement and adaptation of analysis to computationally efficient procedures.

D. CONSTRUCTION OF W A V E FUNCTION CONFIGURATIONS The total wave function must be an eigenfunction of the operators which commute (or almost commute) with the Hamiltonian of the system, and whose eigenvalues characterize, or approximately characterize, the molecular states of interest. These include, for diatomic molecules, the operators of Table 1. TABLE 1 SYMMETRY OPERATORS FOR DIATOMIC MOLECULES Operator-0 ~~~~

9

~

2

yz

9,

Designation of operator

O*

~

~

+

l)* M, 4 iA$

S(S

U"

A*

1

**

Total spin z Component of spin z Component electronic angular momentum Reflection through axis containing nuclei, A = 0 only Inversion through midpoint of molecule-homonuclear molecules only

It is usually convenient, but not necessary, to have each Qi of $, an eigenfunction of the commuting operators. The form for a symmetry adapted m i depends on the method selected for evaluating matrix elements (see Section 11, E) and on the method chosen for generating eigenfunctions of 9''. The determinantal approach uses (24) mi = aik Dik

1

7

h

where Dik are Slater-determinants of space-spin orbitals. The spin-free formulation (Matsen, 1964) gives N

Qi =

1 u ( P >11

4j(rj),

(25)

j= I

where P is a permutation from a set allowed by antisymmetry and symmetry adaptation and U(P) is a coefficient matrix determined by an irreducible representation of the symmetric group for N particles. The space-spin product

J. C. Browne

68

function approach (Cooper and McWeeny, 1966; Harris, 1967a; Kotani et al., 1955) writes (26) where O iis a spin eigenfunction and P permutes both space and spin coordinates. There are alternate forms for the space-spin product eigenfunction (see, for example, Harris, 1967a). Examples of Q i for three-electron doublets are given in Table 11. TABLE I1 THREE FORMS FOR TOTAL SPIN EIGENFUNCTIONS FOR THREE ELECTRONS

Method

@q.b,c

Determinantal method Spin-free method Space-spin product method

+

abc b a ~ cba - cab abc(~,Ba- pact) - cba ' (@a - UEP)- cab ' (.,BE - a@) bac ' - PO(CL) (bca . (,Baa asp,

+ +

+ acb

~

a

I,;:(

etc., are Slater determinants.

* abc za(l) b(2) c(3), bac a and

b(l) 4 2 ) 4 3 ) . etc.

b are spin paired.

It is possible for symmetry adaptation to appear directly in the matrix element calculation rather than as a separate computational step. This is generally the case where spin-free or space-spin product methods are used in the matrix element computation. The determination of eigenfunctions of 0" and i and 9, and 6 p z is essentially trivial but tedious (see, for example, Eyring et al., 1944). The principle problem is the generation of a complete and independent set of eigenfunctions of 9'. This is a much studied problem (Cooper and McWeeny, 1966; Lowdin, 1964; Matsen, 1964; Nesbet, 1961 ; Percus and Rotenberg, 1962; Reeves, 1966; Secrest and Holm, 1964; V. H . Smith and Harris, 1969; Sutcliffe, 1966; and many others). Sutcliffe and McWeeny (1969) give a useful summary of methods for constructing eigenfunctions of 9'. The choice of eigenfunction of 9 'can have a significant effect on the course computation and the results obtained in atomic and molecular calculations. The use of any complete and independent set of spin eigenfunctions for a

MOLECULAR WAVE FUNCTIONS AND PROCESSES

69

given orbital basis yields identical results. If a complete set of spin configurations is to be used in the calculation, the choice of the set is not critical. The projection operator (Lowdin, 1964; Nesbet, 1961; Secrest and Holm, 1964), group theoretic (Harris, 1967a; Kotani et al., 1959, direct diagonalization (Eyring et al., 1944) and valence-bond approaches are all readily programmed for computer application. The use of a complete set of spin eigenfunctions is not, however, always a computationally efficient process since the principal contribution to a wave function from a given set of space orbitals may come from a single spin eigenfunction or a small subset of all possible spin eigenfunctions. The spin-paired or valence-bond approach (calledstructure function in the Matsen spin-free formalism) (see Sutcliffe and McWeeny, 1969, for a recent review of the valence-bond method) generates a set of functions where each Q iis symmetric or antisymmetric with respect to the spin coordinates of specified pairs of electrons. This representation has the advantage of a close correspondence with the physical concepts of atomic shell structure and chemical bonding. This physical correspondence may be useful in the selection of mi.A simple example is the case of an ‘‘ open shell calculation on the Li atom (Brigman and Matsen, 1957). This calculation was made with an orbital basis set of Is, Is’, 2s. The configuration which spin pairs Is and Is’ contributes over 99 % of the wave function. Until quite recently, when the availability of large computers and automatic package programs made the use of very elaborate wave functions possible, wave functions were generally formed into appropriate symmetry states by manual means. Several programs for automatically generating molecular wave functions which are eigenfunctions of the operators of Table I have recently been described in the literature. Gershgorn and Shavitt (1967) use the techniques of spin-bonded functions (see, for example, Cooper and McWeeny, 1966; or Reeves, 1966) to produce eigenfunctions of Y 2 Eigenfunctions . of the other operators are produced by projection operator techniques. Schaefer (1970a) proceeds by producing all possible Slater determinants, D i ,with proper values of A, M , , and g or I I obtainable from the elementary spinorbital product basis. He then diagonalizes the matrix ”

The eigenvectors are then an orthogonal basis set. Schaefer’s programs are available from QCPE. A11 of the major research groups involved in extensive CI calculations have (implicitly or explicitly) similar programs. E. THEEVALUATION OF MATRIX ELEMENTS Matrix element evaluation involves the integrations (@,I HI 0,) = H i j where Qi and Q j satisfy the Pauli principle and are eigenfunctions of Y z ,Y,, and the other operators that commute (or almost commute) with H . The

70

J . C. Browne

I

MATRIX ELEMENT FORMATION

,

SYMMETRIC GROUP METHODS

DETERMINANTAL

6 1 FIG.I . Tree diagram for methods of matrix element evaluation.

evaluation of matrix elements has a very extensive literature. A number of review articles have been written recently describing several different formalisms (Harris, 1967a; Matsen, 1964; Nesbet, 1961). Sutcliffe and McWeeny (1969) discuss a number of the recent developments including matrix elements over pair functions and more general group functions. Figure 1 provides a schematic guide to the formalisms based on whether the antisymmetry condition is met directly by invoking the appropriate representation of the symmetric group or by the use of Slater determinants. The tree diagram branches into symmetric group and determinantal (Lowdin, 1955; Slater, 1929a,b) at the first level. The symmetric group methods further subdivide into the spinfree (Matsen, 1964) and space-spin product (Harris, 1967a; Kotani etal., 1955; and others) formulations. Each method can use any one of the set of eigenfunctions of Y 2described in Section II,D. The symmetric group methods all express the matrix element in terms of sums of integrals over pairs of orbital products, the coefficients in the sum being determined by representations of the symmetric group. The determinantal methods express the matrix elements as sums of integrals over pairs of determinants, the coefficients being determined by the symmetry adaptation phase of the calculations. The complexity of actual evaluation is much greater with nonorthogonal elementary basis sets than with orthogonal elementary basis sets in every formulation. Methods based on the symmetric group methods appear to be intractable for more than eight to ten electrons in a nonorthogonal elementary basis. Recent advances have extended the range of the determinantal methods for nonorthogonal elementary basis sets to perhaps twenty or so electrons. We will use the traditional spin and spin-orbital nomenclature in our discussion. The two basic problems have been to generate a complete and independent set of matrix elements over the spin space accessible to the basis set and to overcome the problems associated with the use of nonorthogonal functions. We will

MOLECULAR WAVE FUNCTIONS AND PROCESSES

71

review only the recent literature from the primary journals and concentrate on that work that is most directly related to computational procedure. The

where P is a permutation taken from the group allowed by the Pauli principle and the constraints that the $ ibe eigenfunctions of Y 2and the other commuting operators for the given orbital products, and U ( P ) is an element from a matrix representation of the allowed group of permutations. The representation matrix U(P) (the entries of the matrix U ( P ) are often, particularly when valence-bond wave functions are being used, called ‘‘ Pauling numbers”) depends on the form chosenfortheeigenfunction of 9’. Matsenand coworkers (Matsen, 1964; Matsen et al., 1966; and Matsen and Cantu, 1969) give procedures for constructing U ( P ) for the spin-paired (or valence bond) eigenfunctions using spin-free methods. Klein et al. (1971) give an economical procedure for computing the Pauling numbers” for the projection operator eigenfunctions of Y 2 Cooper . and McWeeny ( I 966) obtained a representation (for an orthonormal elementary basis) of the first- and second-order spinless density matrices, which is effectively equivalent to the evaluation of H i j for the valence-bond eigenfunctions, in terms of a matrix representation of spacespin product functions. Harris (1967a,b) gives a rather complete set of formulas for the matrix elements over branching diagram and projection operator eigenfunctions of Y 2 .Harris (1967a) gives formulas in terms of sums over elementary integrals that have already taken explicit account of the allowed group of permutations for cases where the elementary basis sets have a high degree of partial orthogonality and also (1967b) for the “open-shell” matrix elements in a completely orthogonal basis. Gerratt and Lipscomb (1967) obtain an expression for the general matrix element over spin-coupled functions via the representations of the permutation group. A nonorthogonal basis is allowed, but only even numbers of electronsareconsidered. Shull(l969) has obtained a particularly computationally convenient form for the evaluation of matrix elements in the valence bond set of eigenfunctions for the singlet case ( N = even, complete pairing) by extending the arguments of Pauling (1933). Orthonormality of the orbital basis set is not required. The determinantal method expresses the matrix elements as Hij

=z

2 aikajr-

(28)

k,l

The problem is then reduced to evaluation of matrix elements over Slater determinants. Lowdin (1955) has shown that for H e , , Eq. (3)
I on> = C C Dmn(r I t> r

t

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J . C. Browne

where r and s index the orbitals of D , , t and u index the orbitals of D,, the D,,,,,(rlt) and Dmn(r,slt, u) are the first- and second-order cofactors of the determinant of overlap integrals ( D , I 0,)= det{(r I t ) } and h ,. =

-:Vi2 - Za/rai- Zb/rbi.

(30)

If an orthonormal orbital basis is used then Eq. (29) collapses to the familiar Slater (1929a,b, 1963) formulas. In a nonorthogonal basis the formation of the matrix element is reduced to the evaluation of the set of first- and secondorder cofactors. If all the cofactors must be evaluated separately, the problem becomes analogous to summing over all permutations and is intractable for large number of electrons. It is well known that first- and second-order cofactors of nonvanishing determinants can be obtained from the inverse of the matrix whose elements are those of the determinant det { ( r l t ) } (Aiken, 1956) provided that det {(rl t ) } # 0. Prosser and Hagstrom (1968a) have shown that cofactors of all orders can be extracted from a knowledge of matrices L and R which diagonalize the matrix S whose elements are those of the determinant det { ( r l t)} LSR = D,

(31)

D being a diagonal matrix and L and R are triangular matrices. This work significantly extends the number of electrons that can be considered using methods in terms of nonorthogonal orbitals. King et al. (1967) suggest the evaluation of (D,IHI 0,) over a nonorthogonal basis through the use of finding unitary matrices U and V such that

U+SV = DIIz

(32)

via solution of an eigenvalue problem S+SV = VD.

(33)

Prosser and Hagstrom (1968b) show that this method cannot be as efficient as their procedure. King and Stanton (1968) show how their method can be optimally arranged as a computational procedure. They conclude that the Prosser-Hagstrom cofactor analysis is probably more efficient because the determination of the triangular matrices L and R is a more rapid computational procedure than solution of the eigenvalue problem Eq. (33). The Prosser-Hagstrom procedures have been found by this author to make feasible on a CDC 6600 calculations with full CI (except for K-shell substitution) in a valence bond framework employing a nonorthogonal basis of ten to fourteen orbitals for molecules with up to twenty electrons. Reeves (1966) and Sutcliffe (1966) have used the determinantal method to obtain complete formulas for the determination of matrix elements over the pair-bonded set of eigenfunctions for the case of orthogonal basis sets. Nesbet (1958, 1961) has given a

MOLECULAR WAVE FUNCTIONS AND PROCESSES

73

procedure based on the use of orthonormal elementary basis sets, Slater determinants, and the projection operator method, which generates a complete and independent set of eigenfunctions of 9’’ and provides an expression for the evaluation of the matrix elements in this basis. Valence-bond, projection-operator, branching diagram, or group-theoretic methods (Matsen and Cantu, 1969) can all be used to find the uikof Eq. (24). The connection between the symmetric group and determinantal methods is easily seen by the relation

where P is a permutation on the indices and ( - 1)‘ is the parity of P, and O,(t) is an indexed spin function a ( t ) or B(t). The coefficients u i k ,the integrals over products of spin functions, and the parity of P are seen to determine to a representation of the symmetric group. Balint-Kurti and Karplus (1969) evaluate matrix elements over valence-bond configurations constructed from Slater determinants of nonorthogonal orbitals by expressing them in terms of sums of determinants over orthogonalized functions. This approach requires the computation of the cofactors of the orbital transformation matrix. It has the advantages for valence-bond wave functions that the matrix element analysis is carried out in the orthogonal framework with the wave function still being expressed in the more rapidly converging localized valence-bond basis. There is the additional advantage that it is not necessary to carry out the transformation of all possible integrals into the orthogonal basis. It appears, however, despite the advances in technique for dealing with the problems raised by nonorthogonality, that the use of nonorthogonal elementary basis sets will not be competitively efficient for systems of more than ten t o twenty electrons. It appears on the basis of present evidence that the evaluation of matrix elements over large nonorthogonal elementary basis sets involving many distinct determinants becomes a computationally limiting factor more quickly than transformation of integrals to an orthogonal basis. F. THEEIGENVALUE-EIGENVECTOR PROBLEM The general eigenvalue problem ( H - SE)C = 0

(35)

where H is real-symmetric and S is real-symmetric positive-definite can be converted to the standard eigenvalue problem

74

J. C. Browne

where B is a lower triangular matrix determined by BSS = I, and H' is real symmetric (Brooker, 1953; Fettis, 1965; Martin and Wilkinson, 1968). Standard methods of numerical analysis based on reduction to tri-diagonal (or diagonal) form (Wilkinson, 1965) suffice for problems of modest size (dimension less than 300) and reasonable condition. A number of tested and verified programs can be found in the computing literature. Stewart (1970a,b) has published a program based on tri-diagonalization by the Householder method (Wilkinson, 1965) and application of the Q R algorithm (Francis, 1961, 1962). Many of these methods have the merit of readily yielding all of the eigenvalues and eigenvectors of the matrix system. Wallis et al. (1969) have studied several methods in the context of quantum mechanical variational calculations and conclude this is generally the most effective procedure. They also developed a simultaneous direct diagonalization procedure on an extension of the cyclic Jacobic method for the general form of Eq. (35). This procedure does not seem to be well known in the numerical mathematics literature. Peters and Wilkinson (1970) have surveyed the standard methods for solving the eigenvalue problem in the form of Eq. (35). If solutions to very large matrices are required, then recourse must be had to iterative methods. Nesbet (1965) has specified a computationally useful scheme for the general problem Eq. (35). This method has been reformulated (Shavitt, 1970) in a format where advantage can be taken of sparseness in the matrix and only a single row of H and S need reside in the main computer memory at a given time for computation to proceed. Matrices of order 12,077 with about 2 x lo6 nonzero elements have been solved (Shavitt, 1970) on a CDC 6400 by an implementation of this method. It should be noted that proofs for convergence of this method can probably only be obtained under special assumptions, such as strong diagonal dominance (Stewart, 1970~).This condition is usually well met in the problems generated by the calculation of molecular wave functions. The iterative methods have the disadvantage of obtaining only a single eigenvalue and eigenvector per convergent iterative cycle. Empedocles (1970) has suggested the use of a quadratically converging steepest descent method for nearly linearly dependent problems. Nonsynimetrical but real matrices arise directly from the method of moments (see Section II,A) or can be generated by the use of the direct transformation (S-' H - S-'ES)C = 0

(37)

on Eq. (35). Reduction to Hessenberg form followed by Q R triangularization (Martin et al., 1970) provides an effective method for the solution of nonsymmetric eigenvalue problems of modest size, 2 300. Programs for this procedure have been given by Gradand Brebner (1968) and Martin etal. (1970).

MOLECULAR WAVE FUNCTIONS AND PROCESSES

75

The Nesbet iterative method can also be adapted to nonsymmetric matrices (Bender and Shavitt, 1970). In summary the eigenvalue-eigenvector problem is not currently a limiting factor in the calculation of molecular wave functions. There are indications (Drake and Dalgarno, 1970; Ford et al., 1970) that the use of resolvent operator techniques based on expansion in the variationally derived set of states (E, -

=

1 IEo - E i I $i>($i

i

may be very effective for the computation properties via second-order perturbation theory. This suggests that methods of solution which readily return all roots for large matrices will become of greater interest in the future. G. CALCULATION OF COUPLING MATRIX ELEMENTS The systematic calculation of matrix elements of the operators that couple eigenfunctions of Eq. (3) has received little attention. The coupling operators due to nuclear motion for Hund's case b coupling of angular momenta (Herzberg, 1950; Pack and Hirschfelder, 1968) are

where H A , couples states of identical electronic angular momentum, A, in the direction of the internuclear line [and also defines the first-order diagonal correction to the fixed-nuclei or Born-Oppenheimer (B-0) potentials] and HA,+' couples states whose A value differs by & 1 (Pack and Hirschfelder, 1968, 1970a; Smith, 1969a). In these equationsp, = i-'R-'(d/dR)R, -Ye is the electronic angular momentum, and H J M ) is related to isotopic mass corrections in the separated atoms. The radiation field couples the electronic molecular states according to selection rules determined primarily by parity (Garstang, 1962; Nicholls and Stewart, 1962). The Breit-Wigner (Bethe and Salpeter, 1957; Pack and Hirschfelder, 1968) relativistic Hamiltonian H , also has nonvanishing matrix elements for many pairs (or sets) of electronic B-0 states. The selection rules are complex (Garstang, 1962). The coupling matrix elements are similar for the time-dependent (classical nuclear trajectory) formulations (Bates, 1960; Bates and McCarroll, 1962) and for the full quanta1 calculation (Bates et al., 1953b; Mott and Massey, 1965). The calculation of ( m I HA,,+,In) presents little difficulty. Matrix elements of HA,+ over even fairly simple wave functions pass smoothly to the correct united atom and separated atom limits and do not vary greatly with

76

J . C. Browne

variations in the wave functions. The form of H A A k suggests an asymptotic behavior of R - 2 for ( m I H A A f , In), where m and n are connected by Y e * . This would in many cases be the asymptotically dominant R-dependent term in the total energetics of the system. Thorson (1969) has shown, however, that this asymptotic behavior is usually neither physically nor computationally significant. The matrix elements over g e 2and H J M ) have been treated by Greenawalt and Browne (1970). They can all be evaluated in terms of well-known auxiliary functions. Matrix elements of HA,, that involve partial derivatives with respect to the internuclear separation are more difficult. These terms, which arise from the action of the nuclear kinetic energy operator on the electronic wave function, are difficult to calculate for the types of wave functions currently being used in molecular structure calculations. The linear coefficients in the expansion of the wave function are functions of internuclear separation. These derivatives can be obtained directly by an extension of the solution of the secular equation (Kolos and Wolniewicz, 1966). It is also generally the case that the orbital exponents (effective nuclear charges) of the exponential-type orbitals are varied if an accurate solution over a range of internuclear separations is desired. Thus, these also are functions of internuclear separation. These derivative terms in the matrix element must be approximated numerically. This is a considerable source of uncertainty in the calculations. Alternatively, one can assume that the approximate configuration interaction wave functions being used in the calculation are exact solutions t o the fixed nuclei problem. It is then possible to obtain the expectation value of the derivative operator in terms of the expectation value ofthe derivative of the potential (Mott, 1931 ; Smith, 1969a).

The operators which result from taking the derivative of the potential are the irregular harmonics.

These same operators appear in the acceleration form of the dipole transition matrix element. These integrals are relatively easy to calculate for typical orbital product configuration interaction wave functions (Kahalas, 1961 ; Pitzer et al., 1962). It is known, however, from the calculations on optical transition probabilities (Schiff and Pekeris, 1964; Weiss, 1963) that the values yielded by the acceleration formulation for optical transition probabilities are unreliable. The matrix element ( m 1 (a2/aR2)In) can be expressed in terms of the second derivative of the potential (Smith, 1969a). This procedure leads

MOLECULAR WAVE FUNCTIONS AND PROCESSES

77

to integrals with singularities which must be dealt with carefully (Pitzer et af., 1962). Alternatively, it can be converted to ( d $ J a R ) I (a$,/aR)) and evaluated by use of the resolvent operator expression

As a further complication, the matrix elements of ( m I (a/aR)1 n ) may approach a constant as R -+ co . This is because a/dR is taken with respect to electron coordinates held fixed in the center of mass of the nuclei coordinate system. This defect can be corrected by adding a " momentum transfer " factor to the elementary basis set (Bates and McCarroll, 1962; Levy and Thorson, 1969; Schneiderman and Russek, 1969). This factor must always be included in any case where the translational velocity of the nuclei is appreciable with respect to the internal velocities of the electrons (Bates, 1970; Bates and Williams, 1964; Ferguson, 1961).Theaddition ofthe momentum transfer factor complicates the elementary integrals but not to a serious degree (Cheshire, 1967; Ferguson, 1961 ; McCarroll, 1961). Evaluation of nuclear motion coupling matrix elements for a basis set of molecular eigenstates to the accuracy of the B-0 energies is within the current computational technology, but has not yet been achieved on any considerable scale. The matrix elements of H a usually require a modified formulation of the matrix element analysis (Bottcher and Browne, 1970; Cooper and McWeeny, 1966; Harris, 1967b; McWeeny, 1965; Sutcliffe, 1966) as well as a different set of elementary integrals. Matcha et al., (1969), Matcha and Kern (1969), and Bottcher and Browne (Bottcher, 1968)(see also Roberts, 1967b) have shown that all of the integrals of Ha can be expressed as linear sums of integrals familiar in the calculation of energies. Very few calculations for molecules, even of diagonal matrix elements, have been made (Bottcher, 1968; Chiu, 1964, 1965; Mackrodt, 1970; Pritchard et al., 1970; Walker and Richards, 1970). The electric dipole approximation to radiative coupling is the only operator with which extensive calculations have been made. Mizushima (1964, 1966) has examined magnetic dipole transitions for some small molecules. Dalgarno et af. (1969), T. C. James (1969), and Karl and Poll (1967) have calculated quadrupole transitions in H,. Several studies have been made of general problems in the context of transitions between molecular states. Hansen (1967) shows that defining the oscillator strength as a product of length and velocity matrix elements f,, = 2/3(0 1 y i 1 m)(O I V iI m ) should lessen the effects of correlation error in the ground state wave function on the calculated f,,. Studies have been made (Kelly, 1964; LaPaglia, 1967; LaPaglia and Sinanoglu, 1966; Marchetti and LaPaglia, 1968) of the defects of independent particle model wave functions as basis sets for the calculation of optical

78

J. C. Browne

oscillator strengths. The actual calculation of dipole matrix elements in the length or velocity formulations between molecular electronic states involves only very simple 1-electron integrals and is an essentially trivial calculation. It is now often done as a routine portion of a molecular calculation (see, for example, Bender and Davidson, 1968a,b; Chan and Davidson, 1968). It is difficult, however, to obtain even moderate accuracy for even such simple molecules as CH or C H + (see, for example, Huo, 1968).

111. Some Results and Expectations for the Future Convenient general surveys of molecular calculations appear regularly in the Annual Reviews of Physical Chemistry. Allen (1969) and Golebiewski and Taylor (1967) have given detailed surveys of all types of calculations. Krauss (1967) gives a detailed summary of ab inirio calculations prior to 1967. Krauss (1970) has surveyed the work on ab initio computation of potential surfaces for small polyatomics. Some of these calculations are now complete enough to be of value for the calculation of rotational and vibrational excitation cross sections although no really detailed studies of potential surfaces of polyatomic molecules have yet been made. There have recently been carried out calculations of moderate accuracy on large sets of states for diatomics of moderate complexity. Schaefer and Harris (1968b) obtained approximate potential curves for 62 states of 0, over a limited range of R. Michels and Harris (Michels, 1970) have done similar calculations for 102 states of N, . These calculations reveal a rich structure of crossings, pseudocrossings, and interactions in the excited state potential curves of these systems. Such calculations, carried to greater accuracy, are likely to be of great value in the analysis of predissociation, photo-dissociation electron impact excitation, and subsequent dissociation, and other phenomena involving interactions of excited states. Davidson and co-workers have carried out calculations for potential curves of selected sets of excited states for LiH and H F (Bender and Davidson, 1968a,b) and BeH (Chan and Davidson, 1968). Brown and Shull(l968) have studied the four lowest 'C+ states of LiH. None of these calculations was carried out for the purpose of forming a basis set for the calculations of any specific atomic or molecular process, but they are illustrative of the scope of computational technology for carrying calculations of sets of molecular states. There have also been obtained solutions of very high accuracy (- 200-400 cm-') for the entire range of R for systems of 3 and 4 electrons. These include He,('3311,) (Gupta and Matsen, 1969), He,+(2C:,,) (Gupta and Matsen, 1967), He,('C,+) (see section IV,A), and He2(3C,+, 'Ig+) (Kolker and Michels, 1969), and LiH ('C) (Bender and Davidson, 1966). Current-day computational technology is capable, only

MOLECULAR WAVE FUNCTIONS AND PROCESSES

79

through routine evolutionary improvements in techniques and computing capability, of producing calculations of this level of accuracy for large sets of states of relatively simple diatomic molecules (- 10-20 electrons). These calculations, if coordinated and planned with the calculations of processes being considered, can provide a basis for the detailed calculation and analysis of a rich variety of experimental phenomena. Computations of potential surfaces for polyatomic systems is sufficiently more laborious as to project approximately a 5- to 10-year lag in the accuracy obtained for larger sets of states of even simple polyatomics. For example, the best calculations currently available for H, (see Krauss, 1970) are of accuracy roughly comparable to that obtained for He, (Phillipson, 1962) nearly a decade ago. A few calculations of sets of molecular states directly for the calculation of processes have been carried out. We now consider this work. Michels et al. (1968) have used calculated potential curves for HF and HF- to estimate detachment cross sections for H - on F and F- on H. Michels and Harris (1968) have also carried out calculations on OH, 0, , and HeNe' which they use to qualitatively analyze possible inelastic atom-atom scattering processes. The next sections deal with examples where calculations of processes have been carried through in detail in a molecular basis or where there has been a detailed interaction between calculations and experiments.

IV. Atom-Atom Scattering There have been a few instances where the calculation of a rate or crosssection for a process in a molecular basis set has-been carried through directly, including the calculation of the molecular basis and the perturbation matrix elements and evaluation of the equations governing the rate process. There have also been instances where molecular calculations and experiments have collaborated to produce detailed comprehension of a process. A few recent examples will be considered.

A. ELASTICSCATTERING The potential curves calculated for the interactions of ground state atoms have played a very interesting role in the interpretation of elastic scattering. Scattering experiments and calculations have proceeded hand in hand to develop an accurate knowledge of potential curves for the interaction of ground state atoms in several cases. Perhaps the most interesting case is the interaction between two ground state helium atoms. There are two very interesting problems connected with the interaction of two helium atoms: the long range Van der Waals attraction, and the nature of the repulsive curve at small internuclear separations. The experimental observations on the nature of the long

80

J . C. Browne

range attractive potential come mostly from bulk properties. Bruch and McGee (1970) have reviewed the determination of the form of the long range potential from analysis of these measurements. They estimate a Van der Waals well depth in the order of 10"-12"K. The most recent theoretical calculationsBertoncini and Wahl (1970) and Schaefer et al., (1970)-calculate well depths in the vicinity of 11" and 12°K. These calculations represent a breakthrough in the direct application of ab initio calculations to the determination of long range forces. Previous efforts to obtain long range forces by direct calculation have been stymied by the difficulty of making calculations sufficiently consistent and sufficiently accurate so that the subtraction of separated atom energies from the energies of the interacting systems did not cause errors orders of magnitude greater than the very small Van der Waals energies of interaction. The success of these calculations is founded upon the fact that correlation error will be independent of R for regions of R where there is negligible interatomic charge distribution overlap. The short range repulsive potential between two ground state helium atoms represents a case where the availability of accurate molecular calculations has led directly to the refinement of experiments and to isolation of inaccuracies in earlier experiments. Amdur and co-workers (Amdur and Harkness, 1954; Amdur et al., 1961) carried out a series of measurements on the heliumhelium interaction by high velocity neutral-neutral scattering. They inverted the elastic cross sections to produce a potential curve for the interaction of two helium atoms over a range of internuclear separations around 0.5A to 0.71A. These interaction potentials were in considerable diagreement with the fairly crude calculations then available. Phillipson (1962) made a 64 configuration calculation yielding a potential curve considerably above that obtained by Amdur and co-workers. The error in Phillipson'scurve was estimated to be nearly an order of magnitude smaller than the difference between the two curves. This discrepancy sparked a number of calculations aimed at lowering and placing more rigorous bounds on the absolute error in the calculated potential curve. Thorson (1963, 1964) examined the possibility of non-adiabatic effects and concluded that they were not significant at the scattering energies involved. Barnett (1967), Klein et al. (1967), Matsumoto et al. (1967) all carried out more accurate calculations on the short range potential interaction. Jordan and Amdur (1967) at almost the same time were carrying out new experiments with improved apparatus. All of these calculated curves agreed fairly well with the interaction obtained by Phillipson. Jordan and Amdur (1967) found that inversion of the measurements produced by their new apparatus yielded a potential curve in the vicinity of 10 eV higher than the previous experiments. The difference in the experimentally derived and calculated curves is now of the order of 2 to 3 volts. The absolute error in the calculated curves is estimated to be not more than 0.3 eV. These energies,

MOLECULAR WAVE FUNCTIONS AND PROCESSES

81

however, are calculated with the fixed nuclei Hamiltonian Eq. (3); and it is conceivable that the 2 to 3 eV error represents both experimental uncertainty and failure of the Born-Oppenheimer approximation at the kilovolt velocities of the experiment. Another case where accurate molecular calculations of the potential curve has led to revisions of experimental interpretation of elastic scattering experiments is the case of a lithium ion being scattered from the helium atom. Zehr and Berry (1967) report a potential curve for the interaction of a ground state He atom with a ground state Li+ ion obtained from inversion of the cross sections obtained in their elastic scattering measurements. Fischer (1968) computed a potential curve for small internuclear separations using a Gaussian basis. Fischer's potential curve diverges sharply from the experimentally derived potential of Zehr and Berry at the smaller internuclear separations. The discrepancy rises to 250 eV at an internuclear separation of 0.2 a,. While Fischer's calculation was not of high absolute accuracy, it would have been expected to yield a relative interaction energy to within 1 or 2 eV. Junker and Browne (1969) made very accurate calculations on the ground state potential interaction between the lithium ion and the helium atom. The absolute error in this curve in the adiabatic approximation is of the order of 0.2 eV over the complete range of internuclear separations. This potential curve agrees very well with that calculated by Fischer, and is thus in disagreement with the potential obtained by Zehr and Berry. Aberth and Lorents (1969) measured the elastic scattering cross sections for Li+ on He for a range of energies and angles. Olson et al. (1970) analyzed this data to produce a potential curve for 0.30 a, i R S 0.80 a,. The resulting potential is in satisfactory agreement with the calculated curves. It seems probable that Zehr and Berry extrapolated their inversion procedure to too small values of internuclear separation.

B. INELASTIC PROCESSES The role played by curve crossings, pseudo-crossings, and degeneracies of molecular potential curves has recently come to dominate discussions of atomatom scattering in the low to moderate energy range. The Landau-ZenerStuckelberg two-state curve crossing theory has long been the staple of theorists for approximate estimation of cross sections in the presence of interactions between molecular potential curves (Bates, 1960; Green, 1966, are representative recent references). The concept of '' diabatic states "(see Section I ) and the corresponding "diabatic" potential curves play a strong role in the qualitative discussions. The first significant quantitative work calculating the effect of coupling of adiabatic molecular states was the calculation by Bates and Williams (1964) that the coupling of the 2pn, state of H 2 + to the 2pC,+ state (important because of the degeneracy of the potential curves as R -+ 0)

J . C. Browne

82

could partially account for the damping of oscillations in the change exchange cross section (Helbig and Everhart, 1965; Lockwood and Everhart, 1962). F. T. Smith et al. (1965) demonstrated that curve crossings (or pseudo-curve crossings) in the 2C,+ states of He,' produced definite perturbations in the differential elastic cross section for scattering of He+ by He. Much of this earlier work has been surveyed by Smith (1969b). (See also Lichten, 1963, 1967a,b.) Coffey et al. (1969) have used calculated potential curves for the HeNe' and a detailed analysis of the differential elastic cross sections (F. T. Smith et al., 1967) for scattering of He+ by Ne to produce a detailed analysis of the potential curves of the He+ + Ne systems. Coffey et al. (1969) give a detailed and valuable discussion of general principles for analysis of scattering data in the presence of interacting potential curves. We now consider some of the recent cases where the molecular calculations have been used directly in the computation of processes. Peek et al. (1968) used the potential curves calculated by Michels (1968) for the 'Zg+, 'Xu+ states of Li,' to predict the low energy exchange cross sections for the scattering of Li+ by Li. Perel et al. (1969) have compared the calculated cross sections to the observed cross sections (the agreement is generally excellent), and analyzed the possible causes of the regions of disagreement. Aberth et al. (1970) have measured the differential elastic cross section and the cross section for Li+

+ L Q S , zS)

-

Li+ i- Li(Zp, 2P).

This data is used, together with Michels (1968) calculated curves, to produce a set of potential curves for the Liz+ molecule including some diabatic crossings, which reproduce the experimental total cross sections as well as the calculated adiabatic curves. Novick and Dworetsky and co-workers (Dworetsky et al., 1967; Novick and Dworetsky, 1969) have discovered a rich spectrum of structure in the total excitation cross sections for He' on He in the low to moderate energy range. Rosenthal and Foley (1969) have shown that such oscillations in total cross section should not occur as a result of single crossings or avoided crossings of molecular states. They show that such oscillations could result fromdouble or multiple crossings (or avoided crossings) which are separated by finite ranges of R . This process is schematically shown in Fig. 2. States 1 and 2 (see Fig. 2) are populated during the inward phase of the collision at R i .There will, in general, be further interaction at R iduring the outward phase of the collision. The channels, both with nonzero flux will proceed on the outward phase of the collision with little interaction until R z R,, where states 1 and 2 again interact strongly. The total cross section would be expected to show regular oscillatory behavior, depending on the relative phase 4 ( E ) of states 1 and 2 at R,, if 4 ( E )is a slowly varying function of impact energy. I f 4 ( E ) is a relatively strong function of E, then irregular

MOLECULAR WAVE FUNCTIONS AND PROCESSES

>-

c3

l x W z W

-I

4 I-

z W

k

83

Uo

Rl RO INTERNUCLEAR DISTANCE

FIG. 2. Schematic diagram for potential curves which would give two interacting inelastic scattering channels.

oscillatory behavior is to be expected. Rosenthal and Foley (1969) carried out calculations on a set of 2C,+ states of He,', which demonstrated the occurrence of double crossings in this set of states. They found that the potential curves of the set of states arising from He+(ls) + He*(ls, n = 2) did not have double crossings, while the set of molecular states coming from He*(n = 3) He' and He*@ = 4) + He' did have the double avoided crossings with the outer avoided crossings occurring in the vicinity of 25-40 a,. This set of potential curves then predicts no oscillatory behavior in the total excitation cross section for He*(n = 2), but the occurrence of oscillations in these cross sections for He*(n = 3, 4). This pattern is confirmed by the observations of Novick and Dworetsky (1969).

+

V. Radiative Processes The range of experimental phenomena that can be approached with a knowledge of the molecular wave functions and dipole transition matrix elements are substantial. They include absorption and emission between discrete molecular states, transitions between discrete and continuum states, radiative deexcitation of excited atoms by binary collisions-with or without molecular formation, and translational absorption of radiation by colliding atoms. Sum rule techniques (Chan and Dalgarno, 1965; Dalgarno and Davidson, 1966; Dalgarno and Epstein, 1969; Victor, and Dalgarno 1969; Dalgarno and Williams, 1964) can be used to calculate molecular polarizabilities which yield Rayleigh and Raman scattering as well as a variety of molecular properties (see, for example, Robinson, 1969, 1970) if accurate dipole transition matrix elements to a numerically complete set of states are known.

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J . C. Browne

Accurate energies, wave functions, and the transition moments among the eigenstates for H,' were tabulated by Bates and co-workers (Bates, 1951a; Bates et al., 1953a). Until very recently, there have been virtually no other calculations of the transition matrix elements between molecular states that cover a sufficient range of internuclear separations and states to allow quantitative comparisons with experiments or prediction of experimental quantities. Further, as pointed out in Section II,F, even wave functions that yield relatively accurate potential curves do not necessarily yield accurate transition matrix elements. Nicholls and Stewart (1962) and Soshnikov (1961) have reviewed the earlier calculations.

A. SPECTROSCOPIC PROCESSES Molecular spectroscopy has long used approximate theories of molecular structure as a basis for the correlation of observed data (Herzberg, 1950). Accurate detailed calculation of molecular spectra has not been possible except for H2' and a few transitions in H, . Rothenberg and Davidson (1967) made some fairly extensive calculations of transition probabilities for the X'C,' + n l n u , n = C, D , . . . , states of H, . These transition probabilities, however, covered only restricted ranges of internuclear separations. Very recently, Wolniewicz (1969) has calculated accurate values for the X'C,' ++ B'C,', X'C,+ c-f C'n,, and B'C,' E, F'C,' transition in H, over a considerable range of internuclear separation. These are the most accurate calculations yet made of the transition moments. Wolniewicz has used these transition moments together with the accurate potential curves for the states calculated by Kolos and Wolniewicz (1966a,b; 1969)to calculate band strengths for selected transitions. Comparison to experiment is difficult since the experimental measurements cannot resolve intensities with the degree of detail of these calculations. The interaction of the hydrogen molecule with radiation is ofparticular interest in astrophysics. It is now believed (Stecherand Williams, 1967) that a primary method for destruction of the hydrogen molecules in the interstellar medium is by absorption from the bound vibration levels of the 'C,' ground state into the discrete vibrational levels of the BIZ,+ and C'n, states followed by decay into the translational continuum of the ground state. Dalgarno and co-workers (Allison and Dalgarno, 1969,1970); (Dalgarno and Stephens, 1970) have used the transition moments of Wolniewicz and Rothenberg and Davidson together with less accurate calculations by Browne (1969a) to calculate accurately the rates for these processes. Dalgarno and Allison (1969) have also compiled the direct photo dissociation cross sections for the X'C,' + BIZ,+, X'C,+ -+ C'n,, and X'C, ---t B"C,+ transitions using the ( X I rl B ' ) values of Browne (1969b). ++

MOLECULAR WAVE FUNCTIONS AND PROCESSES

85

B. SCATTERING PROCESSES

-

The study of binary collision processes leading to the emission of radiation A*+B A*+B

AB+hv A+B+hv

whether leading to molecule formation or to deexcitation of an atomic excited state has not recieved much recent attention since such processes seldom have significant rates under experimental or natural conditions. A particular exception are the chemical reactions in interstellar space. The classic work of Bates (1951b) and Bates and Spitzer (1951) suggested that the binary processes were too improbable to account for the observed ratios of atoms to molecules. This work, however, was based on qualitative and fragmentary information on the molecular transition moments for CH' and CH. Hesser and Lutz (1970) and Smith (1971) suggest that these processes for molecular formation need to be reconsidered in the light of new experimental data for transition moments. Accurate calculations on these systems are needed. Allison et al. (1966) have calculated the cross sections for the process He(ls*'S)S He(ls2s'S)

-

2He(ls2'S)+ hv

which is one of the few for which an experimental estimate of the cross section is available (Phelps, 1955) .This calculation was carried through as a sequence of steps, beginning with the calculation of the molecular wave functions and potential curves, calculation of the transition matrix elements, and evaluation of the cross section. The agreement with experiment is very satisfactory, whereas an earlier calculation of the same process by Burhop and Marriot (1956) using much cruder wave functions was in error by a factor of lo6. Allison and Dalgarno (1963) carried through a similar calculation using the potential curves of Bates and Carson (1956) and estimating the transition moments by perturbation theory for He(ls2s'S)

+ Hf

-

He(lsz'S)

+ H' + hv

with the purpose of comparing the rate of this process with the two-photon decay of He(ls2s'S). Translational absorption, the transformation of photon energy into translational energy, is a function of the dipole moment induced in a pair of colliding atoms by the distortion of the free-atom charge distributions by the interatomic interaction. This may be considered as transitory molecule formation, and the appropriate transition moment is the dipole moment of the ground molecular state of the colliding atoms. Such absorption spectra were first observed by Kiss and Welsh (1959) for pairs of dissimilar rare-gas atoms. This work has been extended (Bosomworth and Gush, 1965;

J . C. Browne

86

Heastie and Martin, 1962) to include detailed line shapes for He-Ar and NeAr mixtures. Levine and Birnbaum (1967) and Levine (1967) have worked out classical and quanta1 treatments of this process which yield detailed line shapes (these authors also review earlier formulations). Futrelle (1968) has pointed out the value of these spectra in estimating the repulsive potential curves of rare-gas systems. McQuarrie and Bernstein (1968) have applied the classical theory of Levine and Birnbaum to the He-Ar system where calculated dipole moments are available from the work of Matcha and Nesbet (1967). Using a rather crude semiempirical potential energy curve together with the Matcha-Nesbet dipole moments, they were able to obtain a rather suprisingly good agreement with the experimental line shapes. Ulrich et al. (1970) have undertaken a detailed calculation of the translational absorption of He-H which is a possible source of opacity in the atmosphere of late-type stars. This system is simple enough so that quite precise calculations are feasible. The HeH ground state (2E.') potential curve and dipole moment were calculated with a 28 configuration wave function of the forni of Eq. (6). The dipole moment of this system is particularly interesting. It is negative (He-H') for large R values (see Brown and Whisenant, 1970) but becomes positive for R < 6.0 a , . Figure 3 shows the dipole moment of HeH('C+) together with the long range values of Brown and Whisnant (1970). The calculated line shape for translational absorption by HeH at 300 KO is given in Fig. 4. Experiments

He+H- (28-TERM WAVEFUNCTION)

c

Oo8T

I .6\

-+

1.4-

=!

z

Y0

5

y

WAVEFUNCTION

'2-

I

1.0-

:::I 0.8

600 800 1000 1200 1400 1600

R

0.6

I

0.0 000 0.00

, I60

\ 240

320

4.00

4.80

hu.)

,

,

I

,

5.60

640

7.20

8.00

R (ad

FIG.3. Dipole moment of ground state '?1+ HeH as a function of internuclearseparations. D(R) = 122/R7 is the long range form. ~

MOLECULAR WAVE FUNCTIONS AND PROCESSES

87

u (CM-’)

FIG. 4. Translation absorption of HeH.A(u) = ct(u)(nHnHe)-l(No/Vo) where a(u) is the absorption coefficient, nH and nHs are number densities, No= Avogadro’s number, Vo = 22413 cm3, and u = w/257c. The units of A(u) are cm-’ amagat-’. The plots are of A ( ~ x) 107.

covering a range of temperatures would be particularly interesting since the dipole moment passes through zero not far from the internuclear separation where the van der Waals minimum occurs. ACKNOWLEDGMENT This research was sponsored by the Robert A. Welch Foundation of Houston, Texas.

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/I

LOCALIZED MOLECULAR ORBITALS HAREL WEINSTEIN and RUBEN PAUNCZ Depar iment of Chemistry, Technion - Israel Institute of Technology Haifa, Israel

and MAURICE COHEN Department of Physical Chemisiry, Hebrew University Jerusalem, Israel

I. Introduction

.. .. .. . . . . . . . . . ................... Principle of the the Method.. Method . . . . . . . . . . . . . . . ....................... Principle of .................... ........ Calculation Procedures. . . . . Properties of the Energy Loca s . .. . . . . . . . . . . . . . . . . . . Applications. . . . . . . . . . ...................

99 99 102 102 102 104

V. Direct Localization Methods. .................................... A. MethodofA Method of Adams . . . . . . . . . . . . . . . . . . A. B. Method of ...................................... VI. Internal and E 'zation Criteria ......................... V1I. The Method o ........ VIII. The Method of Peters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1X. Molecular Orbitals Determined from Localization Models . . . . . . . . . . A. The Method of Weinstein and Pauncz.. . . . . . . . . . . . . . . . . . . B. Method of Letcher and Dunning .............................. ............ X. Localized Orbitals in Expansion Methods A. Method of Certain B. Multiconfiguration XI. Concluding Remarks. References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

116

11. Density Matrix Formalism 111. The Edmiston-Ruedenberg

A. A. B. C. D.

106 108

119 121 122 126 128 128 133

138

I. Introduction The two main methods in the quantum mechanical treatment of the electronic structure of molecules are the valence-bond method and the molecular orbital method. The former is more closely related to chemical concepts, while the latter has the great advantage that it is relatively easy to carry out the 97

98

H. Weinstein, R. Pauncz, and M . Cohen

calculations even when all the integrals are calculated rigorously. In the past few years it has become clear that electronic correlation is of fundamental importance in the treatment of atomic and molecular structure, and has to be introduced through expansion methods such as the multiple configuration or configuration interaction (CI) procedure. However, the great complication of the resulting functions and the relative success of the various " one electron " methods, explain the popularity of the linear combination of atomic orbitalsself-consistent field (LCAO-SCF) approximation as formulated by Roothaan (1951). In the case of a closed shell system, the total wave function is approximated by a single Slater determinant in which each molecular orbital occurs twice, once with CI and once with j? spin function. The molecular orbitals are obtained from the solution of the Hartree-Fock-Roothaan (Roothaan, 1951) pseudoeigenvalue problem and are expected to yield minimum total electronic energy for the given functional form (see, however, Adams, 1961). The molecular orbitals are symmetry adapted (Nesbet, 1965) and they extend over the whole molecule. Ever since the formulation of the molecular orbital method, there have been attempts to find relationships between the molecular orbitals and classical chemical concepts. Thus, although the exact many-electron wave function has been shown to be nonseparable (Primas, 1965), the additivity and separability of measurable chemical quantities might be emphasized by particular approximations to these functions. It was soon realized that the molecular orbitals themselves are not appropriate for this purpose, since they are completely delocalized. However, one can use certain invariance properties of a determinantal wave function and transform to linear combinations of the molecular orbitals (without changing the total wave function) which are of a different nature; they can be localized in different regions of the molecule, and identified as inner shells, lone pairs on isolated atoms, or bond orbitals associated with a pair of atoms. These latter are called localized orbitals. Since they are likely to yield significant electron density only in distinct regions, the total electron density might be expected to be partitioned into separate contributions which then generate localized quantities, such as the bond dipole moment (Coulson, 1942), as a consequence of the chosen form of the total wave function. There are different reasons for seeking localized orbitals, and there are different criteria for determining them. Besides the obvious fact that localized orbitals correspond to chemical concepts, they are important because one might hope them to be transferable from molecule to molecule, and because a set of such orbitals might serve as a convenient starting point in a more elaborate treatment. They are especially suitable for a treatment of correlation because they are localized in different regions of the molecule, and presumably

99

LOCALIZED MOLECULAR ORBITALS

the correlation between electrons that are in different localized orbitals is much smaller than between those which are in the same orbital (LennardJones and Pople, 1951 ; see, however, Sinanoilu and Skutnik, 1968). Special types of localized orbitals might also serve as basis functions in multiconfiguration calculations to speed up convergence. The aim of this review is to present different aspects of the problem of finding localized orbitals. Recently there has been increased use of these orbitals, so that it is worthwhile to summarize the basic principles and to show different possiblities of application. It seems simplest to present the general principles by means of the density matrix approach, and we will first give an outline of this method.

11. Density Matrix Formalism The N-electron wave function of an atomic or molecular system is usually considered, within the formalism of quantum chemistry, to be the most appropriate description of such a system. However, the physical description of large systems by such wave functions always contains a number of degrees of freedom which are often of no interest for some specific problem, and may, therefore, be averaged in an adequate way. A physical quantity 0 is represented in the N-electron configuration space, by the Hermitean operator U,, which is symmetric in the indices of the electrons. The expansion of such an operator is given by

o,

+ c Ui + -1 C’ uij + . . .

= 0,

I

2i,j

and the average value over such an operator is given as

The symbols p l ( x l ’1 x l ) and p2(xl’x2’I x 1 x 2 )are the reduced density matrices ” defined by (Lowdin, 1955a) pl(xl’ I xl) = N

1’

“

one-and two-particle

Y *( 1’2 ... N ) Y ( 12 -.. N ) dxz

dx,

(3)

and p 2 ( x , ’ x , ‘ ~ x , x ,=) ~ ~ ~ S Y * ( l f 2 ‘ 3 . . . N ) Y ( 1 2 3 . . . N ) d x , . . . d x(4) , respectively; 0, and U,, operate on the unprimed variables x1 and x2 only.

100

H . Weinstein, R. Pauncz, and M . Cohen

The usual form of the Hamiltonian operator of an atomic or molecular system within the Born-Oppenheimer (Born and Oppenheimer, 1927) approximation, is

3r = Z ( 0 ) + X(I)+

X ( 2 )

(5)

where

We can make use of the fact that 3r contains terms involving the coordinates of one, or at most two electrons. The expectation value of the energy E=

s

Y*XYdv

(7)

can also be written as E=

HO + Tr [hip11 + Tr W,Z~ 1 2 1 .

(8)

The traces here are consequences of the possibility of constructing a discrete representation of the reduced density matrices and of the operators. This representation is based o n the expansion of the N-particle wave function in a discrete basis. Thus, as shown by Lowdin (1955a), the one-particle density matrix can be expressed in a Hermitean form as ~l(x1’Ixl= )

1 qi*(xl’)p!f’qj(xl>

(9)

i, j

where the arbitary set of functions {q,} is complete, and the pi,!) form a Hermitean matrix. A natural question now arises concerning the way in which the variational methods of approximation for the N-particle wave function can be reformulated in terms of density matrices (Kiang, 1967). Unfortunately, while the Nparticle wave function of an electronic system has been computed for a large number of atoms and molecules within a fairly good approximation, the direct derivation of exact (or approximate) density matrices is usually much more complicated (see, however, Kutzelnigg, 1963; Ahlrichs et al., 1966). Since the calculation of approximate wave functions is much simplified in quantum chemistry by a large number of empirical, topological, and morphological rules, which form the basis of various approximation methods, the best approximations to the reduced density matrices seem to be those obtained from the use of the corresponding approximate wave functions through equations such as (3) and (4) or (9). An arbitrarily chosen basis in the space used to set up a certain approximation will, therefore, yield a representation of the density matrix expressed through its matrix elements, of the form given in Eq. (9). Since any such

LOCALIZED MOLECULAR ORBITALS

101

Hermitean matrix can be diagonalized by a unitary transformation of the basis Xk = ‘ P i Uik, (10) i

the new representation of the density matrix will have the form

P~(x~’Ix~)

=

Ci ni Xi*(xl’)Xi(xl).

The transformation may be chosen such that the and the sum

Xk

(1 1)

form an orthonormal set,

xni=N I

is the number of electrons in the system. The new set of functions is then called the natural orbitals (Lowdin, 1955a) of the system and the ni’s are their occupation numbers. Equation (1 1) then gives the natural expansion of the density matrix and some of its properties will be discussed i n connection with one of the most important and most popular approximation methods used in quantum mechanical calculations on atomic and molecular system-the Hartree-Fock (HF) method. Lowdin (1955b) pointed out that if the total wave function is approximated by a single Slater determinant (as in the H F method), the first order (oneelectron) density matrix determines all the higher order density matrices by means of the relation

where p p ( x l ’ .. . xp‘I x 1 . . . x p ) represents a reduced density matrix of order p , while p(xi’I x j ) are the matrix elements of the first-order density matrix. I n this scheme, therefore, the entire physical situation is determined by the one-electron density matrix alone. Using the idempotency of this matrix, we conclude that the H F one-electron wave functions are the “eigenvectors” of the first-order density matrix. The expansion of p ( x i ’1 x j ) i n such a basis thus gives the well-known natural expansion

with every nk = 1. Fock (1930) made the important observation that an orthogonal transformation among the occupied HF orbitals leaves the form of the first-order density matrix invariant and therefore, through Eq. (13), the

H. Weinstein, R. Pauncz, and M . Cohen expectation values of operators representing physical properties unchanged. This is an inherent property of the single-determinant approximation and a consequence of the fact that all the occupation numbers are equal. In the general case, where the first-order density matrix is given by Eq. (11) with unequal nk’s, we no longer have this degree of freedom. Since the main interest of this work concerns charge density distributions and their localization properties, we shall consider the influence of orthogonal transformations on the first-order density matrix only. According to the statistical interpretation given by Lowdin (l955a) to the diagonal elements of the density matrix, p ( x , I xl) = p(xl) represents the number of particles times the probability of finding a particle within the volume dv, around the point rl having the spin s,, when all the other particles have arbitrary positions and spins. The value of the diagonal element in Eq. (1 1) is invariant to orthogonal transformation, but individual elements are affected, and although the forms of the matrix elements remain unchanged (Dirac, 193l), their detailed composition is altered. Thus, in some regions of the space, particular transformations can lead to increased contributions from certain molecular orbitals uk to the diagonal elements of the density matrix, while the remaining contributions to the sum in Eq. (14) remain small in these regions. This reasoning leads to the construction of certain models of the molecular density and to its arbitrary splitting into localized contributions from certain well-defined orbitals which then become the localized molecular orbitals.” The corresponding orthogonal transformation is then called the “ localization transformation within the given representation. Different approaches to the localization of molecular orbitals will be considered in the following sections and their characteristic features will be compared. Most of the methods use the self-consistent orbitals as a starting set, exploiting the special properties of the Hartree-Fock functional space. However, we shall not restrict our discussion to methods operating in the H F functional domain, but consider also localization methods related to other models and use the H F results only as a reference system. “

”

111. The Edmiston-Ruedenberg Localization Method A. PRINCIPLE OF

THE

METHOD

The basic idea of the method proposed by Edmiston and Ruedenberg (1963) comes from a generalization of the results obtained by LennardJones and Pople (1950) for equivalent orbitals. The functional set was obtained (Lennard-Jones, 1949) by searching for orbitals corresponding to chemical concepts and they were defined so that they transform into each other under

LOCALIZED MOLECULAR ORBITALS

103

the symmetry operations of the group characterizing the molecule. It was observed that certain terms in the electronic interaction energy could be reduced by using equivalent orbitals instead of standard HF molecular orbitals (canonical orbitals). The interelectronic interaction energy expression for a wave function consisting of a doubly occupied single Slater determinant may be written in terms of the one-electron density matrix

E

C - X , say.

(15)

From the properties of the reduced density matrix it follows that both C andX will be invariant to a unitary transformation performed within the HartreeFock functional basis set. If we now decompose C and X into elements of the form Jk/

=

jjI 'Pk(l) I

I 'Pi(2) 1

1

- dt1 dzz

rlz

['Pk21431~1

(16)

and

we may write

and

Since C and X contain some equal elements Jkk

= Kkk,

the total interaction energy expression contains terms of the form Jkk, J k l ,and K k , only. Lennard-Jones and Pople (1950) showed that while equivalent orbitals yield the same values of Eint,C and X as standard molecular orbitals, the sum =

Jkk k

104

H. Weinstein, R. Pauncz, and M. Cohen

can change its value. And since the value of this sum is found to be higher in the equivalent orbital representation for some simple cases investigated, Lennard-Jones and Pople attribute this result to the localization properties of the equivalent orbital basis. The Edmiston-Ruedenberg localization method uses the maximization of the sum D of Eq. (21) as the criterion for a transformation of the standard Hartree-Fock orbitals leading to the localization of the basis functions. This choice is made on the basis of the physical interpretation given to various terms in the interelectronic energy expression. Transformations which maximize the expression in Eq. (21) defined as the “sum of orbital self-interaction energies” are therefore expected to be “ localization transformations.” They also minimize both the remaining “ total inter-orbital repulsions ”

and the “total self-energies of the overlap charge distributions”

The set of transformed orbitals {$i} is then a set of localized molecular orbitals. Since this localization procedure is based on changes in the electronic interaction energy terms due to the transformation, theresultingwavefunctions are defined as energy localized orbitals. The method used to obtain this set of localized orbitals is thus closely related to conclusions drawn from an equivalent-ortibal set, but offers the new possibility of defining localized orbitals in the absence of symmetry. The functions obtained by Edmiston and Ruedenberg are, therefore, characterized by two important features: (I) They form a basis for a representation of the Hamiltonian constructed according to the independent particle model, thus retaining the minimum energy property; and ( 2 ) they are localized in certain regions of the molecular space.

B. CALCULATION PROCEDURES There are two main procedures leading to the determination of localized orbitals through the maximization of the sum of orbital self interaction energies. (1) Once the Hartree-Fock functional space has been determined (usually by an LCAO-SCF procedure), one can perform a series of unitary transformations among the occupied orbitals and, through an iterative procedure, obtain the set of energy localized functions. ( 2 ) Another approach, which is part of a more general formulation of the various efforts to localize SCF orbitals (discussed in Section V), uses a properly set up eigenvalue problem

LOCALIZED MOLECULAR ORBITALS

105

which yields the localized orbitals directly. In this method, D in Eq. (21) takes the form of a “localization potential” which operates on the standard molecular orbitals. In the present section we confine ourselves to the description of an iterative transformation approach which has become very popular and has been widely used in a long series of investigations. The procedure consists of a complete cyclic series of 2 x 2 rotations among the “occupied” SCF orbitals and is based on the observation of Lennard-Jones and Pople (1950) that in a twodimensional functional space, the localized functions can be obtained from symmetry orbitals (such as the standard SCF functions) by constructing normalized linear combinations of the form t+h1 = ‘pl cos y $2

+ (p2 sin y

sin y

= -‘pl

+ ( p 2 cos y.

Edmiston and Ruedenberg find that in such a case D($)of Eq. (21) is given by

D($)= D((P)+ A

+ ( A 2 + B2)’/’ cos 4(y - a)

(25)

where ’4 = [(Pl(P2 I (P1(P21-

$[(PI’

- ‘Pz2 l4?

- (P221

(26)

B = [4312 - VZ2 I (Pl(P21 and 4u is defined by sin 4a = B / ( A 2+ B2)’/’ cos 4a

=

-A/(A2

+ B2)‘/2

From Eq. (25) it is clear that a maximal value for D($) occurs when a The transformation into localized orbitals is therefore written as

= y.

In an N-electron system, the optimally localized orbitals are obtained after carrying out all possible 2 x 2 transformations within the SCF space, repeating the procedure until there is no further increase in the value of D. A steepest ascent variant of this method has been proposed by Taylor (1968). The procedure described here is a modification of an earlier, more complicated, method. Although it involves a large number of linear transformations and the evaluation of many polycenter integrals, it leads to a compact formulation of the localization procedure on a digital computer and to a reasonably rapid convergence towards the localized set.

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H. Weinstein, R. Pauncz, and M . Cohen

c. PROPERTIES OF THE ENERGYLOCALIZEDORBITALS The localized orbitals obtained by this method bear the characteristic features of the chemical concepts of inner shells, bond orbitals, and lone pairs. These electron localization regions may be recognized in the densities obtained from the molecular orbitals which maximize the sum of orbital self-interactions, and are expected to yield the smallest interorbital correlation. Edmiston and Ruedenberg (1965) also state that localized orbitals are more adequate to define the inner shells and outer (valence) orbitals used in various approximations than are the orbitals obtained in a rather arbitrary way by means of the Schmidt orthogonalization procedure. Since “inner shells ” and “valence orbitals ” obtained from Schmidt or SCF procedures are widely used, this observation is important in that it throws light on a common way to misuse results of approximations. Thus, the fact that the total SCF wave function yields a fairly good approximation to the total molecular energy, by no means implies that the basis one-electron orbitals have any definable physical meaning. For example, the definition of an inner shell involves the description of a localized density near a given nucleus, and only a function describing such a density can represent a valid inner-shell orbital. An important advantage in the use of the localized orbitals is the possibility they offer of expressing correctly some physical properties of the electronic density, in a way that is parallel to their definition. Another important property of the energy localized functions is the fact that the inner shells, lone pairs, and bonds they describe, and especially the bent “banana bonds” for BF and N, (Edmiston and Ruedenberg, 1965, 1966), have been obtained from the maximization of the orbital self-interaction energy without recourse to any preconceived ideas concerning a definite model of the electronic density. This property is characteristic of the “intrinsic” methods defined by Ruedenberg (1965) and will be discussed further in Section VI. It should be noted that the existence of such a set of energy localized orbitals which span the same space as the occupied canonical SCF orbitals, expresses an inherent localization property of the SCF space. Since the properties of the equivalent orbitals originally formed the basis ofthe localization criterion used here, it is interesting to compare the “energy localized functions” and the equivalent orbitals for a given system i n the hope of finding some correlation between localization. maximum self-interaction energy, and symmetry properties. Edmiston and Ruedenberg (1966) point out that althoughequivalent orbitals areassumed to maximize the function D in Eq. (21), this assumption is not always justified, and the localized orbitals obtained with the use of this criterion do not necessarily coincide with the equivalent orbitals. An interesting example is the determination of a molecular orbital set obtained from linear combinations of s- and p-type atomic

LOCALIZED MOLECULAR ORBITALS

107

functions. For such a case there is a continuous dependence of the function yielding a “localization value on the “orbital exponent disparity ” of the atomic orbitals. The latter quantity is defined to express the difference in the spatial extension of the s- and p-type functions, and it is concluded that no localized orbital can be obtained from 1s-2p hybridization. However, such hybrid functions remain the equivalent orbitals for the system, while the localized functions are the 1s and 2p orbitals belonging to the C and II irreducible representations of the rotation group. Thus, any hybrid localized orbital will consist only of s and p orbitals with equal spatial extension, and a large local overlap. A possible form of such localized orbitals will therefore be ”

In general, even when equivalent orbitals give a fairly good description of the various localized densities within the molecule, they are not identical with the localized orbitals, and a number of different equivalent orbital sets can be defined for the same molecule. At this point, we wish to emphasize the conclusion drawn by Edmiston and Ruedenberg that the orbitals obtained from the self-interaction energy criterion are better localized than the equivalent orbitals, and that the similarities between inner shells in different molecules are a direct consequence ofthe localization properties of s- and p-type atomic orbitals. Since the 1s atomic function, which gives the main contribution to the inner shell molecular orbital, remains unaffected by the s-p hybridization of higher principal quantum number which generate the localized bond orbitals, the inner shells for similar atoms in different molecular surroundings may be expected to remain unchanged. I n a more general discussion of this question, Adams (1965) points out that equivalent orbitals can be expected to be invariant to the molecular surroundings only in those cases in which the orbitals of the localized charge density considered do not significantly overlap the regions in which the compared molecules differ. However, we must remember that although the equivalent orbitals of a certain region of the total charge distribution are orthogonal to all other equivalent orbitals, this does not necessarily imply that the value of the total wave function will be small i n the regions i n which the orbitals of other groups are large (Mattheiss, 1961). It is therefore reasonable to assume that certain neighboring groups in the molecule will have significant influence on the equivalent orbtials, while a fully localized set of functions should have minimal influence on neighboring groups.

108

H. Weinstein, R . Pauncz, and M . Cohen

Although the localized orbitals obtained from the procedure of Edminston and Ruedenberg do not satisfy these conditions exactly, they are sufficiently invariable to permit their definition as “ chemically similar orbitals,” in which the different atomic contributions to the localized functions are recognizable.

D. APPLICATIONS The Edmiston-Ruedenberg method of localization has turned out to be an important contribution to the investigation of the electronic structure of molecular systems. It has found wide use in the formulation of new approximation methods and models (Cook et al., 1967; Diner et al., 1969a; Kapuy, 1966), the investigation of intramolecular forces and the search for transferable invariant molecular orbitals (Adams, 1965; Magnasco and Perico, 1968). The main drawbacks in the method remain its confinement to the SCF-approximation functional space, and the need for the cumbersome computation of polycentered integrals at each step of the iteration procedure. From the numerous applications we would like to single out an example in which the analytic power of the localized representation isclearly demonstrated. Inaseriesof investigationsof boron hydrides by Switkes et al. (1969, 1970a,b), an ab initio analysis of bonding in electron deficient molecules is undertaken in the localized molecular orbital frame. Starting from a minimal basis SCF calculation for B,H, , B4H10, B,H, , and B,H,,, energy localized orbitals have been determined. These reveal a picture of molecular bonding in which different kinds of bonds are recognized as well as the localized inner shell densities. The self-repulsion energy is maximized by the successive twoorbital transformation procedure proposed by Edmiston and Ruedenberg and a second-order energy test is applied to check convergence. The data given i n Table I illustrate the drastic changes occurring in the two-electron energy terms. In the B,H, molecule (assuming D,, symmetry) four equivalent localized molecular orbitals representing B-H bonds are recognized by negligible contributions from other centers. Similarly, three-centered bonds around the bridge hydrogens are interpreted as electron densities which bond the B(H,) groups together. Density contour maps of the localized orbitals in Fig. 1 show the highly localized character of the different bonds and thus providean excellent tool for the interpretation of molecular bonding. The different types of molecular bonds revealed by the localization of standard SCF molecular orbitals for the B4HI0, B,H,, and B,H, molecules are consistent with the conclusions drawn for electron deficient molecules i n the B,H, case. Of special interest in these molecules are the boron-boron types of bonds for which the three-center theory fails to make full distinction between different possible valence structures. Switkes et al. (1970b) find that

109

LOCALIZED MOLECULAR ORBITALS

these structures are not interconvertible by symmetry transformations and thus have, in general, different self-repulsion energies. The Edmiston and Ruedenberg localization method thus provides an adequate criterion for the choice of a preferred structure for a particular bond. TABLE I TWO-ELECTRON ENERGY ANALYSIS" Term Total two-electron Self-repulsion

Coulomb Exchange Interorbital coulomb

Orbitals

B2H6b

B4H10C

B5H9'

B5Hll'

Canonical Localized Canonical Localized Canonical Localized Canonical Localized Canonical Localized

46.7878 46.7878 5.4312 9.5963 45.7905 37.4603 -4.3339 -0.2688

134.0889 134.0889 9.8042 18.3999 143.3702 134.7777 -9.2813 -0.6889 133.5660 1 16.3811

175.5347 175.5347 1 1.9255 21.7915 186.3872 176.5211 - 10.8524 -0.9864 174.4617 154.7296

183.7597 183.7597 8.6657 22.4877 1 98.51 74 184.6955 - 14.7577 -0.9357 189.8517 162.2078

Energy values are in atomic units. Switkes er al. (1969). Switkes er al. (1970b).

A very similar kind of characterization of molecular bonds in a series of single- and double-bonded molecules has been found by Pitzer (1964) and Kaldor (1967) for the localized molecular orbital sets obtained by the energy localization procedure.

IV. The Method of Boys and Foster The method proposed by Boys (1960) and Foster and Boys (1960) is intended to combine the search for a set of localized functions with the definition of a set of basis functions that can be used in calculations going beyond the independent particle approximation, in a configuration interaction scheme. Such functions should be adequate for a thorough-going comparison of various approximations and could bring to light the characteristics of the models used by different methods. When a localized model is used, Boys proposes the determination of the basis set through conditions imposed on the wave function, which lead to the quasi-invariance of chemically defined fragments of the total electronic density (such as bonds, inner shells, and lone

110

H. Weinstein, R . Pauncz, and M. Cohen

I

I

Z

(b) FIG.1 . Localized molecular orbitals (density for one electron e j a . ~ . ~ (a) ) . B-H,ern,i,,a, bond; (b) three-center bond. From Switkes et a/. (1969). Reproduced by permission.

pairs) to changes in the molecular surroundings. At the same time, it is expected that the localization properties of these functions will make them useful i n a configuration interaction calculation. The quasi-invariant orbitals are obtained from a starting set of basis functions 'piof a single Slater determinant which minimizes the molecular energy ( H F or LCAO-SCF approximation). The new set is determined by the condition that the functions should be least sensitive to changes in the molecule.

LOCALIZED MOLECULAR ORBITALS

111

Assuming that the charge of the nucleus I is changed by the amount 69, and denoting by Xir6gI the corresponding change in the orbital q i , Boys (1960) proposes that the quasi-invariance condition should be reflected in the minimization of

Here K(i) denotes the set of atoms that are distant from center i. These are usually the non-nearest neighbors but may also represent arbitrary points where charges might be introduced later (as in a series of chemical homologs). The invariant functions are thus obtained by a perturbation procedure, the perturbing potential being defined by the changes occurring in the chosen centers. The value of J is calculated at each step, and the procedure repeated iteratively until a converged limit is obtained. It is clear that although the convergence may be influenced by the choice of the starting set, it is only for computational reasons that the SCF orbitals were chosen as a starting point. Although it has not been demonstrated formally that the “ invariant orbitals are also localized, it is probable that this is so. However, Foster and Boys observed that “ i t is still a conjecture that when invariant orbitals for two molecules both containing some particular chemical groups are found, the orbitals for these groups are nearly the same.” Their conjecture remains to be verified. Another method proposed by Foster and Boys (1960) is of considerable significance since the functions obtained display the high-localization properties that are expected to make them “ invariant orbitals.” This method, which yields “ canonicalconfiguration interaction (CI)orbitals,” is much simpler in practical applications, and the authors suggest that the two resulting functional sets are quite similar. The canonical CI method arose originally from the search for an adequate basis set for a multiconfiguration procedure, but the results turned out to contain localized orbitals for which transferability properties from one molecular surrounding to another are expected. Here, localized orbitals are obtained using the transformation invariance of the space spanned by the occupied LCAO-SCF basis functions. Their properties are then used to construct wave functions suitable for the solution of the electronic correlation problem through the multiconfiguration method. The transformation leading to these localized orbitals uses a n interaction energy criterion, which imposes maximal separation of the centroids of the charge described by a given set of molecular orbitals, to yield a set of new functions called “ exclusive orbitals.” Thus, we define the centroids Ria of the orbitals {cp,} as ”

H . Weinstein, R . Pauncz, and M . Cohen

112

in which the {cp,} are supposed to be expanded in terms of some suitable basis set ‘pi =

1 aijqj. i

(32)

The procedure for transforming the standard SCF orbitals (cp,} into exclusive orbitals is then to maximize the product:

l7 =

n [(R, -

i < j

RjJ2

+ ( R , , - Rjy)’ + (Ri, - Rj, ) 1. 2

(33)

Due to the properties of the SCF functional space previously discussed, it is not surprising that the exclusive orbitals so obtained turn out to be localized orbitals that describe densities corresponding closely to the chemical picture of valence structure. For example, the geometrical location of the density centroids found by Foster and Boys (1960) for formaldehyde (HCHO) turn out t o correspond roughly to distinct charge concentrations that may be defined as the oxygen lone pair, oxygen atom core, double bond charge distribution, carbon atom core, and two C-H bonds. Although the invariance properties of the exclusive orbitals in different molecular surroundings are significant, the main aim of this localization method should be considered as the definition of a second set of functions, related to the exclusive orbitals, which will serve in a CI procedure. Now it is generally accepted that replacing orbitals in the ground configuration by functions having considerable density in the same spatial region leads to improved results in a multiconfiguration treatment of the electronic correlation (Lowdin, 1959; Nesbet, 1965; Bender and Davidson, 1966). Thus, Foster and Boys construct the complementary functional set depending on the localized orbitals in the first configuration and having these particular localization properties. It is therefore clear that the starting set for the construction of the new functions (which will actually replace the “ virtual ” or “ unoccupied ” orbitals of the LCAO-SCF method), should have well-defined localization properties, similar to those displayed by the exclusive orbitals and the additional set, called “oscillator orbitals” is obtained by requiring that the oscillator orbitals maximize the dipole moment matrix elements calculated between them and the exclusive orbitals. The physical basis for this criterion is the fact that the magnitudes of the dipole moment matrix elements give a measure of the relative separations of electron distributions described by the individual orbitals. A set of oscillator orbitals xik is therefore defined for each exclusive orbital of the starting set, using the same set of basis functions { q j }

113

LOCALIZED MOLECULAR ORBITALS

subject to the conditions: (Xikl

qj> = O;

(XikI Xjl)

= 6ij6kl

(35)

and m

W =

f

C C1 I((pi(rlXik)12,maximum, i= 1 k=

(36)

where m is the total number of exclusive orbitals i n the starting set, and r is the number of oscillator orbitals defined for a particular exclusive orbital qi . These oscillator orbitals then contribute to the “localization ” of the electronic correlation terms in certain orbitals which describe the localized fragments of which the total charge distribution is supposed to be composed. In a later publication, Boys (1966) proposed a modification of the localization procedures used to obtain both exclusive and oscillator orbitals. While the general features of the functions obtained remain unchanged, the formulation of the new procedure gives a more natural definition of the relation between oscillator orbitals and exclusive orbitals. A hierarchy is established within the group of oscillator orbitals defined for each exclusive orbital which is related to the order in which the functions are determined. In this way, all members of the group of oscillator orbitals are determined. The new formulation of the procedure is based on the minimization of the “ quadratic repulsions of the orbitals with themselves: ”

I = C ( ( p i 2 ( 1) I rt2 1 (pi2(2)) -+ minimum. i

(37)

This criterion is equivalent to maximizing the “sum of squares of the distances of the orbitals from each other,” and is therefore closely related to the original definition of the exclusive orbitals. Defining the centroid of the orbital ( p i as Ri=(qiIrlIqi)

(38)

r l i = rl - R i ,

(39)

and writing

the functional I becomes I=

C (qi2(1>Irt2 I qi(2)2) I

=

(qiqil i

=A

trli - r2J2 I V i q O

- B, say.

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H . Weinstein, R.Pauncz, and M . Cohen

From the definition of R i , the term B vanishes identically. Thus, thislocalization criterion can be interpreted as minimizing the sum of quadratic moments of each exclusive orbital about its centroid. This result emphasizes the localization properties of the exclusive orbitals, since this concentration around the corresponding centroids together with the orthonormality of the orbitals must result in maximizing the sum of distances between centroids o f different orbitals. The physical interpretation of the orbitals derived using the functional I underlines the close connection between this procedure and the method of Edmiston and Ruedenberg discussed previously. In the computer program, Boys uses an alternative equivalent expression of the same functional

I

=

1i (9,qi I ( r l - r212 I

'Pi

qi>

= 2 C (9iIr1219i)-2C(cpiIrl19i)2. i

(41)

Since the first term here is invariant to the unitary transformations used, only the term containing the centroid formula has to be minimized. Consequently, the transformation criterion is dependent on the one-electron density matrix only. It therefore turns out that although the Edmiston-Ruedenberg procedure has clearer physical significance (the charge centroid distances being a somewhat artificial definition), nevertheless the procedure formulated b y Boys is much more economical computationally, particularly when large systems are involved. An important feature of this variant is the definition of the oscillator orbitals, which in the new formulation are obtained from the orthonormalization of a set i n which the functions are

x,, y , , za being coordinates to be measured from the centroid of orbital density with respect to axes taken parallel to the principal axes of the moment of inertia tensor o f the distribution. Thus, the exclusive orbitals can now be regarded as thefundamental members o f a set of functions. A clearer definition of the method for the derivation of a l l members of the sequence is thus obta i ned. As an illustrative application of the Foster and Boys method we mention the papers of Bonaccorsi et al. (1968, 1969). Using a starting set obtained from a SCF minimal basis calculation for a group of acid-anion pairs, these authors found that the Foster and Boys localized orbitals are generally quite adequate for chemical characterization in terms of inner shells, lone pairs, and different kind of bonds (such as the C-N banana bond), but their transferability is unsatisfactory. The definition of the related oscillator orbitals

115

LOCALIZED MOLECULAR ORBITALS

enabled Bonaccorsi et al. (1 969) to perform a configuration interaction calculation using localized orbitals. The expansion in the basis formed from exclusive and oscillator orbitals helped to speed up the convergence by providing a criterion for the choice of configurations withsignificantcontributions. The exclusive orbital and its first oscillator orbital shown in Fig. 2 are seen to

\

,

! !

, /'

(b)

FIG. 2. Maps of (a) the N lone pair orbital of C N - ; (b) its first virtual orbital. The contour values are as follows: 1 = 0.045, 2 = 0.10, 3 = 0.20, 4 = 0.33, 5 = 0.45. The nodal surfaces are represented by a full line with crosses, the full lines refer t o the positive portions of the orbital, the broken ones to the negative portion. The C atom (at left) and the N atom are marked by a dot. From Bonaccorsi er al. (1969). Reproduced by permission.

116

H. Weinstein, R. Pauncz, and M . Cohen

be localized in the same region of space, a characteristic that facilitates the correlation treatment along the lines discussed by Nesbet (1965) and explains the faster convergence when orbitals of this type are used in a configuration interaction treatment. We conclude this section by emphasizing that this method of Boys represents a major breakthrough beyond the independent particle model by using the localization properties of the individual molecular orbitals for the truncation of a multiconfiguration series.

V. Direct Localization Methods A. METHOD OF ADAMS In the procedures described thus far, the localized orbitals have been obtained indirectly, after transformation of standard HF orbitals. It is of considerable interest that they can also be obtained directly, as solutions of a modified eigenvalue problem through appropriate changes in the HF operator. Thus, Adams (1961, 1962, 1965) has derived an eigenvalue equation, the solutions of which take the form of localized orbitals. Starting from a proof that one can release a degree of freedom in the determination of the basis functions for the HF manifold by requiring them to be linearly independent and normalizable but not necessarily orthogonal, Adams (1961) uses the remaining degree of freedom as a localization condition for the resulting set {cpi}. These orbitals are then collected in a molecular matrix function @, the row index being the electronic coordinates (x), and the columns being arranged according to the index i of the functions {cp,}. The density matrix is then given by p = @(l+ S)-'@t (43) where s = @@t -1. (44) The condition that p should represent the H F functional space is expressed in a variational form, the expansion of 6p being expressed in terms of 6@ and 6 s 6p = 6@(1+ S)-'@t + @(l+ S)-'G@t (45)

- p6@(1+ S)-'@+

- @(l+ S)-'G@tp + . . * . Now, if we require that the total energy E be minimized, the variational condition takes the form 6E = [d@'(F@ - pF@)(l + S)-'],,

+ [(l + S)-'(@'F = 0.

- @tFp)6@],,

LOCALIZED MOLECULAR ORBITALS

117

This condition is satisfied if

F@ = pF@,

(47)

E' = (1 + S)-'@"F@,

(48)

and if we define

then we obtain the standard form of the H F eigenvalue equation

F@ = QE'

(49)

Since the set of functions {cp,} constructing the matrix @ can be orthogonalized, the equation for the canonical H F orbitals is obtained after a suitable transformation on @. The canonical H F orbitals are then related to the new nonorthogonal set {cp,} by a transformation Y = @r-'

(50)

where

r = u+(i+ s)?

(51)

This completes the proof that the standard H F orbitals can be obtained from an eigenvalue procedure leading to nonorthogonal functions {cp,} followed by a simple transformation. The important conclusion is that the starting matrix @ can be constructed from certain localized orbitals, and the whole procedure is therefore equivalent to deriving a functional set @which corresponds to some localization model. At the same time, @ minimizes the energy for a H F model Hamiltonian. A possible choice of the starting matrix @ is proposed by Adams (1962) in the form of eigenfunctions of HF-type Hamiltonian operators set up for each of the fragments a, b, . . . of which the complete system is supposed to be built. This results in the formation of corresponding blocks: @ b , @ b . . . in the matrix @, and imposes restrictions on the transformation matrix I'. This defines a new expansion form of the @ matrix in terms of the standard H F orbitals of the complete system

a, = ar,

(52)

r, will have the form of an N x N matrix ( N being the number of electrons in the complete system), having nonzero elements only in the first N,columns. Since, by definition, the blocks @, satisfy the H F equations for the fragments, Fa@, = @,E,

(53)

the matrix F, may be defined by an alternative expression of Eq. (53) in terms of the standard H F solution of the complete system

Y+F,w, = r, E,

(54)

H. Weinstein, R. Pauncz, and M . Cohen

118

which is obtained from Eqs. (52) and (53). The desired solution of Eq. (54) is that which diagonalizes E , , minimizes the energy of the fragment a, and satisfies

r,+r, = 1,.

(55)

The functions thus obtained should therefore be very close to the H F orbitals for the fragments, while their expansion in the functional basis of the complete system transforms them into molecular orbitals. Since the different sets @ a , ab. . . need not be mutually orthogonal, the whole procedure carried out separately for each fragment results in the definition of a final total function a,which minimizes the energy and is composed of basis functions which are localized in different regions corresponding to the fragment definitions. The eigenvalue problem which yields these localized H F orbitals is expressed in a modified formulation of the procedure (Adams, 1962) i n which the localization conditions take the form of the H F operator for the fragment a perturbed by a term which arises from the interactions with the rest of the system. Thus, a perturbing potential of the form “

”

where V, = F

- Fa

(57)

reflects the fact that the fragment function a, is expanded in a basis which extends over the complete system. The density matrix formed from the standard HF functions acts as a projection operator, which extracts from the function only those terms which are defined within the H F functional space. This projection annihilates the perturbation due to V,,,, over a given fragment a when functions expanded in the standard H F basis are used. Adams (1 962) therefore defines VPertas a “ screened interaction potential,” where pv,p acts as a “screening potential” in the region of fragment a. The eigenvalue equation for the localized functions is now written

and the solution is found by iteration. This procedure yields the whole basis of localized orbitals which have the form of linear combinations of standard H F orbitals, when repeated for all the fragments. The “modified H F ” equation (58) is obviously very difficult to solve, and only an approximate solution for the case of the LiH molecule is given by Adanis (1962), based on the simplest configuration of the molecule. However, the similarity between Adams’ method and the orthogonal plane wave concept (Cohen and Heine, 1961; Philips and Kleinman, 1959) may prove very

1 I9

LOCALIZED MOLECULAR ORBITALS

helpful in deriving acceptable solutions to the problem by approximating the complicated screening potentials. Thus, this method seems to be especially suitable for the solutions of the localized H F problem in crystals where the orthogonalized plane-wave method has been found to be successful (e.g., see Kunz, 1969). However, we note that the whole procedure amounts to solving an eigenvalue problem with a total potential which is very different from the standard Hartree-Fock potential (and including some approximate terms), so that the restriction to the H F manifold no longer seems to be justified and may represent a serious drawback.

B. METHODOF GILBERT Gilbert (1964) has demonstratedthepossibilityof introducing a" localization potential" into the H F operator, so providing a more general operator that defines the localized basis directly. In this very general framework, all the various localization criteria which have been used to obtain localized orbitals through transformations of H F orbitals, now take the form of functionals for which a variational principle is used to derive self-consistent equations for localized orbitals. The appropriate functional, constructed on the basis of some physical model, is expressed in terms of a basis set {cp,}, and is required to be stationary

SG

+ S'P~,,. ..

= G['P~

>

'Pn

+ ~'P,I - G[vI,

. . . v,I 3

= 0.

(59)

The condition that the basis functions belong to the H F set is imposed through the H F projection operator PF'Pi

= 'Pi

;

'Pi'PF

= 'Pi'

(60)

and a variational equation is obtained in the form

+

SG/6qi'(x) - yicpi(x) Y],(x) - JpF(x, x')Y],(x') dx' = 0.

(61)

In Eq. (61) SG/6qit(x) are functional derivatives, yi are the discrete Lagrangian parameters coming from the normalization constraints, and q i are the Lagrangian parameters coming from the constraint that the orbitals be Hartree-Fock orbitals. The final equation obtained by Gilbert is similar to the usual H F eigenvalue problem G q n . = C q .J Y ..J ~ (62) i

where

and {qj}are the standard H F orbitals.

120

H. Weinstein, R . Pauncz, and M. Cohen

It is clear that, since the new functions are linear combinations of the standard H F orbitals, a solution can be found for a modzjied eigenvalue problem, containing both the H F operator, F, and any “localization potential” A obtained from the functional G. Such a modified H F equation, proposed by Gilbert, is obtained by choosing (64)

A=-F+G

the whole equation then becoming

(F-

PFFP,

+

=Yiqi.

(65)

The eigenfunctions of the total potential will be identical with the solutions of Eq. (62) within the H F manifold since it is obvious that, within that space, [ F - p,FpF

+ G, GI = 0.

(66)

Gilbert (1964) has also shown that various criteria can be used to define the functionals G to yield the localized functions obtained from Eq. (65) which will be identical with the orbitals obtained from orthogonal transformations of the canonical H F orbitals if no approximations occur in the iterative procedure (see however, Edmiston and Ruedenberg, 1965). Thus, for example, the Edmiston and Ruedenberg criterion might be expressed by the functional R ,

while the Boys and Foster criterion could be based on B, B

=

3

i

] ] q i ( x ) q i * ( x ) l r- r’12cpi*(x’)qi(x’) dx dx‘.

These functionals thus serve to define the modified H F equation (65). Although no practical calculations based on Gilbert’s formulation have been carried through so far, it has led to a generalization of the earlier localization methods, and has proved useful in classifying the nature of the various localization methods in operator form. For example, in the case of Adams’ (1962)procedure, Gilbert shows that if the H F energyof the model fragment a is chosen in the construction of the functional G of Eq. (63), then this particular localization procedure is part of the general formulation. In this case, the physical interpretation of the localizing potential is clear. The operator is Hmode*=

F - PFP

+ PFUP

(69)

where Fis the H F operator of the total system, Fu the operator of the fragment

LOCALIZED MOLECULAR ORBITALS

121

a, and the potential produced by neighboring fragments to a is defined by

Va = F - F,.

(70)

The model Hamiltonian thus becomes : Hrnodcl=

Fa

+ Va - PVaP,

(71)

a form which is identical with the Hamiltonian obtained by Adams. The screening potential is now obtained directly from the physical model rather than from the condition of Hermiticity, thereby emphasizing the connection between Adams’ model and other approximations such as “ potential-well localized orbitals,” “ distorted orbitals,” and “energy localized functions.” However, Anderson (1968) finds the requirement of using the projector p awkward in practice, since it impedes the use of perturbation theory. He suggests instead the use of a non-Hermitean pseudopotential in a modified Hamiltonian yielding localized orbitals which are nonorthogonal (Anderson, 1969).

VI. Internal and External Localization Criteria In the fundamental ideas underlying the various localization procedures discussed in the preceding sections, a common feature will be observed, which makes them all belong to the “group of intrinsic localization procedures within the H F manifold.” The classification of localization methods based on transformations of previously obtained delocalized molecular orbitals into intrinsic” and external ” transformations is due to Ruedenberg (1965) and is based on the form of the criterion used to derive the transformation leading to the localized orbitals. When the criterion arises from an extremum condition for a quantity which does not explicitly require the localization in a particular region of the molecular geometrical space, the procedure is called “intrinsic.” The resulting physical localization is then a consequence of a condition imposed on some numerical term which does not necessarily have any physical meaning, but reveals a property of the functional space in which the term is calculated. The methods of Edmiston and Ruedenberg and of Boys and Foster are thus intrinsic in character, but as we have emphasized earlier, their criteria appear nevertheless to have some physical significance that is, however, not directly related to the localization regions. The orbitals localized in chemically definable regions which result from the transformation of functions defined within the H F space and which have maximal self-interaction, naturally display implicit properties of the approximation space. On the other hand, external criteria require specific geometrical features of the electronic density in accordance with some a priori chosen model. These features may be obtained by requiring the contributions of certain atomic “

“

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H. Weinstein, R. Pauncz, and M . Cohen

orbital from given atoms to be minimal in an LCAO expansion, or by maximizing the overlap with certain wave functions in particular regions. In Gilbert’s (1964) general formulation, there is no requirement that the localizing potential be defined by means of an intrinsic rather than an external criterion. Thus, since the proof of localizability is completely general, the localization in molecular regions defined by a certain physical model might be imposed directly as a transformation criterion. It is therefore of considerable interest to compare the results obtained from the “intrinsic” procedures described so far, with the results of some “external” procedures. In the following sections we examine a number of external localization procedures that still retain the restriction that the transformation is to be performed within the H F manifold. Any external localization procedure will yield a picture of the total electronic density, partitioned into more or less well separated parts, chosen to fit some chosen model of the electronic system. Such a model will be based on an empirical (or speculative) description of the molecular system which corresponds to well-established concepts in physics and chemistry. Confining the localization procedure to some given functional space, such as the H F manifold, expresses the search for a basis set that both fits the chosen model and retains the physical properties of the configuration space. In the case of the H F manifold, the physical conditions consist of minimizing the electronic energy for a model H F Hamiltonian. However, it is clear that complete identification of the basis functions for a given approximate description of the system with the functions generated by another, unconnected, model will not generally prove feasible. The optimal achievement of an external localization procedure will therefore be to make complete use of all those inherent properties of a given functional space that are related to the physical features of the model. This leads to the conclusion that intrinsic and external localization methods leading to “best possible localization ” within a certain space, should yield identical localized orbitals. However, the definition of the externally “best localized” functions is often stated quite arbitrarily, so that the results of external and intrinsic localization procedures are far from identical. A comparison of results obtained from an intrinsic procedure with those from an external one would therefore be of special interest in making precise such concepts as “degree of localization” and “best localized set.”

VII. The Method of Magnasco and Perico The method of Magnasco and Perico (1968) is based on an external localization procedure confined to the H F manifold, and has been applied to a series of LCAO-SCF orbital sets. The transformation criterion which leads

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LOCALIZED MOLECULAR ORBITALS

to the localized orbital set is based on the partition of the total electronic population according to Mulliken’s( 1955) method. In this“ population model,” the density described by a molecular orbital expressed as a LCAO function rk

is given by

where the summations are performed over all atomic orbitals represented by r and s on the atoms k and I , respectively. Srk,,is the overlap integral and N j represents the number of electrons in the j t h molecular orbital. The total orbital density is further decomposed (Mulliken, 1955) into a “partial overlap population,” defined as

and a

“

net atomic population,” defined as

n(j,

rk)

=Nj

ICjrk I

(75)

Unlike partial populations defined by total molecular orbital densities, the n(j, rk) and n(j, r k s L )populations are not invariant quantities with respect to orthogonal transformations which transform the standard SCF wave functions into localized orbitals. Magnasco and Perico (1967) propose the maximization of the numerical values of certain “ localization functions ” constructed from n(j, r k s ] ) and n(j, rk), as the criterion for the localizing orthogonal transformations. The partial populations are related to the localization feature through their interpretation, as given by Mulliken (1955). According to this interpretation, the electronic charge distribution (population) on an atom r is defined as p r

=

c [c I Nq

4

rk

I + 1 crk c r , Srkr,

crk 2

rkrI

I

(76)

where Nq represents the number of electrons in the qth molecular orbital and the summations are taken over all the atomic orbitals centered on atom r. Equation (76) thus represents a truncation of the sums in Eq. (73) to yield a charge density localized in the region of a particular atom. Similar considerations will lead to truncation of the sums in Eq. (73) when thejth molecular orbital generates an electronic density in any defined region of the system. For example, for a molecular orbital defined as a wave function describing an electronic density centered on a particular atom (such as an inner shell density) the partial population must be of the n(j, rk) type, where the summation over atomic orbitals incudes only those atomic functions which are supposed

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H . Weinstein, H. Pauncz, and M . Cohen

a priori to contribute to the specific type of density on that particular atom. Similarly, the “bond region density between two atoms can be regarded as being generated by a partial density which contains an n(j, rks,)-type population with the summations over the atomic orbitals truncated so as to contain only those atomic orbitals involved in the bond. The model used by Magnasco and Perico (1967) thus consists of the definition of inner-shell ( i ) , lone pair (1), and bond (b) molecular orbitals within the basis functional set and the designation of the particular atomic orbitals involved in the generation of the defined densities. From such a model, a set of “local orbital populations P i is constructed, depending on the coefficients of the atomic orbitals in the LCAO molecular orbital expansion. Since local orbital populations are affected by orthogonal transformations, their numerical values can be used to define localizing orthogonal transformations. The “local orbital populations” P i thus become “localization functions,” whose values depend on the LCAO expansion coefficients. Magnasco and Perico define three different forms of localization functions based on the Mulliken population as follows: ”

”

(1) A bond localization function 2Pi = 2 =

2 1 cpic,isp”

p c r i,Cri

n(i,

vAi)+ n ( i ,

~

”

i

+ n ( i , vAi, )

~ ” i )

(77)

where Ti = ( V A i V , B i )is the set of indices of the valence atomic orbitals of the atoms A and B, and the partial populations are defined as

and

( 2 ) An inner shell localization function 2Pi = n ( i , K A i )

(80)

where K A zrepresents the set of inner-shell atomic orbitals centered on atom A . ( 3 ) A lone pair localization function

2Pi = n(i, VAi).

(81)

In order to obtain a set of uniformly localized orbitals, defined as the particular set that yields the optimum localization of all orbitals simultaneously,

LOCALIZED MOLECULAR ORBITALS

125

Magnasco and Perico choose the orthogonal transformation which maximizes the sum (over all molecular orbitals) of the local orbital populations (the localization functions) : 2P = 2 P i . (82)

c i

The computational procedure consists of a series of 2 x 2 orthogonal transformations by means of a rotation matrix used also in the Edminston and Ruedenberg procedure (see Section 111,B above). The optimal transformation in each case is defined by the angle 9 which is obtained from the maximization criterion imposed on the sum 2P of Eq. (82),through the requirement that the total value be stationary

dPld8

=0

It is clear from the definition of the transformation criterion, that the degree and character of the localization obtained by this method, is strongly dependent on the geometrical localization properties of the Mulliken partial electronic densities. While the use of truncated atomic orbital subsets for the partial densities does not always represent a drawback (due to the possibility of using large basis sets with no obvious inconvenience), nevertheless the definition of the inner shell, lone pair, and bond densities remains equivocal. This difficulty becomes clear i n the analysis of results for molecules containing lone pair densities as well as bonds and inner shells, and is expressed by the invariance in the localization functions for orthogonal transformations between lone pairs and bonds on the same atom. This occurs because the localization criterion does not express uniquely the regions in which the specific densities must be localized, but instead obtains the geometrical features implicit in the properties of the s- and p-orbital mixture. The corresponding molecular orbitals are therefore not defined uniquely, and the sum in Eq. (82) remains unaltered by an orthogonal transformation performed within a subset which contains functions of the same type. This lack of uniqueness affects two types of molecular orbitals in particular, since the localization functions of lone pair type depend on essentially the same set of “valence ” atomic orbitals as the bond type partial density. This is emphasized by comparisons with results obtained from other methods (Magnasco and Perico, 1968 ; Trindle and Sinanoglu, 1968). Trindle and Sinanoilu found that, for a series of molecules for which a given step of the iteration procedure yielded equivalent orbitals, these molecular orbitals remained unaffected by any further transformation of the Magnasco and Perico type, and the residual delocalization could not be cancelled by this method. Other work on partial electronic density definitions has shown that the use of Mulliken population functions does not always lead to meaningful

126

H . Weinstein, R . Paimcz, and M . Coken

results (see Cusachs and Politzer, 1968). The equal overlap charges attributed to any two atoms connected by a bond seem as unrealistic as the negative electronic population on an atom, which results in some cases. More seriously, it has been found for some simple examples (Politzer and Cusachs, 1968) that the effect of adding or removing basis atomic functions (even those showing clear geometric properties) does not always have the expected effect on local densities as interpreted from the electronic population methods. However, we wish to emphasize that the method of Magnasco and Perico is one of the simplest localization procedures related to a well-defined functional space of good energetic quality, particularly for cases in which the abovementioned difficulties can be avoided by some additional criterion which distinguishes the different equivalent bonds and lone pairs.

VIII. The Method of Peters The first method proposed by Peters (1963, 1966) consists of iterative orthogonal transformations guided by hybridization considerations. The total N-electron function is approximated by a single Slater determinant for which the energy is minimized by a SCF procedure. The delocalized molecular functions thus represent an LCAO-SCF set of the form c p . = p I Jy .

LJ

(84)

j

where the {xj} are atomic orbitals. The molecular orbital set resulting from Peters' procedure will contain localized functions which have the form of hybrid orbitals defined as lone-pairs and two-electron bonds. Typical forms of these hybrid orbitals are, for a lone pair i on atom a,

i.i(a) = ci(2so) 12~0)+ ci(2pao) I ~ P c J ~ )

(85)

and for a molecular orbital j representing a two-electron bond between atoms a and b, Pj(a; 6)

=

+ cj(2pao)I2P0,) + cj(2sb) I2sb) f cj(2pab) I 2pab) Cj(2S") I 2 S J

(86)

and the localization transformation will simply bring the standard LCAOSCF-MO { q , }to these specified forms. At each step the form of these hybrid orbitals is ,compared with the LCAO expressions of the available molecular orbitals and the orthogonal transformation matrix transforming the delocalized molecular orbitals into hybrid localized orbitals is constructed. Clearly, the properties of the appropriate transformation matrix are specific to the molecule considered as well as to the symmetry properties of the starting

LOCALIZED MOLECULAR ORBlTALS

127

functional set. The orthonormality conditions for the transformation matrix, together with the requirement that the localized functions should contain contributions only from the specified atoms involved in their definition, are used to define a set of linear equations, whose solutions yield the elements of the transformation matrix. As can be seen from a detailed example (Peters, 1963), the matrix cannot always meet all the requirements imposed, and some of them (which are arbitrarily considered less important) have to be discarded. Since it does not seem possible to define an orthogonal transformation that will locai‘ze all the molecular orbitals completely, Peters’ suggestion Is to accept the “almost localized ” molecular orbitals and simply delete the imperfections. By eliminating the contributions of the atoms not involved and thus yielding complete localization, this procedure leads to a set of orbitals which is slightly nonorthogonal, and is presumably not optimal from the point of view of minimum energy. More seriously, since not all secondary contributions can be removed simultaneously, the resulting “ localized orbitals” are not uniquely defined, and Peters (1963) gives four different alternative choices of spurious contributions to be removed. A new method which yields localized orbitals directly from a HF type eigenvalue problem has recently been proposed by Peters (1969a). The tdimensional functional space spanned by the atomic orbitals is divided into two orthogonal subspaces: one is spanned by the n - 1 localized molecular orbitals which are assumed to be known, the other of dimension m = t - (n - 1) is the orthogonal complement. The (n - 1)-dimensional subspace is spanned by all the occupied localized molecular orbitals except the one which one wants to determine. This is called the “fixed space” while the orthogonal complement is the “free space.” One can choose a set of basis functions in this space, and they can be written in the usual LCAO form 1

Peters shows that the total energy of the molecule will be minimized if one sets up secular equations over the Hartree-Fock operator using the u functions for the matrix. In general, the localized molecular orbitals satisfy the nondiagonal form of the Hartree-Fock equation

Here the sum is over the occupied orbitals only. The equation can be rewritten i n the form

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H . Weinstein, R. Pauncz, and M . Cohen

One can utilize the fact that the ith localized molecular orbital is expressed in terms of the u’s

Substituting (90) into (89), multiplying from the left by up and integrating, one arrives at the set of equations

where Fpq is the matrix element of the F operator over up and u q , Spq is the analogous overlap integral. Since by definition of the set, ( u pI $,) vanishes, Eq. (91) reduces to

2 Cqi(Fpq-

Eii

Spq)= 0.

q= 1

The secular equation (92) to be solved here is seen to be of smaller dimension than the standard HF equation, and one of the eigenfunctions obtained is the localized molecular orbital which is sought by the procedure. In the next step this orbital is included in the fixed space and one determines successively the other localized orbitals. Observe that the new “free space” is different from the one used in the preceding step. This procedure is continued until one reaches self-consistency. Besides being somewhat complicated to use and showing slow convergence, Peters’ procedure raises a number of fundamental questions, most of which have also been pointed out by its author (1969b). Thus, it has not yet been demonstrated that the orthogonality and localization conditions are fulfilled simultaneously by the set { t j i } . A definitive answer to these questions may be given by future calculations with the method.

IX. Molecular Orbitals Determined from Localization Models A. THEMETHOD OF WEINSTEIN

AND

PAUNCZ

I. Basis Gf the Method The localization methods proposed by Magnasco and Perico (1967) and Peters (1963) discussed in the previous sections, emphasize the close relationship between the localized description of molecular charge density and the concept of hybridization. The hybridization approach is based on the assumption that an atom retains its identity even when involved in a molecular bond and is associated with the slight changes occurring in its electronic density due to some local bonds in which it is involved. These changes ensure the

LOCALIZED MOLECULAR ORBITALS

129

maximization of electronic charge distribution overlap in the bond regions. It is clear, therefore, that all methods that begin with hybridization models are based on a maximum overlap concept (Murrell, 1960; Del Re, 1963; RandiC and MaksiC, 1965), which in itself bears a local significance. The electronic population generated by a given molecular orbital concentrated in the bond region and expressed by means of an LCAO expansion is therefore the link between the hybridization considerations and the breakdown of the total electronic density into localized contributions. Thus, Polak (1969, 1970) has used the maximization of the projection of localized bond orbitals onto the occupied SCF space as a criterion for constructing hybrid orbitals. The basic idea of the Weinstein and Pauncz (1968) method is to construct the electronic density of a given molecule in a localized representation using the maximal overlap density criterion related to the hybridization scheme and starting only from a structural model of the system in terms of the atomic orbital basis and the molecular geometry. Such a procedure is expected to bridge the gap between related concepts of hybridization and localization. The feasibility of determining a molecular orbital basis which yields such a representation of the molecule is suggested by the results of localization transformations within functional spaces defined by variational criteria. The localizability of the density matrix representation in new defined localized molecular orbital bases obtained by all procedures discussed so far, suggests the possibility of determining a molecular orbital basis set from a criterion based on localization models alone. Since an LCAO expansion of the MO is envisaged, and the procedure is to be free of intermediate energy calculations, the criterion of maximum overlap is used with the localization based on population analysis models along the lines of the method of Magnasco and Perico (1968). The localisation procedure consists of maximizing certain functions of the LCAO expansion coefficients which are related to the electronic charge distributions in certain well-defined regions of the molecule. These functions, based on Mulliken’s population analysis have the general form

where r:” and r:2) represent sets of indices of certain atomic orbitals which are expected to make the main contributions to the electronic density obtained from a given molecular orbital. The atomic orbitals included in the r sets are chosen according to a hybridization scheme to fit a given model of the electronic charge density distribution, partitioned into localized contributions from defined molecular orbitals. This is therefore obviously an external localization procedure, based on results obtained from intrinsic methods.

130

H . Weinstein, R. Pauncz, and M . Cohen

The localization function for an inner-shell type of molecular orbital

therefore includes contributions from atomic orbitals centered on atom A and represented by the set of indices I(Aj).A molecular bond localization function has the general form

where r:.’)and rj2)have been replaced by V ( B j ) and V ( C j ) , representing the groups of valence atomic orbitals on atoms B and C, respectively. The atomic orbitals contributing to the lone pair localization function

are chosen among the valence atomic orbitals on atom D , represented by V(Dj). Defining the localization transformations by means of changes in the basis functions which lead to the maximization of the localization function value of each t,ki in the basis separately, a localization method is obtained which yields a new functional space having localized properties. It is of interest to note that the functional set that results from this localization process, also has a greatly increased overlap with the occupied ” LCAO-SCF space spanned by the occupied SCF molecular orbitals as expressed in the atomic orbital basis. “

2. Calculation Procedure Initially, the localization procedure starts with a set of maximum overlap orbitals defined by Lykos and Schmeising (1961). This set is partitioned into two complementary subsets, one consisting of “ occupied orbitals, the other of “virtual” orbitals, and orthogonal transformations are performed between all possible pairs of orbitals chosen one from each subset. The Nelectron wave function is chosen to be represented by a single Slater determinant and a minimal set of atomic orbitals is used. The transformations obtained by the application of an orthogonal 2 x 2 rotation matrix on the chosen pair is subject to the condition of maximization of the localization functions. The angle of rotation is obtained from the stationary value of the localization function ”

dLFj(9) =o a3 in the form

(97)

LOCALIZED MOLECULAR ORBITALS

131

for a bond orbital, and

for an inner shell or a lone pair. This procedure has yielded some interesting results for a series of hydride molecules (LiH, BH, BH,), emphasizing the connection between the chemical concepts on which it is based and the results of the molecular orbital calculations in their localized representation. The success of any localization method to define a good molecular orbital set depends directly on (a) the quality of the localization procedure involved as expressed by the physical basis of the localization criterion ; (b) the degree of localization achieved; and (c) the actual localization properties inherent in the functional space serving as model. A modified procedure seeks to exploit fully the power of the criterion based on the localization functions defined above. The new procedure uses a numerical optimization of the values obtained for the localization functions with a given basis set through a conjugate gradient procedure based on least squares optimization of the function (Weinstein and Pauncz, 1971). The results obtained with this method are much improved by comparison with the ones obtained by the earlier procedure because the search for maximal localization occurs within the whole functional space and is no longer constrained to the direction of convergence imposed by the orthogonal transformation. The quality of the MO sets obtained for each given molecule is investigated by its degree of overlap with the occupied LCAO-SCF space (expressed in the same A 0 basis), the energy values and the total electronic density contours obtained with the single Slater determinant constructed from these functions. TABLE 11 COMPARISON OF ENERGY VALUES" FROM LOCALIZATION METHOD^ WITH SCF RESULTS ~~

~~

~~

Molecule

Ei,,,,

EscF

LiH BH BHB

-7.87282 -24.85993 -26.31565

-7.96996' -25.0756' -26.337297d

a

Overlap with SCF 0.99158 0.98934 0.99637

Energy values are in atomic units (a.u.). Weinstein and Pauncz (1968, 1971). R a n d (1960). Pipano (1967).

The numerical results obtained for LiH, BH, and BH, are listed in Table 11, and the density contours are displayed in Figs. 3a-3c.

132

H . Weinstein, R. Pauncz, and M . Cohen

FIG.3a. Total electronic density contour for molecular orbital set obtained from localization method for molecules LiH. The densities refer to the plane containing all the nuclei in the molecules.

FIG. 3b. Same as Fig. 3a for molecule BH.

LOCALIZED MOLECULAR ORBITALS

133

FIG.3c. Same as Fig. 3a for molecule BH3

Tt is interesting to note that the partial overlap population values and innershell partial populations obtained as optimal values of the corresponding localization functions, are very close to the bond and inner-shell populations obtained from the localization of LCAO-SCF molecular orbitals (Switkes et al., 1970b) for B-H bonds in some other boron hydride molecules.

B. METHODOF LETCHER AND DUNNING Another method in which the direct solution of the Schrodinger equation of the system is bypassed by the determination of molecular orbitals through a localization procedure has been proposed by Letcher and Dunning (1968). They suggested that a localized representation of a molecular system can be constructed using a “diatomic bond” model. A criterion for the definition of such a representation is considered to consist in the condition that all the atomic orbitals involved in a given chemical bond should possess the same

134

H. Weinstein, R. Pauncz, and M . Cohen

value of angular momentum about the bond axis as they would in a diatomic molecule. An atomic basis determined by this criterion is constructed and a localized molecular orbital representing a bond between centers A and B is then written in the usual expansion form $i

= c(i, sA)IsA)

+ c(i, pA)lpA) + c(i, dA)ldA) + similar terms on B.

(100)

For the determination of the expansion coefficients a parameter fitting procedure is proposed in which parameters such as the “bond polarity coefficient ’’ defined by hi,=

(c(i,sA)I2+ ( c ( i , p , ) l 2 +

...

(101)

are chosen from electronegativity considerations or simple symmetry, and geometrical considerations are used in constructing the functions. Since the procedure does not contain any energy minimization, the manner in which the parameters are chosen is very important. The localized representation thus yields a matrix C of coefficients of the expansion in the chosen atomic basis. Using the properties of Lowdin’s (1950) symmetrical orthogonalization method, the authors propose to transform the localized orbitals into a set of molecular orbitals expanded in a regular Slater-type basis. This transformation will have the form C’ = ca-’’2 (102) where C’ represent the new coefficients, C are the localized coefficients of the orbitals expanded in the old basis, and A is the overlap matrix of the new basis. It is important to note that although Letcher and Dunning do not prove the possibility of retaining the localization properties of the molecular orbitals and the adequate values of the physical constants used as parameters after the transformation, their method is nevertheless based on the assumption that this is actually so. In fact, the nonunitary transformation to which the C matrix is subjected almost certainly delocalizes the orbitals. However, for a series of molecules the procedure has yielded interesting results (Letcher and Dunning, 1968) and has also led to a generalization which enabled Unland et al. (1969) to deal with a more complicated molecule.

X. Localized Orbitals in Expansion Methods A. METHODOF CERTAIN

AND

HIRSCHFELDER

All the localization methods presented so far have been confined (in practice) to the single determinant approximation, which was chosen because of its special properties which simplify the expression of the density operator. In

LOCALIZED MOLECULAR ORBITALS

135

such a functional space the density operator is diagonal, and its form is invariant to any orthogonal transformation between the occupied orbitals. The extension of these localization methods to more accurate approximation is clearly of considerable interest. This generalization is made possible by the definition of the set of natural spin orbitals, which generates a diagonal representation of the first-order density matrix within any convenient functional space. These orbitals also provide the fastest convergence in a configuration interaction treatment (Lowdin, 1955a). Since the one-particle density corresponding to a pure state is obtained by averaging the N-particle density operator, the eigenstates of the one-particle reduced density matrix are connected both with the single particle degree of freedom of the N-particle system and with the overall behavior of the complete system. They may be interpreted therefore as the single particle states of the N-particle system, and it is clear that the expansion of p l ( x l ’ l x l )given in Eq. (14) by the HF basis orbitals is a particular case of the natural spin orbital representation, in which the molecular orbitals are identical with the eigenfunctions of the reduced density matrix p l , and the eigenvalues are all equal to unity. In the more general case, the occupation numbers fulfill the conditions

In this case however, the form of the density operator is no longer invariant under an orthogonal transformation but it may be invariant under some nonorthogonal transformation T, which transforms the natural spin orbitals {xk} into a new set {Gi}related to the old one by

where wz,?/’is a normalization constant. In this new basis the density operator still has the diagonal form

The extension of all the localization methods used earlier for the H F manifold, can therefore be obtained (in principle) by means of a localization transformation T which preserves the diagonal form of the density operator, and defines a set having the desired localization properties. One possible procedure, based on such a general approach, is presented by Certain and Hirschfelder (1968) and used for the definition of localized orbitals in the simple case of H 3 + . These authors find it necessary to use a generalization of the Edmiston and Ruedenberg (1963) localization criterion for the

136

H . Weinstein, R. Pauncz, and M . Cohen

final definition of the actual T matrix, thus widening the scope of the generalized analysis. This criterion is to maximize the component u i i of the total self-energy of the density p ( r ) in the expression

-s-

1 1 p(r>p(r') dr dr' = - C uii + uij. 2 Ir - r'I 2 i i<j

In terms of the orbitals resulting from the transformation of Eq. (104), uii is given as

and the maximization is clearly dependent on the LCAO coefficients which are affected by the transformation T. In order to preserve the diagonal form of the density operator, the transformation must be confined to the space of natural spin orbitals. This is achieved by relating the form of the localized orbitals to the form of the natural spin orbitals obtained from a multiconfiguration wave function. Using a truncated (3 term) expansion of the 2-electron wave function of H3+ in terms of the natural spin orbitals calculated from a 12 configuration function, the localized orbitals are found by defining the transformation matrix T which yields a linear combination of natural spin orbitals having the symmetry properties of equivalent orbitals for the system. Since the matrix is not uniquely defined by the symmetry requirements, the " maximization of self-energy " completes the specification. Another variant of the generalized localization procedure proposed by Certain and Hirschfelder is based on the requirement that the complete N-electron wave function should have the form of a " model function. " Such a model i s defined by the form of the expansion in terms of the natural orbitals basis. As an example of this procedure (Certain and Hirschfelder, 1968) the model function for H3+ has the known valence-bond form

and to obtain the required expansion, one simply has to rearrange the various terms in the natural spin orbital expansion of the total wave function so as to give it the required form. It is quite clear that the two definitions yield different sets of localized orbitals which thus have different localization properties related to the initial form of the total wave functions. It seems likely that the method of Certain and Hirschfelder will prove difficult to apply to many-electron molecules more complicated than the simple H 3 + system, since the definition of the natural spin orbitals becomes very cumbersome, and the simultaneous formulation of

137

LOCALIZED MOLECULAR ORBITALS

the form and symmetry requirements will be much more difficult in such cases. However, the direct relationship between the localized orbitals and the generating natural expansion should make the method of Certain and Hirschfelder very useful for comparison of various approximate wave functions and different localization procedures. B. MULTICONFIGURATION EXPANSIONS In a number of recent calculations, localized orbitals obtained by a variety of different procedures have been used to construct functional basis sets suitable for multiconfiguration treatments of the correlation problem either directly or equivalently through the Rayleigh-Schrodinger or Brillouin-Wigner perturbation theories. The primary aim of all these calculations has been to recover as large as possible a portion of the electronic correlation energy. In a full configuration interaction treatment the choice of the orbitals is clearly of no significance. On the other hand, rapid convergence of the perturbation expansions for the energy has been regarded as an essential criterion for success of approximations based on diferent sets of basis functions. When only the pair correlation corrections to the energy are sought, it appears that the result (essentially correct through second order) is highly sensitive to the form of the SCF orbitals chosen for the zero-order determinant (Bender and Davidson, 1969), although the work of Amos et al. (1969) and Amos and Musher (1971) suggests that some of the discrepancy may be due to the awkward coupling terms which complicate the solution of the equations for the pair correlations whenever localized rather than canonical SCF orbitals are used. Approximate solutions, which allow the equations to become decoupled by neglecting certain contributions from distant ” orbitals or geminals, have yielded satisfactory results in a number of calculations (Kapuy, 1958, 1961, 1968). The calculational procedure is based on a separated pair expansion, with the geminals subjected to conditions of strong orthogonality. Kapuy (1958, 1968) shows that the necessary orthogonality condition is satisfied most easily when the geminals are localized. A careful comparsion would be highly instructive. Systematic use of Rayleigh-Schrodinger perturbation theory through higher orders, using “excited configurations” built up from hybrid “bonding and antibonding localized orbitals yields a number of interesting results, not all of them encouraging (Diner et al., 1969a,b, Malrieu et al., 1969; Jordan et al., 1969). It turns out that most of the correlation energy can be recovered, but the perturbation sums do not converge uniformly, there being significant cancellations between the energy improvements in second and third orders. “

”

138

H . Weinstein, R. Pauncz, and M . Cohen

(Diner ct al., 1969b). Furthermore, the best bonding orbitals are not necessarily those built up from hybrids which optimize the overlap in the bonds in the zero-order (fully localized) determinant (Jordan et al., 1969). On the other hand the exclusive and oscillator orbitals proposed by Foster and Boys have been shown to yield rapid convergence of both RayleighSchrodinger and Brillouin-Wigner perturbation theories in some recent studies of OF, , NO2-, and CN- (Bonaccorsi et al., 1969). Some very recent multiconfiguration calculations (Levy, 1970) have served to reemphasize the intuitively obvious fact that physical localization properties must result naturally as the number of configurations is increased (or a perturbation expansion is carried through to sufficiently high order) whether or not the basis orbitals satisfy any localization criteria.

XI. Concluding Remarks In the preceding sections, we have seen that the abstract notion of localization cannot be deduced from first principles (see, however, Metiu, 1970), and this explains the great variety of procedures that have been proposed for determining localized orbitals. Our review of the various methods has sought to emphasize the underlying principles and the connections between them by means of a few illustrative examples. As we have seen, the localized basis sets that result from different procedures are suitable for widely different purposes, such as (1) linking chemical concepts with orbital descriptions (2) improving the starting basis set used in expansion methods, (3) analyzing the correlation problem, and (4) defining new approximation models. Since each procedure is intended to achieve one particular end, we cannot expect all these localization features to be optimal for orbitals resulting from different methods. In spite of this, there will often be a high degree of resemblance between orbitals obtained by different methods (LCvy et al., 1970). REFERENCES Adams, W. H. (1961). J . Chem. Phys. 37,89. Adams, W. H. (1962). J . Chem. Phys. 38, 2009. Adams, W. H. (1965). J . Chem. Phys. 42,4030. Ahlrichs, R., Kutzelnigg, W., and Bingel, W. A. (1966). Theor. Chim. Acta 5, 289. Amos, A. T., and Musher, J. I . (1971). J . Chem. Phys. 54, 2380. Amos, A. T., Musher, J. I., and Roberts, H. G . F. (1969). Chem. Phys. Lett. 4, 93. Anderson, P. W. (1968). Phys. Rev. Lett. 21, 13. Anderson, P. W. (1969). Phys. Rev. 181, 25. Bender, C. F., and Davidson, E. R. (1966). J . Chem. Phys. 70, 2675. Bender, C. F., and Davidson, E. R. (1969). Chem. Phys. Lett. 3, 33. Bonaccorsi, R., Petrongolo, C., Scrocco, E., and Tomassi, J. (1968). J. Chem. Phys. 48, 1500.

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Bonaccorsi, R., Petrongolo, C., Scrocco, E., and Tomassi, J. (1969). Theor. Chim. Acta 15, 332. Born, M., and Oppenheimer, R. (1927). Ann. Phys. (Leipzig), 84, 571. Boys, S. F. (1960). Rev. Mod. Phys. 32, 296. Boys, S. F. (1966). In “Quantum Theory of Atom, Molecules, and the Solid State” (P.-0. Lowdin, ed.), p. 253, Academic Press, New York. Certain, P. R., and Hirschfelder, J. 0. (1968). Chem. Phys. Lett. 2, 274. Cohen, M. H., and Heine, V. (1961). Phys. Rev. 122, 1821. Cook, D. B., Hollis, P. C., and McWeeny, R. (1967). Mol. Phys. 13, 553. Coulson, C. A. (1942). Trans. Faraday SOC.38, 433. Cusachs, L. C., and Politzer, P. (1968). Chem. Phys. Lett. 1, 529. Del Re, G. (1963). Theor. Chim. Acta 1, 188. Diner, S., Malrieu, J. P., and Claverie, P. (1969a). Theor. Chim. Acta 13, 1. Diner, S., Malrieu, J. P., Jordan, F., and Gilbert, M. (1969b). Theor. Chim. Acta 15, 100. Dirac, P. A. M. (1931). Proc. Cambridge Phil. SOC.27, 240. Edmiston, C., and Ruedenberg, K. (1963). Rev. Mod. Phys. 35, 457. Edmiston, C., and Ruedenberg, K. (1965). J . Chem. Phys. 43, S, 97. Edmiston, C., and Ruedenberg, K. (1966). In “Quantum Theory of Atoms, Molecules, and the Solid State” (P.-0. Lowdin, ed.), p. 263. Academic Press, New York. Fock, V. (1930). Z . Phys. 61, 126. Foster, J. M., and Boys, S. F. (1960). Rev. Mod. Phys. 32, 300. Gilbert, T. L. (1964). In “ Molecular Orbitals in Chemistry, Physics, and Biology” (P.-0. Lowdin and B. Pullman, eds.), p. 405, Academic Press, New York. Jordan, F., Gilbert, M., Malrieu, J. P., and Pincelli, V. (1969). Theor. Chim. Acta 15, 211. Kaldor, U. (1967). J. Chem. Phys. 46, 1981. Kapuy, E. (1958). Acta Phys. 9, 237. Kapuy, E. (1961). Acta Phys. 13, 461. Kapuy, E. (1966). J. Chem. Phys. 44,956. Kapuy, E. (1968). Theor. Chim. Acta 12, 398. Kiang, H. (1967). J . Math. Phys. 8, 450. Kunz, A. B. (1969). Phys. Status Solidi 36, 301. Kutzelnigg, W. (1963). Theor. Chim. Acta 1, 327. Lennard-Jones, J. E. (1949). Proc. Roy. SOC.,Ser. A 198, 1 . Lennard-Jones, J. E., and Pople, J. A. (1950). Proc. Roy. Soc., Ser. A 202, 166. Lennard-Jones, J. E., and Pople, J. A. (1951). Proc. Roy. Soc., Ser. A 210, 190. Letcher, J. H., and Dunning, T. H. (1968). J. Chem. Phys. 48, 4538. Ltvy, B., Millie, P., Lehn, J. M., and Munsch, B. (1970). Theor. Chim. Acta 18; 143. Ltvy, R. (1970). Int. J . Quantum Chem. 4, 297. Lowdin, P.-0. (1950). J. Chem. Phys. 18, 365. Lowdin, P.-0. (1955a). Phys. Rev. 97, 1474. Lowdin, P.-0. (1955b). Phys. Rev. 97, 1490. Lowdin, P.-0. (1959). Aduan. Chem. Phys. 2, 207. Lykos, P. G., and Schmeising, H. N. (1961). J . Chem. Phys. 35, 288. Magnasco, V., and Perico, A. (1967). J . Chem. Phys. 47, 971. Magnasco, V., and Perico, A. (1968). J . Chem. Phys. 48, 800. Malrieu, J. P., Claverie, P., and Diner, S. (1969). Theor. Chim. Acta 13, 18. Mattheiss, L. F. (1961). Phys. Rev. 123, 1209. Metiu, H. I. (1970). Phys. Reu. A 2, 13. Mulliken, R. S. (1955). J. Chem. Phys. 23, 1833. Murrell, J. M. (1960). J . Chem. Phys. 32, 767.

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Nesbet, R. (1965). Advan. Chem. Phys. 9, 321. Peters, D. (1963). J. Chem. SOC. London p. 2003, 2015, 4017. Peters, D. (1966). J. Chem. SOC. A p. 644, 652, 656. Peters, D. (1969a). J. Chem. Phys. 51, 1559. Peters, D. (1969b). J. Chem. Phys. 51, 1566. Philips, J. C., and Kleinman, L. (1959). Phys. Rev. 116, 287. Pipano, A. (1967). Ph.D. Thesis, Technion-Israel Inst. of Technol., Haifa. Pitzer, R. M. (1964).J. Chem. Phys. 41, 2216. Polak, R. (1969). Theor. Chim. Acta 14, 163. Polak, R. (1970). Znt. J . Quantum Chem. 4, 271. Politzer, P., and Cusachs, L. C. (1968). Chem. Phys. Lett. 2, 1. Primas, H. (1965). In “ Modern Quantum Chemistry” (0. Sinanoglu, ed.), Part 2, p. 45. Academic Press, New York. RandiC, M., and MaksiC, Z. (1965). Theor. Chim. Acta 3, 59. Ransil, B. J. (1960). Rev. Mod. Phys. 32, 239. Roothaan, C. C. J. (1951). Rev. Mod. Phys. 23, 69. Ruedenberg, K. (1965). In “Modern Quantum Chemistry” (0. Sinanoglu, ed.), Part 1, p. 85. Academic Press, New York. Sinanoglu, O., and Skutnik, B. (1968). Chem. Phys. Lett. 1, 699. Switkes, E., Stevens, R. M., Lipscomb, W. N., and Newton, M. D. (1969). J . Chem. Phys. 51, 2085. Switkes, E., Epstein, I. R., Tossel, J. A., Stevens, R. M., and Lipscomb, W. N. (1970a). J. Amer. Chem. SOC.92, 3837. 92, 3847. Switkes, E., Lipscomb, W. N., and Newton, M. D. (1970b). J. Amer. Chem. SOC. Taylor, W. J. (1968). J. Chem. Phys. 48, 2385. Trindle, C., and Sinanoglu, 0. (1968). J. Chem. Phys. 49, 65. Unland, M. L., Dunning, T. H., and Van Wazer, J. R. (1969). J . Chem. Phys. 50, 3208. Weinstein, H., and Pauncz, R. (1968). Symp. Furuday SOC.2,23. Weinstein, H., and Pauncz, R. (1971). To be published.

GENERAL THEORY OF SPIN-COUPLED WAVE FUNCTIONS FOR ATOMS AND MOLECULES J. GERRATT Department of Theoretical Chemistry, University of Bristol Bristol, England

I. Introduction .................................................. 11. Properties of the Exact Electronic Eigenfunction .................... 111. Construction of the Spin Functions .............................. IV. The Spin-Coupled Wave Functions ..................... V. Calculation of Matrix Elements of the Hamiltonian .................. VI . The Orbital Equations ..................... VII . Symmetry Properties of Functions . . . . . . . . . . VIII. The Hund's Rule Coupling ...................................... IX. The General Recoupling Problem and Bonding in Molecules . . . . . . . . X. Conclusions .................................................. Note AddedinProof ............................................ Appendix A. Proof of Relations (59)-(62). ......................... Appendix B. Matrix Elements of Spin-Dependent Operators Appendix C. Proof That the Orbital Equations Are Invariant under X . Appendix D. Proof That the Operators F(") and F P )Are Invariant under Unitary Transformations of the I#,, and $,, Sets of Orbitals . . References ....................................................

141 144 147 152 154 163 168 180 194 206 201 207 21 1 21 3 215 219

I. Introduction The objects of any useful theory of natural phenomena are threefold: to give quantitative answers that are in reasonable agreement with experiment, to provide a logically self-consistent physical model, and to predict correctly the results of further experiments. The Hartree-Fock molecular orbital (HFMO) theory of the electronic structure of atoms and molecules may claim to be a substantial success on all these three counts. The theory still possesses a number of shortcomings, however. As is wellknown, it gives an inadequate account of chemical binding both qualitatively and quantitatively. Thus according to the HFMO model, molecules in their ground electronic states (except in very special circumstances) dissociate into 141

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J. Gerratt

ions, which is in contradiction with observation. The calculated binding energies are only between 30 and 50 % of the observed values, and occasionally the theory predicts no binding at all (e.g. see Wahl, 1964). The open-shell HFMO theory (Roothaan, 1960) can sometimes be particularly erroneous, yielding numerical values for molecular properties that are 100% in error.’ In particular, the open-shell model fails to account even qualitatively for the phenomenon of spin density at the nuclei of molecules. These failures are recognized as being due to the absence in the theory of any correlation between electrons with opposing spins, particularly between the two electrons in the same orbital. It would seem, then, that the simplest way of introducing some correlation among the electrons is to construct a wave function in which all the orbitals are distinct, and so avoid the double occupancy characteristic of the HF function. This was done for the helium atom by Eckart (1930) in which he replaced the H F function 41s(r1)41s(r2) by a function of the form q51s(rl)4~s(r2) 41s(r2)4’&1) and obtained a considerable lowering of the energy. Very encouraging results were also obtained by Coulson and Fischer (1949) who constructed a similar function for the H, molecule.2 This program, however, has proved to be extremely difficult to carry out for a general atomic or molecular system. The prototype of this kind of approach is, of course, the valence bond (VB) method, but the complexity of its general formulation is such that it has received very little attention over the last 30 years, and it still awaits the compact formulation analogous to that achieved for the HFMO method (Roothaan, 1951). There has been some recent revival of interest in the method, however, especially since it was found to be the natural vehicle for the interpretation of the finer details of NMR spectra (e.g. see Barfield and Grant, 1965). Significant progress was made by Hurley et a/. (1953) who, starting from the HFMO function, replaced each doubly filled orbital function 4s(r1)$p(r,) by an electron-pair function of the form 4p(r1)4s’(r2) 4p(r2)4p’(r1).The pair function is formally similar to that arising in the VB treatment, but the + p are not assumed to be atomic orbitals and so the resulting wave function is more general. The 4p are expected to be more localized than molecular orbitals, but no precise analysis of their possible symmetries was given. A particularly attractive feature of this work is that because of the orthogonality assumed between the different electron-pair functions, the expression

+

+

This is the case in the calculation of the dipole moment of the CO molecule in the a3rI, state (Huo, 1966). As pointed out in their article, the Coulson-Fischer function is, in fact, equivalent to the Weinbaum function (Weinbaum, 1933). This last, although constructed from just a 1s orbital on each nucleus, yields a binding energy that is 85 % of the observed value. In contrast, the HFMO function gives 77% of the binding energy.

SPIN-COUPLED WAVE FUNCTIONS

143

for the expectation value of the energy is hardly more complex than that obtained for the HFMO function. The variational equations determining the 4fi were given shortly afterward by Hurley (1956), but there seem to have been no numerical applications of the method. The problem of constructing a wave function with all orbitals distinct was approached from a different point of view by Lowdin (1955; also see Pauncz, 1967). He showed that this could be accomplished by the action of a simple projection operator upon a Slater determinantal wave function. Denoting the total number of electrons in the system by 2n, it can be shown that the resulting wave function is constructed so that the first n electrons are coupled together to give the maximum possible resultant spin, S = i n . The last n electrons are similarly coupled, and the two subsystems are then coupled to give a resultant spin S = 0, 1 , . . . This differs from the Hurley, Lennard-Jones, and Pople function in which the electron spins are coupled together in pairs to form singlets, the total resultant spin then necessarily being a singlet. An ingenious attempt to construct a theory intermediate between the HFMO and VB approaches has been made by Linnett (1961). The formalism is essentially that of VB, but the orbitals used are two-center ones, containing one or two adjustable parameters. The theory is still somewhat intuitive, but the results obtained from it compare extremely well with other semiempirical calculations (e.g. see Empedocles and Linnett, 1964). The object of this article is to develop a theory that is as general as possible and to examine its consequences thoroughly. In particular, an extensive analysis of the spatial symmetry properties of the orbitals is given. This is of some importance since much of the physical content of the theory reveals itself through this. The various approaches discussed in this Introduction then appear as different special cases. Some progress in this direction has been made in a recent series of papers by Goddard (e.g., see Goddard, 1967a,b, 1968a,b), and his work is referred to at various points. His results, however, have been disappointing, with total energies differing very little from the corresponding HFMO results. This is due t o the particular formalism used, which obscures the physical significance of the different possible coupling schemes for the orbitals. The manipulation of the matrix elements of the Hamiltonian in the general theory is much aided by the use of the coefficients of fractional parentage technique. This concept was originally introduced by Racah (1942a,b, 1943, 1949) in connection with the calculation of matrix elements of operators in the theory of complex configurations in atomic spectra, and applied by Jahn to the shell model for nuclei (e.g., see Jahn, 1950, 1951, 1954). In an interesting series of papers, Kaplan showed how the technique could be profitably applied to molecular problems (Kaplan, 1963, 1965a,b, 1966a,b, 1967a,b). A summary of the formulas derived in this paper by this means has already been

144

J . Gerratt

given (Gerratt and Lipscomb, 1968; Gerratt, 1969). Some of the material in Sections 11 and 111 has appeared in various places in the literature, but it is collected together here to develop a self-contained argument.

11. Properties of the Exact Electronic Eigenfunction This section comprises a review of some of the properties of the exact electronic eigenfunction, and also serves to introduce much of the notation used. Consider an atomic or molecular system consisting of N electrons and A nuclei. We choose a system of coordinates with origin at some arbitrary point external to the molecule with which to characterize the positions of these particles. The positions of the nuclei will be denoted by the vectors R j ( J = 1, 2,. . . , A ) , and those of the electrons by rlc, (p = I , 2,. . . , N ) or simply by p. The Born-Oppenheimer approximation is assumed (Born and Oppenheimer, 1927; Born and Huang, 1954), since we shall not at present deal with degenerate electronic states. We may thus define the following electronic Hamiltonian:

Atomic units (a.u.) are employed throughout. The nuclear repulsion term has been omitted in (1) as it is of no immediate importance. Spin-dependent terms have been omitted for the moment but will be considered later in this article. The notation rpJ, r p yis used to denote the scalar quantities Ira - R, 1, and I rp - rv1, respectively. The Hamiltonian thus consists essentially of a sum of one-electron terms and a sum of two-electron terms, as has been explicitly indicated in the second line of (1). The eigenfunction of H is denoted by Y , so that H Y = EY. Y is a function of both the set of N spatial coordinates r l , r 2 , .. . , r,, and the set of N spin coordinates cl, o2 , . . . , oN :

Y = Y(rl, r 2 , . . . , r,; cl,0 2 ,. . ., 0),

(2)

Since H does not contain any spin terms, it commutes with both the operator corresponding to the square of the total spin angular momentum, and with its component . Thus

s2,

s,

s2]= 0 ;

[If,

[H,

s,] = 0.

(3)

SPIN-COUPLED WAVE FUNCTIONS

145

From this it follows that Y can be chosen simultaneously to be an eigenfunction of and $,, as well as of H , and hence may be labeled by two quantum numbers S and M :

s2

The Pauli principle states that if Y s Mrepresents a physical state, then it is antisymmetric under any simultaneous permutation of space and spin coordinates py,yM = Y({PsM-' rl, rz, . . . r,; o i 2 0 z ,. . . GN))

(5)

=EPYSM

In this equation, P is an operator that permutes the space and spin coordinates, and E~ denotes the parity of P ; thus E~ is equal to + 1 or - 1 according to whether P is an even or odd permutation. An operator that permutes only the spatial coordinates is written P r , and one that affects only the spin coordinates as Pa,

p

= p'p" = pap*.

(6)

We observe that H is completely symmetric under all permutations of the spatial coordinates, [ H ,P']

= 0,

for all P E 9,

(7)

The symbol 9,is used to denote the group of all permutations of N objects (9, is variously referred to in the literature as the symmetric group or the permutation group). From this it follows that the eigenfunctions of H may be chosen to form bases for irreducible representations of 9,. 3 The most general form that Y s Mmay have is then given by Wigner (1959):

In this equation, the set of spatial functions (D& ( k = 1, 2, . . . ,f s N )form a basis for an irreducible representation of 9,. The matrices of this representation are written as U s x N ( P the ) , values of S and N being sufficient to determine the particular representation when dealing with electrons. They may We shall be much concerned in this paper with the properties of these irreducible representations. An excellent reference is the article by Kotani ef a / . (1963), upon which, moreover, a good deal of my notation is based. The articles by Jahn (1954; Jahn and van Wieringen, 1951) are also useful, and the subject is treated exhaustively in Hamermesh 1962). In addition, Coleman (1968) has written a very helpful elementary introduction to the subject.

146

J. Gerratt

be taken to be real and orthogonal, and are of dimension f s N ,this being given by ( 2 s + 1)N! (9) f s N = ( : N + s + l)!(JN-S)! Hence for any permutation P‘,

in which the arguments r l , r 2 , . . . , rN have been omitted for clarity. The superscript N on Us-“(P) will also be dropped when there is no risk of confusion. Each function @ ,: is an eigenfunction of H a n d belongs to the same eigenvalue E , as the total wavefunction Y : H@&

k

= Em,”,;

=

I , 2, . . . ,fsN .

(11)

The @,”k may be taken to be normalized and orthogonal:

I @,”,>

<@:k

(12)

=

It would thus appear from Eq. (1 I ) that the energy level E isfsN-fold degenerate. However, this “permutation degeneracy cannot be removed by the application of an external field, since the complete set of the @,”k functions are needed to define a state. The spin functions O,”, appearing in Eq. (8) are all eigenfunctions of and 3,: ”

s2

$0,”.M : k

=

S(S

+ l)O:,

M;k

S z O , ” , M M : k = M O t , M (; kk = , 1 , 2 , . . . , ff)

(13)

in which the arguments of the functions have again been omitted for clarity. The spin functions form an orthogonal set, <@,”, M ; k

I@,”, M ; I>

= 6kl 9

(14)

and they too transform irreducibly into linear combinations of one another under a permutation of the spin coordinates. However, in order to satisfy the Pauli principle, the representation thus generated must be of the form puO,”,M ; k

= &P

c I

ufk(p)O,”,

M ;I .

(15)

The matrices ePUs(P)are said to constitute the “dual representation.” Consider now a quanta1 operator 0 that contains no spin-dependent terms, but is otherwise arbitrary. Its expectation value with the eigenfunction YSMis given by

147

SPIN-COUPLED WAVE FUNCTIONS

(y,SM

I I y.SM)

= (fSN>-

1k

<@;k

10 I

=

I I @;k) (I

= 1,

2,

. . ., f s N ) .

(16)

No spin functions appear on the right-hand side of this equation. The eigenfunctions (DsNk, in short, carry all the information about the spin properties through their permutational symmetry. Equation (16) is the basis of the “ spinfree quantum chemistry approach employed by Kaplan (1963) and by Matsen (1964). Note that the second equality in (16) follows essentially because 0 is totally symmetric and hence commutes with all permutations P‘; it is the analog of the result that in systems with spherical symmetry, the expectation value of a scalar cannot depend upon the MJ quantum number: ”

( d M j 10 I CI’J’MJ’)= 6

j j T

6,,M,,(ctJljOll~r’J).

Should the system possess, in addition, any spatial symmetry, i.e., should there be some point symmetry group 9 under which H is invariant, then each @& function can be further chosen to form a basis for an irreducible representation of 9. In that case, each will require two extra labels A and p, to denote the irreducible representation and the particular member of the basis, respectively. Thus for any operation R in ’3,

where d,, is the dimension of the representation A. Note that as long as a spin-independent Hamiltonian (1) is used, the operations R of 9 affect only the spatial functions @;A (Herzberg, 1966; Wigner, 1959).

111. Construction of the Spin Functions The form of Y S MEq. , (8), leaves both the @:k and @”, M;k functions indeterminate to within a unitary transformation. There is thus an infinite number of possible bases in spin space for the O”, M ; k , and some of the most important of them are considered in this section. The most commonly used basis is that known as the Young-Yamanouchi basis (Yamanouchi, 1936, 1937; Corson, 1951 ; Jahn, 1954; Hamermesh, 1962; Kotani et al., 1963). The functions in it are so ubiquitous in this work that I shall refer to them simply as standard functions.” One starts with the spin functions for a single electron, @a), where g = f or - 4,and “

()=M

if

g = 12 .7

S=fl

if

g = - L 29

J. Gerratt

148

and builds up the N-electron function by coupling the spins successively according to the rules for coupling angular momenta in quantum mechanics. Thus N

S + f; *; M - 2 ; 2 1 M - a; k cl(bN) + ( S + 12", *2 M + l . 2 - 21 I SM)@:;,', M + f ;k P(ON)? ?

S')@t;t,

@s, M ;k = (

3

k N @s,M;k

=(S

- 3; f; M

=

1 , 2 , . . .,j:Ti,

(18)

I SM)@:zi, M - f ; I cl(bN) + ). - T I SM>@,N-,, M - a ; I P(UN>,

- 1.1 29 2

+ ( S - 1.1. 29 2 2 M

k

1

9

= jg;f

+ 1;

1 = 1,2, .. .,f g 1 ;

(19)

In these formulas, the symbols (S,S,M,M, I S M ) denote the appropriate Clebsch-Gordan coefficients. In these particular cases they are given by (e.g., see Brink and Satchler, 1962)

(S

+ f ;f; M + +; - ) I S M )

=

[

2s +2

The process described by Eqs. (18) and (19) is imagined as follows: The first electron always has a spin f and on coupling a second electron to this, one obtains either a singlet S = 0, or a triplet, S = 1. On coupling a third electron, one necessarily obtains from the two-electron singlet a doublet, S = f , but from the two-electron triplet one may obtain a quartet, S = 3, or another doublet. There is thus a large number of ways of coupling together the spins of N electrons to obtain a given resultant spin, S. The totality of spin functions constructed in this way is conveniently visualized with the aid of the wellknown "branching diagram" (Van Vleck, 1932; Corson, 1951) and is reproduced in Fig. 1. In this the resultant spin S is plotted against the number of electrons, N . The integer fsN is seen to be the number of different ways of starting from S = f , N = 1, on the diagram and arriving at a given resultant S, N ; each circle in the figure contains the value offsN for that position. We observe that fSN

= f;;;

+ f:$

SPIN-COUPLED WAVE FUNCTIONS

149

3

t S

2

I

-N-FIG.1. The branching diagram.

The index k on each function Og, resultant spins :

may be interpreted as a series of partial

k=(S,S,...S,...S,_,), where S, is the resultant spin of the function after coupling p electrons. S, is, of course, always equal to f , and it is not necessary to specify S, as this is just the total resultant spin S. As an example, we consider a system of four electrons with resultant spin S = 1. In this casefi4 = 3, so that there are three independent spin functions characterized by k = (41+), (+1+), and (fog). These are illustrated in Fig. 2. The permutation symmetry properties of the standard functions follow directly from their definition by Eq. (18) and (19). Combining these with Eq. (15) it may be seen that in this particular basis, the matrices U S * , ( P for ) those permutations P" that do not affect the Nth electron are in fully reduced form. This is illustrated diagramatically in Fig. 3. The standard basis may thus be succinctly characterized as that basis in which 9,is reduced to 9N-l x Y, ( P ) are known, (Kaplan, 1965b, 1966a). Thus assuming all the the entire representation corresponding to the N-electron system may be

150

J. Gerratt

-N+

-N+

-N+

FIG.2. Standard spin functions for N

= 4,S = 1 .

determined provided that the matrix US3N(PN-1N) is known. This last may be determined from the following equations (Kotani et ul., 1963):

pi-1N @ SN, M ; SiSz . . . S + IS++ = @ SN, M ; SlSz ...Sf IS++

Gabriel (1961) has used these equations to construct a method of computing the Us(P) systematically on a machine (also see Mattheiss, 1958). Another basis of some importance is one in which two standard functions of N , and N 2 electrons, respectively, are coupled together as follows: @:,M;SlSzklkz

(S,S2MlM,ISM)@,N:,M,;k,@,N,iM~;,,

=

M I ,MZ

(M,

+ M 2 = iM);

3

+ N, = N.

Nl

(21)

In this basis the matrices of permutations of the form P a p p ,where P” is in Y Nand 1 P Din Y N,,appear in fully reduced form, and the basis may thus be characterized as the one in which Y N +Y N xI Y,, . It is instrumental in the derivation of the molecular coefficients of fractional parentage (Section V), and is also important when discussing the dissociation of a molecule into its constituent atoms, and in formulating theories of intermolecular interactions. The standard basis and (21) are connected by an orthogonal transformation which may be written @ ,!

M ; SlSZklkz =

1( S k 1

s1s2

klk2)@:,

M:k

*

(22)

k

The coefficients (Sk I S,S, k , k , ) are kind of generalized Clebsch-Gordan coefficient for the symmetric group. They may be calculated by methods of

151

SPIN-COUPLED WAVE FUNCTIONS

Kaplan (1959) and Horie (1964). This problem has also been considered by Littlewood (1940) and Hamermesh (1962). In general we observe that the simple interchanges PI,, P,, , . . . , P N - i N (or PN-2N-1)commute with each other and that (P,-l,)2 = E. This shows that there is a basis in which all the matrices of these permutations are simultaneously diagonal : I / ~ , ( P , - ~ ,= ) +6,,;

p = 2, 3,

. . . , N or

N - 1.

(23)

This corresponds to the construction of the N-electron spin functions by first coupling together the electrons in pairs to form singlets or triplets, and then coupling the pairs together to form the desired resultant, S M . This basis, which reduces 9,to 9,x 9,x ... x Y 2 ( x 9'J, has been termed by Jahn as the "PB-lp-diagonal basis," and we shall adhere to this terminology. It was introduced by Serber (1934a) in an extension of the vector model for atomic spectra, and in a further paper he generalized the method to calculations on molecules (Serber, 1934b). In particular he showed how the point symmetry of a molecule could be elegantly treated within the same framework. The functions in the P,_,,-diagonal basis are labeled according to a scheme such as k

3

( . . * ( ( ( s ~ ~ s ~s 5~6 ))sS6 , ~s7,)S, , . * . S N - 2 ; ~ ~ N - , N ) S , for N even, (('.* (((S12S34)s4; S 5 6 ) s 6 ;

S78)S8

."

SN-3;

SN-2N-1)sN-1; sh')S, for N odd

in which the s,-~, may be 0 or 1 according to whether electrons p - 1 and p are coupled to form a singlet or triplet. Thus one has P , " - I , ONs , M ;=k ( - l p - 1 p + ' N p = 2 , 4 , . . ., N(or N - 1) O,,,;,, for all functions in this basis. Mattheiss (1959) has given a method for constructing the representation matrices by induction, similar to that used for the standard basis. If N is assumed to be even for the moment, then a basis of singlet spin functions may be constructed by coupling all possible pairs of electrons to form singlets and then coupling +N of the pairs at a time to form N!/(2"(3N>!) total spin functions. Of course not all of the functions constructed in this way are linearly independent, but Rumer (1932) has given a graphical method for selecting a linearly independent set from these. This has received considerable attention in the literature as this is just the valence-bond basis. However, the functions in it are not orthogonal and this compounds the difficulties in its use, these being already sufficiently severe due to the nonorthogonality of the spatial orbitals. We shall not consider this basis any further, and confine our attention entirely to orthogonal ones.

152

J. Gerratt

IV. The Spin-Coupled Wave Functions Let us now assume that one has chosen an arbitrary spatial function Qo(rl, r 2 , . . ., r,) according to some physical model, or some other preconception one may have about the molecule. How does one construct from Qo an acceptable approximate wave function in the form of Eq. (8)? The problem is essentially that given some function Qo without any particular permutation symmetry, to construct from it a basis that transforms irreducibly under 9,. This is in fact a standard group-theoretical problem, and our object is achieved by the use of the following (fSN)’operators 02 :

where the sum runs over all permutations in 9,. We shall refer to these operators as “Young operators” after Jahn (1954). They are, except for a trivial normalization factor, identical with the usual Wigner projection operators (Wigner, 1959). It is not difficult to show that the function (oskQo) has the necessary permutation symmetry. Thus for any permutation P‘, fS N pr(osk

@O)

=

1

n= 1

u%p)(O~k

(25)

@O).

The set of functions (uskQo), I = 1 , 2, . . . ,f s N , forms for each value of k a basis for an irreducible representation of 9,. An approximate eigenfunction Yg,M ;k is then given by

yz,

fS N M ;k = ( l / f S N ) ” ’

1(di

@O)@,

I= 1

M ;I .

(26)

In this equation, the spin functions O”, are, of course, those which transform according to the representation cPUs(P)which is dual to the one used to construct the wsk in (24). From Eq. (26) we see that from an arbitrary spatial wave function Qo one obtains in general a total of f S N approximate eigenfunctions, y:, M:k , one for each value of k. Equation (26) may be put into a somewhat more familiar form as follows: From (26) and (24),

SPIN-COUPLED WAVE FUNCTIONS

153

where d is the usual antisymmetrizing operator:

s = N-1x!

Ep

P.

p

(Recall that P = PrPu.) The two alternative forms (26) and (27) for Yg,M ; k are both important. We shall not develop an exact theory any further, except perhaps to remark that any spatial function of arbitrary complexity may be expanded in a suitable complete set of functions : @o =

c

bu@ou,

a

which may then be substituted into (27). From this we conclude that upon refining any Q0 sufficiently, the approximate function "2, M ; k will converge upon the exact eigenfunction. Instead, we confine ourselves to the approximation where Q0 is chosen to be a product of N spatial orbitals: @ON

= &(ri)#z(rd* . * $ N ( r N ) ,

where now a superscript N is added to presently. The wave function

Q0

(29)

for reasons that will become clear

Ys,M ; k = ( N ! ) ' i Z d ( @ z @ : , M ; k )

(30)

will be termed a " spin-coupled wave function," since the N orbitals may be regarded as being coupled together according to the particular scheme k.4 The most general wave function obtainable from a function (DoNis then a linear combination of all the possible spin couplings: f SN "S, M =

dSk Y S , M ; k k= 1

where in the second equality of (31) we have introduced the normalization factor, Azk, for convenience later: Atk

=

(YS,

I

M ; k YS, M : k ) .

(32)

The linear coefficients cSkor dSkare found by solving a secular equation whose The superscript 0 on Yg. M : k is now dropped, since we shall confine our attention entirely to approximate wave functions.

154

J . Gerratt

order, in general isfsN. However in all cases where the molecule has any symmetry, the actual number of spin-coupled functions in (3 1) will be considerably less than this (see Sections VII and VIII). The presence of a number of different spin couplings in (31) has important physical implications. Thus in the simplest nontrivial case, that of the Li atom in its ground ' S state, there are only two terms in (31), corresponding to k = ($1) and ($0) in the standard basis. It has been shown that although the k = ($1) coupling contributes a negligible amount to the total energy, its presence is vital in accounting for the spin density at the nucleus and thus for the magnetic hyperfine splitting observed in the optical spectrum of the atom (Gerratt, 1969). The form (31) is even more important in the proper description of molecular structure, for although a single spin coupling may give an adequate description of a molecule in its equilibrium configuration, a linear combination of different couplings is necessary to predict the correct dissociation products.

V. Calculation of Matrix Elements of the Hamiltonian We consider first a single spin-coupled function Y s ,M ; k . The expectation value of the Hamiltonian ( I ) given by this function is: (YS,M ; k

I I YS, M ; k )

=

!(&(@ON@:,

= ( N !/fSN) =

M ;k ) ( O S k @ON

1

I I @ON@:, I I @ON)(@:,

C Ulk(P)(P'@oN1 H 1 Q O N ) . P

M ;k )

I

M;1 @ ,:

M;k )

(33)

Similarly, the normalization factor (32) is given by AEk

=

C UkSk(P)(P'@oNI @ o N ) , P

(34)

and the total energy given by this wave function is thus Es,

= (Afk)-

'C P

Ufk(P)(P'@ON

I I@ON).

(35)

This result is easily generalized to the general wave function (31), and we obtain for the total energy of the sytem: Es =

C WS(P)(P'@oNI H I@ O N ) , P

where

(36)

SPIN-COUPLED WAVE FUNCTIONS

155

The linear coefficients CSk occurring in the wave function are determined subject to the condition that

It is not possible to manipulate expressions (35) or (36) further as the oneand two-electron integrals do not appear explicitly. In order to proceed, we describe a general method first used by Racah (1943) in the theory of complex atomic configurations. We confine ourselves to a single coupling at first, since the results may be easily extended to the general case. Instead of the permutations P' which operate upon the electron coordinates, it is convenient to introduce a second set of operators, denoted by P', which permute the orbitals. Thus compare

Pr@oN = $i(Pi)42(PJ* * * 4 N ( P N ) ,

and

P'@oN = 4P,(l)4P2(2)* * 4PN(N). From this we see that P r P = E,so that p r = pr-1.

(39)

In general, the two sets of operations commute and we have

e'(P*@,")

=p.p- lQ0N.

= P' e ' Q o N

(40)

Now since YS,M ;k is completely antisymmetric in all the particle coordinates, we may write for the expectation value of H : (yS, M: k

IH I

yS, M ; k ) = N(yS,

I I yS, M : k )

M:k hN

+fN(N - l)(yS,M;klgN-lNI~S,M;k).

(41)

In order to make use of (41) we seek a form for Ys, k in which the last one or two electrons are explicitly decoupled from the others. In general, this may be accomplished by adopting the basis of spin functions (21) in which Y Nis reduced to Y N x, Y N.2Now any permutation P of Y Nmay be written as a product of three permutations

P = RTQ, where R is a permutation of .YN1 , T a permutation of 9',,, and Q one of the cosets in the expansion of 9,in its subgroup Y N x, Y N 2This . subgroup is of order N , ! N , ! , so that there will be N! N , ! N,!

156

J. Gerratt

permutations Q (including the unit permutation Q = E ) . There is, in general, considerable freedom of choice for the Q. Then for any permutation P, M ; SiSzklkz

= RaT"Q"@!,M ; s I s 2 k l k 2

c c c1

= RUT"

Si'Sz'ki'kz'

=

Si'Sz'ki'kz'

11

N

&Q U : ~ ' S z ' k ~ ' k z ' ; S i S z k l k z ( Q ) @ S , M ; S1'Sz'kl'kz'

&Q U : ~ ' S l ' k ~ ' k ~ 'S; I S Z ~ I ~ Z ( Q )

12

&R U::k'I'(R)ET

Uf$z'(T)@:

M ; S1'S2'11/2

.

Hence yS,

M ;k

yS, M ; S i S z k ~ k z

(Q!)'Iz &(@@ '; ,;: x ( ( N 2 !) '', d (@;'@;if, M 2 ;k 2 * ) } U ~ l ' S z i k l ' k z ' ;s 1 S 2 k l k 2

M ] ;k l ' ) }

(42) Equation (42) is the one we seek. I t expresses Y s ,M ; S , S z k l k z in the form of a sum of products of two spin-coupled functions, the first being a function of N , electrons and the second of N , electrons. The configurations and 02 appearing in them have the following meaning. We write (DON

where

= @;f@;2,

157

SPIN-COUPLED WAVE FUNCTIONS

@ti= 41(1)42(2)

* ' '

4N1(~1),

and @:2

=~

N I l(Nl +

+ 1 ) 4 N 1 +2(N1 + 2)

* * *

4N(N)*

The permutation Q' interchanges orbitals between the two configurations and so that

Q*@)o" Hence @gl and

(43)

= @Ni@N2 Q Q

a;'

are the two configurations that result after the operation and a?. Essentially, Eq. (42) is a rederivation of Jahn's (1954) result. Thus if the function Ys,M ;S i S 2 k l k 2 is written in the alternative form (26) one may then easily obtain from (42) the relation

Q' has interchanged a certain number of orbitals between

S mktk

S

= m S ~ r S z ' k i ' k ~ S' ;i S 2 k i k z

which is equivalent to Jahn's expression (37). In analogy with his work, the coefficients in the expansion (42)

could be termed " molecular coefficients of fractional parentage " (mcfp). The calculation of the expectation value of H is now effected by substituting (4 2) into (41). We consider first the one-electron term. For this we put Nl = N - 1, and N , = 1 . The permutations Q' ( N in number) are chosen to be the set

F p ~/J.=1,2 ; ,..., N , and we obtain y S , M ;k

= yS, M ;S l k l

x { ( N - ~ ) ! " z ~ ( @ ~ ~ ' ~ ~ i ~ ~ l ; ~ l ' ) } (45) ~ a ( ~ ) ~ (

In this expression M 2 ( = aN)= 4 or -3, and there is no summation over S,' or k,' since S,' is 4 and so k,' has only one value. @:-' denotes a configuration of N - 1 orbitals derived from the first N - 1 orbitals of aON, but with 4preplaced by 4N: @ -:

= 41(1)42(2) ' *

4N(/J.)

* * '

4 N - l(N

- l).

158

J. Gerratt

In general we shall employ the notation .

to denote a configuration of N - p orbitals in which 4 , , 4 , , 4 u ,4 r ,. . . , have been replaced by 4 N , 4 N - 4 N -2 , $ N - 3 , . . . , 4 N - p - respectively. However, if N - p < p , then Q ..!;: is defined by [cf Eq. (43)] - P:NP:N-,CN-2...@gN = @;v;”.... . . 4 , ( N - 2 ) 4 , ( N - 1)4@(N). We now have N(yS,

M;k

I k N I yS, M ; k )


l@:-1)(4,1~14”).

The expression inside the curly bracket may be simplified as follows. Since P E , 4 p N - 1 , it follows from the properties of the standard basis that U,”,‘,;:; ‘ ( P )is identical to U ~ ; , N k I . , ; S I , k , , ( P )(see Fig. 3).

FIG.3. Structure of the U s . N ( P matrices ) (P

E

9,- in the standard basis.

so that we may write N

where the coefficient H(pv I Skk) is defined as

Expression (46) has been derived for a standard spin-coupled function, but

SPIN-COUPLED WAVE FUNCTIONS

159

on reflection it may be seen that it remains true whatever basis is used. This is because there is always a unitary transformation connecting the standard with any other basis. Thus, for a spin-coupled function in some arbitrary basis @;," ~,the matrix element Ufk(PVNPPpN) in (47) is replaced by

c

k, 1

( r 1 k)Uzl (

P ~ NP P p N ) ( z

I r,

where ( r I k), (I1 r ) are the coefficients in the transformation between the two bases. But this expression is equal to

u: ( p v N

ppph'>

which is just the corresponding element of Usin the new basis. We now proceed to the two-electron term. In this case Nl is put equal to N - 2 and N 2 to 2 in Eq. (42). This means that S , = 0 or I , and hence if S2 is specified, k , is also fixed. There will now be $N(N - 1) permutations p, and there is considerable freedom of choice for the set. We consider here two possibilities v = I, 2, . . . , N - 1 Q' = c N i 5 6 N - 1 , /* = v + 1 , v + 2, . . . , N and for N 2 4,

pr

Qr=Ip* 2

pN

vN-1,

v = l , 2 , ..., N - 1 / * = 1 , 2,..., N ; /*#v.

(49)

The use of one or other of the two sets (48) and (49) leads to slightly different expressions for the total energy, each of which is useful. Employing the set (48) we have yS, M ; k

yS, M ; S l S z k 1

J . Gerratt

160

As in the one-electron case, we observe that

Hence we obtain

(53) Again we observe that although (51) was derived for spin-coupled wave functions in the basis Y N - x, Y 2 ,the expression remains the same for wave functions in any basis. The total energy of the molecule given by this wave function is then

N

+

-f c {J,(pvot I Skk)(pv I9 lot> + K,(pvat I Skk)(VP 19 I at>).

p>v= 1 o>r= 1

(54) In this expression, the integrals ( 4pI h I 4 v ) , ( 4p4v1g I 4u4r), etc., have been written as ( p 1 h I v ) , ( p v 191a t ) for clarity, and the H , J , and K coefficients trivially redefined so as to include the normalization factor Aik:

I I J,,(p~otI Skk) = ( A : J 1 J ( p ~ ~ o tSl k k ) K , , ( p ~ aItSkk) = (Afk)-'K(pvot I Skk). H,,(pV S k k ) = (Afk)-'H(pV Skk)

(55)

Equation (54) has the familiar form of a sum of three contributions to ESk:a one-electron term, a " Coulomb " term, and an " exchange" term. However, it should be noted that for N 2 4 one may easily prove the following equalities between the J,, and K, coefficients: J,,(pvot I Skk) = K,,( vpot I Skk)

I

= J,,(atpv Skk) = K,,(pvto

1 Skk)

(56)

161

SPIN-COUPLED WAVE FUNCTIONS

These symmetries are paralleled by similar ones between the two-electron integrals : ( VPIglra) = ( P V I S l o ~ ) =

(57)

and using these two sets of relations, we may write ESkin the form N

ES k

=

1 Hn(W I S k k ) ( P I

Iv>

p, v = 1

+ 4a,

N

c N

J,,(PVUT

v = 1 a, r = 1 (afr)

I S k k ) ( W I g I as>,

( N 2 4)

(58)

(a+v)

in which the K,, coefficients do not now appear. Expression (58) is also obtained when the set of permutations (49) is employed in (50). The H , and J,, coefficients are just elements of the one- and two-electron density matrices in explicit form. They consist only of ratios of overlap integrals, and for very small N ( N 5 4, say) may be computed as they stand. However, it should be noted that a complete hierarchy of such coefficients may be constructed: 1 = (AEJ N

=

c

a= 1

H"(PJISkk)(Pl a>

( P = 192, . .

' 9

N)

N

=

2

u,r = 1

J,(pvotlSkk)(~la)(vI~)

(p, v = 1, 2 , . .., N ;

# v)

(afr)

N

c

=

a , r, q = 1 (of r f v )

-

Q,(pv~or~lSkk)(~~o)(v/s)(~.~~) (p, v,;-= 1,2, . . . , N ; p f v Z 4

..., etc.

(59)

where

and ( p 1 o), ( v I r ) , etc., are just overlap integrals between the orbitals, e.g.,

Relations (59) are proved in Appendix A. From them it follows that N

H,,(pa I S k k ) =

1 J,(pVaT I S k k ) (

V 17)

for all v

= 1,

r= 1 (r+o)

2, . . . , N ( # p ) ,

(61)

162

J . Gerratt N

J , , ( ~ v a tI S k k ) =

v=

Q,,(PJ~ov I S k k ) ( i I r l ) I

(rl+a, T )

for all J.

=

1 , 2 , . . ., N ( # v , Z p ) , (62)

Equations (61) and (62) may be used to construct the H,, and J,, coefficients systematically by recurrence starting from some conveniently small p-electron density matrix ( p 5 4, for example), and working upwards. In connection with this, it might be mentioned that in an interesting series of papers, Arai (1962, 1964, 1966) has used similar relations to construct what are essentially the H,, , J,, , Q,,, . . . , etc., coefficients by a similar recurrence technique. He was concerned with solids in which ofcourse N-+co, but showed that the H,,, J , , Q,,, . . . , etc., may all be written as an expansion of the form

which is rapidly convergent. The coefficients f p ( p v ) are products of overlap integrals, selected and ordered according to the particular permutation P ; they may be systematically computed with the aid of diagram techniques. These considerations may be immediately extended to the most general spin-coupled wave function (31). In this case the total energy is given by N

E,

=

c H A P I I h I v> + 2 c c J,(WJtl

8,v=

1

-

S)(P

1

N

N

8 , v = 1 u,r = I

W I * V

I

I9 ot>,

( N 2 417

(63)

(rc#v) (u#r)

in which the H,,(pv 1 S ) and J,(pvat 1 S ) coefficients are now defined in terms of the W s ( P )coefficients (37); for example,

etc. These more general coefficients also satisfy a set of relations similar to (61) and (62). Thus, for example, N

H,,(paI S ) =

1 J,,(pvatl S ) ( v l t )

for all v

=

1, 2, . . . , N ( # p ) .

(65)

r= 1 (r+u)

The expectation values of arbitrary one- and two-electron operators are given immediately by Eq. (58) and (63). Thus for example the expectation value of an operator N

163

SPIN-COUPLED WAVE FUNCTIONS

given by a single spin-coupled function is just N

1 K ( P V I SkkK I* I f I v).

p, v= 1

Spin-dependent operators may also be treated within this same framework, and the calculation of the expectation value of an operator of the form N

c

p= 1

f p h

is given in Appendix B.

VI. The Orbital Equations is of the form

From Eq. (31) the wave function

c c

/SN

yS, M =

d S k y S , M ;k k= 1 ISN

=

cSk(Afk)k= 1

M ;k >

1’2yS,

where

y,,M ;k

=

JN!

M ;k )

&(@ON@:,

and @ON = d)1(1)d)2(2).’ ’ 4 N ( N ) . We wish to vary the total energy E, given by this function with respect to the linear coefficients dSkand with respect to the orbitals d)p comprising Q O N . Note that for this purpose it makes no difference whether the linear coefficients to be varied are the csk or the dSk, and since this yields more convenient equations we choose in fact to vary the d S k .However, when we come to discuss the contribution of each spin-coupling to the total wave function then we must do so in terms of the csk. The variation is effected subject to the conditions that < ~ S , M l Y S , M )=

1,

and (plp)=l

f o r a l l p = 1 , 2,..., N.

For this purpose, we introduce the Lagrange multipliers q, and N

p s = ~ ~ s , M l ~ - ~ s l ~ s , CE,(C(II*). M ) p= 1

E,

and vary

164

J . Gerratt

Then

This variation is set equal to zero. Since the variations in the dSkand C#I# are assumed to be independent, it follows that their variations may be put separately to zero. In addition, the variations in d,*, and ds, , Y,: M ; and Y s , may be regarded as independent so that from the first line of (67) we obtain two equivalent sets of equations. The first of these is

1d s d y s , 1

M;k

IH - ‘IsI y s , M ; I >

=0

for all k

=

1> 2 , * .

. 7

fsN,

(68)

which only has solutions if det {( yS, M ; k 1 H - ‘ I S I yS, M ; I > >

= 0.

(69)

If so desired, one may work with the normalized spin-coupled wave functions in which case instead of (68) and (69) one obtain^:^

1cs,(A;

(Ys,M ; k I H - qs 1 Ys,M ; I > = 0

1

for all k

=

1, 2,

...,fsN, (70)

deL{
I

- ‘ I S I yS, M ; I>>

= O.

(71) The advantage of these last two equations over (68) and (69) is that the diagonal elements in the determinant are all of the same order of magnitude with the off-diagonal elements much smaller, and when expanded out they yield expressions in terms of the coefficients H, and J, introduced in Section V. It is clear that the parameter qS may be identified with the total energy E s . The remaining parts of (67) involve variations of the orbitals only, and these yield the equations

1 k,I

or

I ” (yS, M ; k

N

d S k dS1(6yS,

M; k

I

- ES

I yS, M ; 1 )

=

1

p=

111)

I

N

(72)

The nonlinear equations for the d,, recently derived (Ladner and Goddard, 1969) are unnecessarily complicated, since the conditionz, d& = I imposed is just that the ‘F,,+,; are orthonormal. A transformation to such a basis may always by achieved by diagonalizing the As, matrix.

165

SPIN-COUPLED WAVE FUNCTIONS

Entirely equivalent equations result from varying the Y s , I and show that the E,, must be real. Let us consider first the case of just a single spin-coupling k . Equations (72) now reduce to N

(Afk)-

(6yS,

M ;k

1 - ESk I yS, M ; k )

=

c

Ep(6p

p= 1

I p).

(73)

Now

and substituting this into (73) we obtain the series of equations s

-1


M ;k(bpL)

I

- ESk

1 YS,

M ;k )

I p)?

= 'p('p

for all p = 1, =, . . ., N

(74)

where Ys,M ; ( 6 p ) denotes a function in which orbital @, is varied The lefthand side of (74) is now expanded out with the aid of Eqs. (58), (61), and (62), and after some manipulation one obtains the result N v= 1

u, I = 1

or 7," = (1 - dpY),where 6,, is the more usual Kronecker delta. Equation (75) is valid for N 2 6. The generalization to any number of electrons requires a slight redefinition of the J , and Q, coefficients, and this is described in Appendix A.

166

J . Gerratt

Upon equating the coefficients of dc$jl under the integral sign on both sides of (75), we obtain the equations for the orbitals in the form F F l p ) = c,,lp)

for p

=

1, 2, ..., N .

(77)

The operator F i k kis defined by N v= 1 N

N

N

(78) An expression for the orbital energy cp can be obtained from Eq. (78) with the help of (61) and (62): N

E , = Hfl(PP

I Skk)Esk - C 1Ydjl YTjl J n ( P P 7 I Skk)(a 1 I 7 > o,7=

1

N

N

It is not possible to relate the total energy &k in any simple way to a sum over the e,, . This is in marked contrast to the Hartree-Fock case for which one has the relation

and from which in turn it is possible to relate the c r F ) to the ionization potentials of the molecule (e.g., see Roothaan, 1951). As will be shown in the next section, the orbitals c$,, determined by (77) do not have the same symmetry properties as the Hartree-Fock orbitals, and in general are more localized. One would therefore not expect the c p to be directly related to any molecular property such as ionization potentials; without a radical readjustment of the symmetries of the remaining orbitals, the wave function for the ( N - 1)-electron system does not possess the correct molecular symmetry. This is no doubt a more realistic description than that given by the HartreeFock theory, but it means that ionization potentials are only obtainable in this theory by a separate calculation. It is best to regard the cjl simply as the energy of the corresponding electron in the presence of all the other electrons.

SPIN-COUPLED WAVE FUNCTIONS

167

Note that we may easily obtain an explicit expression for the F:kkby writing

These results are all easily extended to the general spin-coupled case (72) for which we obtain

The Hn(pvI S), etc., coefficients appearing here have been defined by Eq. (64). An explicit expression for the F; operator may be immediately obtained in the same manner as in (80). A complete determination of the wave function YsMthus involves the solution of Eqs. (81) for the orbitals and (70) for the linear coefficients. In practice one employs a cyclic procedure in which a first guess at both the 4 r ' s and csk's is made, and then (70) and (81) solved alternately. The situation is similar to that in the multiconfiguration self-consistent field (SCF) calculations (e.g., see Manning, 1954; Das and Wahl, 1966; Hinze and Roothaan, 1967; Veillard and Clementi, 1967). The form of the orbital equations (77) or (81) shows that an electron assigned to an orbital +r may be interpreted as moving in an average field F:" or Frs due to all the other electrons in the molecule. We observe that the operator is different for each orbital, so that each electron experiences a different average field. This contrasts with the Hartree-Fock equations (Roothaan, 1951)

PIP) = & y ) l p ) ,

p = 1, 2,

. . ., 3 N

J. Gerratt

168

in which each electron experiences the same average field. The fact that the orbital operators in the spin-coupled case depend upon the orbital index p has profound consequences for the symmetry of the orbitals, a topic to which we now turn.

VII. Symmetry Properties of the Spin-Coupled Wave Functions We assume that the molecule under consideration belongs to some point symmetry group 9, i.e, that the Hamiltonian of the molecule is invariant under all the symmetry transformations R of 9 :

[H, R]= 0

for all R in 9.

Then, as pointed out in Section 11, the exact eigenfunctions of H may be chosen to form bases for irreducible representations of 9. We now impose this requirement on the spin-coupled wave functions, i.e., that they too should possess this spatial symmetry. Hence we need to add to Ys, two additional labels A and p, to denote, respectively, the particular irreducible representation of 9, and the particular member of the basis of the representation. Then for any operation R of 9,

[cf Eq. (17)]. Degenerate electronic states, however, occur strictly only in linear molecules as otherwise the degeneracy is removed by a Jahn-Teller splitting (Jahn and Teller, 1937; Herzberg, 1966). We thus simplify the problem in this section by confining our attention to the case where Y s , Mbelongs to a nondegenerate irreducible representation of 9. In short, we consider only C states of linear molecules and A or B states of polyatomic molecules. Hence for any R of 9, where l Ris just a phase factor,

The requirement (84) leads to certain symmetry conditions on the orbitals and which we examine in this section. The analysis of the problem given here owes much to Roothaan’s approach (Roothaan, 195 I ) , and indeed the restriction of Ys, to a nondegenerate representation of 3 is the analog of his restriction of the MO wave function to closed shell systems.

41ccomprising Ys,

SPIN-COUPLED WAVE FUNCTIONS

169

From Eq. (84) we obtain

We now suppose that the orbitals 41,d ) 2 , . . . , g N may be supplemented by an infinite set 4 N + ld ,) N + 2 , . . . , such that two together form a complete set of functions. The members of this complete set do not have to be orthogonal to one another; it is sufficient for them to be linearly independent. Hence any one-electron function may be expanded in this set, and in particular we expand the function R4,:

Substituting this into ( 8 5 ) , we have

Now this equation is an identity. However the left-hand side of it is a sum over all possible configurations of orbitals, while the right-hand side involves just the ground state configuration 4,d2 . . . d ) N . Thus we may collect together all the terms on the left-hand side which involve only the ground state configuration and equate just these to the right-hand side. The contributions from all other configurations on the left must identically give zero. The ground state configuration occurs whenever the numbers ( Y ~. .Y. yv) ~ are just a permutation of (123 . . . N ) . Then in such a case

in which P is the permutation

J . Gerratt

170

This may now be equated to the right-hand side of (87) and after some rearranging, we obtain the equations dSk

[1

usk(p)D~l

I

' ' '

for all 1 = 1, 2, . . . , f s N .

D V N ~=][ R ds,

(88)

PEYN

Equations (88) constitute a set of conditions to be satisfied by the coefficients d S k for Y S M to have the correct symmetry. We now consider the excited configurations that occur on the left-hand side of (87). In particular, we examine all the terms that give rise to the singly excited configuration

4,42***4,...4N in which orbital

1...,

V I , V2,

( = 1 . 2 ,..., p ,

4,(p > N) replaces D ~ 1i D ~ 2 2

. ' ' Dvrp

'

4,(p 5 N). The sum of all such terms is

'' D

~

~

N

VN

..., N )

[T

dSkd'N!d(4v~dvz

"'

4 v P ."

~VN@;,M;L)]

The condition that this expression is zero leads to

C

PtYN

USk(P)D,, 1

for all 1, k

. . . DYrP. . D V N N= 0 *

=

I , 2, . . . , fs".

(89)

The summation in this equation is over all permutations of the numbers ( I , 2, . . . , p - I , p , p + I , . . . , N ) . Now each term in this would be exactly the same as the corresponding term on the left-hand side of (88) except for the presence of coefficient D,, in place of a coefficient D p , . Hence for (88) and (89) to be compatible, it is necessary that

D,, = 0

for all ;I= 1 , 2, . .., N

p

=N

+ I , N + 2, . . .

(90)

Substitution of this into (86) gives N

R4,=

1 DV,(R)4"

v= 1

which shows that the orbitals of Y s ,+,, are transformed into linear conibinations of themselves under a symmetry operation R. Thus the orbitals generate an N-dimensional representation of 9 as is emphasized by the insertion of R into D,,in (91). Equation (91) is a necessary condition for the symmetry condition (85) to be satisfied. Generally, however, it is not a sufficient one as may be seen by substituting (91) back into the first equality in (85). For when this is done we obtain

171

SPIN-COUPLED WAVE FUNCTIONS

in which the numbers vl, v 2 , . . . , vN are not all different. Thus the left-hand side of (92) includes configurations with one or more doubly occupied orbitals, and we require all such terms to be zero. Now the conditions that the configurations 4l243d)4***4N and 4Z24344.**dN2 for example, should be absent lead to equations of the form

1

ufk(p)Dv,

lDvz2

‘

‘

Dv,vN

=

for all 1, k

=

1, 2, . . . ,f s N ,

(93)

P€Y N

in which (v1v2, . . . , vN) is a permutation of the numbers ( I , 1, 3, 4, . . . , N ) in the first case, and of the numbers (2, 2, 3, 4, . . . , N ) in the second. (There will hence be only + N ! distinct terms in the summation in each case.) Each term in (93) would once more be the same as the corresponding term on the left-hand side of (88) except that here some vp = vq = 1 or vp = vq = 2 in each term, whereas in (88) vp = 1, vq = 2, or vice versa. In general, equations of the form (93) and (88) may only be satisfied simultaneously if D y p p D v q= q0

for all p , q

=

I , 2, . . ., N

if v,, = vq = 1 or 2 or p = q ; in other words, the first two rows of the matrix D(R) each have only a single non-zero element, and these occur in different columns. By repeating this argument for any two configurations with one orbital doubly occupied, 4142

’ . . 4 p* * . 4 p .. * 4 N

7

we conclude that D(R) has in each row and column only one non-zero element which we denote by (,JR). An exception to these considerations occurs when in a particular spin-coupled wave function, a number of orbitals, n say, are coupled together to form the maximum possible spin, i n . Such a wave function is invariant under any linear transformation of these orbitals among themselves, so that an equation of the form (91) is in this case sufficient for the satisfaction of the symmetry condition (85). Spin-coupled wave functions of this kind are discussed separately in the next section. We have established the general result that under an operation R of the group, an orbital 4 p is transformed into some other orbital of the configuration, 4” say, multiplied by a phase factor

W V= Cfl”(RM,, >

where in order to preserve the normalization of the orbitals I ( J R ) I = 1. The orbitals may thus be divided into sets, such that the effect of an operation R upon YS,+,is to permute the members of the sets among thenselves. We may therefore associate with each R of 9 a particular permutation P , of the orbitals. The theory of permutation representations is well-known, and a

J . Gerratt

172

complete account may be found in Burnside (1911; also see Hall, 1950; Altmann, 1958). However, the essential parts of the theory which concern us here are as follows: Let the group B contain a sub-group X . Then B may be written as an expansion in the left-cosets of 2, % = [XE, x A >

X B r . . . ,

XpP]

(94)

where X E= 2, X A= A X . . . , etc., and A , B, . . . , P are a suitably chosen set of elements of B not contained in 2.The cosets (94) generate a permutation representation of 9, for if they are multiplied on the left by any element R of 3 they are permuted among themselves:

R X E = X R ,R X A = X R A ., .., R X p = X R p . The dimension d of this representation is obviously equal to the number of cosets in (94). If 99 and X are both finite, this is given by d = n,/n, where n, and n, are, respectively, the orders of 9 and Af [it is in fact unnecessary for the two groups to be finite, but only that the expansion (94) be finite]. The same permutation representation is generated by a set of orbitals

4, 4 A i

4B7...?4P

each of which belongs to the corresponding nondegenerate representation of the respective subgroups

2, A X A - I ,

B X B - ' ,..., P X P - ' ;

thus

R$J = ( ( R ) 4 R$JA= [ ( A - ' R A ) 4 ,

for all R in 2, for all R in A X A - ' , . . . , etc.

= From this it follows at once that the orbitals are all real, and so the CpU(R) 1. This is a slight generalization of the equivalent orbital " representation introduced by Lennard-Jones and Hall (Lennard-Jones, 1949a,b; Hall and Lennard-Jones, 1950)6 in which the [ J R ) = + 1 for all R. The equations determining the orbitals must also possess the same symmetry and that this is indeed so is shown in Appendix C. One thus needs to solve the equations for just one of the members in a set forming a permutation representation as the others in it differ only in their orientation in space, and possibly, in their sign. Altmann (1958) has given a general method for constructing orbitals with this kind of symmetry.

But it should be noted that the equivalent orbitals are always orthogonal, whereas this is not the case here.

SPIN-COUPLED WAVE FUNCTIONS

The orbitals

4pof Ys,

173

hence are symmetry orbitals not of the point group

9, but of some subgroup of it. If 9 has more than one subgroup, as is often the case, then there is a distinct permutation representation of 9' for each

nonconjugate subgroup. There will therefore be a choice of symmetries for the orbitals, the best wave function being a linear combination

the superscript (A,) on Ys, indicating a particular permutation representation. In practice, however, we employ physical arguments to pick out a single symmetry A,. Should the calculation extend into a region where the symmetry of the orbitals may be expected to change to that of another representation, Aprsay a small number of excited configurations from within the original symmetry A, can be added to allow for this. A simple example of these considerations is afforded by the H, molecule. If we consider the singlet states, the (unnormalized) spin-coupled wave functions are of the form y 0 , o = J 2 ! d(4142 @, 0;1) = J,(4142

+ 4 2 4l)Jf(.P

- Pa).

The symmetry group of the molecule is D,, , and this has only one suitable subgroup, C,, , for which Dmh

=

f OhCmu >

or D,h

=

c,, f ic,, .

Thus for the ' C l ground state there are two choices for the symmetries of dI and 4 ~In~the. first case 4' may be chosen to be totally invariant under C,, and 4,under d h C,, Oh (=C,,), so that 4' and 6,are both o orbitals that overlap strongly and are converted into each other by the operation In the second case, both 4'and 42 are totally invariant under Dmh, 9 and 2 thus coinciding. These two possibilities are illustrated in Fig. 4a and b. Denoting the wave function constructed from the CT orbitals of Fig. 4a by Y ( ' ) we see that this predicts the correct dissociation products, but collapses to the Hartree-Fock function for helium, He(ls2; IS) in the united atom limit. On the other hand, the function constructed from the two O~ orbitals of Fig. 4b, Y(') does not dissociate correctly but does collapse to a spin-coupled wave function for helium He(ls, Is'; 'S).The most general fiinctionwould be of the form y(Z,+) = cly(u+ 0,o

c,

y(2)*

174

J . Gerratt

*(I)

FIG.4. Possible symmetries for the orbitals of H z ('X:). (a) The orbitals are invariant under Cmu;(b) the orbitals are invariant under Dxh; (c) the orbitals are invariant under C,, and uhCrru;' as in (a), but possessing nodes.

175

SPIN-COUPLED WAVE FUNCTIONS

One might say that Y ( l )introduces longitudinal correlation between the electrons, and Y(’’axia1 correlation; there is no angular correlation in this model. From a physical point of view, one would predict that should be an adequate wave function for all internuclear distances from R N Re to infinity. Actual calculations (Davidson and Jones, 1962) show that the potential energy curve given by Y ( l )remains below that of Y ( ’ ) for all R down to R 2: 0.8 a.u. ( R e = 1.4 a.u.) at which point the two curves cross. The binding energy of the molecule according to Y ( l )is 0.15207 a.u. which is 87% of the observed value’ (the Hartree-Fock SCF wave function gives 77 %). One may also construct a pair of a orbitals with reflection symmetry similar to those in Fig. 4a, but each possessing a node in planes perpendicular to the internuclear axis. Such orbitals are denoted by 6 and are illustrated in Fig. 4c. A wave function formed from them, Y ( 3 )is, not expected to lead to a stable state of the molecule but to a repulsive Cg+ state. However, if one wished to extend a calculation with Y(l) into the region below R = 0.8 a.u., then a linear combination of four configurations ( l a , l d ) , (1 G, 16’), (20, 207, and (26, 26’) yields a wave function with the alternative symmetry The first pair of configurations in this correspond to those of and Y ( 3 )respectively, , with the second pair also having similar properties but based upon orbitals with higher energies. We now examine for the spin-coupled wave functions the consequences of the fact that the orbitals form a permutation representation of $9. The form of the matrix D(R) causes all terms in the summation over P in (88) to vanish except for the one permutation P, with which D(R) is associated. Equation (88) reduces to
d S k u,”,(pR)

’’‘

=l

R dSL

9

(95)

in which the phase factors [,,(R).-- l A A ( Rappearing ) on the left-hand side are those associated with the orbitals q5p, 4 ” ,. . . , 4Awhich are not permuted by the operation R under consideration. Nondiagonal lpV’s always appear as part of a cycle and hence give unity. Thus for example, let P , permute the orbitals 4@,+”, and q5r as follows:

(any permutation P , may always be written as a product of such cycles). This will give rise to a factor

lp,a i v l

’

CAT

i,,

The function I in a recent calculation by Morrison and Gallup (1969) using a basis set of six Slater orbitals is equivalent to a linear combination of Y’”“) and Y”(*), and gives a binding energy for HI of 0.15828 a.u. (91 % of the observed).

176

J. Gerratt

in front of the wave function, and which is just unity. We may identify with this product of CPw(R)’s,and obtain IdSk U ~ ( P R=) dsl,

for I

= 1,

2, . . . ,f s N ,

k

CR

(96)

and all R in 3. This equation implies that the dSkcoefficients form an eigenvector of the Us(PR)matrix with eigenvalue 1 for each R in 3. In addition of course, the dsk’s are solutions of the secular equation (68). For this to be so, all the Us(PR)matrices must commute among themselves (which, in fact, turns out to be the case), and also with the matrix ( Ys, + ,;, k I H - E, I Y,, l) of the secular equation-with which they never do. The only other alternative is for the Us(PR)matrices to be diagonal and identical with the unit matrix E: Only those permutation representations are possible f o r which the matrices Us(PR)= E.8 This is a very severe condition and is not always fulfilled. A particular permutation representation, Ap , chosen for the orbitals on physical grounds may not always lead to spin-coupled wave functions in which Us(PR)= E for all R of 3.In such cases we must seek linear combinations of the Y,, ,+,; that do have this property. This may be accomplished by using the following projection operator

where the superscript (A) designates the symmetry ( A or B ) of the total wave function; (97) is the analog of the Young operator (24) for point groups. We consider first a single spin-coupled function for simplicity, and one can then quickly extend the results to the general case. We start from a function Y, in which the orbitals form a particular permutation representation A p . It is necessary to produce from this a linear combination of functions, with fixed coefficients for which the matrix element Ufk(PR) = 6,, for all 1. Such a symmetry-adapted’’ spin-coupled function is denoted by YitL; and is given by “

1

the coefficients gjt’ in this equation being defined by

For a single spin-coupling k , the condition is

177

SPIN-COUPLED WAVE FUNCTIONS

It is clear that the function (98) is merely multiplied by a phase factor l Runder any operation R of the group. The g$‘) coefficients are easily determined from a knowledge of the relevant Us(PR)matrices; Melvin’s factored form of the projection operator (97) is most convenient for this purpose (Melvin, 1956). The number of independent symmetry adapted functions that may be constructed in this way may be obtained by reducing the representation of Y formed by the [ J R ) . * lI,(R)US(P,) into its irreducible components: r(Ap)

=

1

a(A)rcA)

A

The number of times a particular A or B representation occurs, d A )or dB), is then found from a table of characters of the relevant representations.’ This procedure serves as a useful check on the calculations (98) and (99). In general, however, one can, with a little physical insight, choose a basis of spin functions that is most nearly symmetrized” for the Ap under consideration, and thus reduce the number and complexity of thegj:’ coefficients considerably. The entire theory developed so far now goes through for the YitL; functions exactly as before. Thus the normalization integral for these functions is given by “


k)

S(A2)

= Akk

= (o‘A)yS,

I

M ; k o(A’yS,

= Jn9
M;h

M ; k)

I yS, M ; k)

where in general the VfiA)(P)coefficients are defined by

Note that although this expression can in fact be written as

1n

ufh’(

’

P i P),

the form (101) is more convenient for actual computations. The total energy of a symmetry-adapted spin-coupled function is denoted by Eic) and is given by N

Ei;) =

1 H,,(pV SAk/i)(p I h 1 v)

8 ,\ ’ = 1 4

N

(B+v)

k

(c+p)

Character tables of the syinnletric group have been given by Littlewood (1935, 1940), Kaplan (1969), and Lyiibarskii (1960).

178

J. Cerratt

in which " symmetry-adapted " coefficients H,,(pv I SAkk), etc., are defined precisely as before. Thus, for example, H,(pvISAkk) = [A;i")]-'

C

~ ~ ~ A ) ( P " ~ P P ~ ~ ) ( P(103) r ~ ~ - l l ~

P E Y N - I

[compare Eqs. (47) and (55)]. From Eq. (102) orbital equations may be derived of the same form as those previously given in (77) or (81), except that the H,,(pv I Skk), J,,(~voT1 Skk), . . . , etc., coefficients are replaced by their analogs, H,,(pv 1 SAkk), J,,(pvar I SAkk), . . . , etc. As an example of these considerations we take the ammonia molecule in its ' A , ground state. Ignoring the two Is electrons on the nitrogen atom, we treat the system as an eight-electron problem. The wave functions in some basis of spin functions are therefore of the form 0;k

= J8!&(6162

* *.

6 8 O08,O;k ) .

(104)

The symmetry group of the molecule is C,, , and this contains the subgroups C, and C,, either of which could, in principle, be used to form a permutation representation for the orbitals. However we now imagine a process in which the molecule is formed by bringing up three hydrogen atoms from infinity to the nitrogen atom but in such a way so as to preserve the C,, symmetry throughout. During this, the three hydrogen 1s orbitals h,, h , , and h, must each correlate uniquely with one of the orbitals of the molecule at equilibrium with no discontinuous change in symmetry. Hence we may assign three of the orbitals in the molecule to this symmetry: they are totally invariant under the subgroups C, , C,' , and C," , respectively, and are permuted among themselves by the other operations of C,, . The nitrogen atom in its ground state has the configuration (2s2s'2p3) giving the term 'SU.In a fieldof C,, symmetry these orbitals become (2a,2aI'; 3a,e(')e('))(using labels for the orbitals appropriate for this group), and give a 4A, term. We might therefore by the same arguments as before assume these symmetries for the orbitals in the molecule and indeed we shall return to this point in the next section. But we observe that three orbitals, one of symmetry a , , and two of e, may be combined among themselves to form three orbitals n , , n, , and n3 forming the same permutation representation as the h,, h,, h, above, and furthermore the 4A, wave function is invariant under this transformation. On the basis of these arguments we assign the following symmetries to the orbitals of (104):

a , , a,', n , , n 2 , n 3 , A,, h,, A , ,

(1 05)

which are also just the ones that lead to the formation of a strong bond between the N atom and each of the H atoms, with large overlaps between the pairs (n,h,), (nzA,), and (n, h3),respectively.

SPIN-COUPLED WAVE FUNCTIONS

179

We now form the wave functions (104) and for this purpose first of all consider the P,-,,-diagonal basis with orbitals (105) in the order a1a1’n1h1n2 h, H , h, . For simplicity we assume that the (a,al’) pair of orbitals, which constitute a “lone pair” form a singlet, so that the problem is effectively further reduced to one of six electrons. The five functions in this basis are labeled according to Serber (1934b), and the necessary Upk ( P R )matrix elements that occur on application of the projection operator C O C A 1 ) to them are taken from or calculated from the same paper. The result is y(Ai)

0,o; 1

Ye;;

2

J6 3

= -[Yo, 0 ; 2 = J6Yo, 0 ; 5

+ y o , 0 ; 3 + y o , 0 ; 41 (1 06)

.

If we reduce the representation of C , , formed by functions (104) with orbital symmetry as in (105), we obtain Ap(C,) = 2 A ,

+ A , + E,

from which we see that we have correctly obtained two A , states. The same result is obtained quickly and elegantly using the basis for which Y 6+ 9,x 9,and the orbitals now ordered according to (105). The spin functions are labeled by the indices ( S 1 S , k , k 2 )in which S, = p or 4,and k , , k , are (51) = 1 if S, = +, and (31) E 1, ($0) = 2 if S, = f . We observe that the groups C,, and Y 3are isomorphic, so that a function y O , O ; S ~ S l k l k ~= J 6 ! d ( n l n 2 n 3 h l h 2

h3°06,0;S~Slklkz)

( 1 07)

is the form of a product of two basis functions for irreducible representations of C , ” . Thus the function with S , = 4 corresponds to a product of two A , states, the indices k , , k , labeling the two basis functions and similarly the function with S , = 4 corresponds to a product of two ,E states. Hence in order to obtain a total wave function with A , symmetry, one simply puts k , = k , in each case and sums over this index obtaining

The two sets of functions (106) and (108) are of course connected by an orthogonal transformation of the kind described by Eq. (22). However, from a physical point of view, the P,-,,-diagonal basis (106) is the appropriate one to use when considering the equilibrium properties of N H , , since in this basis the function Y&{A,, corresponds to three electron pair bonds and is expected to make the overwhelming contribution to the total wave function. But when problems involving the dissociation of the molecule are considered,

J . Gerratt

180

one should transform to the 9,x 9, basis (108) since in this limit the wave function goes to the function Yb$ll in the basis, which of course predicts the correct dissociation products.” The transformation between the two bases just described thus furnishes a description of molecular dissociation as a smooth recoupling of spins: from the pair bond function “FA!, in (106) to the “atomic” function YbfA! in (108). The methane molecule forms an interesting but more complex example, but for consideration of this we need to make use of some results of the theory developed in the next section, and to this we now turn.

VIII. The Hund’s Rule Coupling A great deal of the theory developed so far shows obvious similarities to both the spin valence theory of Heitler and Rumer (Heitler, 1934), and to the the valence bond theory of Slater and Pauling (e.g., see Eyring et a / . , 1944). The fact that the orbitals form a basis for a permutation representation, however, means that sets of degenerate orbitals-a feature so characteristic of MO theory-are necessarily absent. It is thus not clear how, for example, a closed shell atom such as neon, which is described in the Hartree-Fock theory by the configuration ls22s22p6, would be described by the present theory, and it is to this problem that we now address ourselves. When one thinks of the process of filling the 2p shell in atoms, one of Hund’s rules tells us that the state of lowest energy occurs when the spins of the electrons couple together to form their maximum possible resultant. This implies that as the shell is being filled, the f i s t three electrons couple to form the resultant S = t , and hence the last three must do likewise. The two subsystems of three electrons each are then coupled together to give a zero total spin. The occurrence of orbital degeneracy thus leads to the consideration of the following type of spin-coupled wave function

& W P + P- PoP+’P-’Po’@60,0:++1I). (109) The superscript (0) on Y indicates the value of the L quantum number, and the spin function is a member of the basis in which ,4a6 -+ 9,x Y , , ~ o(0) , o ; + + I I=

@60,

0 ;+ + I l ( G 1 ,

O2

9

=

..‘

9

a6)

(?dMlM2

loo>@~,M,;

02

a3)@i,M2;

I(a4

9

O5

9

06).

Mi,M2

(M,+Mz=O)

This particular function is in fact identical to the first member of the standard basis for which k = 1 = (tltl+).In the Hartree-Fock theory, the p orbitals l o The function ‘I’”b:d.)2 of (108) in the dissociation limit reduces to a specific linear combination of the zDuand 2Puterms of the nitrogen atom.

SPIN-COUPLED WAVE FUNCTIONS

in (109) are such that p i form

E

const.

181

p+ , po’ = po so that this function goes over to the

J6

! z~(p:p!p~~@j,o; ;otot)

in which pairs of doubly filled orbitals are coupled to form singlets. Thus the physical insight afforded by Hund’s rule disappears on introducing pairs of identical orbitals. This is not so in the more general theory, however, and in this section we examine the properties of spin coupled functions of the general form yS,M;fn+nll=

J G j w 4 1 + 2 ... 4 n + 1 + 2

*..

$neS2:M;+n+nll)

(110)

in which the total number of electrons in 2n, and S = 0, 1, 2, . . . , n. More complicated cases in which functions of the form (1 10) constitute just a part of a more general spin-coupled function will often occur, but for the sake of clarity we concentrate first on (1 10). From its construction, it can be seen that the spin function in (110) is completely symmetric under any permutation of the arguments ol, 0 2 , .. . , on among themselves, and of the o n + 2 ,.. . , oZnamong themselves. This means that the total wave function (110) must be antisymmetric under any permutation of the set of orbitals 41,4 2 , . . , 4namong themselves, and similarly under the $ 1 , t+h2,.. . , $, among themselves. (I shall henceforth refer to the two sets of orbitals as “the qhp ,” or “the t,hp ,” as the case may be). The consequences of this property may be seen by subjecting the 4pand $ p to linear transformations among themselves :

Substituting this into (1 10) we obtain

The antisymmetry within the two orbital sets does not allow any orbital to appear more than once, otherwise this expression goes to zero; the numbers ( v l v z , . . . , vn), (Al,A2 , . . . , A,) are therefore permutations of (12 . n) and so the expression reduces to det(A)det(B)J(2;)!d(4,’

1 . .

$n‘$l’... $,’O~,”M;tnt,,ll) (111)

which except for a trivial constant, is just the same as the original function (1 lo). The transformations A and B may be chosen so as to orthonormalize

182

J . Gerratt

the 4 p and $ p among themselves and without any loss in generality we may always assume this to be the case:

<4,14v>=

= 6,";

P9

" = 1,

2, . . ., n.

(1 12)

From (1 11) it then follows that the wave function (1 10) is invariant under any unitary transformation of the two orbital sets among themselves, so that the 4, and $,, are determined only to within such a transformation. It is well known that this degree of freedom in the definition of the orbitals may be used to introduce yet more orthogonality among them. Let the overlap matrix between the 4, and $,, be denoted by A(aB):

A'!:

=

< 4,I $,>.

Now two unitary transformations A and B of the found so as to diagonalize (1 13) :

(1 13)

4,, and $,

may always be

AtA("P)B = 1, where 3, is such a diagonal matrix (Amos and Hall, 1961 ; Lowdin, 1962). Hence in this transformed basis we have in addition to (1 12)

< 4, I *,>= 4d,,

(1 14)

The matrix A is, in fact, one which diagonalizes A("P)A("P)t,and B one that diagonalizes A(ap)tA(ap).The extensive orthogonality relations (1 12) and (1 14) simplify the calculation of the total energy considerably, enabling one to derive explicit expressions for the Hn, Jn, etc., coefficients (e.g., see Pauncz, 1967; Goddard, 1968a). The total energy given by the wave function (1 10) is, according to ( 5 8 ) , 2n

,,,=I 2n

2n

+ t 1 a ,2r = l J I l ( P V 0 T I Sll). p,v=l (!I#\,)

(115)

(UfT)

For the sake of simplicity, the index ( i n i n 1 1) of the spin coupling has been abbreviated to 1 since this will always be the first member of the Y,x Y n basis. In order to simplify subsequent manipulations, we introduce the following conventions in notation. The symbol 5, ( p = 1, 2, . . . , 2n) is used to denote an unspecified orbital from either set,

5, = 4, = *,,-,,,

for p s n, for p > n .

183

SPIN-COUPLED WAVE FUNCTIONS

The configurational wave function 4 , 4 2 . . . b,, form

*

$,, is written in the

@,ba)n@(D)n, 0

where

We use the symbol @I")" to denote a configuration of the first set of orbitals from which 4c is missing and is replaced by $,,:

@'I"'"= 41(1)42(2) . . . . . 4n(n), and in general @:a?b,,. denotes a configuration in which 4c, 4", 4u,4 r , .. . , $ n b )

*

are replaced by $,,, $ n - 3 , . . . , etc. Similar definitions apply to the $,, set; thus @Lo)"-' denotes a configuration from which orbital is missing: (DID)"-

1

- $l(n

+ 1)$2(n + 2) . . . $ n b ) . . . $ n -

I(2n - 11,

a configuration where I)~-,,,$,-,, t,bT-.,,,. .., are reand @$IT::,p placed by $,,, $n-lr $ , - 2 , ~,b,,-~,. . . , $ , , - p + l , respectively. The H,, and J,, coefficients appearing in (1 15) are thus given by

in which @in-

1

5 n,

= @,l")n@&D)"-

1

if

= @,ba)n@(D)n-

1

if n < p < 2n.

c

Similar definitions are used for the J,, coefficients. The variational Eqs. (73) assume the form 1, v = 1

( 1 19) in which the Lagrange multipliers E?;, EL!) have been introduced so as to preserve the orthonormality of the 4, and $ p during the variation. The d') and dD)are Hermitian matrices which may be diagonalized by appropriate unitary transformations of the two sets of orbitals, and we thus derive the orbital equations

J. Gerratt

184

In deriving (120) we take full account of the orthogonality relations (I 12), so that in contrast to (78) we now have

x

I $v-n)T

n

x

p = 1, 2,

r

l4vh

..., n , (121)

2n

p =n

+ 1 , n + 2 , . . . , 2n,

( n 2 3), (122)

in which the Hn , etc., coefficients are now given in accordance with Eqs. (I 17) and (1 18). Equations (120), however, make no use of the further orthogonality relations (114). To do so one might, in the first instance, add the following terms involving extra Lagrange multipliers to the right-hand side of (1 19):

and hence derive orbital equations of the form

n

F:?, I$~-, )

=

1 [ E ~ ' ? ~ ~ I $ ~ ) + Y~-,,~E~!~~I$~)I,

v= 1

p =n

+ 1, n + 2, .. ., 2n. (123)

The invariance of the left-hand side of (123) under unitary transformations of the 4" and $, may, as before, be used to diagonalize, the dQ)and d B ) matrices. but off-diagonal elements from dup)would remain in both cases.

185

SPIN-COUPLED WAVE FUNCTIONS

The equations could be reduced to pseudo-eigenvalue form by using appropriate coupling operators introduced by Roothaan for this kind of situation, and generalized by Huzinaga (1964). It turns out, however, that as a consequence of the symmetry properties of the orbitals, this procedure is unnecessary. It will be shown presently that in most cases of interest Eq. (1 14) can be made to hold good automatically, without introducing any extra Lagrange multipliers. As a first step in this direction, we define the following two operators, F ( a ) ,F ( p ) which do not depend upon the orbital index p :

c Fp', P ' = 1 Fp2nl$K-n)($K-nl= c F y ) . n

1 F!"'I 4K)(q5Kl=

F('I)=

K =

1

K =

(124)

1

2n

Zn

K=n+

n

1

Ken+

(125)

1

The F',"', F',B' operators in these equations are defined by Eqs. (121) and (122), respectively, with the help of (80). From these definitions it follows immediately that F c a ) M p= ) q?l4,>

=Ep14p),

F ( p ) l $ p - n= ) FLp2nl$,-n)=

p = 1 , 2) . . . )n,

E L ~ ~ , , I $ ~ - , , =) , ~n + 1, n + 2, ..., 2n.

(126)

Thus the 4p and $, are each determined by a single effective operator. The transition from Eqs. (120) to (126) is similar to that in open-shell HartreeFock theory (e.g., see Birss and Fraga, 1963); a result equivalent to this has been obtained by Goddard (1 968a). We turn now to a consideration of the spatial symmetry properties of the orbitals comprising the wave function (1 lo). For this purpose we consider first of all the permutation that interchanges the two sets of numbers (12.. .n) and (n In 2...2n) . Denoting this by Q T ,

+ +

it can be shown that Q:'OH,"M;

= ( - l)n-SOi[tM;

(128)

Equation ( I 28) follows from the symmetry properties ofthe Clebsch-Gordan coefficient ( t n t n M I M 2 I S M ) linking the two subsystems of the spin function. Then the effect of interchanging the two orbital sets in the wave function is given by

J G j ! d ( $... , $ n 4 1 . . . 4n'OHy,t,; 1 ) = C b?l(~,>J(2n)!d ($1 . . . 4 n $ 1 . . . IcIn G ~ , t , ; i > I

= (- 1)'J(2n)!

a ~ ( .4. .4,, ~ . . . $n 02:~;

( 1 29)

186

J . Gerratt

The wave function is thus simply multiplied by a factor (- 1)' upon making this interchange. This property in certain circumstances has important effects upon the symmetry properties of the orbitals. Proceeding now as in the previous section, we impose the requirement that Ys,M ; belongs to a nondegenerate irreducible representation of the point group 9, RyS,M;

in 91,

1 z= CRYS,M; 1

(130)

I.e.,

&2n)!

d ( R 4 , . . . R$,, R$, . . . R$, OinM;

= CRX2n)! d ( 4 1

' ' '

4 n $1

* . ' $ n @::M;

1).

(131)

We suppose that the functions R4, and R$, may each be expanded in one of two complete orthogonal sets of functions ofwhich the functions 4I . . . +,, and i n each case form the first I I members: $1 ...

Substituting these expansions into the left-hand side of (131) we obtain

c rn

VI,

..., v n =

m

1

2.1,

2 ...,A,=

D:, . . . Din" 1

Now Eq. (131) is an identity so that (133) must also be such. We may therefore pick out all those configurations 4 v l . . $,,n . . . $vn on the left-hand side of (133) that consist of just the ground state orbitals, and equate just the sum of these to the right-hand side. This procedure leads to the equations for all 1 = 1, 2, . . .,fin,(134) in which P ( a )is the permutation corresponding to the numbers (vlv2...vn), and Pea) the one corresponding to (1112 .. .An). The sums in ( 1 34) are over all such permutations. The sum in (133) over all the remaining, excited configurations must be identically zero. By picking out all the terms involving a particular excited configuration in which 4,, say, is replaced by 4 p and $" by t,hq ( p , q > n), we are led by the same arguments as used in the previous section to the equations

187

SPIN-COUPLED W A V E FUNCTIONS

showing that the two sets of orbitals of the ground state configuration each transform linearly among themselves under an operation R and generate representation matrices D'(R) and D"(R), respectively. Since the 4, and $, are both orthonormal, these matrices may be taken to be unitary. In addition to this, we observe that matrix elements U c ( P )are, for P = P(")P(p) as in (1 34), only nonzero for 1 = 1, so that this equation becomes

5,

=

1

US1(P("))D:,, . . . D;,, USl(P'p')Di'l . . . Din,,

P("1, P(P1

= det

(D') det (D")

= +1.

These results are consequences of the two expansions (132), but there is no reason why these should not be reversed since both of them form complete sets. One then has

c B&

v= 1

c B;[, 4 A . m

00

R4p=

*v;

W P= A=

1

(136)

Substitution of these into Eq. (131) now leads to

Vl,...,

5

v,=l

A1

f

,...,1,=1

B:l . . .B", v n B;,

' * '

Bin,,

and by reasoning similar to that given above, we derive in place of (134) and (135) the results

and

1

U,l(Q,P'a'P'B))B~,l...BknnB;Ill. . . B i n n= d l l c R .

P("1. P ( B 1

(139)

The representation matrices B'(R), B"(R) now generated may also be taken to be unitary, and Eq. (139) reduces to

5,

=

Uf,(Q,) det (B') det (B")

= +I.

In Appendix D it is proved that the orbital operators F(") and F ' p ) are each invariant under any unitary transformation of the 4, and $, sets among themselves, and that the two operators are converted into each other by the operation (2,. One thus has two choices for the symmetries of these orbitals:

I88

J. Gerratt

In the first case the two sets may each transform into themselves according to Eq. (135) under all operations of the group. This means that the 6, and $ p are eigenfunctions of totally symmetric operators so that the two sets may each be taken to form bases for irreducible representations of Y. On the other hand, the choice of Eq. (138) admits the possibility that under some operations R, the two sets of orbitals are interchanged by the permutation Q, and then transformed linearly among themselves. One may therefore take Q, to correspond to some operation of the group (which we will simply denote as Q, to avoid confusion), and so a special kind of permutation representation is generated. For this to be possible, Y must contain a subgroup X l l z ,such that

9 = Xilz + QrXi/z

( 140)

7

and hence the order of X l l zmust be exactly half that of 9. The subgroup is thus known as a " halving subgroup." In such cases the 6, and $, transform irreducibly according to (135) under all operations R in the subgroups X'l,z and Q; fl112 Q,, respectively, and are interchanged by Q,. This choice of symmetry therefore depends upon whether or not Y possesses a halving subgroup. Should 3 not possess such a subgroup, then the orbitals are forced to be symmetry orbitals of Y. Table 1 lists all the common point groups that TABLE I COMMON POINTGROUPS POSSESSING HALVING SUBGROUPS

g C Zh c 4 h C6h DZh

D3d D 4 h D 6 h o h

Order of Y 4 8 12 8 12 16 24 48

21,2 C S

Example Trans-dichloroethylene

s4 c 3 h CZ" CS"

C4",D Z d D3h,C6u

Td

Ethylene Ethane (staggered conformation) Cyclobutene Benzene SF6

possess a halving subgroup; it should be noted that D,, , the point group of the benzene molecule, in fact possesses two such subgroups. It is worth summarizing briefly the work so far in this section. The two sets of orbitals, 6, and $, that comprise the Hund's rule coupled wave function (1 10) are determined by the pseudo-eigenvalue equations

SPIN-COUPLED WAVE FUNCTIONS

189

The two sets may either be taken as bases for irreducible representations of 9,or if 9 possesses a halving subgroup, they may be chosen to form a special kind of permutation representation: the 4, transform irreducibly under X1,,, and the $, irreducibly under Q; X1,,Q,, the two sets being interchanged by the operations of 9 corresponding to Q I . As an example of these considerations we take a closed-shell atom such as neon. We choose the following spin function: 10

@O,O;SISlkll

=

1

(S1S1M1M2

I o O ) @ ~ ~ , M i ; k i @ ~ l M1~ ;

(141)

MI,M2 (MI+M2=0)

in which the outer six electrons are coupled by a Hund’s rule function, this function itself being coupled to some arbitrary spin function for the four inner electrons. The intermediate spin S, may be 0, 1, or 2, the function with S1= 0 being the most important since the others correspond to highly excited inner shells.” The symmetry group of the atom is, of course, the full threedimensional rotation-reflection group, K , , and this has no halving subgroup. The outer electrons must therefore be described by symmetry orbitals of K , of which the lowest in energy are the two sets of 2p orbitals (2p,, 2p+ , 2p-), and (2p0’, 2 p + ‘, 2p-’). The physical picture that emerges from these considerations is of two slightly different sets of 2p orbitals orientated toward each other so as to minimize their mutual repulsion. One would write the most general function of this form for Ne as

giving a total of six terms in the sum, this number being reduced to two if S, is restricted to zero.12 An interesting example of the use of the Hund’s rule coupling occurs in the six n-electrons of the benzene molecule. We denote these orbitals by 41, 4 2 , .. . , 46. The point group of the m o ~ e c u ~iseD6, which possesses two halving subgroups, D,, and C,, , so that we now have three possible choices for the symmetries of the orbitals. If the orbitals are chosen to have D,, symmetry, then the two sets of functions (4142 &) and (44 4546) each span the A,” and E” representations of this group and are interconverted by the operations C , or 0 ” ’ .The general form of these functions is such that they Functions such as (141) with S, # 0 may however become important in describing hyperfine structure in atoms such as N(4S)and 170(3P). I Z An alternative possible description would be to use a single Hund’s rule function for all ten electrons, thus obtaining two sets of five symmetry orbitals Is, 2s, 2p0, 2p, , and Is’, 2s’, 2p0’, 2p,’. However this function is not expected to furnish a useful description of the ‘ S ground state of neon as it corresponds to a most “ unphysical ” mode of coupling the spins: One would obtain from this a buildup principle that predicts 6S(ls2s2p02p+2p-) as the ground state of the B atom!

190

J . Cerratt

alternate in amplitude around the ring; the orbitals of the first set show peaks of maximum amplitude at, say, carbon atoms I , 3, and 5, while those of the second set show similar peaks at carbon atoms 2, 4, and 6. This is just the “alternate molecular orbital” (AMO) method of Lowdin and Pauncz (e.g., see Pauncz, 1967) which was first proposed on intuitive, physical grounds and is now seen t o be a natural consequence of the general theory. If we adopt the 9,x 9,basis of spin functions, then besides the Hund’s rule coupled function, with index (3:;fl ;+l), the other ones compatible with the D,, symmetry of the orbitals are (++;fl ; f l ) and (f+;+O;tO). The most general function for the six orbitals is therefore a linear combination of these three coupling^'^ (Tamir and Pauncz, 1968). If the orbitals are chosen to have C,, symmetry, each set spans the representation El and A , . The two sets are concentrated on opposite sides of the plane of the carbon ring and are interconverted by the operations G~ or i. This choice of symmetry thus introduces some vertical correlation between the electrons and is of the same kind as that introduced in Dewar’s split p orbital method (e.g., see Dewar, 1963). The method was originally proposed on intuitive grounds also, and is again seen to be a consequence of the general theory. The third possibility is for the two sets of orbitals to form bases for the full symmetry group so that each spans the representations E l , and A , , of D,, . In this case one set of orbitals will be concentrated close t o the ring of carbon atoms, and the other set further out. A kind of “radial” correlation is thus introduced. Which of the three choices for the symmetry of the orbitals provides the best description of benzene can, of course, only be decided by actual calculation. Having done this, some of the correlation provided for in the other two choices may then be introduced by a limited configuration interaction calculation. The presence of orbital degeneracy and the Hund’s rule coupling has a characteristic effect on molecular binding and as an example of this we consider the H F molecule. We disregard the four most tightly bound electrons and concentrate attention upon the six outer ones. As the internuclear distance “

”

“

”

l 3 It should be noted that much of the simplicity of the A M 0 scheme for this molecule depends upon the use of complex orbitals, for in such a basis the matrices of the E representations in both D6,, and D3hhave the form of permutation matrices introduced in Section

where 6 exp (7743). The spin coupled function ($&;$O;&O) is then invariant under operations in DBhsince these are equivalent to permutations PiZP,,with the phase factors 5 cancelling out. If one converts to real orbitals this attractive simplicity is impaired to some extent since either one must use complex spin functions, or this last spin coupling becomes a linear combination of nine configurations (including doubly occupied orbitals) with fixed coefficients.

VII, e.g.,

191

SPIN-COUPLED WAVE FUNCTIONS

R becomes very large, the molecule is appropriately described using the standard spin functions since these reduce 9,to 9,x 9,. Hence as R + co ,

in which h stands for a Is orbital of the hydrogen atom. The function (142) represents a fluorine atom in its ground ’P state (with M , = 0) coupled to a ’ S hydrogen atom to form a ‘Z+ state of the molecule. Now as R decreases towards its equilibrium value, the molecule is better described by wave functions of the kind :

in which the orbitals I n + , I n - , 30, 1n+’, ln-’, 30’ correlate with the separated atom orbitals p + , p- , po, p+’, p-’, and h, respectively (see Section IX). The spin function is of the type

xi,, ,

,

where is a four-electron Hund’s rule function, and @i2, Mz; may be a singlet or triplet function. This set of functions is chosen so as to preserve the Hund’s rule coupling for the n orbitals in the molecule, since the interatomic interaction may only break down this coupling for the 0 electrons. The overall spin S is of course zero, so that in (143) S, must be equal to S, . The function with S , = 0 then describes the HF molecule as having two closed shells, consisting of the (30, 30’) and ( l n + l n - l n + ’ 1 ~ ’ orbital ) sets-a description similar to that afforded by the MO theory. Hence in passing from the separated atom wave function (142) to the molecular function (143), one must allow not only for the deformation of the orbitals, but also for the recoupling of the spins that occurs, with the (30, 30’) pair forming a singlet and being primarily responsible for the bonding. The total wave function for the molecule must therefore be written as a linear combination of two functions yo.0

=

c

s1=0,1

~sI~o,o;sIs1113

(145)

since one can show that these two components form the separated atom function (142); the contribution from the term with S, = 1 is no doubt negligible at equilibrium as far as energy is concerned. The symmetry group of the molecule, C,, , possesses no halving subgroup so that the n orbitals occurring in these wave functions are symmetry orbitals of the full group. (Indeed C,, possesses no subgroup at all which would give a finite expansion (94), and hence all orbitals not coupled by Hund’s rule

J . Gerratt

192

functions must be CJ orbitals). The two sets ( l n + , 1z-) and ( lz+', 171-') are therefore not interchanged by any symmetry operation. Finally in this section we re-examine from a group-theoretical point of view the overlap matrix Acas)Eq. (1 13), between the 4, and $, sets. Let us assume first that the two sets form bases for irreducible representations of the whole group. Thus for any R of 9,

and

R$,

=

z

[A- 'D'"'(R)Al,,

$r,

(146)

in which r and r' denote two irreducible representations of Y. Note that one cannot, in general, assume that the two sets transform identically, but only that they form equivalent bases as expressed in (146). Then

By using the orthogonality theorem for irreducible representations, the term inside the brackets is reduced to

Jr

where fr is the dimension of the representation I-. Hence

= S,,.const.

A,, .

(147)

Then as long as no representation r is spanned more than once by the 4, and $, ,the condition that the A(ap' matrix should be diagonal is that A should be diagonal, i.e., that the 4, and $, sets transform identically. If the orbitals are constructed so as to fulfill this condition, then Eq. (1 14) is automatically obtained. There is then no need to introduce extra Lagrange multipliers to preserve this orthogonality property during the variation process as in (1 23) and pseudo-eigenvalue equations of the form (126) are thus always obtainable directly. As an example of this we consider a closed-shell atom, in which the two

193

SPIN-COUPLED WAVE FUNCTIONS

sets of functions in a given shell, I, are orientated at some angle towards each other: 4 n 1 m = Rn,(r) Yfrn(6,4)

4’).

= R;,,,(r) YLfrn,(6’,

If we assume that the $’s are at an orientation relative to the the Euler angles a, p, y, then

4’s specified by

1I ‘

and therefore (4nrm

I +n,f,m,>

= sii,
(148)

Maximum orthogonality between the sets of functions is achieved by rotating the two coordinate systems until they coincide. The angles a, b, and y are then zero, and the DtL. (a, p, y) factor reduces to Smn1,. Note that this orthogonality may also be obtained with less restrictive conditions under which /3 is 0 or 271. This means that the two sets of functions are referred to a common z axis, the one coordinate system being rotated about that axis by an angle c( + y relative to the other. This is true only of bases constructed from spherical harmonics, however, (eigenfunctions of j 2 and 2,) and does not hold for a basis of real orbitals. Thus in the case of the 71 electrons of the H F molecule discussed above, the two sets of orbitals ( l ~ +I T, - ) and (In: , 171): may be rotated through some angle x , say, relative to each other, but even so there isoverlaponlybetween the In,, Irr+’and I n - , 1 ~ ’ o r b i t a l s . If real orbitals (In,, 1zy) and ( 1 71,’ , I ny’) are used, maximum orthogonality between the two sets is obtainable only if c( = 0, i.e., if the two coordinate systems coincide completely. This remarkable simplification also occurs if 9 possesses a halving subgroup JE‘l,zand the 4, and $,,are chosen to be invariant under P l 1and 2 Q;1Af‘li2 Q , , respectively. This is because is always an invariant subgroup of 9 so that JE‘l,2 and Qr-’Pl,2Q, are in fact identical. Then since the and $, are invariant under the same subgroup and are interchanged by Qr, they must transform identically. Thus letting y, y’ label the irreducible representations of . f l l j Z ,we have for any R of this subgroup,

+,,

R4,, = Hence and

c Dg)(R)4A.

194

J . Gerratt

If we denote Q;’RQ, by T-an also in &?,/2-then

= const

operation which by the reasoning above is

d,, .

We see that if no representation y is spanned more than once by the 4Po r $ P , then the orthogonality (1 14) is always obtained automatically as a consequence of the symmetry properties of the orbitals. The truth of (150) can most easily be seen in the case of TI orbitals of homonuclear diatomic molecules. The symmetry group is D,, , and = C,, . Four TI electrons could therefore be accommodated in two sets of orbitals (n+, n-), ( T I + ’ , n-’), invariant under C,, and interchanged by the operation ch. It is clear, however, that the only overlaps will be between t h e n + , n,’ and T I - ,n-’orbitals. The orthogonality ( 1 14) is a basic feature of the A M 0 method for the benzene molecule, but it has always been assumed that this is due to the special construction of the AMO’s from MO’s. However Eq. ( I 50) now shows that in fact the orthogonality arises quite naturally as a property of a Hund’s rule coupled function.

IX. The General Recoupling Problem and Bonding in Molecules In a number of molecules one must, in order to give a n adequate account of the bonding, combine the orbital symmetries allowed by the Hund’s rule coupling with the permutation representations discussed in Section VII. This process is best described by a n example, and the ‘ A , ground state of methane is particularly suitable for this purpose. We ignore the two Is electrons of the carbon atom and treat the molecule as a n eight-electron system. We imagine the molecule as being formed by first coupling four tetrahedrally placed hydrogen atoms to give a prescribed resultant term, 2 S r + 1ofr za n, (H)4 complex and then combine this with a n identical term of the carbon atom to yield a total function with symmetry ‘ A , . The 9,x 9,basis of spin functions is most convenient for the description of this process since the groups T, and 9, are isomorphic. Thus the wave

195

SPIN-COUPLED WAVE FUNCTIONS

function for the (H)4 complex is given simply by

J4!d(h,h,h3h,0,4,,M2;k2)

(151)

in which the h, ( p = 1, 2, 3, 4) are 1s functions of hydrogen. For S2 = 0, this function gives a ‘ E term, the two values of k , labeling the basis functions of this representation, and similarly S , = I and S , = 2 give 3F1and ’ A , terms, respectively. As the C-H internuclear distances decrease towards the equilibrium value, each of the functions 17, goes over smoothly to one of the orbitals of the molecule, four of which must therefore form this same permutation representation. Thus each of these four orbitals (which for simplicity we continue to denote by h,) is completely symmetric under all operations of one of the four conjugate C3, subgroups of Td. Turning now to the carbon atom, we see that we must construct ’ A , , 3F1, and ‘ E terms of it since only these may lead to a ‘ A , state of the whole molecule. The lowest configuration of the atom (in a tetrahedral field) is of the type (al ’ f Z 2 ) , which may be formed from both the primed and unprimed sets of orbitals, a,, u,’,f$’), andfi(”’, the indices 2, p, v, . . . , etc., now being used to label the basis functions of representations. From this configuration we may obtain a 3F1(the ground state) and a ‘ E term as follows: y , (l F, M l ) ,r ; 3 =

c

(F2 ~ , ~ ~ I F , ~ ) J ~ ! ~ ( ~ l ~ l ~ ~ ” ’ f , ( ~ ) ~ ~ , M , ; t o t - >

a. P

in which the quantities (AA’pvl A”z) are coupling coefficients for point groups analogous to the Clebsch-Gordan coefficients in the rotation group (values of them for all the 32 point groups have been tabulated by Koster et al., 1963). If we now by the same arguments assign the symmetries a,, a,’, and f 2 to the remaining four orbitals in the molecule, we may combine functions (151) and (152) to form the following total wave functions:

YL:~);,

=

1

( ~ 2 ~ , ~ ~ l 1 ~ z ) J 8 ! d ( ~ l a i ~ ~ ’ ~ l fh ;3 h ‘ ”4 0 ) :h0 I; 0h02 2~),

A,,,

(153) in which the index k , has in each case been replaced by z to emphasize its role as a label for the basis functions of representations. However, neither of these functions (153) is expected to lead to a useful description of methane at equilibrium since they cannot correspond to four C-H bonds and do not have a closed-shell structure (in the sense of a single

J. Gerratt

196

configuration whose orbitals span complete representations). Rather, as is well known, we should consider the next lowest lying configuration ( a 1 f z 3 ) , from which we may obtain the terms ’ A , (the lowest state of this configuration), 3F1,and lE:14

Y y 2 , ;1 = J4!d(f,“”~”3’a10,4,Ml,tlt)

x

f i !. r Q ( f p p p ) u l O : ,

(154)

0;+ot).

The three f, orbitals of the 3F1term are here coupled according to the scheme {(fZ)’E ; f,;Fl}, while those of the ‘ E term are coupled according to {(f2)’ F, ;fi; E } . We now assign the symmetries a,, fz,f,, and f 2 to four of the orbitals of the molecule and derive the complete wave functions ‘Yy’d,3

d ( f : ” f p f j 3 ’ a , h 1 h , h, h, @:,o;

=JS!

2211)

z [z

%%;4 = A , P . v , x

JS!

x $!

( F , F , AP I Eo)(EF2 gv IF,?)]

0

d(f~A’f;(p)f~v)a 1 h 1 h, h3 h4 0:.

0;1 1 3r)

(155)

d ( f ~ A ) f ; ( p ) f i ( v ) h, a , hh3l h4 @, o; O O Z r ) .

The most general wave function for methane obtainable from the two configurations of carbon under discussion is then a linear combination of the five functions of (153) and (155): y‘A” 0,0

-

5 dk k= 1

ybtb’;

k

.

Of these it is seen that only the function Y y $ ;3 , derived from the 5 A z state of carbon, is a single configuration and most treatments of this molecule are based upon it. Yb.f$, is recognized to be just a Hund’s rule coupled function and hence it is invariant under linear transformations of both the a 1 f z 3and l 4 A different selection of primed and unprimed orbitals will also give 3F1and ‘ E terms of this configuration, but the ones given in (154) are those expected to yield the lowest energies.

197

SPIN-COUPLED WAVE FUNCTIONS

h,h2 h, h4 sets among themselves. Thus one may transform the h1h2h, h, so as to form a second a1f23 set, the function now assuming the usual Hund’s rule coupled form. If the two a l f i 3 sets are further made to coincide, Ybf,): reduces to the MO wave function with the configuration (a12f26) for the occupied orbitals (cf. Herzberg, 1966). On the other hand, we may transform the a ,f 2 3set of Ybf,); so as to form a set of tetrahedrally orientated orbitals t,, t, , t3 , t , , which therefore form the same permutation representation as the h,, and base our treatment of the molecule upon the configuration

,

( 4 t 2 13 t4 hIh2 h3 h4).

We now proceed in a similar way to that described for the NH, molecule at the end of Section VII, and obtain in place of (155) the three functions:

%Y; 3’ = yo,0; 2 2 1 1 (A YO,

b);4,

+ ~ 0 , 0 ; 1 1 2 2+ ~ 0 , 0 ; 1 1 3 3 = y o , 0;001 1 + y o , 0 ;0022

=~

y(Ai) 0 , o ; 5’

(156)

o , o ; l l l l

*

This set should provide an excellent description of methane for calculations at or near the equilibrium nuclear configuration. For this purpose, however, it is more convenient to transform to the P,-,,-diagonal basis with the orbitals in the order t1h1t2h2t3h3 r4h4. Using the methods of Section VII one derkes

J6

y(Ai) 0,0;4”- -

3

0 ;6

Yyb),5’’= 2J6Y0,,;

+

14,

0;8

+

0;9

f

0; 1 1

+

0; 12

+

0;1 3 )

(157)

where the functions on the right-hand side have been ordered according to Serber (1934b). The function Ygb!,sf, in this basis corresponds to four tetrahedrally directed bonds between the carbon and the four hydrogen atoms, and is expected to make the overwhelming contribution to the total function. However, the 9,x 9,basis (156) has the advantage of emphasizing the fact that according to our present treatment, the molecule dissociates to the 5 A 2(or ’SU)state of the carbon atom. This state which lies at 33,735.2 cm-’ ( = 4.2 eV) above the ground state is thus precisely the often-discussed “valence state” of carbon (e.g., see Herzberg, 1966). The full treatment of methane, Eqs. (1 53) and ( 1 55) is somewhat simplified if we allow the primed and unprimed orbitals a,, q’,f 2 , and f 2 ‘ to coincide

198

J. Cerratt

and thus give some doubly occupied orbitals. The total wave function then reduces to the Hartree-Fock function for carbon in its (s2p2)3 P g r ~ u n state d as the C-H internuclear distances increase. In this approximation, Eqs. (1 53) and (1 55) are generalizations of functions first constructed by Kotani and Siga” (1937). An equivalent complete treatment was carried out by Voge (1936) using instead the set of tetrahedral orbitals t , , t,, t , , t 4 , and indeed three of his functions coincide with the set (1 57). These two early calculations used a minimum basis set of four hydrogen 1s functions, and Slater 2s and 2p orbitals for the carbon atom. At that time all overlap and multicenter integrals had to be neglected throughout. The difficulties experienced in formulating an adequate description of methane are fundamentally due to the fact that four of the orbitals of the molecule in its equilibrium nuclear configuration ( t , t , t , t 4 ) , must go over into the orbitals (a,a,’f,(”lf,’”))of the carbon atom as the hydrogen nuclei are removed. The passage from molecule to separated atoms thus involves an actual alteration in the symmetry of these four orbitals and one needs at least two configurations to describe this. Such problems, happily, do not occur in diatomic molecules since the orbitals in them may always form the same permutation representation as in the separated atoms. We may therefore extend the theory more comprehensively. Thus having discussed in the previous two sections the possible symmetries of the orbitals, we are in a position to draw correlation diagrams for the orbital energies in diatomic molecules in a similar manner as in MO theory (Herzberg, 1950). This is shown in Fig. 5 for the homonuclear case. The orbital energies of the separated atoms are drawn on the left-hand side of the figure, and are denoted by Is, Is, 2s, 2s‘, . . . , etc. The orbitals in the molecule are assumed to possess C,, symmetry so that primed and unprimed pairs of orbitals are reflected into each other by the operation a h .Thus for the internuclear distance R very large, the l a and lo’ orbitals are expressible as Is, + %lsband Is, + Als,, respectively. At the other extreme when R -+ 0, this choice of symmetry causes the orbital pairs to become identical with the (la, la’) pair, for example, collapsing to the configuration 1s’ of the united atom. The united atom orbital energies plotted on the right-hand side of the figure represent therefore just the familiar Hartree-Fock orbital energy levels each of which may accommodate t ~ i welectrons. The orbitals in the separated atoms and in the molecule, of course, may contain only one electron each. Because of this choice of symmetry for the orbitals, the correlation diagram is in fact similar to the one in MO theory that refers to heteronuclear diatomic molecules. The g, u symmetries characteristic of D,, , are absent so l 5 These authors also considered the interaction of thefZ4 configuration the presence of which, although of very high energy, restores to the total wave function the invariance under any linear transformation of the a, andf2 orbitals.

199

SPIN-COUPLED WAVE FUNCTIONS

3s-’

-3P

United atom

FIG.5. Correlation diagram of the orbital energies in a homonuclear diatomic molecule, assuming C,, symmetry for the orbitals.

that the Iso’, 1s,’ orbitals of the separated atoms give risetoapairofmolecular spin-coupled orbitals that eventually collapse into a united atom 2s orbital (instead of a 2p0 as in the case of D,, symmetry). This pair is denoted by (16, 16’) in which, it will be recalIed, the tilde indicates the presence in each orbital of a node in a plane perpendicular to the interatomic axis. Such nodes must necessarily be present, at least when R becomes small, since the united atom limit of 2s is orthogonal to the united atom 1s orbital. The (16, 16’) pair in a first approximation may therefore be written as (lsa’ - Rls,’, Is,’ - Also’). Similar considerations hold for the higher orbitals. Figure 5 has, of course, only a symbolic value and serves merely to give an indication of the expected order of the orbital energies, particularly toward the left-hand side of the diagram. The energies near the united atom limit are probably less

200

J. Gerratt

useful since if one were interested in the region of very small R,the choice of full D,, symmetry rather than C,, for the orbitals (or C,, with some configuration interaction) may be expected to give a more useful description of the system.

2

.3dJ

"-3d

-3P' P

3

'

:

+ \.-

3 -s/ -3 s

Is:.

United atom

FIG.6. Correlation diagram of the orbital energies of a heteronuclear diatomic molecule.

A similar correlation diagram may be drawn for the heteronuclear case as shown in Fig. 6. One has no choice here, for the symmetry group of the orbitals this necessarily being C,,;, so that except for the doubling of each orbital energy the diagram is the same as the one in the M O theory. The united atom is now a spin-coupled function, so that the passage to this limit does not introduce any nodes in the molecular spin-coupled orbitals, as in

20 1

SPIN-COUPLED WAVE FUNCTIONS

the homonuclear case. Since in the absence of orthogonality constraints there are no nodes in the ns, 17s‘ orbitals in the separated atoms, one expects that there will be none either in any of the (no, no’) orbitals of the molecule. This discussion of how the orbital energies correlate at the two extremes, though useful in some respects, still leaves open the question of how the orbitals are coupled. If the total wave function is to be written as the most general linear combination of spin functions consistent with the symmetry of the orbitals, then the choice of any complete basis will lead to an identical wave function. From a physical point of view, however, some bases of spin functions are obviously more “physical than others-in the sense that the coefficient cSkof a single spin-coupled function in such a basis is z 1, and all others are negligible to a first approximation. Contributions from the other spin couplings in the basis may then be taken into account to describe, say, hyperfine interactions, or the recoupling of the electrons that occurs as the molecule dissociates. In what follows I put forward a number of physical arguments to arrive at a basis of spin functions that one may reasonably expect to be particularly useful for the discussion of electronic structure and bonding in diatomic molecules. There is no difficulty in extending these arguments to polyatomic molecules. Consider a diatomic molecule in the limit as R co . In this limit, the wave function for the molecule must describe essentially two atoms in prescribed states suitably coupled to give a molecular state. A basis for which 9, Y N xa Y N bwhere , N , , and N , are the numbers of electrons associated with the two atoms A and B, respectively, would therefore be a sensible choice for the spin functions. As R decreases toward its equilibrium value, the interaction causes some of the electrons to recouple, but we may expect that this will be confined mainly to those electrons in the valence shells of the two atoms; in comparison, the coupling of the electrons in the two atomic cores remains relatively undisturbed. As a physical model therefore, it is reasonable to divide the electrons of the diatomic system into four sets: n(t’ electrons in the core of atom A , n:”’ in the valence shell of A , n:”) in the core of B, and n$‘) in the valence shell of B , where ”

--f

--f

and of course,

the total number of electrons in the system. I n conformity with this, the basis of spin functions is chosen to be one in which Sf’, -+ YnlC,, x Yn,c,, x Y,,c,~ x Y,,,,,, . Such a basis involves the coupling of four intermediate spins, and it is necessary to specify both the coupling modes inside each of the four

202

J. Gerratt

components, and the mode o f coupling between them. We therefore adopt the following notation for the functions in this basis: O ~ , ( ~ ) h 2 ( o ) k , ( b ) k z o ( ( s y ) s(S?) y ) ) SSay; ) s , ; SM). (158) This indicates that two standard functions, which we write (SI“),k‘,“))and ( S y ) ,k y ) ) ,associated, respectively, with the core and valence shell of atom A are coupled to give a resultant spin S a . Analogously, (SI”’, kib)) and (Sib), k:”) give a resultant spin Sh for atom B, and S,, S b then form the resultant S M for the whole system. The molecular wave function as R -+a may thus be written y ~ ~ ~ ) k 2 ( 0 ) k , ( b ) k 2 ( b ) ( ( S ( P ) S ( 2 n ) )(s:b’s:b’)Sb; S,; SM)

=

a!

(s\b’s$b’)sb;SM)],

~[(D~’(D~’~~b’(DD(2h”,(~~)kz(”]kl(b)‘kr(b)((S(P)S(;1))Sa;

( 1 59) in which a?’, (I)$‘), my),(Dy) describe the orbital configurations in the core and valence shell of A and the core and valence shell of B, respectively. It should be emphasized that the set of funtions (159) with all allowable values of k(:’, kp’, kib),k:“’, s?),SY), S,, Sib', Sib),,and Sb forms a complete basis for the given S M . From a physical point of view, however, by far the most important members of the set are those in which the two cores form closed shells so that S y ) = S\b)= 0 , and Sy’ = S o , Sib) = Sb. Let us now attempt to follow what happens t o functions of the form (159) as R decreases. In order to put (159) into a suitable molecular form, it is first necessary to recouple the intermediate spins S(la),SYl, S:”, and Sib) as follows. According t o the usual theory of angular momentum (e.g., see Brink and Satchler, 1962), we may write the spin functions ( I 58) O,”,( “ ) k * ( “ ) k l [ b ) k r ( b ) ( ( S : ( I ) S y ’ ) s , ; (S\b’S‘,b’)Sh; S M ) = O ~ ~ ( “ ) k l ( f ~ ) k 2 ( ~ ) h 2 ( b ) ( ( S : a ) S : b ) ) S : ( l b ) ;(s‘;”s~b’)s~”’; SM) S

,

1s2

( u b ),

( ub )

s (s~’sy’)s,; (s\”S~’)sb; s).

x ( ( S y ’ S ~ ’ ) S y b ’(; S y ’ S ~ b ’ ) S ~ h ’I ;

( 160) The spin functions occurring on the right-hand side of (160) now describe states in which the two cores are coupled t o form a resultant Sy),and the two valence shells to a resultant S y b )with an overall spin S as before. The transformation coefficients, which we denote by ( a b ; ah; S I aa; bb; S ) , for short, are directly related to the Fano X function o r Wigner 9 j symbol:

( a b ;ab; Slaa; bb; S )

+ 1)(2S:””’+ 1)(2S,,+ i)(2Sb + I)]X

= [(2S‘,‘“‘)

SPIN-COUPLED WAVE FUNCTIONS

203

This transformation is particularly simple if S y ) = Sib’ = 0 since the lefthand side of (161) in this case is just unity, and there is no summation in (160). The wave functions (159) may therefore be put into the form 1Ib ) k 2 ( b ) ( ( S y ’ S f ’ ) S a

Ifli:L)k21‘L)k

x =

; (S‘,b’Sp’)Sb; S M )

((sy’sy)syb’;(S:”’S:h’)S:”h’;

C

SM)

( 0 6 ; a b ; S ~ U Ubb; ; S)JN!d[@,‘,“’@\”’@~)@:b’

S I ( U b ) ,s 2 ( U b )

x @,”,

(U)k

(b)k,(.),,(b)((s~’s~b’)syb’

;

(sy’s$,b’)s~b’ ;S h f ) ] .

(1 62)

Now the recoupling of the electrons in the valence shells that occurs as the x Y , , , , b ) ] coupling, interaction becomes strong breaks down the [Yn2(=) and a single set of spin functions ( S p b ) k)-which , may possibly include Hund’s rule coupled functions where appropriate-is more suitable. The transformation to such a basis is made in a similar way as is described by Eq. (22): @:l(u)k

,(b)k2(u)k2(b)(

=

(S‘p’S:b’Syh’; ( S y ’ S ~ ’ ) S : “ b; slbf) ’

1(Syh’/I S~’S:h’~~’k:h’)~~l((,),l(b)~((S(P’S~b’)s~b’; Syb’ ; S M ) . (163) I

The spin functions appearing on the right-hand side of (1 63) are members of the basis in which Y , + Y,,,,,)x Y , , , , b ) x y n 2 +( ” n2) ( b ) , so that there are now only three intermediate spins: those of the two cores, Sy’ and S?), and that of the complete valence shell of the molecule, Sfb’. Each of the orbitals occurring in the wave functions (159) or (162) correlates uniquely with some orbital of the molecule as prescribed in Figs. 5 and 6. The orbitals in the core configuration @y’@y),however, merely deform on molecule formation, whereas those in the molecular valence shell in addition alter their ordering during the recoupling. This is expressed by writing @:“)@(b) 2

= Q@(ab)

(164)

in which denotes the configuration of the molecular valence shell, and Q is the permutation associated with the recoupling process. Substituting (163) and (164) into (162) we obtain for the wave functions (159): @(Oh)

204

J . Gerratt

The set of functions occurring on the right-hand side of Eq. (165) is proposed as a particularly useful basis for the description ofdiatomicmolecules : \Ykl(u)k,(b)k((S(P)S\b))S(Pb); S y b ) ;S M ) - JN! &[@,(P)@{b)@("b)@N ki(")ki(b)k ((S'")S@))S(Pb); 1 1 Sf");SM)]. (166) The wave functions in this basis are characterized by two cores whose spins SY) and Sib' are coupled to a resultant Syb),and by a molecular valenceshell function with a resultant spin SYb'; the two spins S(leb) and S y b )then couple to give an overall spin for the molecule of S. A useful first approximation is to take those functions in the set (166) for which SY) = Sib)= 0, as these will undoubtedly make the major contributions to the total energy and spin-dependent expectation values. Finer interactions that occur particularly in molecules where S # 0 may be dealt with by taking into account further functions with non-zero values of the core spins. In a similar way, a correct description of molecular dissociation is obtained by using sufficient functions in a linear combination so as to cover those values of k , SYb),and Syb)that appear in the summations in (1 65). As can be seen, the determination of these values is a purely group-theoretical problem and can always be solved beforehand. The set (166) is already symmetry-adapted for heteronuclear diatomic molecules. In the homonuclear case, however, one must ensure that the functions are multiplied by f 1 under the operation o h . For this purpose it is necessary to take in the first instance the linear combinations Jf[\Yk,(.)k,(b)k((S'p)S:b))S~b);

& (-

S(;lb);S M )

"b)+Slcu)+s,(b)tnlca, I)S1(

y k

I( b)kl

(a)k((

S\b)Sy))Syb) ; Syb'; S M ) ] ,

which are multiplied by f 1 when the two core configurations Q'f) and Q\b) are interchanged. The valence shell functions must then be symmetry adapted separately as described in Section VII. As a simple example we consider the Li, molecule in its 'Xi ground state. According to Fig. 5 the electron configuration of the molecule is lolo'li7li7' 2 ~ 2 0 ' In . this particular case we might regard all six electrons as belonging to the molecular valence shell, so that as R + a3 we should use the 9,x Y , basis of spin functions. The wave functions in this are therefore of the form. ~ & ~ ~ ; s klr ks z z=

J6!

~(toti72alo'1i7'20'@~,~~s~sz k,kz),

( 1 67)

the superscript (a) indicating that these correspond to the asymptotic form of the molecular wave function. If we denote the members of the set (167) by just their indices (S1S2; k , ; k,) for a moment, then the functions adapted to the : C representation are (33.11. 2 2 , , '1)

, (1 2 21 , : I , . -:I ), J-[(-1 ;;; -; I ; -i0)

+ (11.' 0 . 'l)], 2 2 , 2

>

2

and

(2121,." 2 0 9. LO), 2

205

SPIN-COUPLED WAVE FUNCTIONS

while J+[(++; + I ; $0) - (++; +O; +I)] belongs to the C: representation. For the molecule at equilibrium we use the standard basis with the wave functions Y o , o ;= k J k ! d(lola'lf7lf7'2a2a'0:,

( 168)

O;k),

the symmetry adapted functions in this being

In order to carry out the transformation between the two sets (167) and (168), we need, according to (165), the matrix (Sk 1 SlS2 klk2) connecting the standard and Y 3x 9, bases. This last is given by [($$; 41; &I)(++;31 ; +I)($+; $1 ; 40) (++; g). '12 ) ('1. 2 2 , 2 '0. = [(!1311) (11' 1 ' ) (10111 2 2 2 2 2 2 2 2 2)($~$0+)(+0+0+)1 2

9

LO)] 2

and with the help of this transformation we obtain, for example, the result that Y'b;J;tt-22 = -Yb ' ?,;': + -1 WE,+) (171) 2 2 0,0;4'

J3

Thus by using a linear Combination of just the two functions on the righthand side of (171) in a calculation on Liz, one obtains a wave function that predicts Y g , o ; i i 2as 2 the dissociation product, which in fact corresponds to two lithium atoms in their ground states and with (lsls') cores coupled to form singlets. Except in very special cases,16 the use of a single spin-coupled function to describe the molecule gives as dissociation products two lithium atoms in a mixture of doublet and quartet states." l 6 Thus if the orbitals are put in the order lal6la'ld'2a2a', the use of Equation (160) shows that 'Vb%TJ of (169) dissociates correctly to ' V ~ ~ o ~ i f Z Z . The Liz wave function, for example, of Goddard (1968b) goes as R+oc to the linear combination - $ ( i ; ; $ l ; &l)+&,(JA; $ 1 ; & l ) - d $ [ ( J J ; $ l ; $ O ) + ( ~ ~ ; $ O ; ~ l ) ] +z ~z zl ,( ' l ~ $0; $0) with the orbitals ordered as in (167).

206

J . Gerratt

Lastly in this section we remark that most of the energy of a molecule is associated with the core configurations @?’ and @Ib)of (166) and that their contribution remains more or less unchanged upon going to the constituent atoms. The actual binding energy is mainly confined to the molecular valence shell configuration which undergoes major changes upon dissociation involving both massive deformations of the orbitals and a recoupling of the spins. In order, therefore, to focus attention upon these last processes, one might replace the cores by simpler functions, and (166) provides a suitable framework for doing so.Thus and mib)could be replaced by closed-shell configurations with doubly filled orbitals of either molecular or atomic form. A less restrictive simplification would be to introduce orthogonality constraints between certain of the orbitals. This would greatly reduce the labor of computing the energy contributions from the cores while retaining a realistic model. The theory would also then be directly applicable to a much larger class of molecules. The symmetries of the orbitals concerned play a central role in such approximate treatments, since these essentially determine which overlaps may be reasonably neglected and which may not. This subject will be pursued further in subsequent articles.

X. Conclusions The theory of spin-coupled wave functions has been developed in this article, and its consequences for our understanding of the electronic structure of atoms and molecules pursued at some length. One of the most important features of the theory is the great flexibility imparted to it by the use of many different bases of spin functions. The transformations between such bases provide a useful insight into some of the processes that occur when a chemical bond is formed. In addition, this formulation makes it possible to discuss both the equilibrium properties of molecules and their dissociation into atoms within the same framework. The spin-coupled theory incorporates the strengths of both the VB and MO theories and avoids many of their weaknesses. In the case of diatomic molecules there is no difficulty for example in describing the BeH molecule (X’C’), or in predicting that the ground state of the oxygen molecule is a triplet-both of which are celebrated failures of the simple V B theory. For polyatomic molecules, the theory provides a description in terms of localized bonds where necessary such as for NH, and CH,, or in terms of delocalized electrons as in the case of benzene. However, the practical feasibility of the theory ultimately rests upon the calculation of the J , ( p v ~ ~ j S k kcoefficients, ) Eqs. (52) and (53). For this purpose the recursion formulas given in Appendix A put these calculations on a systematic basis, and it should be possible to apply the full theory to

SPIN-COUPLED WAVE FUNCTIONS

207

systems with up to ten electrons. But it should be emphasized that however much computer time it takes to calculate the necessary Us*"(P)matrices, this task once done need never be repeated and the results may be stored and re-used for any N-electron system with spin S and of whatever spatial symmetry. (The transformation to any desired spin basis is a minor operation.) Indeed because of their method of construction, the appropriate parts of these matrices may be used for almost any system with smaller N . In addition, the presence of degenerate sets of orbitals in a molecule extends the range of the theory considerably since the orthogonality among them arising from the Hund's rule coupling makes the calculation of the J,, coefficients very simple. Thus a calculation on the F, molecule with 18 electrons should be feasible as eight of these are accommodated in 71 orbitals. The theory can of course be applied to larger systems, but in order to do so, it is clear that some approximations in the form of orthogonality constraints must be applied. For this purpose, the form of the full theory and particularly the symmetry of the orbitals offers a guide to the way in which this should be done. One might foresee that the use of the most general linear combination of spin functions is by far the most important factor in obtaining a good description of different molecular properties, and that orthogonality constraints which one might introduce play only a minor role. NOTE ADDED IN PROOF The recursion scheme described by Eqs. (59)-(62) has now been applied to some 4electron systems, and makes the calculation of all the H,, J , , and Qmcoefficients so fast as to be trivial. ACKNOWLEDGMENT This work was begun while the writer was a member of Professor W. N. Lipscomb's research group at Harvard University, and he would like to thank him for the many pleasant and enlightening discussions they had and for his constant encouragement.

Appendix A. Proof of the Relations (59)-(62) We start from the expression for the normalization integral for an arbitrary spin-coupled function:

where

J. Gerratt

208

Now, for any P E 9, (@ON

I pr@ON)= ( P U N @ON 1 P,, =

Pr@ON)

(@;-l~,(N)~P;,F@O~)

('4-2)

We substitue this into (A-I), and decompose P into cosets of S,-l. For this purpose we choose the N interchanges P,, (c = I , 2, . . . , N ) and obtain

N

=

1 H(pol S k k ) ( p l e ) ,

rJ=

for all p

=

1, 2, . . . , or N .

(A-3)

I

But instead of (A-2), we may write (@ONlpr@ON) = (Pp,PvN-l@ONI = (@F:2dv(N

-

FvNPVN-]Pr@ON)

1)4,(N)Ip ~ N - I p L N p r @ O N ) ?

in which p > v = 1, 2, . . . , N . We substitute this into Eq. (A-I), and now decompose 9,into cosets of 9 N -x 2 9,. Thus any P E 9,may be written in the form P Q P r JPrwi ~ =PQP,N-IP~N,

where c > T = I , 2, . . . , N ; P is now a permutation of Y N P and 2; Q [ E , PN-IN]. Then

=

N

We write this equation in the form hi

where D(')(pv; cz I Skk) is an element of the two-electron density matrix, and is defined as follows:

(A-5)

In this equation a and b are, respectively, the larger and smaller of c and z, and Q-' = E if c > T , Q-' = P N - l Nif CJ < T . The element O(,)(pvv; o z l S k k ) is therefore just the coefficient J(pvoz I Skk) if c > T , orK(pvc.rI S k k ) if c < T .

209

SPIN-COUPLED WAVE FUNCTIONS

On comparing Eqs. ( A - 3 ) and (A-4), we see that N

H(pa I S k k ) =

C D”’(pv; 7=

1

c r I~Skk)(v I T ) .

(A-6)

(7+a)

Thus if all the elements of the two-electron density matrix are known, it is unnecessary to calculate the one-electron density matrix separately, since the H ( p a I Skk) are all given by Eq. (A-6) If N 2 4 , then for cr < T = P;r

@;a-

(q72,

and D‘”(pv; C T 1 S k k ) is always equal toJ(pvar1 Skk) for all values of cr and T. Equation (A-6) then becomes identical with Eq. (61) of the main text. Continuing in this way, we decompose 9,into cosets of Y N - x, 9, , and obtain A;h in the form:

which we write as A;h

I

=

N

a, 7, q= 1

D‘3’(pvA;

I S k k ) ( i I q)(v 1 T > ( f i l

OTr?

(A-8)

O).

(q+r+a)

DC3)(pvA; O defined by

T S ~ kk)

D‘3’(pv2; O

T ~S k k )

is an element of the three-electron density matrix and is

I

= P E Y N - 3

u,”,(paN P b N - I P c N - 2

PQ - ‘ P A N -

2 PvN- lp,tN)
I

; :@

’),

(A-9) in which a is the largest, b is the middle, and c is the smallest of cr, respectively. The permutation Q - ’ is given by Q-I

=

N‘

N”

N

N-1

N-2

T,

q,

210

J. Gerratt

Comparing (A-4) and (A-8), we see that N

(g # 7 # rs)

If N 2 6, then

and substituting this into Eq. (A-9),it can easily be shown that

D'3)(pvA;0 7 q I Skk) becomes identical with the Q(pvia7q I Skk) coefficient defined in Eq. (60). In this case, Eq. (A-10) is the same as (62). Equations (A-3), (A-6), and (A-10) form a set of recurrence relations that can be extended as far as is necessary, simply by decomposing Y Ninto cosets of L f N - p x 9,(p = 1, 2, 3, . . . , N - 2). Thus a p-electron density matrix with elements of the form D(P)(p1p2. . .p p ; v1

v2

*

..V P I S k k )

involves just N - p electrons. On putting p = N - 2, one obtains a density matrix whose elements all have the simple form aA + bA2, where A is an overlap integral, and a and b are matrix elements of the form u,"(Pv, NPvzN - 1 . . * P,,, ,v - p + iPP,N- p + 1 . . .Pp , N ) . These may all be computed once and for all and stored sequentially on magnetic tape. It should be noted that the L)(,,)(pl. * .p,,; v1 . . . vp 1 Skk) possess a very high degree of symmetry among the indices, and it is in fact only necessary to store those for which p1 > p 2 > " * p P , v1 > v 2 > * . . > v p , plus certain permutations of these indices. From the D ( N - 2 matrix, ) one forms a list of the D ( N - 3 )matrix elements by using equations of the form (A-lo), followed by D(N-4),. . . , D ( 2 ) ,and D ( l ) . In this way, one avoids having to compute separately ( N - 2)! terms for each D(')element, and ( N - I ) ! terms for each ~ ( l element. ) Formulas such as (75) in the main text become applicable to any number of electrons if the J,, & , .. . , etc., coefficients that occur in them are simply replaced by Dh2),DL3'. . . , respectively, the subscript n denoting a normalized density matrix element, e.g., D P ) ( p v ; O T [S k k ) = ( A f k ) - ' D ( 2 ) ( p v a; p I S k k ) .

21 1

SPIN-COUPLED WAVE FUNCTIONS

Appendix B. Matrix Elements of Spin-Dependent Operators We confine our attention for the present to one-electron spin-dependent operators of the form

Then we have (yS, S ; k

I

c f , ~I

j z p yS, S; k)

=N(yS,

I

I

S ; k f N j z N yS, S; k).

(B-1)

P

Using the fractional parentage decomposition (45) this becomes N

T - i -

L L ,

-i-

0 , V = 1 S i ’ k i ki”

L

M I ,U N (MI +u,v=S)

in units of h. Now the quantities $ { ( S , ‘ . 1. s - +; 2 r 3

+ p s y - (&’; +; s + 1. - -; 1 w2> 29 (S,’=S++,

s-4)

are essentially the elements of the matrix of i z N : ( @ F , s ; S l l ~1 3 ; N I o t , S ; S l k l )

and substituting explicit expressions for the Clebsch-Gordan coefficients, we see that the elements of this matrix are given by -S 6f1kl

611kl+,

(2s)’

f,”;: +

I , , k , I f,”;:; 5

ll, kl

5

fSN.

It will be denoted by x ~ . and ~ ,its form is illustrated in Fig. (B-I)

212

J . Gerratt

FIGB-1. General form of the matrix

We note that since both the commute. Let us define a matrix

xS and

x ~ * ~ .

US(P) matrices are block-diagonal they

WQiPQz) = US(Qi ) X ~ U ~ ( P ) U ~ ( Q Z )

(B-4)

so that

in which the integral (4,,1 f I & v ) is written ( p l f I v ) for clarity. If we now define a general normalized coefficient T , ( ~I v S k l ) = (A:k A;)-’

1 Tzi(PvNPI‘,,)(

Pro:-

I @-:

’)

(8-6)

P E Y N - I

then we have finally that S

-1

(Akk)

(yS, S; k

1 1

f p $zp

li

I

N

Tn(pv I S k k ) ( p IJ’I v>.

y S , S; k ) =

(B-7)

p. v = 1

The T,(pv I Skk) coefficients may be constructed from a recurrence scheme similar to that described for the H , coefficients, Eqs. (61) and (62). Note for S = 0, the entire TS matrix (B-4) is zero.

213

SPIN-COUPLED WAVE FUNCTIONS

Appendix C. Proof That the Orbital Equations Are Invariant under X Preliminary Remarks. Consider a symmetry operation R of 9. In general R is a transformation of the electronic coordinates, so that R@ON= "1(r1)42(r2).

4,(r,)I

*.

= (R41(rl))(R4Z(r2))

''*

(R4N(rN))

= ~ $ ~ ( R - ' r ~ ) 4 , ( R - ~ r--*q5,(R-'rN). ,)

(C-1)

Now in section VII it has been shown that the orbitals form a permutation representation so that, for example,

4i(R-'ri) = iiR1+RI(ri) where 4RIis some other orbital in the set d1, 4 , , . . . , 4, and factor and equal to & 1 . Hence (C-1) becomes =[ R

R@ON

i l RisI a phase

4R1(~1)4Rz(~2) ' * * ~RN('N)

in which R,, R , , . . . , R, is some permutation of the numbers 1 , 2, . . ., N , and the phase factor i Rarises fromjust those orbitals 4p, 4,,, . . . , .. . etc., which are not transformed into other orbitals by R. i R =

c,,(R)cvv(@. . *

inm =

*

1.

The effect of R on @! is hence just the same as an operator mutes the orbitals. Thus we may write R@! = i R P R0;= eR(PA)-'@;.

i RH R that per(C-2)

Consider now some matrix element of an arbitrary one-electron operator f(r) formed by two orbitals 4pand $,,. The value of this integral cannot be changed by any transformation of the coordinate system, and in particular by a transformation corresponding to a symmetry operation R:

(4p(r)lf(r)14v(rD

=

(4p(R-1r)lf(R-1r)14v(~-1r))

= cpR,

If('>

CVR,<+R,(~)

I $Rv(r))?

(C- 3)

since the operatorf(r) is assumed to be totally symmetric under all operations R. The truth of (C-3) is obvious if one thinks for example of the CH, molecule in the simple VB theory, Denote the four sp3 hybrid orbitals on the C atom by t , , t , , t , , t , and the four hydrogen Is orbitals by h,, h,, h,, and h,. Each f, and its partner hp(p = 1, 2, 3, or 4) is totally symmetric under all operations of one of the four C3"subgroups of T,, and is carried into some other member

214

J. Cerratt

cpv,

of the set t v , hv by all other operations not in C 3 , . The phase factors etc., are all unity in this case, and Eq. (C-3) states, for example, that the overlap integrals ( t i I hi), ( t z 1 h2), ( f 3l h 3 ) , (t4 I h4)

are all equal, which is obviously true. It also follows from (C-3) that the effect of R upon a function f(r)$Jr), where f(r) is again a totally symmetric but otherwise arbitrary operator, is given by R(f(r)4p(r))

=

~pR,f(r)4R,(r)’

(C-4)

Turning now to the orbital Eqs. (77)

FFk4, = E p 4p

3

it has to be shown that for any operation R of some subgroup A? of 9,

the last equality occurring since, by hypothesis, representation of X . Now,

+,, belongs to a nondegenerate

N v= 1

We consider only the first term of this, the effect of an R of X upon it being given by N

x fJrNMV(rN))N

(C-6)

in which f p v ( r Nstands ) for h(rN) - y p y ESc and the subscript I . . .)N indicates that there is no integration over the Nth coordinate. Note that the phase factor { J R ) occurs on the right-hand side of this equation because orbital + p is missing from the configuration Q N - ’ ,whereas all orbitals are present in @ t - 1 f p Y ( r N ) 4 y ( rHence N ) . instead of obtaining I l RI = 1 as the phase factor, one obtains I I 2 / { p p ( R= ) { J R ) . We now observe that the part

cR

215

SPIN-COUPLED WAVE FUNCTIONS

Ip R

@-:

I

' ~ ~ V ( ~ N ) ~ ( ~ N )p )R N

41(l)

'

..

4N(')

' ' '

4 N - l(N

-

l)&v(N)dv(N))N

may be written as

I4 R I ( 1 ) 4 R 2 ( 1 ) ' .

*

$RN(v)

'..+RN-I(N

-

1).fpRU(~)4R,,(~))N

using Eq. (C-4), and that the summation Xu may be replaced by CRYsince there is a one-to-one correspondence between v and R v .The integral in (C-6) is therefore unchanged by a permutation p i ' of all the orbitals comprising the integrand, and we obtain N

K

c

v=l

Hn(P I s w . f f l v ( r ) 4 v ( r ) = cpp(R)(Azk)-l

1

ufk(pvN

1

pppN)(pr@:-'

v=1 PEYN-1

~ ~ V ( ' N ) ~ V ( ~ N ) ) N

Similar arguments hold for the second and subsequent parts of F,Skk4, so that Eq. (C-5) is proved, and F F is invariant under all operations in 2.

Appendix D. Proof That the Operators F a and ) Fp) Are Invariant under Unitary Transformations of the b p and $fl Sets of Orbitals We consider first the operator F'") defined by Eq. (124) and which we write in the form

c 2n

- Es,

v=n+l

+

H,(Kv I S11) I $v-n)(4K I other terms

1.

(D- 1)

Note that the invariance used to arrive at the orbital Eqs. (120) was merely that

Ft""I 4p'>= ! q I 4J,

in which the primes denote the transformed functions. However we now wish to show that F'"" = pa', and for this purpose we write out (D-1) as follows:

216

J. Gerratt

Our task is to show that this operator remains invariant under the transformations

4,

n

DVp4,,’;

DtD = E ;

fB , , + , ~ ;

B ~ B =E;

= v= 1

+p

=

v= 1

p = 1 , 2 , ..., n .

03-3)

The determinant of the transformation matrix D is given by det (D) =

c

EPDV, PEY,

1Dv*2. .* DV,?,

(D-4)

where the permutation P corresponds to

Since for any P E 9’,,U s l ( P )= c p , the determinant may be written in the form

is decomposed into 9,-, x 9’,, where 9’,-, consists of all permuIf 9, tations of the numbers ( 1 2 e . e ~- 1 p + 1 . . - n ) ,

C

PEY,

c n

=

v=l

C

for any p

PP,,

=

1,2, . . . , n ,

PEYn-l

then Eq. (D-5) may be written in the form

In this equation the symbol 9

(3

denotes the minor of det (D) formed from

the latter by deleting the vth row and pth column. Equation (D-6) is equivalent to the expansion of a determinant in terms of its cofactors, as can be seen if U,,(P,,) is replaced by (- 1)”’. Higher minors of det (D) are written

denoting that the vth, oth, zth,. . . , etc., rows and the pth, Ath, qth,. . . , etc. columns are missing. Equation (D-6) may be generalized to read

217

SPIN-COUPLED WAVE FUNCTIONS

x 9,, Analogous relations exist between higher minors. Thus if Y P-+n9,-, we may derive from (D-5)

c9

det(D)=f

v, u = I

i

=

")

DvpDd9i" P

v,a= 1 (v*a)

i

~S1(PY,)~Sl~PUJ~

(D-8)

where 8 denotes the complementary minor of 9. Comparing with (D-6) we obtain

which may be generalized as in (D-7) to give (D-10) In addition, from the fact that = Dy*, = det

[D-'I,,

(Dt)9

we derive the useful relation

13

UsI(Ppv),

(D-11) Expressions similar t o (D-7)-(D-10) hold for the expansion of det (D) down one of its columns. We now consider the terms (D-2a) and (D-2b) and concentrate attention first upon the transformation of the configuration a :)''. Under the transformation (D-3) this becomes n

= v , , v2.

c

.._,v n =

(DvllDvz2 1

' . . DvK-

IK-

1

DYh-+lK+

1

.

' '

Dv,n>

( v h - missing)

x 4;l(1)4;z(2)

. . . Ic/n(K>

'

. . 4:,(n>.

(D-12)

Upon substituting this into (D-2a) and (D-2b) we see that none of the orbitals 4:, , 4bz,.. . , etc., may be repeated. Thus a particular term in the summation (D-12), characterized by the numbers ( v 1 v 2 v n ) , is just a

218

J. Gerratt

permutation of (12.s.p - 1 p + 1 n) in which orbital sing. Each term of (D-12) is therefore of the form 4il(1)d):2(2)

' ' ' $ ! I ( . ) ' ' '

4:"(n>=

'LK

4:i(1)d)12(2)

* * '

is mis-

4:"(n> $h) $n'(n> ' ' *

= pLK Qrd~'(1)d)z'(2)' "

p = 1, 2,

= PL,Q'@?'",

4p',say,

*.'

. . . , n,

(D-13)

in which Q is the permutation that brings the #,,' orbitals in this configuration to their natural order. The summation over all vl, v 2 , . . . , v, is thus equivalent to a sum over-all permutations Q. Substituting now (D-12) in (D-2), and with the help of (D-13), we obtain F'a'=x(D,*,1D,*,2

"'Dy*,fl)(A?l)-'

*"DV*,-~K-lD?K+IKfl

Q X

[

p,

D:K

(pr@F)n@(D 0 b2"

Thus (D-14) simplifies to F(") = det (Dt)(A7

CU?I(PVZ~PQ-~P~KPK~~) P

V,K,A=l

I

1 @ f ) n @ ( B0 )n-

1

>hl4">(GI

SPIN-COUPLED WAVE FUNCTIONS

219

which is precisely of the original form (D-2). A similar proof goes through for the configuration @,%)n-l in (D-2a) and for at)”in (D-2b) leading to an expression such as (D-15) but multiplied by det (D). Since D is unitary, det (D’) det (D) = 1, so that at least for those terms of F ( “ )appearing in (D-2), the form is invariant under a unitary transformation of the 4,. Precisely the same proof now follows for the transformation of the $, , and the same invariance holds for the other terms making up F‘“’. The demonstration of the truth of this last statement is again essentially the same as carried through so far, though some tedious algebra involving the higher minors of det (D) is necessary. Equations (D-8) to (D-11) are required for this purpose. We therefore conclude that the entire F‘“’ operator is invariant under unitary transformations of the 4, and $,. It should be noted that the sum over K in (D-I) is essential for the proof; in other words, the individual operators F:’ do not possess this invariance property, but only their sum. The application of the operator Q, [Eq. (127)] to F‘”) simply changes this operator into F ‘ p )as can be seen by inspection of, for example, Eqs. (D-2). REFERENCES Altmann, S. L. (1958), Proc. Cambridge Phil. Soc. 54, 197. Amos, A. T., and Hall, G. G. (1961). Proc. Roy. SOC.,Ser. A 263, 483. Arai, T. (1962). Phys. Rev. 126, 147. Arai, T. (1964). Phys. Rev. 134, A824. Arai, T. (1966). Prog. Theor. Phys. 36, 473. Barfield, M., and Grant, D. M. (1965). Advan. Magn. Resonance 1, 149. Birss, F. W., and Fraga, S . (1963). J . Chem. Phys. 38, 2552. Born, M., and Huang, K . (1954). “Dynamical Theory of Crystal Lattices.” Oxford Univ. Press (Clarendon), London and New York. Born, M., and Oppenheimer, R. (1927). Ann. Phys. 84, 457. Brink, D. M., and Satchler, G. R. (1962). “Angular Momentum.” Oxford Univ. Press (Clarendon), London and New York. Burnside, W. (191 1). “Theory of Groups of Finite Order.” Cambridge Univ. Press, London and New York. Coleman, A. J. (1968). Advan. Quantum Chem. 4, 83. Corson, R . M. (1951). “Perturbation Methods in the Quantum Mechanics of n-Electron Systems.” Blackie, London. Coulson, C. A,, and Fischer, I . (1949). Phil. Mag. 40, 386. Das, G., and Wahl, A. C. (1966). J . Chem. Phys. 44, 87. Davidson, E. R., and Jones, L. L. (1962). J. Chem. Phys. 37, 1918. Dewar, M.J.S. (1963). Rev. Mod. Phys. 35, 586. Eckart, C. (1930). Phys. Rev. 36, 878. Empedocles, P. B . , and Linnett, J. W. (1964). Proc. Roy. Soc., Ser. A 282, 166. Eyring, H., Walter, J., and Kimball, G . (1944). “Quantum Chemistry.” Wiley, New York. Gabriel, J. R. (1961). Proc. Cambridge Phil. SOC.57, 330.

220

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Gerratt, J. (1969). Annu. Rep. Progr. Chem. A 65, 3. Gerratt, J., and Lipscomb, W. N. (1968). Proc. Nar. Acad. Sci. US.59, 332. Goddard, W. A., Jr. (1967a). Phys. Rev. 157, 73. Goddard, W. A,, Jr. (1967b). Phys. Rev. 157, 81. Goddard, W. A., Jr. (1968a). J. Chem. Phys. 48, 450. Goddard, W. A., Jr, (1968b). J . Chem. Phys. 48, 5337. Hall, G. G. (1950). Proc. Roy. Soc., Ser. A 202, 336. Hall, G . G., and Lennard-Jones, J. E. (1950). Proc. Roy. Soc., Ser. A 202, 155. Hamermesh, M. (1962). “Group Theory.” Addison-Wesley, Reading, Massachusetts. Heitler, W. (1934). M u r x Handbuch d . Radiologie 11, 485. Herzberg, G. (1950). “Molecular Spectra and Molecular Structure. I. Spectra of Diatomic Molecules.” Van Nostrand, Princeton, New Jersey. Herzberg, G . (1966). “ Electronic Spectra of Polyatomic Molecules.” Van Nostrand, Princeton, New Jersey. Hinze, J., and Roothaan, C. C. J. (1967). Progr. Theor. Phys. Suppl. 40, 37. Horie, H. (1964). J . Phys. SOC.Jap. 19, 1783. Huo, W. (1966). J. Chem. Phys. 45, 1554. Hurley, A. C. (1956). Proc. Roy. Soc., Ser. A 235, 224. Hurley, A. C., Lennard-Jones, J. E., and Pople, J. A. (1953). Proc. Roy. Soc., Ser. A 220, 446. Huzinaga, S . (1964). IBM Res. Paper RJ 292. IBM Res. Lab., San Jose, California. Jahn, H. A. (1950). Proc. Roy. Soc., Ser. A201, 516. Jahn, H . A. (1951). Proc. Roy. Soc., Ser. A 205, 192. Jahn, H. A. (1954). Phys. Rev. 96,989. Jahn, H. A,, and Teller, E. (1937). Proc. Roy. Soc., Ser. A 161, 220. Jahn, H.A., and van Wieringen, H. (1951). Proc. Roy. SOC.,Ser. A 209, 502. Kaplan, I. G. (1959). Z h . Eksp. Teor. Fiz. 37, 1050. [Sou. Phys.-JETP 37, 747 (1960).] Kaplan, I. G . (1963). Liet. Fiz. Rinkinys 111, 227. Kaplan, I. G. (1965a). Teor. Eksp. Khim. 1, 608. Kaplan, I. G . (1965b). Teor. Eksp. Khim. 1, 619. Kaplan, I. G . (1966a). Teor. Eksp. Khim. 2, 441. Kaplan, I. G. (1966b). Z h . Eksp. Teor. Fiz. 51, 169. [Sou. Phys.-JETP 24, 114 (1967).] Kaplan, I .G. (1967a). Teor. Eksp. Khim. 3, 150. Kaplan, I. G. (196713). Teor. Eksp. Khim. 3, 287. Kaplan, I . G. (1969). “ Simrnetriya Mnogoelektronnikh Sistem.” Nauka, Moscow. Koster, G. F., Dimmock, J. O., Wheeler, R. G., and Statz, H. (1963). “Properties of the Thirty-Two Point Groups.” M.I.T. Press, Cambridge, Massachusetts. Kotani, M., and Siga, M. (1937). Proc. Phys.-Math. SOC.Jap. 19, 471. Kotani, M., Amemiya, A., Ishiguro, E., and Kimura, T. (1963). “Table of Molecular Integrals,” 2nd Ed. Maruzen, Tokyo. Ladner, R. C., and Goddard, W. A., Jr. (1969). J . Chem. Phys. 51, 1073. Lennard-Jones, J. E., (1949a). Proc. Roy. Soc., Ser. A 198, 1. Lennard-Jones, J. E. (1949b). Proc. Roy. SOC.Ser., A 198, 14. Linnett, J. W. (1961). J. Amer. Chem. SOC.83, 2643. Littlewood, D. E. (1935). Proc. London Math. SOC.39, 2150. Littlewood, D. E. (1940). “The Theory of Group Characters and Matrix Representations of Groups.” Oxford Univ. Press (Clarendon), London and New York. Lowdin, P. 0. (1955). Phys. Rev. 97, 1509. Lowdin, P. 0. (1962). J. Appl. Phys. 33, Suppl., 251. Lyubarskii, G. Y. (1960). “The Application of Group Theory in Physics” (Transl. by S. Dedijer). Macmillan (Pergamon), New York.

SPIN-COUPLED WAVE FUNCTIONS

22 1

Manning, P. P. (1954). Proc. Roy. Soc., Ser. A 225, 136. Matsen, F. A. (1964). Advan. Quantum Chem. 1, 60. Mattheiss, L. F. (1958). Quart. Progr. Rep., Solid State and Molecular Theory Group. M.I.T., Cambridge, Massachusetts, October. Mattheiss, L. F. (1959). Quart. Progr. Rep., Solid State and Molecular Theory Group. M.I.T., Cambridge, Massachusetts, October. Melvin, M. A. (1956). Rev. Mod. Phys. 28, 18. Morrison, R. C., and Gallup, G. A . (1969). J . Chem. Phys. 50, 1214. Pauncz, R . (1967). “Alternant Molecular Orbital Method.” Saunders, Philadelphia, Pennsylvania. Racah, G. (1942a). Phys. Rev. 61, 186. Racah, G. (1942b). Phys. Rev. 62,438. Racah, G. (1943). Phys. Rev. 63, 367. Racah, G. (1949). Phys. Rev. 76, 1352. Roothaan, C. C. J. (1951). Rev. Mod. Phys. 23, 69. Roothaan, C. C. J. (1960). Rev. Mod. Phys. 32, 179, Rumer, G. (1932). Goettingen Nachr. p. 371. Serber, R. (1934a). Phys. Rev. 45, 461. Serber, R. (1934b). J. Chem. Phys. 2, 697. Tamir, I., and Pauncz, R. (1968). Int. J . Quantum Chem. 2, 433. Van Vleck, J. H. (1932). “The Theory of Electric and Magnetic Susceptibilities.” Oxford Univ. Press, London and New York. Veillard, A , , and Clementi, E. (1967). Theor. C h h . Acta 7, 133. Voge, H. H. (1936).J. Chem. Phys. 4, 581. Wahl, A. C. (1964). J. Chem. Phys. 41, 2600. Weinbaum, S. (1933). J . Chem. Phys. I, 593. Wigner, E. P. (1959). “Group Theory.” Academic Press, New York. Yamanouchi, T. (1936). Proc. Phys.-Math. SOC.Jag. 18, 623. Yamanouchi, T. (1937). Proc. Phys.-Math. Soc. Jap. 19, 436.

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DIABATIC STATES OF MOLECULESQ UASISTATIONAR Y ELECTRONIC STATES THOMAS F. O’MALLEY Physics Department, University of Connecticut Storrs, Connecticut

I. Introduction .................................................. 11. Mathematical Preliminaries ...................................... Born-Oppenheimer Approximation ... 111. Molecular Ground States-The One-State Problem-The Stationary Adiabatic Representation . IV. The Na CI Two-State Pr V. Charge Exchange in Heliu VI. Dissociative Recombination and Attachment-The Quasistationary Representation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Simple Resonant Scattering-Quasistationary State Formalism B. Application of the Quasistationary State Basis to Dissociative Attachment. ................................................ VII. Slow Heavy-Particle Collision Theory-Extension of the Quasistationary Representation to Rydberg States. .................... A. Negative Molecular Systems. ...................... B. Neutral and Positive Molecular Systems ........................ VIIJ. Summary and Conclusion. ....................................... References. ......

+

223 225 226 228 230 232 236 239 241 243 244 244 245 248

I. Introduction “What are the forces that two slowly moving atoms (in the range of vibrational to kilovolt energies) exert on each other?” It is basically this question that the so-called diabatic states of molecules, through their diabatic potential energy curves, are meant to answer. To speak of the diabatic (i.e., not adiabatic) states, it is first necessary to say something about the competing adiabatic states, since it seems to have been assumed for about a generation that the answer to the above question was obvious-“ atoms move along the adiabatic potential energy curves as long as they are going much slower than the valence electrons.” These “ adiabatic ” 223

224

Thomas F. O’Malley

states, with which the reader is assumed to be familiar, are simply the eigenstutes of the electronic Hamiltonian, H e . The adiabatic potential energy curves are the corresponding eigenvalues of H e , defined for each internuclear separation, R . They are, in fact, the potential energies which would exist for the atoms ifthey were not able to move, or in the limit (often ruled out by the uncertainty principle) of sufficiently small internuclear velocity. Now if these adiabatic potential energy curves generally did approximate the actual force acting between two atoms with small butjnite velocities (as had been taken for granted), then there would be nothing more to be said on the subject. It is only because one set of experiments after another has shown that the forces are often very different from the adiabatic, even at very small velocities that a number of workers have become increasingly concerned with defining the actual forces, or more basically the corresponding electronic states, currently known as diabatic states, that give rise to these forces. An important recent effort at finding such a definition is contained in the work of Felix Smith (1969), which is recommended at this point for its excellent introduction to the problem. It is discussed further in Section VIII. The plan of this article is to try to suggest the direction in which nature points for the definition we seek of the electronic states. This is done by a brief and judicious (though very incomplete) review of a handful of developments in the history of the definition of electronic states, each fashioned to meet a specific experimental result. After introducing some of the necessary mathematical concepts in Section 11, the traditional “adiabatic” states are then presented in Section 111 as the answer to the problem of defining the permanent or stationary states, especially the ground state, of a molecule. All other definitions are seen to arise in the context of the two or many-state problem that is generated by collisions either among atoms or between an electron and a molecule. The Na + Cl problem is seen in Section IV to have suggested, in the earliest days of quantum mechanics, the ionic and covalent states as the natural expansion basis for the molecular wave function for this system. More recently the experimental results on symmetric charge exchange have led, once more, to the rejection of the adiabatic states in favor of single configuration molecular orbital wave functions as an expansion basis, as discussed in Section V. Finally, an understanding of the dissociative attachment and recombination of electrons with molecules is shown in Section VI to have required the expansion of the wave function, again not in adiabatic electronic states, but in a quasistationary representation of the electronic Hamiltonian, H e . Fortunately this last representation seems to be general enough to be able to embrace the earlier single configuration representation and covalent-ionic representation and even to degenerate into the conventional adiabatic representation in the less interesting case of a one state problem.

225

DIABATIC STATES

11. Mathematical Preliminaries The motion of any molecular system is taken to be described by the solution of the Schroedinger equation ( H - E)Y

= 0,

(1)

where H i s the total Hamiltonian for the system. Although most of the results of this work, since they concern electronic states, apply equally to diatomic or polyatomic systems, the explicit discussion will be limited to the case of diatomic systems both for simplicity and to conform to historical development. For our purpose, the Hamiltonian is most conveniently divided into

H z= TR + H e ,

(2)

where TR, the nuclear kinetic energy operator, is given by

T,

=

- (h2/2M)VR2.

(3)

M is the reduced mass of the two nuclei [ M = M AM , / ( M A + M E ) ]and R is the internuclear radius vector. The electronic Hamiltonian, H e , is defined as the remainder of H , i.e.,

where the sum is over all N electrons, m is the electron mass, and r i A , r i B , and r i j are the distances of electron i from nucleus A or B or from electronj, respectively, and 2 is the nuclear charge. The point of the division ( 2 ) is, of course, to separate the nuclear kinetic energy operator TR, whose electronic matrix elements are expected to be infinitesimal because of the mass ratio m / M , from the rest of H. For later use let us now consider that we have any basis set of electronic functions ($i(r: R)}, where the vector r is used conventionally to signify all the electronic coordinates. The functions $ iare “adiabatic” in one of the many senses of the term, namely, that the internuclear coordinate R is considered as a parameter only rather than a dynamic variable. The $ i are otherwise undefined for the present purpose. The full wave function for the diatomic system may be expanded in the basis set { $ i } each multiplied by a nuclear motion wave function xi(R), as y ( r , R) =

1 $ i ( r ; R)xi(R) = c i

i

$i xi

1

(5)

Substituting the expansion ( 5 ) for Y into the Schroedinger equation (l), multiplying on the left by each $i*in turn and integrating over all electronic

226

Thomas F. O'Mulley

coordinates produces the formal set of coupled equations for the nuclear wave functions xi(R)

where Vjk(R) = ( 4 j I He I 4 k )

are theelectronic matrix elements of H , , integrated over electronic coordinates, and T' and T" result from the action of the Laplacian operator on the product 4i xi (V'4x = 4vzx + 2 v 4 vx xV'4), '

+

Tij = -2(h2/2M)(4i1VR!4j). V,

(7)

TI'. 1J = -(h2/2n/r)(4i(V,'(4j).

(8)

[We note in passing that the matrix element in the T' term contains the V, operator, which is odd in parity and has no rotational invariance. One consequence of this is that its diagonal matrix element vanishes on the left-hand side of Eq. (6). Another is that it couples electronic states of different angular momentum, and can therefore cause transitions between such states at sufficiently high collision velocities as has been discussed by Bates and Williams (1964).] BORN-OPPENHEIMFR APPROXIMATION Notice that no approximations have been made in deriving Eq. (6) for x,. It is this fact that makes it useless for most practical purrposes as it stands. The one all-important fact for molecular systems, and the indispensable basis of of our understanding of molecules to date, is the huge ratio M/tn of nuclear to electronic mass, which runs from a minimum value of 2000 for H to 30,000 for atmospheric gases and over 400,000 for the heaviest atoms. The principal consequence of this tremendous mass ratio is the qualitative difference that it causes between the fast and highly quantized electronic motion, on the one hand, and the ponderously slow yet generally semiclassical movement of the nuclei, on the other. The mathematical counterpart in the exact Eq. (6) of this qualitative difference between electronic motion ( 4 , )and nuclear motion (x,)is of course the so-called Born-Oppenheimer upproxmmfion, (sometimes also called the adiabatic approximation), i.e., the recognition that under all reasonable conditions the T' and T" terms in (6), resulting from the operation of TR on the electronic dt, are totally negligible compared with the other terms in the equation. If one accordingly makes the Born-Oppenheimer approximation Neglect T' and T"

(9)

(B.O.)

Eq. (6) becomes ITR

+

vii

( R )-

xi

(R) =z -

XJ.

v i ~ I # l

(10)

DIABATIC STATES

227

In these simpler equations the potential energies for nuclear motion as well as the coupling terms are given by the electronic energies V i j (although we have not yet explicitly defined the electronic basis (qhi} and hence the V i j ) . Let us examine the justification for the Born-Oppenheimer neglect of the T‘ and T“ terms in a different manner than usual, but one which is more appropriate to our present interest in collision problems over a fairly wide range of energies. Consider the absolute magnitude of the operators TR , T‘ and T“ of Eqs. (3), (7), and (8), each operating on a x i . For the sake of estimating the rough magnitude, we will concentrate on the radial component d/dR of the gradient operator V,, assuming that the rotational velocities are not much larger than the radial. We first note that IVRxI rr K x , where the wave number K is related to the nuclear kineticenergy E by K = ( ~ M E / ~ Z Nowthemagnitude ~)”~. of V R 4 i , measuring the parametric variation of the 4 with R has no such simple expression. However, experience shows that normal, well-behaved electronic states d o not vary greatly over distance much smaller than the atomic unit of distance, a,. In other words, the assumption

5

IvR4il

aO-’14il

(1 1)

is an extremely reasonable one, barring any anomolous impulsive jumps in 4i (an exception which can always be avoided by a rational choice of 4i, as will be seen). With the assumption (1 I), the magnitude of T‘ and T“ operating on xi follows immediately and may be compared with Txi T X = EX

1 T’x I 5 2(Ry

(12) *

E>”2(m/M>”2I x I

lT”xI 5 RY m/MIXl

(13) (14)

where E is the local nuclear kinetic energy [E - Vii in Eq. (6)], Ry = h2/2maO2 is the Rydberg unit of energy (13.6 eV), and m/M is the infinitesimal electron to reduced nuclear mass ratio. We see from (14) that under the reasonable assumption (1 I ) the T” operator is independent of energy and in the range lo-’ to eV, depending on M, so that it may be neglected at all energies. The energy dependent T‘ term is likewise seen to be very small except in the limit of high kinetic energy. An estimate of what energies are “high” o r ‘‘low’’ in this context can be obtained by noting that T‘ attains the value of 2 Ry (1 atomic unit) for a kinetic energy of (M/nz) Ry, or in other words when the nuclei are moving with an atomic unit of velocity ( r o = e’/h). Accordingly the region of nuclear velocity belowe2/h(E M/n7 Ry),in which the nuclei are moving much more slowly than the valence electrons, is appropriately referred t o as the “adiabatic region,” i.e., the region in which the BornOppenheimer approximation can normally be expected t o be good (but this does not imply that the motion will be along the adiabatic ” potential curves). “

Thomas F. O’Malley

228

In the remainder of this work, we shall limit our interest to this “adiabatic region ” of small internuclear velocities (c 6 e 2 / h )in which the Born-Oppenheimer approximation is expected to be accurate, and we will therefore be working with Eq. (lo), as did all the authors mentioned in the historical development. To anticipate a possible objection, there are times when the Born-Oppenheimer approximation is reputed to be totally bad, even at low velocities, namely, when the “ adiabatic” electronic states (see next section) suffer an “ avoided crossing thereby making the assumption (1 1) invalid for those states. It will become clear however, that this problem always arises from a bad choice of the basis functions $ i (in particular, the “adiabatic” representation is frequently bad), and that there always exists some obvious and reasonable definition of {$i} that avoids this artificial difficulty and preserves the essential distinction between nuclei and electrons, which is lost when the Born-Oppenheimer approximation is abandoned. ”

III. Molecular Ground States-The One-State Problem-The Stationary Adiabatic Representation The problem that has been at the center of molecular physics since the discovery of quantum mechanics is the understanding of the permanent bound electronic states of a molecule, together with their low lying vibrational and rotational levels, as revealed by the spectroscopic observation of transitions between these stationary states. In the last section, a formal expansion ( 5 ) of the molecular wave function in an arbitrary basis set { 4 i }was substituted into the Schroedinger equation and the exact coupled equations (6) for the vibrational wave functions xi(R) derived. The basic Born-Oppenheimer approximation then reduced these to the form [TR

+ Vii(R) - E]xi(R) = - C Vij

xj.

j t i

(10)

Now consider what happens if we define the heretofore arbitrary electronic function + i to be the stationary eigenvalue of the electronic Hamiltonian H e (4), i.e.,

H e 4:d = V::(R)q5;d

(15)

or equivalently, in matrix form, V;;

=

(+,?dlHel$;d) = V : ! ( R ) a i j .

(16)

This is the diagonal representation of H e in stationary states, the so-called adiabatic states. The superscripts “ ad ” refer to this conventional designation

229

DIABATIC STATES

of adiabatic (in the sense that R enters only parametrically). It should more properly be called the stationary adiabatic representation, since there are countless other adiabatic but nonstationary representations (see following sections), but the terminology is firmly established. With this representation (16) of H e in the so-called adiabatic states, Eq. (10) for the nuclear motion reduces to the very simple form

Equation (17) was derived by Born and Oppenheimer. The electronic energy eigenvalue Vid(R)becomes the potential energy for nuclear motion, in agreement with a qualitative picture of the electrons adjusting themselves rapidly to the slowly changing nuclear position. The essential element of Eq. (17) is that x i are totally uncoupled, so that the states, i, are permanent. This is what makes the diagonal or stationary adiabatic representation of H e perfect for describing the ground and low-lying electronic states of molecules, as observed by spectroscopists, which are in fact permanent (at least in between radiative transitions.) Equations (1 7), in other words, describe a one-state problem. A further interesting property of the stationary adiabatic representation is the famous noncrossing rule of von Neumann and Wigner (1929), which states that two potential energy curves, V,?:(R) and V$(R) may not cross if they have the same symmetry (spin, parity, angular momentum). This rule can be derived rigorously and straightforwardly (von Neumann and Wigner, 1929; Landau & Lifschitz, 1958) from the definition ( 1 5). It has generally been overlooked however, that the noncrossing rule holds true only for the diagonal representation of H e , but not for nondiagonal ones, so that for one thing it is an artificial mathematical construct and not in any sense a law of nature. Before leaving the one-state problem of determining the permanent bound states of molecules and the appropriate set (4yd}of electronic functions, a word of caution is appropriate. I t seems that the overwhelming success of @d in describing the one-state problem, combined with the preoccupation of molecular physicists with the one-state problem for more than a generation, have gradually led to the incorrect impression, embodied in textbooks and throughout the literature, that &d of the diagonal representation is the electronic state, and that the noncrossing rule is a law of nature. This kind of misunderstanding is perpetuated by the use of adjectives such as first order” or “approximate” to characterize any other choice of 4 i , as well as by the name pseudo-crossing” which is often misapplied to the actual crossing of the corresponding potential energy curves, because the true curves of course cannot cross.” It is to be hoped that these nonsensical phrases will soon die out. “

“

“

230

Thomas F. O’Malley

In the next three sections, selected examples of tbvo-state and many-state (as opposed to one-state) problems are considered. It will appear from the development that there are choices of the electronic basis functions (pi which successfully describe the results of experiment in each case much more faithfully than do the “adiabatic” functions I $ : ~It. will emerge that these alternate definitions of electronic states, which have come to be called diabatic states, lead to alternate nondiagonal representations of H e .

IV. The Na

+ C1 Two-State Problem-Covalent

and

Ionic States

We recall the basic question to which we addressed ourselves at the beginning of this work, namely “ what are the forces acting on a pair of atoms (or ions) as they collide?” For the Na + C1 system, this question was a major preoccupation of Zener (among others’) more than a generation ago. The natural answer he found provides perhaps the best generalization of the onestate problem in the situation where there are t ~ ’ physically o meaningful and distinct electronic states rather than just one. If a Na and a CI atom are moving in each other’s vicinity, their electronic state may be described accurately at large and moderate values of R as either covalent (Na + Cl) or as ionic (Na’ + Cl-), with the corresponding electronic wave functions (pcov and (pion. Energetically it is known that the covalent configuration is slightly lower in energy when the atoms are very far apart, but at most finite separations the Coulomb attraction between ions makes the ionic configuration more stable, as shown in Fig. 1, where the expectation values V,,, = ((pcovl HeI (pCnv)and Vion= ((pion I H,I (pion) are plotted against R. This is a good example of electronic states of the same symmetry (’C for both) which violate the noncrossing rule. They may do this because they are not (nor should they be) the stationary eigenvalues which diagonalize H e . Rather in this two-state space the He matrix

which is a two-dimensional representation of H e , is nondiagonal since the off-diagonal elements, though very small, are not vanishing or even negligible. [They are much larger than T‘ or T“, so that T G Vcov,ion< 1 Ry.] What Zener did in describing collisions in this system was both masterful and marvelously simple. He chose the wave functions (pcovand (pion of the two I See, for example, London (1932) or Berry (1957) for a much more conventional treatment of the same problem.

23 1

DIABATIC STATES

10

20

30

Internuclear distance, R ( 0 . u . )

FIG. I . Electronic energies V,,, and Vionof the ionic and covalent ' C states of NaCI, plotted against internuclear distance. R, is their crossing point.

physical states directly as the electronic basis for expanding the full molecular wave function Y in the fashion of Eq. ( 5 ) y ( r , R)

= 9covxcov(R>

+ 4 i o n Xion@).

(19)

For these 4's, the Born-Oppenheimer approximation is accurate, as it should be, so that the two relevant nuclear motion wave functions 4,,, and satisfy the general equation (lo) in the simple form

We see that each element in the electronic matrix H e of Eq. (18) plays a transparent role in the nuclear motion described by the x's. The diagonal V's are the potential energies for elastic motion in that particular state, while the nondiagonal elements of Vcov,ion provided the coupling between the two states. This nonvanishing coupling is what makes the present Eqs. (20) essentially different from the corresponding Eq. (17) of the last section, which explicitly rules out transitions. Zener (1932) then went on and calculated, to a reasonable semiclassical approximation, the probability of a transition from one state to the other from the coupled Eqs. (20). First he found that the probability was only appreciable in the neighborhood of the point R, of Fig. 1, where the two potential energy curves Vcovand Via, crossed, in accordance with the FranckCondon principal of conservation of nuclear momentum. He then derived the

232

Thomas F. O’Mulley

explicit formula for the probability P of a transition from one state to the other during a passage through the crossing point R , ZJ = 1 - e-,* z 26

(21)

where

with all the matrix elements V i jas well as the internuclear velocity u evaluated at the crossing point R , . Equation (21) is the famous Landau-Zener formula (Zener, 1932; Landau, 1932; Stuckelberg, 1932). It predicts that the probability of leaving the covalent (ionic) state will be small when 6 is, i.e., at all but the very lowest velocities. The success of the representation (18) of He for describing the motion of the Na + C1 system, leading to the simple coupled equations (20) and the approximate but very useful Landau-Zener formula (21), naturally forces one to consider the possible extension of the nondiagonal representation ( 1 8) of He and the resulting equations of this section to describe elastic motion and transitions in other colliding systems which may also be described mathematically by crossing potential energy curves as in Fig. 1, but where the physics does not allow the convenient division of electronic states into covalent and ionic as in the present case. In the next two sections, two further classes of collision processes are treated in which the experimental facts also demand a nondiagonal representation of H e , and theoretical considerations suggest the appropriate, and progressively more general, nondiagonal, or diabatic representations which successfully describe the motion of the system.

V. Charge Exchange in Helium-Single Configuration Diabatic States More recently the charge exchange experiments of Ziemba and Everhart (1959) on helium (He’ He -+ He + He+) revealed a striking oscillatory structure as a function of velocity in the moderately low u region of 0.1 to 1 x e2/h.In the light of the then existing impact parameter description’ of this process, which expressed the probability of the reaction in terms of the lowest gerade and ungerade potential energy curves V gand V,,for the He,’ molecular system, the experiments in effect constituted a fairly direct probing of the forces between the He’ and He atoms, thus enabling nature itself to provide an answer to our question “what are the actual forces acting between slowly moving atoms ?

+

”

*See Firsov (1951), Kohn (1953), Bates et al. (1953).

DIABATIC STATES

233

Successful theoretical agreement with the experiments was first obtained through the impact parameter description by Ziemba and Russek (1959), who interpolated between published values of the potential curves V , and V, which had been given at R = 0 and co . The potentials of Ziemba and Russek for this system (He,’) accordingly provide, at least in a numerical sense, the answer to our question about the forces between the He and He+. But the numbers must be analyzed to determine first whether or not these potentials are the “ adiabatic” potential curves of Section 111, and second, if they are not how are the electronic states and their wave functions 4 characterized in terms that might provide a generalization of the covalent and ionic electronic configurations of the last section? This needed analysis was done most effectively by Lichten (1963), who in the process gave the name diabatic to the alternative he discovered to the adiabatic states. Lichten analyzed the relevant He2+electronic states in terms of molecular orbitals (MO) and then went further and asked whether a single configuration wave function could be made to suffice, at least roughly. For the ground state of the system, the ’C, , there was no problem. It had already been established that the lowest configuration with this symmetry, i.e., 1 ag’, lau is a reasonable approximation to the ground state at all R’s (for example, it goes from He + He’ at R = 00 to Be’ Is2, 2p at R = 0), needing only the 1 or 2 eV correction or shift which would be furnished by configuration interaction. ’C,. On the other hand, the ’Cg state presented a problem. At R = 03 the initial and final electronic state in the charge exchange reaction is He(ls2) + He’(1s). The only single configuration ’1, state which can be constructed from these orbitals is the log, (IOU)’ configuration

4 , = lag, (lau)2.

(23)

In the united atom limit ( R = 0) however, the configuration 4 , goes to the Is, ( 2 ~ configuration ) ~ of Be+, a doubly excited autoionizing state lying about 8 Ry (over 100 eV) above the ground state, and above all the discrete states of Be’, high up in the ionization continuum. Small R. The lowest ‘C, configuration for He,’ is the (lag),, 20g,

which begins with the (Is)’, 2s configuration of Be+ at R = 0. But the 4 , state dissociates into He’ + He*(ls, 2s) after its potential energy curve V , , crosses the energy V , of the other configuration (See Fig. 2). Its dissociation limit therefore rules out 4, as the ,C, state at large R . What then is the *C, electronic state which describes the motion ofthe atoms as observed in the low velocity charge exchange experiments? The true believer in the “adiabatic” definition of electronic states (as described in

,

Thomas F. O’Malley

234 r

-12

-

I

P W C

-20t / Internuclear distance, R (a.u,l

FIG. 2. Electronic energy (without the singular Z A Z B / Rterm) of some low lying ’X, configurations of He2+. Vl and V2 are the energies of the 2C, configurations l a g , I ou2) and C$2(lag2,2ag), which cross at R , . V,, is the 2C, “adiabatic” energy, the lowest eigenvalue of the 2 x 2 He matrix. V , is consistent with the charge exchange experiments while V,, is not.

Section 111 in the context of the one-state problem) will tell us that one musf take the linear combination of dl and 4’

which diagonalizes H e and defines its lowest eigenvalue Vad at all R’s. Vad is plotted (schematically) in Fig. 2 along with the single configuration energies V , , and V , , . To jump right to the outcome, Lichten examined the impact parameter result both with the adiabatic electronic state 4 a d and with the single configuration state 4,. He found that the calculation with the adiabatic state completely failed to describe the observations, predicting a total lack of charge exchange in the velocity range studied. On the other hand, the single conjguration state lag, (IOU)’ with wave function 4’ reproduced the observations in fairly good detail ! Nature had therefore decided against the adiabatic state, a surprising result to those who believed in the unique position of the adiabatic states for all slow nuclear motion. In the present case She decided in favor of the single configuration state 1ug, (1 au)’. The situation is similar to that of Na C1 in the last section where Zener found that the ‘C motion was described most effectively by the expansion ( 1 9) of the total wave function Y in the physically simple basis 4ionand 4,,,, which are also single configurations, but in the valence bond rather than MO description.

+

23 5

DIABATIC STATES

The total Y for the present 2C, motion of the He,’ system might similarly have been written YJ = 4lXl(R)

+ 4 2 XZ(R).

(26)

With the initial condition for this problem [He+ + He(ls2) incident], the Landau-Zener formula (21) could again be drived and would show that transitions to state 2 are probable only at very much smaller velocities, making the x, term in (26) negligibly small. Hence the observed success of the b1 term. In qualitative terms, the preference of Nature for following the simple 1-configuration term (whether valence bond or MO) in preference to the highly artificial “ adiabatic ” state, which involves major rearrangement of electrons from one configuration to the other over a small range of R , can be explained as follows. If the nuclei were indeed moving infinitely slowly, then there would be available all the time necessary for them to rearrange their configuration (say from 1 to 2 in Fig. 2) while moving a small distance in R. However a major rearrangement of electronic structure, being difficult, must be improbable, i.e., sloii~[which is reflected quantitatively by the smallness of the offdiagonal electronic matrix element Vijz in the numerator of the LandauZener parameter 6 (22)]. Therefore even a very small nuclear velocity L‘ is enough to move the system beyond the crossing point in less time than it takes the electrons to make this very difficult rearrangement. The electrons therefore tend to retain the same overall configuration (whether ionic, covalent, ag,au2, etc.) at almost any velocity. I n contrast, it seems that minor contractions or expansions of the electronic cloud are made easily and rapidly at any velocity which is small compared to electronic velocities. Conclusions. The conclusion of Lichten’s analysis of the helium charge exchange reaction is in agreement with Zener’s analysis of the Na + C1 collisions-namely, that the appropriate electronic basis functions to be used as the expansion basis in the general expansion (5) are those corresponding to a simple physical configuration of the electrons, whether an MO configuration or a valence bond configuration. Lichten went further and pointed out something that had apparently gone unnoticed, at least since Zener’s work, namely, that a set of physically simple 4i’s may constitute a representation of the He operator, but a nondiagonal representation (in contrast with the stationary “adiabatic states which make up its diagonal representation). The fact of being nondiagonal is no argument against it, because He is only a part of the full Hamiltonian H . The laws of quantum mechanics require only that the full Hamiltonian, H , + T, , be diagonal. Diagonalizing He is therefore good only if it helps to diagonalize H , as it does in the one-state problem (Section Ill), and bad if it hinders the diagonalization of H , as it does for the two-state problems we have examined. ”

Thomas F. O’Malley

236

Finally, the single configuration electronic states, Cpl and Cp2, etc., were named by Lichten cliabatic ” states, as opposed to the stationary “ adiabatic states. He pointed out some of the main properties of the diabatic single configuration states: “

”

1. 2. 3. 4.

They are purely electronic states, related to the operator He only. However, they are not the stationary adiabatic states. They do not diagonalize H e . They violate the noncrossing rule of von Neumann and Wigner, in contrast to the stationary adiabatic states. 5. They define a representation (nondiagonal) of He for a given symmetry (spin, angular momentum projection, parity). 6. They are in some cases autoionizing and possibly doubly excited at small R . 7. However, they normally dissociate to stationary atomic states at large R. The identification of the single configuration diabatic states which seem to be followed by atoms colliding in nature appears to be an important step in understanding atomic collisions. As described here, however, they do have some minor drawbacks. For example, if the single configuration concept is taken literally in doing an ab initio calculation, it is known to be limited in accuracy to a couple of electron volts, requiring that the energy be shifted ” by this amount when applied to the physical problem. The ionic and covalent states of Section 1V also have their special difficulties if pursued very far. While these valence bond states are accurate (by construction) at large R , if one attempts to push them to small R , the essential distinction between the two configurations fades away in the small R limit, when the electron is equally close to either nucleus. In the next section, both the problem of accuracy and the problem of continuation to small R are pretty well eliminated when the entirely different problems of dissociative attachment and recombination suggest the use of general quasistationary states, or a quasistationary representation, to play the role of diabatic states, thus generalizing the results suggested by this and the preceding sections. “

VI. Dissociative Recombination and Attachment-The Quasistationary State Representation I n the last two sections, a consideration of the Na + CI and of He+ + He collisions suggested the picture that even slowly moving atoms in collision tend to assume electronic states which maintain the same overall “configuration” (Zener called it “character”), such as covalent or ionic in the Na + C1

DIABATIC STATES

237

case or single configuration MO's as in the Hef + He case, rather than the (sometimes very unsimple) eigenstates of H , which are called " adiabatic " states. In this section, these ideas will be confirmed and the possibility introduced of a much more systematic mathematical definition from an apparently very different source-the study of the dissociative recombination (DR) or attachment (DA) of electrons to molecules, i.e., e+AB+

and efAB

-

-

A*+B

(27)

A-+B

(28)

where A* indicates some definite possibly excited state. A bit of history will be helpful in introducing the mathematics which has been found to describe these reactions effectively. In 1950 the dissociative recombination process (27) was put forward by Bates (1950) to explain electron loss in the atmosphere, and was described by a simple picture, but one which has remained intact through later more sophisticated analyses. The picture is best shown by the potential energy curves of Fig. 3. When an electron collides with a molecular ion AB', the DR reaction is described as taking place through an electronic transition from the initial state AB' + e (shown as the shaded continuum in Fig. 3) to the final electronic state AB, of the total system, which then dissociates along its potential energy curve to the final

Rc Internuclear distance, R

FIG.3. Electronic energy curves for complete set of states of a typical neutral molecular system AB, as invoked to explain dissociative recombination. The shaded region is the continuum (e AB+). The dissociating state AB, is an autoionizing quasistationary state crossing through the continuum for R < R, and through the discrete Rydberg states ABR, for R > R,, with no regard for the noncrossing rule. This set of states makes up the quasistationary representation of He defined in the text.

+

23 8

Thomas F. O’Malley

products A* + B. It was noted that the state AB, was not an ordinary state of the system AB (certainly not a stationary eigenstate of H e ) , but rather an autoionizing state (otherwise called a resonant state or a quasistationary state), in that it could spontaneously reemit the captured electron, returning to the original configuration AB’ + e. Accordingly the reaction (27) was written more explicitly

-

(29) indicating that, after capture into AB, , autoionization competes with dissociation. From this model, a supposedly rough formula for the cross section was derived, which has since been confirmed in large part (mostly for DA) and extended by later analyses. A long step toward the quantitative understanding of dissociative attachment and recombination was made by the mathematical model of Bardsley et al. (1964). These authors considered the attachment problem (28), which is essentially the same as recombination (except that the presence of the one extra electron screens the Coulomb field of AB so that first the Rydberg states of Fig. 3 are absent, making the problem simpler, and second, AB, and A* become AB,- and A-, respectively). Starting with the same physical picture, described by potential curves like those of Fig. 3 but without the Rydberg states, they focused on the motion of A- and B in thejinal state AB,-, asking, as we have done, what are their mutual forces, or, more generally, seeking the full equation of final state motion. Noting that the (final) dissociating state is a resonant state, which is degenerate with the electronic continuum where the capture takes place, they chose the complex resonance formalism of Kapur and Pieirls to describe the final state electronic energy curve, postulating as the equation of motion e+AB+

[TR

AB,

A*+B

+ E,(R) - $iT(R) - E]z,(R) =

-U

where x, is the desired wave function for the nuclear motion of A- and B in the state AB,- ; E, - $iris the complex Kapur-Pieirls energy, generalized to be a potential energy curve, varying with R ; is the width of the state (here the width for autoionization back into the initial reactants AB + e), which is finite in the shaded region of Fig. 3 where autoionization is possible; and finally, the inhomogeneous term a is a “ source term,” accounting for the initial capture ( e + AB --+ AB,-) into the state. Starting mathematically from (30) and using the Born-Oppenheimer and other good semiclassical approximations and some formal manipulation, they finally derived a physically transparent formula relating the cross section to the parameters of the initial and final potential curves AB and AB,- and to the homogeneous solution zrhof (30), i.e., [TR E,(R) - +ir(R)- E]x,*(R) = 0. (31)

+

DlABATIC STATES

239

The Bardsley, Mandl, and Herzenberg formula for DA, which expresses the cross section as the product of an electron capture cross section, a FranckCondon overlap factor, and a survival factor (for depletion of the state through back autoionization), has proved extremely useful in the semiempirical analysis of experiments, and equally important the mathematical description of the reaction by Eq. (30) allowed completely ab initio calculations to be attempted for the simplest system H 2 - (Bardsley et al., 1966). A still more complete mathematical description of dissociative attachment has since been given by an independent approach (O’Malley, 1965, 1966), starting again from the same picture of the physics as used by the above authors, but this time going back to the Schroedinger equation (1) and deriving everything from the beginning. The physical insight that the essential transition was from an initial scattering continuum state (AB + e) to a final autoionizing or resonant state AB, was used in the following way. We have seen that, given any electronic basis {4i},the full wave function for a molecular system may be expanded in this basis as in Eq. (5) with each 4imultiplied by a nuclear motion function x i , as was done with different bases in Sections 111, lV, and V of this paper. Now does the physics of DA, as just described, suggest an appropriate basis of electronic states ?

A. SIMPLE RESONANT SCATTERING-QUASISTATIONARY STATEFORMALISM Simple resonant scattering problems (for example electron-molecule scattering), where there is a transition from a scattering continuum into a quasibound resonant state (and then back to the same or another continuum state), have been found to be described most effectively by a basis of states consisting of a bounded resonant state function 4, and a continuum of “potential scattering” state wave functions 4p,i. Of the different possible ways of defining such a resonance-potential scattering basis, the most general and useful for our purpose, is given by the quasistationary state forrnali~rn.~ One starts with some quasistationary state wave function 4, and its energy

where the energy E , is an approximation to the energy of the observed resonance and 4, is a bounded wave function describing the temporarily bound electron. We call the Hamiltonian “ He’’ with the molecular application in mind. Lippmann and O’Malley (1970) contains a summary of the formalism. Note that the meaning of P and Q has been interchanged from that use here.

Tliomas F. O’Malley

240

The potential scattering functions 4,,, which complete the basis are then defined as the eigenstates of H , in the space of states orthogonal to 4,.In other words, if the projection operator P is defined by P =1-

then the

4p,

Q,

Q = I4i-)<4rI>

(33)

are eigenstates of the operator P H , P, or P(He - E p ,

i)f‘4p,

I

= 0.

(34)

Note that we said the E , was simply an approximation to the resonant energy (say Er). This is not a problem because the completeness of the basis {4,, q5p, i}allows the exact scattering wave function to be expanded formally in this basis [Y,, = 4, J c ( E ~ ) dEi]. ~ ~ , The Breit-Wigner formula then follows directly3 and shows a resonance at the unique energy

+

E, = E ,

+ A@,).

(35)

The level shift

corrects for any approximation in the initial E , . Likewise the exact width the resonance follows from 4, and 4 p by = 2nI (4rI He I 4

p ) I ’,

of

(37)

where 4 p is the 4p, for the resonant energy E,. It was the basis {4,, 4,,, i}for the space of electronic functions which was used by O’Malley (1966) in Eq. (5) to expand the exact molecular wave function for dissociative attachment and later for the more general problem of diatomic collisions (O’Malley, 1967). It has since been suggested by Lippmann and O’Malley (1970) that for either a general theoretical description or for the sake of semiempirical application of the resulting formulas, it is desirable for the arbitrary or approximate quasistationary state energy E , of (32) to be equal to the unique and physically observable resonant energy E, of (35). They accordingly defined a special quasistationary state representation by putting the constraint on the 4, that it make the level shift A vanish at E = E,, i.e.,

for Er = (4r I He I 4,).

The solution of this eigenvalue problem makes the diagonal matrix element ($,I H e /4,) = E, unique in addition to the already unique nondiagonal ($, I H,I 4 p )which determines r.

24 1

DIABATIC STATES

B. APPLICATION OF THE QUASISTATIONARY STATEBASISTO DISSOCIATIVE ATTACHMENT

‘To return to the more complicated problem of DA, the optimum mathematical treatment of the problem was found by expanding the full wave function Y according to Eq. (5) but now with the quasistationary state basis of He (39) {4i) { 4 r > 4 p . i> used as an expansion basis, i.e., +

(40) ‘ ~ ( rR, ) 4r(r; R)x,(R) + Si + p , i(r, R)xi(R) [where is used according to convention to denote an integral over the continuum as well as a sum over any (possible) discrete states]. Following the development of Section 11, the expansion (40) reduces the Schroedinger equation to the form (6), and finally to (10) when the basic Born-Oppenheimer approximation is made. In the present representation, the electronic Hamiltonian matrix V i j which provides both the elastic potential and the coupling between states in (lo), takes the form4 1

si

Er(R) vlj

(4i I H e l 4 j ) =

1

J‘l(R)

V , ( R ) E p0, ( R ) T/,(R) V,(R)

...

T/,(R)

V3(R)

Ep2 0( R )

0

0

Ep3(R)

0

...

...

...

!!I.

(41)

... ...

where Vi = ( 4 r I He I 4 p i > . This representation of He is also written more formally (O’Malley, 1965) H e = (QHe Q

OH,‘)

PHeQ

PH,P

The matrix (41) or (42) produced by the quasistationary state basis (39) is therefore a nondiagonal representation, the quasistationary representation of H e , which is also called the quasiadiabatic representation because it is truly adiabatic but quasistationary rather than stationary. The coupled equations (10) with the representation (41) become explicitly

+ E ~ ( R-) EIXLR)= - S i

Vi x i

(43a)

+ E,i(R> - E]xi(R) = - Vi x r .

(43b) Equations (43) are the basic equations for the DA problem, determining the vibrational-rotational functions xr and x i (e.g., xr describes dissociation along the final potential curve A B r - ) . [TR

O’Malley (1 967). Demkov ( I 966) was also led to the same matrix for a special class of problems.

242

Thomas F. O’Malley

The starting point, Eq. (30), of the Bardsley et al. (1964) mathematical model of DA is easily derived from the above Eqs. (43), as was done by O’Malley (1966), if one first solves (43b) formally for xi in the form xi = xo 6(E - E,,) + ( E - TR - E,, + ;&)-I V ,x, (where the first term represents the incident wave in DA, i.e., e + AB in the vibrational-rotational state xo). This is then substituted into (43a) to give immediately

The integral correction term to E,, in (44), a manifestly nonlocal operator in R, was found nonetheless to reduce to the - qir term of Eq. (30)

-+ir = -

i 7 r ~ ( ~ r ~ ~ e ~ ~ , ) / z

(45)

when one treats the vibrational spectrum of the Green’s function semiclassically5 and further remembers that the electronic level shift A(&) is being defined to vanish, as in the Lippmann and O’Malley (1970) paper. [Actually it was the directly useful homogeneous counterpart (3 1) of the equation that was obtained in the original derivation, after formally relating the matrix element for the direct process to that for the inverse reaction (A- + B + AB e) which is described directly by the homogeneous equation.] Thus the starting point (30) of the Bardsley et a/. ( 1 964) model of DA has been derived directly from the Schroedinger equation with the help of the electronic expansion basis (39), so that their valuable formula for the cross section again follows, completing the solution of the problem. To summarize, we have seen in this section, after reviewing some of the history of the dissociative attachment problem, that a full ab initio mathematical description, tailor made to suit the physics of the reaction, follows if one expands the full wave function Y through Eq. (40) in the electronic basis consisting of the resonant state and potential scattering continuum wave functions (4, and the 4,,i ) defined in the quasistationary state f o r m a l i ~ m . ~ The resulting equations (43) following the Born-Oppenheimer approximation are then solvable in good semiclassical approximation,‘ thus completing the description of the problem. The analogy to Zener’s treatment of the Na + CI problem is obvious, as it is, to a lesser extent perhaps, to Lichten’s description of the He+ + He

+

The detailed reduction, suppressed in O’Malley (1966) for brevity, isgiven in O’Malley and Taylor (1 968). Neither the Born-Oppenheimer approximation nor the semiclassical approximation is strictly necessary in a general treatment. If one wishes, for example, to assess the magnitude of the error introduced, either the T’and T” terms or the quanta1 solution of the nuclear motion equations can always be brought back. It is just that it is rare in atomic physics that one gets the chance to make really good approximations such as these.

DIABATIC STATES

243

problem. In each case the full wave function Y has been expanded in an electronic basis suggested directly by the physics involved. In the first case, the finite basis consists of the ionic and covalent states. In the second, a truncated basis of single configuration molecular orbitals is used. In the present section, the complete basis of resonant and potential scattering states is employed. It should be observed that the present representation of He does not suffer either from the problem of inaccurate (shifted) energy levels as did the single configuration representation, nor of lack of definition at small R as did the other. [Actual numerical calculations (O’Malley, 1969) on H, and He,’ suggest further that quasistationary states AB, identifiable with both Zener’s valence bond and with Lichten’s MO configuration are easily produced in practice.] In the next section, the extension of the quasiadiabatic representation of H , to treating the problem of slow heavy-particle collisions is very briefly reviewed and discussed.

VII. Slow Heavy-Particle Collision Theory-Extension of the Quasistationary Representation to Rydberg States In the last section, a mathematical machinery was presented for dealing with a certain class of reactions (dissociative attachment and recombination) by expanding the wave function in a quasistationary electronic basis. It was pointed out (O’Malley, 1967), following the solution of the DA problem, that if one considers the process of DR run backwards (i.e., A* + B --* AB’ e), a simple examination of the potential energy curves of Fig. 3 shows that one is led directly into the general problem of two slow atoms colliding, with the possibility of making transitions to any of the discrete or continuum states crossed by the curve AB,, thus resulting in almost any state of excitation, charge distribution, or association. The coupled equations derived in the last section, in particular the coupled equations (43) for the nuclear functions xi, therefore already describe this whole class of heavy-particle collisions, as was discussed in some detail. We therefore take as the basic wave function for a broad class of heavy-particle collisions, Eq. (40) of the last section which we repeat here:

+

The symbols mean the same as in the last section, only now the discrete sum implied in is at least as important as the integral over continuous states. The coupled equations (43) for the xi are likewise identical and need not be repeated. Since it was seen that the definition of the $ p , i was automatically determined by (34) and (33), given =,, the problem of defining the diabatic states in

si

Thomas F. O'Malley

244

general reduces to defining the particular diabatic or quasistationary state 4,. In the remainder of this short section, therefore, we consider briefly the extension of 4, from the continuum or shaded region of Fig. 3, where it is natural and straightforward, to the region of discrete curve crossings.

A. NEGATIVE MOLECULAR SYSTEMS A negative system AB- is the easiest in which to define and E,(R). In the shaded region of Fig. 3, AB; is an autoionizing resonant state. While the quasistationary state formalism allows an arbitrary definition of 4, , we mentioned in the last section that the special definition of Lippmann and O'Malley, which defines the unique resonance energies E, of H e , has obvious advantages, and this special definition is assumed here. In summary, E, and 4r are determined by the simultaneous eigenvalue equations Er(R) = ( 4 r

I H e I 4,)

(47)

=0

(48)

and A"Er(R)I

with A given by Eq. (36) and with E,(R) and all other quantities defined with R as a parameter. Now negative systems (i.e., singly negative) differ from the neutral system depicted in Fig. 3 and from positive systems in one important way. Since an electron taken far away from the residual core sees the screened charge of a neutral system rather than an attractive Coulomb field, there are no Rydberg states for negative systems. The discrete Rydberg states of Fig. 3 are therefore absent for a negative system AB- , making it much simpler. The definitions (47) and (48) define the state AB-, for R < R,. But for R > R,, the state is no longer autoionizing. It is not difficult to convince oneself that Eqs. (47) and (48) simply define a true bound state or eigenstate of He in this region ( R > R J , so that AB; becomes just the lowest adiabatic state of the system between R , and cc , given this definition. [Note that no discontinuity is introduced at R, since the same definition (47) and (48) is used. What happens is that r goes continuously to zero at R,.]

B. NEUTRAL AND

POSITIVE

MOLECULAR SYSTEMS

Neutral and positive molecular systems have discrete Rydberg states as shown in Fig. 3. The definition of AB, in the autoionizing region is no different from that given by (47) and (48) for negative systems. However when R > R,, the potential curve of AB,, as drawn in Fig. 3 cuts through the Rydberg states, violating the noncrossing rule. Therefore it is definitely not an adia-

DIABATIC STATES

245

batic state, even in this region, but rather a diabatic state (which is the whole point of this work !). The way in which one may define the diabatic state AB, with wave function 4, and energy E,(R) uniquely in the region in which it crosses the discrete states is discussed in some detail in Section V of Lippmann and O’Malley (1970). In brief, one analytically continues the state from the region R < R, down into the region R > R, in the manner done in quantum defect theory, either by continuing the H e matrix elements V i jthemselves, or by analytically continuing the Green’s function which occurs in the definition of A before applying the definition (38), and proceeding as in Section VI to construct the matrix V i j of (41). Note that, unlike the atomic quantum defect theory, the analytic continuation of E, = E,(R) appears to be more naturally and simply interpreted as an expansion in R rather than in E. Such an analytic continuation should take one down in practice to roughly the lowest curve crossing betwen AB, and the lowest Rydberg state. The diabatic state AB, is therefore now defined exactly and uniquely from R = 0 to R, and on down to the general neighborhood of the lowest curve crossing. Going further, once one gets well beyond the lowest curve crossing, the fact that Eqs. (47) and (48) as they stand define the adiabatic state may again be used in the large R region to R = 03. This leaves one small gap in the definition of the diabatic state AB, (and hence the whole quasiadiabatic representation of H e ) for neutral and positive systems. This gap is in the general neighborhood of the lowest curve crossing. Although any number of ad hoc procedures suggest themselves for getting through this region and connecting the analytically continued quantities to the adiabatic ones at large R , no elegant solution has been proposed. This gap in the proposed general definition of the diabatic states is left as a presently unsolved problem, and any general or elegant solution for filling it would be appreciated.

VIII. Summary and Conclusion With the question in mind “what are the forces acting between two atoms which move with small but finite velocity?” we considered the equations of motion of the atoms and in particular the expansion of the full wave function in various bases or representations of H e . Through a brief and perhaps somewhat subjective history, we saw that, while the eigenstates of H , (the socalled adiabatic states) are perfect for describing the problem presented by spectroscopy of a single permanent molecular state, an entirely dzflerent kind of basis or representation was needed for describing the two-state or nianystate problems presented by collisions-a number of which were reviewed.

Thomas F. O’Malley

246

These different electronic bases or representations which successfully describe collision problems have come to be called diubatic states. In looking first at the Na + CI collision problem and then at He’ + He charge exchange, we saw that both were described faithfully, if sometimes roughly, by a basis of physically simple “ single configuration wave functions, whether valencebond or molecular orbital. Moving to dissociative recombination and attachment, again the proper basis was seen to be a physically simple one consisting this time of resonance and potential scattering functions. This last basis, when defined by the quasistationary state formalism, had the advantage of allowing one to define the potential energy matrix (and thus the forces we are seeking) not only with arbitrary precision but even in a sense uniquely, in that the level shifts could be guaranteed to vanish [Eq, (48)] when the states are coupled. It is this latter quasistationary state representation that is proposed as a general definition of the diubutic or collisional electronic states of molecules, diatomic and polyatomic. Although it was not emphasized in this limited treatment, it is clear that the He’ + He states discussed in Section V, and probably the Na + C1 states as well, are special cases of the quasistationary representation of Sections VI and VIII, which is therefore capable of supplying the needed accuracy in the first case and small R definition in the second. We therefore feel that we have fairly well answered the question as to the forces between slow atoms. In qualitative terms, they are the forces between physically simple configurations,” whether ionic and covalent, single MO configurations, or resonant and potential scattering configurations. Quantitatively we have put forth the special quasistationary state formalism of Lippmann and O’Malley as the way of dejning them explicitly in a mathematical sense, and of guiding attempts at ab initio electronic calculation. With the electronic basis {4i} defined by (39), the end result is the coupled equations (10) or more explicitly (43) for the nuclear motion, with the forces now known. There is a great deal of fruitful work now being done (too much to which to give adequate references), all of it starting with equations of the form (10) and then attempting to solve the coupled nuclear motion equations better or more generally than was done by Landau and Zener. We therefore end where the others begin! It does seem, however, that the almost totally neglected electronic problem, which we have attempted to illuminate, forms a necessary basis on which all this other work on the nuclear motion problem must rest. I n between the definition of the diabatic electronic basis, which we have considered, and the solution of the coupled equations there are a number of additional related points which might be worth mentioning. First we have consistently made the Born-Oppenheimer approximation because it is a good and accurate approximation (when the 4iare at all reasonably defined) and ”

“

“

”

247

DIABATIC STATES

embodies the essential difference of molecules from atoms. There are a few instances, however, of transitions (albeit slow ones) which can only occur through the coupling term T'of Eq. (7), even when the 4i are optimallychosen. An important example is that of transitions between ekctronic states of different angular momentum component A , The different symmetries imply that the electronic matrix element Vikmust necessarily vanish in any representation, so the T' must mediate such slow transitions, as was discussed by Bates and Williams. To include this kind of transition, the T' term in Eq. (6) should be reinstated so that (10) is generalized to LTR

+ Vii

( R ) - Elxi (R) = -

c

j# i

Vij x j

-

c

k#i

Tik X k .

(49)

The terms in the last sum have the different subscripts k to suggest that, whereas the electronic statesj all have the same symmetries as the state i (otherwise the V i j would vanish), the states labeled k which are reached via T' are generally of different symmetries. Some more should be said about Felix Smith's (1969) definition of the diabatic representation, which was mentioned in the introduction. His representation has all the qualitative features of the diabatic states which we have described. What he does briefly is the following. The exact equations (6) contain both potential coupling terms Vij and kinetic coupling terms Tik and Ti;. Smith's representation is in a sense the exact opposite of the so-called adiabatic representation. Whereas the latter representation, by diagonalizing the H e matrix, arificially constrains the nondiagonal potential coupling terms Vij to vanish, Smith's diabatic representation constrains the kinetic coupling terms T i j and T,; of Eqs. (7) and (8) to vanish for all i , j , and R [except for the rotational coupling already alluded to (Eq. 49)]. As a result, the anomalously large wave function gradients which are generally found in the adiabatic representation near an " avoided crossing " are, happily, eliminated, and in fact replaced by zero gradients. The Born-Oppenheimer approximation (Eq. 9) is converted into an identity in this representation by postulating that the gradient matrix elements (4ild/dRI4 j ) vanish identically. [In contrast, the presently proposed definition of the diabatic representation merely prevents these matrix elements from becoming anomalously large (in atomic units) and relies on the smallness of the M - ' factor and of the velocity in T' and T"to make the Born-Oppenheimer approximation a good one.] Equation (10) results in either case. It would be interesting to see an extensive concrete study of the Smith representation (as well as the presently proposed one) to see how the definitions work out in practice. For example in those regions away from curve crossings, where the motion is predominantly elastic, one would like to verify that the elastic motion is actually described accurately by the appropriate diabatic potential curve.

248

Thomas F. O’Malley

Another study, by Levine et a/. (1969), touched peripherally on the diabatic state problem through the interesting decoupling approximation for the twostate Landau-Zener problem. This approximation seeks to define a pair of best ” elastic scattering states by effectively minimizing the entire coupling [the right-hand side of Eq. (6)] for each energy. One might expect the resulting decoupled states, constructed according to this criterion, to approach diabatic type states in the relatively high energy limit. Although some numerical study was done, it stopped short of actually constructing the best decoupled states, so that comparison can not be made. Finally, it may not have been emphasized enough that the quasistationary state representation of H e , suggested here as the appropriate mathematical definition of the diabatic states, is certainly not cut and dried or in a state of completion, even as presented in Section VII, but still requires a good deal more exploration and especially more concrete study. For example, the gap already mentioned (Section VII) in the formal definition of 4,. in the neighborhood of the lowest curve crossing presents an obvious challenge. The application of the method to systems which, like NaC1, have ionic states, or to systems with two or more closely spaced resonant states of the same symmetry, for example, will certainly present interesting and perhaps new problems. And the building of further connections between the quasistationary state definition and the more successful computational methods for resonances, such as the stabilization method (Eliezer et al., 1967) and others, might be one of the most fruitful advances in this area. “

REFERENCES Bates, D. R. (1950), Phys. Rev. 78,492 (L). Bates, D. R., and Williams, D. A. (1964). Proc. Phys. Soc., London 83, 425. Bates, D. R., Massey, H. S. W., and Stewart, A. L. (1953), Proc. Roy. SOC.Ser. A . 216, 437. Bardsley, J. N., Herzenberg, A,, and Mandl, F. (1964) In “Atomic Collision Processes,” (M. R. C . McDowell, ed.), p. 415. North Holland Publ., Amsterdam. Bardsley, J. N., Herzenberg, A,, and Mandl, F. (1966). Proc. Phys. Soc., London 89, 321. Berry, R. S. (1957) J. Chem. Phys. 27, 1288. Demkov, Y. (1966) Dokl. Akad. Nauk SSSR 166,1076 [English trans]: Soviet Phys.-Dokl. 11, 138 (1966)l. Eliezer, J., Taylor, H. S., and Williams, J. K., Jr. (1 967). J. Chem. Phys. 47, 21 65. Firsov, 0. B. (1951)Zh. Eksp. Theor. Fir. 21, 1001. Kohn, W. (1953). Phys. Rev. 90, 383. Landau, L. (1932). Phys. Z. Sowjetunion 2, 46. Landau, L. D., and Lifschitz, E. M. (1958). “Quantum Mechanics,” p. 261ff. AddisonWeslley, Reading, Massachusetts. Levine, R. D., Johnson, B. R., and Bernstein, R. B. (1969). J . Chem. Phys. 50, 1694. Lichten, W. (1963). Phys. Rev. 131, 229.

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Lippmann, B. A , , and O’Malley, T. F. (1970), Phys. Rev. A 2, 2115. London, F. (1932).Z. Phys. 74, 143. O’Malley, T. F. (1965). Proc. 4th Int. Conf: Phys Electron. At. Collisions, p. 97. Science Bookcrafters, Inc., Hastings-on-Hudson, New York. O’Malley, T. F. (1966). Phys. Rev. 150, 14. O’Malley, T. F. (1967). Phys. Rev. 162, 98. O’Malley, T. F. (1969). J . Chern. Phys. 51, 322. O’Malley, T. F., and Taylor, H. S. (1968). Phys. Rev. 176, 207. Smith, F. T. (1 969). Phys. Rea. 179, 11 1 . Stiickelberg, E. C. G. (1932). HeIv. Phys. Acta 5, 369. von Neumann, J., and Wigner, E. P. (1929). Phys. 2. 30, 467. Zener, C . (1932). Pvoc. Roy. Soc. Ser. A 137, 696. Ziemba, F. P., and Everhart, E. (1959), Phys. Reu. Lett. 2, 299. Ziemba, F. P., and Russek, A. (1959). Phys. Rec. 115 922.

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SELECTION RULES WITHIN ATOMIC SHELLS B. R . JUDD Department of Physics, The Johns Hopkins University Baltimore, Maryland

I. 11.

.............................

C. Extensions D . Quasi-spin ....................... ........ ........ 111. Irreducible Representations ............ A. The Wigner-Eckart Theorem . . . . . . . B. Designations ........................................... C. Commutators .......................... D. Explicit Constructions ........................ E. Definition through Matrix Elements. ...........................

256 258 259 263

H. Quasi-spin Assignments ............ IV.

A. TheGroupR(3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Selection Rules for Radial Integrals. .

V.

A. Direct Application . . . . . .

..........................

VI.

Conflicting Symmetries ...........................

VII.

VIII. IX .

References. . . . 25 I

214

252

B. R. Judd

I. Introduction Null matrix elements are fascinating objects. They are important in simplifying perturbation theory, thereby making feasible many calculations that would otherwise be impracticable. They indicate the existence of an underlying structure possessed by the states and operators that form the matrix elements. If this structure has a physical basis, a class of vanishing matrix elements usually leads to selection rules that can be given a direct physical interpretation. If, on the other hand, the structure is of the type exploited by Racah (1949) and described by him in terms of continuous Lie groups, the null matrix elements may merely simplify the analysis without being susceptible of experimental verification. Both types of null matrix element will be discussed here. As more and more calculations in atomic shell theory have been performed, however, a new feature has emerged. This is the existence of matrix elements that vanish for no apparent reason. A calculation may be carried out in which the positive and negative parts of some final sum exactly cancel. Such matrix elements are especially intriguing, because it is difficult to suppress the feeling that they are the indicators of some as yet unknown structure in the atomic shell. Moreover, we may expect that this new structure, if found, would simplify the calculations of the nonvanishing matrix elements-quantities which are more mundane than their null companions, but vital in numerical work. Considerable progress has been made in recent years on the interpretation of the unexpected zeros. Not every one of them is necessarily understood, of course; but a large proportion of them have yielded to analysis, thus making it appropriate to include an account of them here.

11. Groups A. DETERMINANTS

To begin with, some brief account must be given of Racah’s use of Lie groups. It may be wondered why such groups should be needed at all. Any state of lN can be defined by a linear combination of Slater determinants, and these expressions can be used to calculate the matrix elements of any operator. In preparing to follow this procedure, two difficulties become apparent. The first is simply that the number of required determinants runs into the hundreds for many states o f f 5 , f ‘, and f ’. The second stems from the problem of defining a state. For all but the simplest configurations, it is not enough merely to give I (the azimuthal quantum number); N (the number of electrons); S, L (the total spin and total orbital angular momentum quantum

SELECTION RULES IN ATOMIC SHELLS

253

numbers, respectively); and M , and M L (the quantum numbers corresponding to the projections along the z axis of the angular momenta S and L). For example, there are two D terms in d 3 , and so two orthogonal linear combinations of Slater determinants can be constructed for any state of d 3 for which S = 4 and L = 2. Racah’s groups provide a natural way to choose such an orthogonal pair. It is a remarkable fact that Condon and Shortley (1935), knowing nothing about Lie groups, but guided by a strong sense of what is mathematically convenient, picked for a pair o f zD states precisely those linear combinations of Slater determinants that correspond to two different irreducible representations of R(5), the rotation group in five dimensions. Their invitation “ Let us choose one [state] at random ” has proved to contain more substance than met the eye in 1935. Perhaps it should also be mentioned here that Slater (1960) has given the determinantal expansions of all the states of d N for which M , = 0, and Tan Wei-Han et al. (1966) have extended the calculations to thefshell (giving the expressions for M , = L, however). Each linear combination of determinants calculated by these authors corresponds exactly to a sequence of irreducible representations of various Lie groups. Of course, Racah introduced these groups to enable states to be well-defined without having to specify their determinantal composition. The example of the ’D terms suggests that the separations made on the basis of group theory are the most natural and convenient ones. B.

RACAH’S

SCHEME

The basic step taken by Racah (1949) was to introduce the rotation group R(21+ 1 j whose space of 21 + 1 dimensions is spanned by the 21 + 1 orbital functions of a single electron 1. Included among the matrices of 21 + 1 rows and 21 + 1 columns that describe the transformations of this group are those that transform the basic orbitals in a way that corresponds to an ordinary threedimensional rotation of axes. It follows that R ( 3 ) is a subgroup of R(21+ I). We write R(21+ 1 ) 2 R(3). The matrices that describe R(21+ 1) are not the most general that preserve the orthonormality of the basic orbitals, however. Matrices of the latter kind describe the transformations of U ( 2 1 f l ) , the unitary group i n 21+ 1 dimensions. If we restrict our attention to those matrices whose determinants are + 1 , the special unitary group SU(21 + 1 ) is obtained. We can write U(21+ 1)

3

SU(21+ 1)

3

R(21+ 1 )

3

R(3).

(1)

Forfelectrons, Racah showed that the exceptional group G, of Cartan ( 1 894) can be inserted i n this sequence: U(7) 3 SU(7) 3 R(7) 3 G,

3

43).

(2)

B. R. Judd

254

It is well known that rotations in ordinary three-dimensional space can be carried out by a suitable combination of the components of L. Thus the three components L,, L,, and L, can be taken as the generators of R(3). To remind ourselves of these generators, we shall write this group as R,(3) from now on. The generators of the other groups are most suitably described by first defining the single-electron tensor operators w ( ~ whose ~ ) , amplitudes are given by (nl /I d K k 11 nl) ) = {[k][~]}”’, where, as usual, [ X I = ( 2 x + 1). When K = 0 it is convenient to introduce tensors d k )= J 2 w ( O k ) ;their amplitudes are given by (nI (1

dk)/I nl) = [k]’I2.

For the configurations l N for which N > 1, the following definitions are made: N

N

the sums running over the various electrons i. The generators of the groups introduced by Racah (1949) are given in Table I in terms of the V(k).The order of each group (i.e., the total number of generators) is also specified. For any group, the generators possess the basic property that the commutator of any two of them must be expressible as some linear combination of the members of the entire collection of the generators of the group. Transformations among the basic 21 + I orbitals of an electron I induce transformations in the many-electron states of I N . The states of these configurations break up into irreducible representations of the groups in question. Because of the rather general character of the transformations associated with U ( 2 l + 1) and S U ( 2 l + I), each irreducible representation of either group merely corresponds to all those states that possess the same spin S . There is thus no point in using the representations of U(21+ 1) or SU(2l + 1) as labels. The usefulness of Racah’s classification scheme therefore lies in the existence of R ( 2 l + 1) and, for f electrons, of G, as well. It is traditional to use the symbols W and U for the irreducible representations of R ( 2 l + 1) and G,, so a state o f f ” is defined by writing

if” WUTLM, SM,). The additional classificatory symbol T is required in a few instances when a given L value occurs more than once in U . Finding the most natural choice for T is an example of the inner multiplicity problem; solutions were specified by Racah (1 949) for all U occurring in f” .

255

SELECTION RULES IN ATOMIC SHELLS

TABLE I GENERATORS OF GROUPS Group

Order

Generators 1, . .., 21)

(21 i1)2

(k

4 4 1 1 1)

( k = 1,2, . . . ,21)

1(21+ 1) 14

( k = 1 , 3 , 5 , . . . 21-1)

= 0,

)

3

(=L[3/1(1+ 1)(21+ l)]"')

3

(= S d 2 )

+

(k =o, 1,. . ., 21)

(21 112 t 3 2)' (41i-

( ~ = 0 , 1 k; = 0 , 1 ,

+ 1)(41+ 3 ) + 112 (21 + 1)2

..., 21)

+ k odd) ( k 0 , 1 , .. .,21; - k

(21

(K

(21

=

2 9I k)

14 14 3

( k = I , 3 , . . . ,21- 1)

4214- 1)

C. EXTENSIONS There have been many developments in atomic shell theory since Racah's work. It is nevertheless true to say that the sequence of groups established by Racah remains the central core around which all subsequent group structure is built. Some extensions are rather obvious. For example, the transformations described in Section 11, B all take place in the orbital space. However, we may easily perform transformations in the spin space by introducing R,(3), the group whose generators are the components of S. Since the spin and orbital spaces are independent, the combined group is a direct product such as R,(3) x R,(3). An important subgroup of this group is RJ(3),whose generators are the components of the total angular momentum J. At the other extreme, we can take U(41+ 2) as the embracing group R,(3) x U(21+ 1). The generators of U(41+ 2) are included in Table I. The various relations between group of subgroup can be established by inspection, since the generators of a subgroup must be included in those for the group. It is clear, for example, that CJ(41+ 2) 3 Rs(3) x U(21+ 1 )

3

R,(3) x R(21+ 1).

256

B. R . Judd

Alternatively, we may introduce the symplectic group Sp(41+ 2) as an intermediary : U(41+ 2) 2 Sp(41+ 2) 3 &(3) x R(21+ 1). Although Racah never examined Sp(41+ 2) in detail, he made implicit use of it through the concept of the seniority number, u. It turns out that every u corresponds to an irreducible representation of Sp(41+ 2) (e.g., see Judd, 1963). An idea of Shudeman (1937) can be used to give another group structure. The electrons are separated into two classes: class A, for which all spins point up (all m, = -I), and class B, forwhich all spins point down (all m,y= -4). The group U(41+ 2) now decomposes in a different way (Judd, 1967a):

U(41f 2) 3 uA(21 + 1) X uB(21 f 1) 3 RA(21 f 1) X RB(21 + 1) RA(3)

RB(3)

RL(3).

(3)

Forfelectrons, G2A x G2B can be inserted in the sequence; and we may, of course, combine the A and B spaces at any point to produce the groups of Racah's sequences ( I ) and (2). For example, c2B

c2.

The generators of some of these groups are given in Table I. It is clear that we have only to add corresponding generators of G,, x G,, to obtain the generators for G, , and corresponding results hold for the other direct products in the sequence (3). As we shall see later on, the separation of the total space into a product of a "spin-up'' space and a "spin-down'' space turns out to be particularly effective in accounting for some of the more intractable of the unexpectedly null matrix elements. D. QUASI-SPIN It was mentioned in Section 11, C that Racah's seniority number u corresponds to irreducible representations of Sp(41+ 2). However, the modern way to handle seniority is to reduce all problems in which u enters to problems in angular-momentum theory. A detailed description of how this works has been given elsewhere (Judd, 1967b). In brief, we first define the double tensor at of rank 1 in the orbital space and rank s(= 4) in the spin space. Its component uLsml,when acting on the vacuum state lo), creates the singleelectron state I Im, sm,) : I lm, ~ ~ 1 . v ) E a L s m i I 0). To get a double tensor from the corresponding annihilation operators umVml, we take for the components the quantities

257

SELECTION RULES IN ATOMIC SHELLS

Three scalars in both spin and orbital spaces can now be readily constructed from pairs of creation and annihilation operators: Q+ = ${[s][l]}”2(atat)(00), Q-

=

-${[s][l]}1’2(aa)‘00’,

’ + (aat)(oo’}.

Q = --I{[ s][I]}’/2{(ata)(00

The commutation relations satisfied by Q + , Q - , and Q, can be found in a straightforward way, since the individual annihilation and creation operators must satisfy the familiar anticommutation relations for fermions. It turns out that these commutation relations are identical to those satisfied by S,, S - , and S, (where S , = S, + is,,, etc.). To bring out the analogy with S, the vector Q is called the quasi-spin. It is not difficult to show that the eigenvalues of Q,, which are called M,, are -$(21+ 1 - N ) . Furthermore, the shift operators Q , add or subtract pairs of electrons coupled to ‘ S states, and this is precisely how Racah (1 943) defined a string of states possessing a common seniority. The maximum value of M , is +(21+ I - v ) , and this is taken to be Q. So the couple ( Q M Q ) conveys the same information as (Nu). To illustrate these ideas, we may take the two ’ 0 terms of d 3 . The two irreducible representations of R(5) derive from two irreducible representations of Sp(l0) that correspond to li = 3 and I . So, for one ’ D term, Q = - M , = 1. For the other, Q = 2 and M , = - 1. The state for which Q = - M , = 2 occurs in the configuration d ’ , and is simply the ’0 term that corresponds to a single d electron. The string of states with fixed u, S, and L, but variable N , comprises a quasi-spin multiplet. The presence of annihilation and creation operators greatly widens the opportunity for introducing new groups. For example, the generators of uA(21+ 1 ) can be rewritten as

(ata)by’ + (ata)b:’

( k = I , 2,

. . . , 21 - 1 : - k I q

I k).

If these operators are augmented by

(atat)‘,y’,

(aa)yik

( k = I , 3, . . . , 21 - 1; - k 5 q 5 k),

the group R,(4/ + 2) results. The terms of l N with either N even or N odd and for which all m, = +,form a single irreducible representation of RA(41+2). This group is, in itself, not particularly interesting. However, it has recently been shown (Armstrong and Judd, 1970) that we can write, for both spin-up and spin-down spaces, R,(41+ 2) 3 R,(21+ 1) x R,(21+ I), R,(41+ 2) 3 R,,(21+ 1 ) x R,(21+ 1).

258

B. R . Judd

r)

The generators for a particular R,(21+ 1) (where 0 =A, p, v , or are given in Table I in terms of the following quasi-particle creation operators:

Pd

= J&71,2,,

- (-1)1-qal,2,-q19

v:=J+[u+-1/2,q

rl; =

J+[at1,2.q

+ (-1)f-4a-,,2,-q1, - (-1)J-q%,2,-q1.

By adding the corresponding generators together, we may readily verify that R,(21+ 1) x R,(21+ 1) x R,(21 + 1) x R,(21+ 1)

13

R(21+ I),

the rotation group on the right being the original (21 + 1)-dimensional rotation group introduced by Racah (1949).

111. Irreducible Representations A. THEWIGNER-ECKART THEOREM Suppose a matrix element that we wish to evaluate is written in the form (yRil O(R'i')I y"R"i"). The symbols, R, R', and R" stand for irreducible representations of a group 9 ; the labels i, i', and i" specify the components of the representations (and may themselves be irreducible representations of a subgroup of 9 ) ; while y and y" describe the actual atomic system in which R and R" occur. The symbol 0 (standing for "operator") is of the same nature as y and y". According to the Wigner-Eckart (W.E.) theorem (O'Raifeartaigh, 1968), the matrix element above is equal to (Rqi(R'i', R"i")(yR// O,(R) 11 y"R"). tl

The first factor here is a Clebsch-Gordan coefficient, and is independent of y and y". The second factor is a reduced matrix element, and is independent of i, i', and i". The sum runs over as many terms as the number of times that R occurs in the decomposition of the Kronecker product R' x R". This number is denoted by c(RR'R"), and the various equivalent representations R are distinguished by q. Although the majority of selection rules i n physics are examples of the equation c(RR'R") = 0, others depend on nonvanishing values of c(RR'R"). For this reason, the W.E. theorem has been quoted in full here.

SELECTION RULES I N ATOMIC SHELLS

259

B. DESIGNATIONS The actual process of attaching labels Ri and R“i“to states of atomic shells is a lengthy operation. However, the problem was solved by Racah for 1 < 4, and a complete tabulation for these configurations has been made by Nielson and Koster (1963). For example, we can rapidly find the WUT labels (as well as u and hence Q ) for all the states of f N ; and the states themselves can be considered to be uniquely defined by the accompanying tables of coefficients of fractional parentage (cfp), or by the determinantal expansions of Tan Wei-han et ul. (1966). For the purposes of this article, we shall therefore consider R , i, R”, and i” as being given. The labels R’ and i’ that are associated with the operator 0 are not so readily obtained. They are scattered in the literature, and have been calculated in a piecemeal fashion. This is partly because the R‘i‘ to be attached to a given 0 depends on the group that is used to define the states, and also because only the simplest operators correspond to a well-defined R‘. Usually an operator must be decomposed into parts before assignments of R‘ and i’ can be made. It seems appropriate therefore to describe in a general way the methods by which group-theoretical labels are attached to operators. As a continuing source for examples, we select the f shell. The states of this shell are sufficiently complicated to make the group-theoretical techniques essential, and the simplifications that result are all the more striking.

C. COMMUTATORS The most direct way of assigning group-theoretical labels to operators is to establish a correspondence between operators and states. Suppose the generators of the group 3 are written as Xu.Then, if the coefficients a,, in [ X , , O(R’r’)]=

c

uor0(R’r)

r

are identical, for all

0

and

Y,

to the coefficients buri n

X,lyR”r‘)= 1b,,(yR”r), r

it must follow that R‘ = R“. This conclusion is little else than a tautology, since the two equations above are precisely the ones from which the transformation matrices are derived. The method works quite well for operators that are sums of single-electron operators. This is because the correspondence

can be established for all groups for which the generators include only those tensors W C h for h ’ which h- + k is odd (see, for example, Judd, 1963). Examples

260

B. R. Judd

of such groups are Sp(41f 2), R ( 2 1 f I), G, , R,(3), and R,(3). Thus, the fact that the terms ID,'G, and ' I o f f 2 form basis functions for the irreducible representation (200) of R(7) (see Nielson and Koster, 1963) means that the tensor V'*), V4)and V(') form in R(7) an irreducible tensor transforming according to the representation (200). The components Vq(k) are proportional to the spherical harmonics Ykq, and they enter any calculation in which an electric potential is expanded in spherical harmonics. The perturbation of a crystal lattice on an embedded ion is often envisaged in this form (Elliott and Stevens, 1953). The quadrupolar hyperfine interaction involves V2), and thus can be labeled by (200) forfelectrons. Tensors W'sL' for which S + L is odd correspond to the forbidden terms of 12, and must be labeled accordingly. For example, the dipolar part of the magnetic hyperfine interaction involves W"'), whereas the interaction of the nuclear magnetic moment with the purely orbital motion of a circulating electron involves L. Put equivalently, this last is W'''). These tensors correspond to the forbidden states 3 D and 'P of 12, which can be shown to belong to the irreducible representation (2oooO00) of Sp(14). This description can therefore be attached to the magnetic hyperfine interaction. Further details are given in Table 11. It is convenient to make the abbreviation (a1a2* . . a, 0

. . ' 0) = (a1a2. . . a,)

(4)

for irreducible representations of Sp(41f 2), thereby avoiding the writing of long strings of zeros. The scalar (0 . * * 0 ) is abbreviated to (0).

D. EXPLICIT CONSTRUCTIONS The method of the previous section tends to become unsystematic when two-particle operators (such as the Coulomb interaction) are studied. What is worse, the actual algebra involved in working out commutators, which is by no means trivial for the example above, becomes excessively tedious for such operators; and we do not even know whether suitable states can, in fact, be found to parallel the transformation properties of the operators. These difficulties are avoided if operators with well-defined group-theoretical properties are constructed explicitly. As an example of this approach, we select the Coulomb interaction between electrons in the f shell. The description here follows quite closely the original calculation of Racah (1949). The starting point is the expansion of r;' as a sum over scalar products of spherical harmonics by the well-known additicn theorem (Condon and Shortley, 1935). These scalar products are related by various normalization factors and Slater integrals Fk to the products ( v ! ~ ). v y ) ) where (forfelectrons) k = 0, 2, 4, and 6. The first term (for which k = 0) shifts configurations as a whole, and we drop it. Now, the tensors v ! ~ )(for k = 2, 4, and 6) satisfy the same commutation relations with respect to the generators of G, as the

SELECTION RULES IN ATOMIC SHELLS

26 1

TABLE I1 DESCRIPTION OF OPERATORS FOR f

Interaction

Operator

Irreducible representations KUWUKk O f R Q ( ~ )Sp(14), , R(7), Gz, Rs(3), R L ( ~ )

Crystal field Spin-orbit Quadrupole hyperfine Magnetic hyperfine Coulomb

Orbit-orbit

Contact spin-spin Dipolar spin-spin

Spin-other-orbit

tensors V‘k’ (for k = 2, 4, and 6). From Table I, we see that they transform according to (20) of G,. The tables of Nutter (1964) or of Wybourne (1970) yield

+

(20) x (20) = (00) (10)

+ ( I I ) + 2(20) + 2(21) + (30) + (22) + (31) + (40).

The branching rules of Racah (1949) show that, of all the representations on the right, only (OO), (22), and (40) contain scalars i n R,(3). The three linear corn binations

1h ((20)k + (20)k I UO){vjk’ v ~ ’ } ‘ ~ ) ’ ,

262

B. R. Judd

for U = (00), (22), and (40), therefore provide three operators to which welldefined labels U can be attached. The isoscalar factors ((20)k + (20)kl UO) are special cases of the coupling coefficients that invariably arise in the construction process; they are the same as the Clebsch-Gordan coefficients of Section 111, A except that the RL(3)-dependence has been factored out so as to form a scalar product between vik)and vy). Two methods of calculating the isoscalar factors are available to us today. We may either refer to the tables of Nielson and Koster (1963) and use the W-E theorem to extract the isoscalar factors that we need; or we can use the tables of two-particle cfp of Donlan (1970) and use equations such as

((20)k + (20)k 1 (22)O)= A ( f 2 ' k , f ' k I f 4 ( 2 2 ) I S ) , where A is independent of k. To find A, we have only to apply a normalization condition on the isoscalar factors. The operators corresponding to U = (40) and (22) are called e2 and e3 by Racah (1949), who combined the operator for which U = (00) with the constant term (for which k = 0) to give the two additional operators e, and el. The entire Coulomb interaction is written as

e, Eo

+ elE' + e2E 2 + e3 E 3 ,

the operators e i having well-defined group-theoretical labels with respect to G , and, as it turns out, to R(7) as well. The parameters E' are linear combinations of the Slater integrals F k . This method of explicit construction could, of course, be used for the operators such as V ( k or ) W(I2)that were treated in Section 111, C. It is only necessary to regard them as coupled pairs of creation and annihilation operators. As we might expect, the tensor at transforms like the irreducible representation [lo . . * 01 of U(21+ l), which also labels the 21 + 1 orbital functions of a single electron 1. Similarly, a belongs to [00 ... 0 - 11, which corresponds to a hole in the complete shell. It is easy to show that

) to [lo . . . 0 - 11 of U(21+ l), except the scalar so all tensors W ( K kbelong W(Oo),which belongs to [OO . . . 01. The representation [lo . . . O - 11 does not occur in 1' (whose terms are labeled by [110 ... 01 and [20 ... O]), and the condition of Section 111, C that the generators W C Kof k )a group possess ranks for which K k is odd now appears vital in excluding U(21+ 1) from the list of groups for which a correspondence can be drawn between operators and the states of12.

+

263

SELECTION RULES IN ATOMIC SHELLS

E. DEFINITION THROUGH MATRIX ELEMENTS The explicit construction of operators labeled by irreducible representations can often be avoided by a judicious use of orthogonality relations and available tables of spectroscopic coefficients. We illustrate this approach with the dipolar spin-spin interaction H,, , which is given by

1 where fi is the Bohr magneton. First, standard tensor algebra can be used (e.g., see Innes, 1953) to obtain

for equivalent electrons. The symbols M k stand for the radial integrals of Marvin (1947); the reduced matrix elements are given explicitly by

Since the 3-j symbol i n this equation vanishes when t is odd, the sum over k in Eq. ( 5 ) runs over even k only. We specialize now to f electrons. The spin-spin interaction is a twoelectron operator, and is thus completely defined if all its matrix elements in f are known. Since its spin rank is 2, allelements vanishexcept thoseinvolving the triplets, for which S = 1. These belong to just one irreducible representation of R(7), namely (1 10) (Racah, 1949). From previous discussions it is clear that the tensors w ( l k )for even k belong to (200) of R(7). The tables of Nutter (1964) give

(200) x (200) = (000) + (110) + (200) + (220) + (310) + (400). The branching rules of Racah (1949) and Shi Sheng-Ming (1965) indicate that D states arise in the decomposition of (200), (220) (three times), (310) (twice), and (400). This specifies the distribution of tensors with orbital ranks of 2. We would except a total of 7, since there are 7 ways of forming an angular momentum of 2 from couples (kk') for which k and k' can be 2, 4, or 6. However, not all of these have to be considered. This is because ~((110)(310)(110))= ~((110)(400)(110))= 0, as can rapidly be verified from Nutter's tables. The Clebsch-Gordan coefficients that occur when the W.E. theorem is used are therefore zero,

264

B. R. Judd

and the corresponding matrix elements vanish (see Section 111, A). This means that the operators associated with (310) and (400) can be discarded. We can therefore write

H,,

= h,

+ h , + h3 + h,,

where the component parts hi correspond (in order) to WU = (200)(20), (220)(20), (220)(21), and (220)(22). The group G, gives a neat separation of the three tensors belonging to (220). We now have the problem of actually constructing the matrix elements of the hi.First, we separate out the dependence on J by writing (Trees, 1951a)

(f2SLJIhiI f’S’L’J’)

L

= 6(J, J’)( - 1) S ’ + L + J ’

L‘

s

2)

( f 2 s L 11

t;22) i l f 2 s r ~ r ) .

The reduced matrix elements of the t!22’ are not calculated directly. Rather, we write ( f 2 S L 11 2\22) Ilf2S’L’) = U i ( f 2 S Ljl zi IlfZS’L‘), and construct the zi with arbitrary normalization. It is easy to make a start, since zi corresponds to (200)(20)0, which are the labels for V‘”. Hence we can write

(f’3L 11 z1 Ilf’ 3L’)= 3J(14/5)(f2 3L 11 V(’)l l f 2 3L’), the prefacing coefficient being chosen to avoid fractions. The reduced matrix elements of V”) can be read off from the tables of Nielson and Koster (1963); their tensor U(’) is Vc’)/J5. The reduced matrix elements of z1 are entered in Table 111. Consider now z 3 , which belongs to (220)(21). The P and H terms of (1 10) belong to (1 1) of G, , while the F term spans (10) by itself. Nutter’s tables yield c((l1)(21)(1 I)) = c((10)(21)(10)) = 0, TABLE 111

REDUCED MATRIXELEMENTS OF THE SPIN-SPIN

3P 3P 3F ’F 3H

3P 3F 3F 3H =H

9 343 -22/14

d22 -dI43

-9 62/3

-414 22/22 d143

INTERACTION FOR f z

63

48dF 112d14 16V‘Z

-Id143

0 1/43 0

-3/2/z 0

1 0 0 0 9/2/1E

SELECTION RULES IN ATOMIC SHELLS

265

which means that all matrix elements of z 3 between the P and H terms vanish, as does the diagonal element for 3F. Only two matrix elements survive, namely those connecting 3 F to 3P and to ’ H . These can both be calculated (to within a normalization) by using the orthogonality relations between the Clebsch-Gordan coefficients that enter when the W.E. theorem is used. For us, these orthogonality conditions are equivalent to L,L’

and the ratio between the two nonvanishing matrix elements of z 3 is immediately obtained. They are entered in Table I11 with an arbitrary normalization. The reduced matrix elements of z4 can be handled in a similar fashion. Only z2 remains. The orthogonality conditions are not quite sufficient to determine all of its matrix elements. A supplementary equation can be obtained from the Clebsch-Gordan interchange ((I 1O)LMLI (220)(20)DML;(1 I0)PML”) = ( - 1)“((1

1O)LMLI ( 1 IO)PM,”; (220)(20)DML),

where x is independent of L, M,, M I , and ML”.We have merely to synthesize examples of the W.E. theorem in which the above Clebsch-Gordan coefficients appear and then take ratios: ((1 10)(10)P II 2 2 II (1 10)(11)P)- ((1 10)(11)P I1 W ( l 1 )11 ( 2 2 0 m w ) ( ( l l o ) ( l o ) F ~ ~11z(110)(11)P) 2 - ((110)(10)F~1W(l1)ll (220)(20)0)’ The ratio on the right can be found from Nielson and Koster’s tables forf4; it turns out to be (7/16)J3. All matrix elements of z 2 can now be calculated to within a normalization. The actual numbers are set out in Table 111. Until now all normalizations of the zihave been arbitrary. This has been possible because their prefacing coefficients a, have been at our disposal. Having fixed the z, , we are now in a position to determine the ai. We compare the matrix elements of H,, , as given, on the one hand, by linear combinations of the a , , and, on the other, by an explicit calculation for f 2 (Jucys and Dagys, 1960). We find

a , = 4(55M0 - 44M2 - 50M4)/165, a2 = (66M2 - 175M4)/770, a3 = 8(143M2 - 175M4)/77, a4 = -3(286M2 + 175M4)/22. These coefficients specify the amplitudes of the various components z i in terms of the radial integral Mk.The decomposition of H,, into its grouptheoretical parts is completed.

266

B. R. Judd

F. EFFECTIVE OPERATORS One interesting aspect of the analysis of H,, remains to be discussed. In Table 111 there are five possible matrix elements for a given zi;yet only four values of i appear. A fifth column can thus be constructed, orthogonal to the other four in the sense of Eq. (6). It is entered in Table 111 under the heading zo . The usefulness of zo lies in the fact that any scalar two-particle hermitian operator with spin and orbital ranks of 2 can be expressed as some linear combination of the five zi(0 5 i I 4). Such a set of operators would be useful if it was decided to take configuration interaction into account by introducing effective operators to reproduce the observed energy levels of some configurationfN. An actual calculation f o r f 3 has been carried out by Crosswhite et al. (1968). The use of effective operators in atomic spectroscopy dates from the work of Trees (1951b, 1952) and Racah (1952), who considered scalar operators i n both spin and orbital spaces to correct the positions of entire LS terms. It is not difficult to assign irreducible representations to effective operators. Take, for example, zo . The triplet states off' belong to (1 lo), for which (110) x (110)= (000)+ (200) + (111) + (211) + (220). (see Nutter, 1964). Not all representations on the right are of interest to us. The Kronecker product of two identical representations (corresponding here to a bra and a ket) can be separated into a symmetrical and an antisymmetrical part. For nonvanishing matrix elements, the former has to be associated with a hermitian operator, the latter with an antihermitian operator. The existing tables (Judd and Wadzinski, 1967) indicate that the product (1 10) x (1 10) decomposes into a symmetrical part comprising (000), (1 1 l ) , (200), and (220); and an antisymmetrical part comprising (1 10) and (21 I). Since all zi correspond to orbital ranks of 2, we seek the D states in the symmetrical parts. From the branching rules of Racah (1949), we obtain one in (200), three in (220), and one i n (1 1 I). The first four describe z , , z, , z 3 , and z,; hence zo must belong to (1 1 I). Its G, description is unambiguously given as (20). The assignment WU = (1 11)(20) was first made to zo by Armstrong and Taylor ( 1 969).

G. SYMPLECTIC ASSIGNMENTS The question of the quasi-spin ranks and the irreducible representations of Sp(4[+ 2) that can be assigned to operators requires a special discussion. In some cases we can proceed as in Section 111, E. For example, we have already stated in Section 111, C that the tensors w ( ~for ~ )K + k odd belong to

SELECTION RULES IN ATOMIC SHELLS

267

(2000000) of Sp(14); and since those for which IC = 0 cannot be coupled to a spin rank of 2, all matrix elements of H,, must correspond to (20 0) x (20 * .. 0),which decomposes into

(0)

+ (11) + (22) + (31) + (4).

[The abbreviations given by Eq. (4) have been made.] Decompositions of this kind as well as branching rules involving symplectic groups can be found from the work of Flowers (1952) or from Wybourne (1970). Of the six representations in the direct sum above, only the first four are needed to label nonvanishing matrix elements. Of these, only (22) contains '(ZOO) and 5(220) when Sp(14) is reduced to R,(3) x R(7). It follows that H,, can be assigned the single irreducible representation (22) of Sp( 14). In other cases, however, the tensors w ( ~ missing ~ ) from a complete irreducible representation of Sp(14) sometimes make it impossible to assign a single irreducible representation of Sp( 14) to an operator. Take, for example, the part e3 of the Coulomb interaction (see Section 111, D for its introduction). It is composed of scalar products of the type ( ~ i ( ~* )vjk') (where k = 2, 4, 6), and corresponds to WU = (220)(22). We can equally well form an operator 52 belonging to (220)(22) from those scalar products for which k = 1 and 5; this is at our disposal to combine with e3 to form an operator to which a unique irreducible representation of Sp(14) can be assigned. In fact, R was normalized by Racah (1949) in such a way that e3 + 52 should possess properties of a particularly striking and elegant kind. Although Racah was unaware of the fact, he was nevertheless constructing an operator that has turned out to belong to the single irreducible representation (1111) of Sp(14). The actual matrix elements of e3 are found by calculating those of e3 + 52 and then subtracting the contributions from Q. It is not difficult to do this (see Racah, 1949).

H. QUASI-SPIN ASSIGNMENTS It was shown in Section II,D that a state with a seniority u [and hence a unique irreducible representation of Sp(41+ 2)] corresponds to a unique quasi-spin, Q. The connection for operators between the $441 + 2) labels and the quasi-spin ranks K is more complicated. A direct way to approach the problem is to examine the commutation relations satisfied by Q and the operator in question; but this method is reasonably simple only if K = 0. A quicker technique is to first consider R,(3) x $441 + 2) and then embed this direct product in R ( 8 1 t 4), the group whose generators are all possible commutators [a', a'], [a', a], and [a, a]. It can be shown that the kets of I N with N even span the irreducible representation ($4... 4) of R(81+ 4),

B. R. Judd

268

while those for odd N span the complementary irreducible representation (11 2 2 . . .1 - i)(see Judd, 1968a). The same is true for the bras. To see this, we need only note that the products (++ . . . + + + ) x (+$...I+ 1 2 -2)

contain the scalar (0 . * . 0), whereas

do not (Murnaghan, 1938). The complete expansion of the former is ( 0 ~ ~ * 0 ) + ( 1 1 0 * ~ ~ 0 ) + ( 1 1 1 1 0. *~* ~+(11 * 0 ) +... 1 f 1).

The W.E. theorem immediately tells us that an operator must transform like some superposition of these representations of R(81+ 4) if it is to have nonvanishing matrix elements when sandwiched between a bra and a ket characterized by the same N . The process of picking the appropriate representation (1 1 . . . 10 * * . 0) of R(81f 4) is easier than one might at first imagine. This is because the tensors at and a together transform like (10 . . . 0). To understand this, we have only to note that the generators of R(81+ 4), when commuted with the tensors at or a, can only yield components of these same tensors; so we have a representation of dimension 81 + 4 that is clearly not reducible to 81 + 4 scalar parts. The next simplest representation after (0 . . . 0) is (10 * . . 0), which has precisely the requisite dimension and must hence describe at and a. In the notation of second quantization, an operator that is a sum of singleelectron operators (like the ordinary spin-orbit interaction) is a sum of terms, each one of which is a component of a'a. For an operator that is a sum of single-electron operators (like the Coulomb or spin-spin interactions), we must take sums over the components of atataa. In general, an operator that is a sum of r-particle operators requires r creation and r annihilation operators. Its description for R(81+ 4) can only involve those representations that arise when the multiple Kronecker product (10 . . . 0)2ris decomposed and which, at the same time, are ofthe type (12x041+2-2x ), where xis an integer (including zero). However, the maximum x is determined by r. In the stretched case, we have x = r (e.g., see Racah, 1951); so, in general, x _< r. This means, for example, that the tensors W ( K kbelong ) to (1 10 . . . 0) and (0 . . . 0) of R(81 + 4). Having established the possible representations of R(81+ 4), we have only to obtain the branching rules for the reduction

R(81

+ 4)

-+

RQ(3)x Sp(41+ 2),

and the quasi-spin ranks k to be attached to irreducible representations of Sp(41+ 2) follow immediately. The actual calculation is simplified enormously by first embedding R(81+ 4) in U(81+ 4); each representation (12x04J+2-2x 1

269

SELECTION RULES IN ATOMIC SHELLS

1 of of R(81+ 4) is equivalent to the irreducible representation [12x041+2-2x U(8t + 4) when 41 + 2 > 2x (e.g., see Jahn, 1950). The decomposition to study becomes u(81 4) + RQ(3)x Sp(41 2),

+

+

and this is identical to the isospin separation for a nuclearj shell, i.e., to

U(4j

+ 2)

+ &(3)

x Sp(2j

+ I),

provided we write j = 21 + 4. This general problem has been treated by Flowers (1952), and we have merely to take over his results, extending them where necessary. They are set out in Table IV for r 2. These cases follow immediately from Table 6(c) of Flowers (1952). TABLE IV BRANCHING RULESFOR THE REDUCTION R(81+ 4) + RQ X Sp(41+ 2) Irreducible representation of R(81-t 4)

Irreducible representations [Ku] of Ro(3) X Sp(41 2)

+

It is remarkable that many symplectic representations are associated with a single quasi-spin rank K . The magnetic hyperfine interaction corresponds to r = 1 and the irreducible representation of Sp(41+ 2) is (2). From Table IV, we see that it is a quasi-spin scalar. On the other hand, the two-electron operator e3 + C2 (described i n Section 111, G), which belongs to (1 1 1 1) of Sp(14),corresponds to K = 2. In contrast, a two-electron operator transforming according to (11) is a superposition of three operators belonging to K = 0, 1, and 2, respectively. A separation can be effected by analyzing the dependence on N and u of the matrix elements of such an operator (Judd, 1966).

I. TABULATIONS The operators of interest i n energy-level calculations for f” are listed in Table 11. They are described by irreducible representations of RQ(3),Sp( 14), R(7), G, , Rs(3), and R,(3), the assignments being made by the methods outlined in Sections 111, C-G. Not all operators in common use can be assigned

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unique irreducible representations of all these groups. If two or more irreducible representations of a group are required, they are put in brackets. However, the operator z, of the spin-other-orbit interaction has been explicitly decomposed into z 1 2 ,zI3,and z14 (Crosswhite et al., 1968), and for this reason these last operators are listed separately. Since we are interested in deriving selection rules, the actual definition of the operators listed in Table I1 is not important: the group labels tell all. Readers interested in complete descriptions are referred to Racah (1949) for the Coulomb interaction, Judd (1966) for the orbit-orbit, Armstrong (1968) for the contact spin-spin, Judd and Wadzinski (1967) for the dipolar spin-spin, and Judd, et al., (1968) for the spin-other-orbit. As mentioned in Section 111, F, spectroscopic term analyses often require the introduction of effective operators to reproduce the influence on f” of other configurations. Often the effects of configuration interaction are absorbed by parameters (such as the ai)associated with intraconfigurational contributions; and sometimes the effects of configuration interaction are the more important. The operators in Table I1 listed as “orbit-orbit”, for example, play their most important role when representing the coupling of configurations by the Coulomb interaction. The importance of effective operators means that Table I1 is not necessarily complete. We have already discussed (in Section 111, F) the usefulness of z,, (which, incidentally, corresponds to [Ka]E (2)( 11 11)). Of other effective operators, the most important are the three-electron scalars. A discussion of their properties has been given elsewhere (Judd, 1966). Only a few of the groups of Table I are used in setting up the entries of Table 11. This is not to say that the others are not important; in fact, we shall see that they are crucial in explaining some otherwise mysterious null matrix elements. Nevertheless, it is fair to say that the six groups RQ(3),Sp(14), R(7), G,, R,(3), and R,(3) are the most directly useful for work in the f shell. They have developed naturally out of Racah’s original analysis, and they can be readily used in conjunction with the various published tables of matrix elements.

IV. Generalized Triangular Conditions A. THEGROUP R(3) We have now reached the point where we can begin to study selection rules. The discussion of Section 111 may appear to have delayed matters unduly, but, in fact, the assignment of the appropriate group-theoretical labels to an operator is often the central problem. Once this has been done, the rest is easy-at least in many cases.

SELECTION RULES IN ATOMIC SHELLS

27 I

The most important and most frequent source of selection rules stems from the condition c(RR’R“)= 0, implying that the irreducible representation R, which labels the bra of a matrix element, does not occur in the decomposition of the Kronecker product R’ x R”, where R’ and R“ are irreducible representations labeling the operator and ket, respectively. This principle was mentioned in Section 111, A, and it has already been used in Section 111, E to construct operators having prescribed transformation properties. In its most familiar form, it refers to an R(3) group. The representations R and R“ are angular momentum quantum numbers, and R‘ is the rank of the operator taken with respect to the angular momentum in question. Now, R‘ x R“ is particularly easy to decompose (see Weyl, 1928); using the notation for RL(3),we can write (L’) x (L”)= (L’ + L”)+ (L’ + L“ - 1)

+ ... + (IL’ - L’”).

(7)

So, unless the representation ( L ) occurs on the right-hand side of Eq. (7), it follows that ( y L I O(y’L’)1 y”L”)= 0. The symbols y, y’, and y ” represent additional quantum numbers that may be necessary to define the states and the operator. It is to be stressed that the condition c(RR’R”)> 0 does not imply that the matrix element is non-zero; in fact, we may very well have c((L)(L’)(L”)) = 1 in the illustration above and yet the matrix element may still vanish. For example, if the y s are the representations of some other group, it could easily happen that ~ ( y y ’ y ’ ’ )= 0. The group RL(2) is an obvious example of this; we have merely to write y E M L , and the condition ~ ( y y ’ y ” )= 0 becomes M L # M i + ML”. The condition that ( L ) occurs on the right in Eq. (7) is equivalent to the condition that it is possible to form a triangle whose sides are of length L , L‘, and L“. It is therefore referred to as the triangular condition. The ease with which Clebsch-Gordan coefficients for R(3) can be manipulated algebraically has made it possible to introduce quantities such as 6-j, 9-j, 12-j symbols, etc., in which the angular momenta L, L’, and L“ appear (Edmonds, 1960). Such symbols vanish if the relevant triangular conditions are not satisfied; this is brought about by the appearance in the denominator of the factorials of negative numbers. As examples of selection rules that derive from the triangular conditions or R,(3) and RL(3),we cite the conditions AS, AL = O,+ 1 for the spin-orbit and spin-other-orbit interactions, and

A S = 0,

AL = 0, f 1, + 2

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B. R. Judd

for the quadrupole hyperfine intcractions. In these equations, A S is the difference between the spin quantum numbers in bra and ket, and AI., is defined analogously. Matrix elements satisfying the above equations are not necessarily non-zero; but those that do not satisfy them certainly are zero. Since the spin-spin interaction is a quasi-spin scalar, we have A Q = 0, or, what is equivalent, Au = 0. Trees (1951a) was the first to notice that the spin-spin interaction is diagonal with respect to seniority. Again, since e3 + R corresponds to K = 2 (see Table Il), its matrix elements must vanish when it is set between bras and kets that correspond to Q = 0 or Q = 4. Racah (1949) stated that e3 + R vanishes for states of seniorities 7 and 6 (to which Q = 0 and Q = $ correspond), but no explanation was then given. Many other examples of the triangular conditions can be constructed for the operators listed in Table 11.

B. GENERALIZATIONS The simple form of c(RR'R") for the group R(3) is quite exceptional. In general, we have to refer to tables for the decomposition of the Kronecker products. When the tables of Nutter (1964) or of Wybourne (1970) are not adequate, the decompositions must be worked out by the standard techniques of group theory (Littlewood, 1950). It is worth noting that, for all the groups whose representations appear in Table 11, the numbers c(RR'R")are invariant under any permutation of R, R', and R".This is not true in general: exceptions occur for unitary groups and for rotation groups in an even number of dimensions. Examples of the use of the condition c(RR'R") = 0 have already been given i n Section 111, C for G, and R(7), and in Section 111, G for Sp(14). Other examples are easy to find. A rather remarkable one for G, depends on thevanishing of c(( 1 I)( lo)(1 I)). This means that the matrix elements of the tensor V(3) vanish when the tensor is set between the 3 H terms off'. The vanishing of c((l1)(21)(1 I)) has already arisen in the discussion of z 3 in Section 111, E; put in the form c((1 1)(1 1)(21)), it indicates that matrix elements of the spinorbit interaction vanish between states labeled by ( I I ) and (21). This is a very common selection rule in , f N . McLellan (1960) has assembled tables of c(U(1 I)U") and c( W(l IO)W''), from which it is a simple matter to read off all the selection rules for the spin-orbit interaction i n the f shell. It is not usually necessary to work out the numbers ~(aa'a")for Sp(41+ 2), since it is almost always easier to obtain equivalent selection rules from considerations of quasi-spin.

c. SELECTION RULESFOR RADIALINTEGRALS The usefulness of the triangular conditions is not confined to the groups of Section 11. Armstrong (1970) has recently given a particularly elegant

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SELECTION RULES IN ATOMIC SHELLS

explanation of a selection rule for hydrogenic eigenfunctions that appears to have been first noticed by Pasternack and Sternheimer (1962). The rule runs (n/lr-klnl’)=O

( k = 2 , 3 ,..., I I - - / ’ l +

1).

(8)

According to Armstrong, the appropriate group is O(2, I), whose ClebschGordan coefficients are algebraically very similar to those of R(3). It turns out that, fork I I + I‘ + 2,

where A is independent of n. The range of k of Eq. (8) corresponds precisely to the violation of a triangular condition on the triad ( I , k - 2, 1’). A similar analysis can also be given for integral r k with positive k . The fact that radial properties can be cast in this form in highly suggestive, and opens the way for many developments in atomic shell theory.

V. Generators A. DIRECT APPLICATION It occasionally happens that operators of physical interest are the generators of groups. For example, the components of L are the generators of R,(3), and the magnetic hyperfine operator is formed from the generators of Sp(41 2). The matrix elements of such operators must be diagonal with respect to the irreducible representations of the groups for which they serve as generators; for otherwise the representations would not be complete. The selection rules implied by this statement are much more stringent than those that would be determined by assigning irreducible representations to the generators and then using the methods of Section IV. For example, the magnetic dipole operator is L + 2S, so magnetic dipole radiation satisfies the selection rule AL = A S = 0. The methods of Section IV would merely give AS, AL I 1. Some of the most striking examples of this kind of selection rule are provided by V‘3) [a generator of R(7)]and by V(5) [a generator of G, and R ( 7 ) ] . Most of the zeros that appear from a casual glance at the tables of Nielson and Koster (1963) correspond to the vanishing matrix elements of these operators. Although V(3) and V‘5’ do not appear in Table 11, and hence do not in themselves directly represent operators of physical interest, they are of great importance in many ancillary problems of atomic shell theory, such as the calculation of isoscalar factors and fractional parentage coefficients.

+

B. PROPORTIONAL MATRIX ELEMENTS It sometimes happens that an operator of interest has the same description with respect to a sequence of groups as a generator of one of the groups of

B. R.Judd

274

the sequence. If the W.E. theorem can be used to relate matrix elements of the operator to those of the generator, then some of the severe selection rules obtaining for the latter are carried over to the former. For example, we see from Table I1 that both the spin-orbit interaction H,, and the orbital angular momentum L correspond to WUk E (1 10)(11)(1). Suppose we are interested in the matrix elements of H,, within the representation (30) of G, . From Nutter's tables, we quickly find that c((30)(11)(30)) = 1, so the sum over y~ in the W.E. theorem reduces to a single term. This means that the matrix elements of a particular component L, of L are proportional to those of the part WAi ') (for any n) of H,, . If the dependence on q is removed by passing to reduced matrix elements (Edmonds, 1960), we can say that the reduced matrix elements of L are proportional to those of Hso. Now the former are diagonal with respect to L , and so we can conclude that ((30)LI Hsol (30)L') = 0

( L # L').

This is stronger than the ordinary selection rule on H,,, which merely states AL 5 1. The method works here only because we are studying matrix elements diagonal in U ; for the off-diagonal case, all matrix elements of L vanish (since L is a generator of G J , and the proportionality cannot be established. From the point of view of group theory, the example described above is similar to a rather familiar selection rule on H , , , which runs as follows: All matrix elements of H,, taken between those states of l ~ ' l. ~ . z for which all m, = 3 vanish if off-diagonal with respect to L. An elementary proof can be readily constructed (see Judd, 1968b). From our present standpoint, it is only necessary to note that (i) the collection of terms for which all m, = span the irreducible representation ( 1 1 . . . 10 . . . O ) of R(21, + 21, + . . . + 21, + p); (ii) H,, and L both belong to (1 10 . . . 0) of this group; and (iii) that

+

c((l1 ... 1 0 ~ ~ ~ 0 ) ( 1 1 0 ~ ~... ~ 0l O). (. . 1O )1) =

1.

The reduced matrix of H,, for the terms i n question is proportional to the reduced matrix of L, and the selection rule on H,, follows immediately. C. COMMUTATORS

Generators can sometimes be used very effectively to explain isolated null matrix elements. The classic example is the vanishing of

((1 I ) P I/

v(,)I/ (21)F).

To understand this, we consider the commutator [V('), V(5']. If this commutator is evaluated, we obtain various tensors V ( k for ) which the maximum k is 6 ; hence the matrix elements of the commutator must vanish if it is set between the states ( I l)P and (21)L (where L means here a state with a n orbital

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275

angular momentum quantum number equal to 8). But the commutator can be evaluated by introducing intermediate states; and so

c*

$ ~ ( 1 ~ (' 21 1 ) ~ ) = 1 (( 1 l)P I v'5' I @)(@ I v(2)1 ( 21)L).

((1 1)piv(2)i~ d?

(9)

(Here and elsewhere we do not include quantum numbers in the description of the bra and ket if their values are incidental to the analysis.) In Eq. (9), $ and CD must belong to (21) and (I I), respectively, since V ( 5is) a generator of G , . The triangular conditions limit $ to (21)F. As for @, there is no I state in (1 1) (Racah, 1949), so the sum on the right-hand side of Eq. (9) vanishes. Thus ((1 1)pI

v ( y ( ~ I ) F ) ( ( ~ Iv)(F5 I) 1 ( 2 1 ) ~=) 0.

At least one of the matrix elements in this equation must be zero. Without further analysis, it is impossible to be more precise; but one would not expect conditions to be placed on generators by arguments of this sort, since one somehow feels that their matrix elements are more fundamental than those of other operators. Readers who are not prepared to accept an intuitive argument of this kind can verify from tables that the matrix element of V(5' is indeed non-zero ; and the required selection rule follows immediately. This method works well if there is an obvious choice of operators and states. More frequently, one is confronted with a wide range of possibilities, and a successful outcome depends largely on one's ingenuity and patience. Most examples (though by no means all) of this method are for matrix elements for which the rank k of the operator and the angular momenta L and L" describing the states are "stretched"; that is, the triangle with sides equal to L , L", and k reduces to a straight line doubled back on itself. This situation is important for reducing the number of possible sums of the type running over $ and CD i n Eq. (9). Armstrong and Taylor (1969) give several examples of selection rules obtained by this method for spin-spin matrix elements in f '. It is perhaps worth mentioning here that the use of the generators of a group is merely to reduce the sums over intermediate states and is not crucial in itself. It may also be useful to prescribe the components of the tensors. Both of these points can be illustrated by giving an explanation of the null multiplet splitting of 2Pinf3. We consider

The commutator appearing here can be evaluated by straightforward methods (see Judd, 1963); it turns out that it is expressible as a linear combination of the components I.V(.,"; for which h- + k is odd. The only possibility satisfying all the triangular conditions on the ranks of the operators is WL'f,), and this

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B. R . Judd

cannot connect a P state to an S state. The matrix element above is thus zero. Turning now to the question of intermediate states, we see at once that W&') has a vanishing diagonal matrix element within 4S,and so we obtain ( f 3

'P,+ , I I w g y l j 3 2

~

+, i ) ( f 3 ' P , +, 1 I wii;:lf3 ,

+,

4 ~ , 0) = 0.

The vanishing of either of these matrix elements would constitute a surprising selection rule on the spin-orbit interaction; in fact, the second one is finite and so we can conclude that ( f 3 'PIH,,I f 3 ' P ) = O .

Forf3, there should thus be no separation of 'P3,2 and 'PI,' near the limit of Russell-Saunders coupling.

VI. Conflicting Symmetries A. METHOD The Clebsch-Gordan coefficient (Rqil R'i', R"i") that enters in the W.E. theorem (see Section I11,A) is described in terms of the irreducible representations R , R', and R" of a group 3.Since the groups that we have been using are members of a sequence of groups, it is often possible to replace i, i', and i" by the quantities rj, r'j', and r"j",where the rs are the irreducible representations of a subgroup &? of %, and thejs label their components. The ClebschGordan coefficient now appears as (Rqrjl R'r'j; R"r"j").

Suppose it happens that R' E R" and r' 3 r". The Kronecker products R' x R' and r' x r', being formed from two identical representations, can be broken up into their symmetric and antisymmetric parts. Suppose, further, that R occurs in the symmetric part of R' x R' while r occurs in the antisymmetric part of r' x r ' . It is clearly impossible to have a nonvanishing Clebsch-Gordan coefficient, for, if it were not, we should be able to construct a function that was both symmetric and antisymmetric under the interchange of its components. It only remains to feed the null Clebsch-Gordan coefficient into the W.E. theorem, and a vanishing matrix element is immediately obtained. To apply these ideas, it is essential to be able to decompose such Kronecker squares as W x Wand U x U into their symmetric and antisymmetric parts. The situation for R(3) is straightforward. The representations ( k ) that occur in the decomposition of (k') x ( k ' ) are symmetric if 2k' - k is even and antisymmetric if 2k' - k is odd. This is obvious from a consideration of the allowed terms of the electronic configurations 1' (for k integral) o r j 2

SELECTION RULES IN ATOMIC SHELLS

277

(for k half-integral). Other groups are not so easy to treat. A method based on an iterative procedure has been used to bisect the products W x W and U x U for all Wand U that occur in f" (Judd and Wadzinski, 1967). Smith and Wybourne (1967, 1968) have pointed out that the separation of a Kronecker square into its symmetric and antisymmetric parts is a particularly simple example of a plethysm. The mathematical literature on the subject, which stems mainly from the work of Littlewood (1950), can thus be exploited to make the calculation easier. As an example, consider the following matrix elements of the spin-orbit interaction: ((21 1)(20)LIH,,I (1 10)(11)L"). Matrix elements of this type occur in all configurations f" for which 4 I n < 10. Spin labels are irrelevant for the discussion here, and they are suppressed. (This is in keeping with our general policy to prevent clutter in the bras and kets.) From Table 11, we note that H,, has the same WU description as the ket, so that any calculation of the matrix element must necessarily involve the isoscalar factor ((21 1)(20)I(110)(11)

+ (1 lo)( 11)).

From the table of bisections (Judd and Wadzinski, 1967), we find that (21 1) occurs in the antisymmetric part of (1 10) x (1 lo), whereas (20) occurs in the symmetric part of (1 1) x (1 1). It follows that all matrix elements of the spin-orbit interaction between states labeled by (21 1)(20) and (1 10)(11) are zero. The method of conflicting symmetries has been used by Judd and Wadzinski (1967) to study the selection rules for the components hi of the spinspin interaction. A complete listing has been provided for the configurations

f ". B. THEHALF-FILLED SHELL Virtually no mention has been made of the group R(2). This is not only because it is trivially easy to treat, but, more significantly, because R,(2) and R,(2) imply the existence of an axis in ordinary three-dimensional space. In studying free atoms, no direction is to be preferred to any other, and such an axis can only be a mathematical artifice. The aim of the algebra of Racah (1942) is to remove all considerations of this kind from the analysis. The situation is different, however, for RQ(2), since Me is equivalent to -+(2l+ 1 - N ) (see Section 111, D). Chemists can separate atoms with different numbers of electrons, and so the quasi-spin axis takes on a physical significance. Consider, then, a matrix element of the type (RMQ I O(R') I R"MQ),

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278

where 0 is any operator of Table 11 (and hence corresponds to A M Q = 0). The bra ( R M Q Jtransforms under RQ(2)in the same way as the ket I R - M Q ) ; so, for I MQI > 0 we cannot use the method of conflicting symmetries. However, if MQ = 0, corresponding to the half-filled shell, we are confronted with a remarkable situation, since bra, operator, and ket all belong to the identity representation of RQ(2). This representation must occur in the symmetric part (the only part) of the square of itself, and hence all matrix elements of the type above vanish if R“ = R‘ and if R occurs in the antisymmetric part of R’ x R’. An obvious choice for the group whose representations are labeled by R, R’, and R“ is RQ(3).In this case, the W.E. theorem runs (QMQIO(K)IQ”MQ)= B(QMQIK0, Q”MQ>, where B is independent of M Q .As the arguments above indicate, the ClebschGordan coefficient must vanish if the three conditions K = Q”, M , = 0, and 2K - Q be odd are satisfied [the last coming from the demand that (Q) occur in the antisymmetric part of ( K ) x ( K ) ] .We can, of course, repeat the analysis by coupling bra and operator (in which case Q = K and 2K - Q“ must be odd) or by coupling bra and ket (in which case Q = Q” and 2Q - K must be odd). This last condition tells us that operators with odd K (such as the spin-orbit interaction) vanish when taken between states of the same seniority in the half-filled shell. However, if MQ = 0, the Clebsch-Gordan coefficient vanishes merely if Q + K Q” is odd (Edmonds, 1960), and this condition embraces all of those stemming from arguments of conflicting symmetries. Since the ranks K of all operators of interest to us are listed in Table 11, it is straightforward to work out any number of selection rules. For e3 + Q, for example, we obtain Au = 0, f 4 (withinf’, of course). Selection rules for the tensors W(Kk)were first obtained by Racah (1943)by other methods.

+

C. RESIDUAL ZEROS A number of methods for explaining selection rules have now been described. Although we cannot be absolutely sure, there seems little else that can be done with the groups of Table I1 to account for null matrix elements. There nevertheless remains a small but significant residue of zeros in the tables of Nielson and Koster (1963). As a concrete example, we take the matrix elements of V(6) in f4. (In Nielson and Koster’s notation, V(6)=&3UC6).) If we exclude all cases that violate the selection rule A S = 0 or the triangular condition on the triad (L, 6, L”), there remain 368 matrix elements. Of these,

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SELECTION RULES IN ATOMIC SHELLS

TABLE V EXCEPTIONAL NULLREDUCEDMATRIX ELEMENTS OF

v"' I N f

21 are zero. These 21 are listed in Table V; category numbers are assigned to them to aid the subsequent discussion. In category I are two matrix elements that are zero because the generalized triangular condition is not satisfied. The operator V(6) belongs to (200) of R(7) (see Table II), and it is obvious that c((000)(200)(220)) = 0. The five matrix elements in category 11 can be understood in terms of the method of Section V, C . The first four have a common origin: the commutator [Vc6), V'')] is set between the bra of the matrix element and the ket I (220)(22)'S). The combination V(6)V(5)picks out (220)(22)'H as an intermediate state, and the reversed form V(5)V(6)contributes nothing because (21) of G, does not contain an I term. The fifth null matrix element can be explained by taking [VC6),V(3)] and the same ket as before. The seven matrix elements in category 111 can all be accounted for by the method of conflicting symmetries (Section VI,A), either applied to the pair of groups R(7) and R,(3), or to the pair G, and R,(3). There remain the seven null matrix elements i n categories I V and V. Similar residues occur when the zi coming from the spin-spin interaction and the spin-other-orbit interaction are evaluated (Armstrong and Taylor, 1969; Crosswhite and Judd, 1970). As will now be seen, a large proportion of such null matrix elements receive a natural explanation if we use the scheme of groups based on the separation of the electrons into two groups according to their spin orientation.

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B. R . Judd

VII. Oriented Spins A. METHOD The separation of the total space of a configuration into a spin-up space (space A) and a spin-down space (space B) has been described in Section 11, C . Suppose 9, and YB are two groups whose generators differ only in that those of 9, refer to electrons in space A, while those of YB refer to electrons in space B. Suppose, further, that the group whose generators are formed by taking sums of the corresponding generators of 9, and YB is written 3. For example, we could have 9,

R,(21+ I),

3~ R ~ ( 2 + 1 I),

R(21-b l),

in the notation of Section 11. Once M , is specified, any state described by the groups of Table I1 can be expanded as a sum over states of the type I ( R , R,')R"i), where R,, R,' , and R" are irreducible representations of Y, g B ,and 9, respectively; and where i denotes a component of R". Extending the language of R(3),we could say that R , and R,' are coupled to R". Now, it might happen that a state of interest to us--such as I f"(211)(21) LM,)-could be expressed as just one state of the type I (RARB')R"i)if Ms were chosen judiciously. If the bra and the operator of an entire matrix element also required only one term in their expansions, then selection rules could be obtained by examining the generalized triangular conditions on the representation in the A and B spaces separately. As we shall soon see, this is not so extravagant a possibility as might appear at first sight. All the techniques of Section 111 are available to us in assigning representations of 9, x gBto operators of physical interest. If an operator possesses a spin rank K greater than zero, then several alternative expressions can be constructed, depending on the number N ' of electrons the operator is required to move from the B space to the A space. Although all aspects of the spin are separated out from the start, it is important to recognize that the ClebschGordan coefficient ( S M , I K N ' ,S"M,") must necessarily enter the calculation of a matrix element. If M,, N ' , and M i are chosen unwisely, this coefficient could vanish. Such a possibility would make it pointless to perform any analysis on the associated orbital part of the matrix element. As an example, consider the equation (f4(211)(30)1v(3)~f4(21 i)(30))= 0, which can be checked by reference to the tables of Nielson and Koster (1963). The methods of Sections 111-VI are ineffective here. If we take M , = 0, there are two electrons in the A space and two in the B space. The only

SELECTION RULES IN ATOMIC SHELLS

28 1

representations of G , , and G,, available are (1 1) and (10).From the tables of Nutter (1964), we find (30) can be formed in only one way, namely by coupling (1 1) i n the A space to (1 1) i n the B space. Turning to the operator VC3),we note that it is a sum of one-particle operators, and also that it belongs to (10) of G, . Thus it must be some superposition of (10) x (00) and (00) x (10). It follows that the entire matrix element can be described as <[(I 1) x (1 i)i(30)1 [(lo) x (00)

(00)x (lo)i(lo)i [(I 1) x (1 1)1(30)).

Irrespective of the choice of sign, the entire matrix element must be zero because c((l1)(10)(11)) = 0 (see Section IV, B). The alternative sign would correspond to Wh’ 3 ) ; but this matrix element would necessarily vanish because (10110, 10) = 0. We cannot, therefore, come to any conclusion concerning ( p ( 2 1 i)(30)1 w ( yf4(21 i)(30)); but, in point of fact, matrix elements of this kind are non-zero, provided the ordinary triangular conditions on the orbital angular momenta are satisfied (Karazija et al., 1967). B. EXTENSIONS To simplify the presentation, the various methods of obtaining selection rules have been illustrated with examples that are straightforward and largely free from complications. In practice, however, we may often have to combine methods. Consider, for example, the matrix elements in category IV in Table V. They can all be written as

[(wx (20)1(20)61 [(I 1) x (10)1(21)~”),

([(I 1) x (1 1)1(30)~1

but it is not obvious why some of them vanish. To find the source of this property, we must use the fact that ( I I ) and (30) derive from (1 10) and (21 I), respectively. It happens that (21 1) occurs in the odd part of (1 10) x (110) (Judd and Wadzinski, 1967); so, if L is even, the angular momenta LA and L , that derive from the two (1 1) representations in the bra cannot be equal, by the argument of conflicting symmetries. The branching rules for (1 1) yield P + H (Racah, 1949); but it is at once clear that the matrix element for which L , = 1 must be zero, since, in the B space, the triad (1, 6, 3) does not satisfy the triangular condition. Hence the matrix element above must be proportional to

( ( P W LI (06)6 I (PF)L”) for even L. The triad (1, 3, L”) in the ket violates the triangular condition if L“ > 4, and all the matrix elements of category 1V are accounted for at a single stroke.

B. R.Judd

282

The methods become more involved for configurations such as f or for the two-electron operators zi . It is sometimes necessary to introduce Sp(14) into the arguments of conflicting symmetry. Most of the zeros unaccounted for by Armstrong and Taylor (1 969) yield to a concentrated application of the principles described here (Bauche, 1970), but a sense of dissatisfaction is felt as the methods becomes less transparent. Even so, the methods are very much more direct than the original calculations, which involve detailed sums over parent states.

VIII. Special Cases A. PROBLEM TRANSFER If we use the W.E. theorem to account for null matrix elements, then the central problem often reduces to showing that some Clebsch-Gordan coefficient or isoscalar factor is zero. These quantities exist in their own right, however, and not merely as an adjunct to atomic shell theory. This being so, it is not unreasonable to imagine that they might occur in problems of a different kind; and, in this different context, perhaps the vanishing of one or more of them might be virtually self-evident. For this idea to work, the appropriate problem in which the Clebsch-Gordan coefficients or isoscalar factors recur has to suggest itself. Usually, this amounts to replacing one intractable problem by another. One rather striking example of this approach can be described, however. It concerns the vanishing withinf" of all matrix elements of the type

where the operator and states are labeled by irreducible representations of G, and R,(3). The representation (21) occurs frequently i n f";and since no fewer than seven operators i n Table I1 correspond to Uk = (1 1)(1), examples of this null matrix element are continually arising. There appears to be no way of understanding this result in terms of the methods previously described, although its general validity can be readily proved (Judd, 1963). Instead of thinking in terms offelectrons, let us shift our attention to the treatment of the p + h shell by quasi-particles. As indicated in Section II,D, the quasiparticle approach allows us to factorize the spin-up space and the spin-down space. It turns out that the L values of the terms of the configurations pxhY for which all m, = and both x and y are even can be obtained by evaluating the Kronecker square

+

[(D)+ ( F ) + (GI

+ (HI + ( K ) + (QI2?

(1 1)

SELECTION RULES IN ATOMIC SHELLS

283

the representations in the brackets being precisely the structure in RL(3) of the representation (21) of G, , The detailed reasoning for this is given elsewhere (Judd 1970): put briefly, it is simply that the procedure of factorization requires as a basis either representation (34 4 & 3) of R(21+ 21' + 2); and, for (I, 1') EE (5, l), the representation (++ * * * -I f) decomposes into (21) of G, . The important point to emerge here is that (21) plays a fundamental role in the p + h shell, in contrast to its unexceptional character in the f shell. The next step is to note that the states ofg"hY (with all m, = 4 and both x and y even) can be obtained by coupling the states of p x to those of hY. The former arise from the Kronecker square [($)I2; the latter from [(5/2) + (9/2) + (15/2)12 (Armstrong and Judd, 1970). Thus the states D,F, etc., of (21) can be considered as arising from the couplings (1/2, 5/2), (1/2, 9/2), and (1/2, 15/2). Such a possibility cannot occur within thefshell. A vector operator TY) (like 0;) or Ti') [like (0&)(')] belongs to (1 1) of G, , so the matrix element (10) reappears in the form

+

((l/Z 5/2)FI Tj')l

9/2)G).

This is obviously zero because, if I = 1 , the triad (5/2, 0, 9/2) in the space of the h electron does not satisfy the triangular condition; and if I = 5, the triad (5/2, 1,9/2) again does not satisfy the triangular condition. One final point has to be made. From the tables of Nutter (1964), we find c((21)(11)(21)) = 2, so we cannot be sure that the matrix elements within (21) of a particular operator TIL)are proportional to the same ClebschGordan coefficients as, for example, those of W("), evaluated in the f shell. This is simply because the parameter q of Section III,A can assume two values. However, the (21) matrices for TY) and Ti') are in fact linearly independent; so the reduced matrix elements are zero for either v ] , and our demonstration is complete. As presented here, it is little more than a sketch. Hopefully, it has given some idea of the possibilities open to us if we are prepared to step outside the immediate context of a problem.

B. INTRACTABLECASES At the present time, there remain a few null matrix elements for which no reasonably simple and direct explanation has yet been given. The single matrix element in category V of Table V is a case in point. It often happens that the techniques of Section V, C can be used to relate one exceptional zero to another. For the example in hand, it may readily be shown that we may deduce

284

B. R.Judd

for (LkL’) = (246), (642), (426), and (624), provided we assume that the equation holds for (264) or (462). The available tables of matrix elements are often helpful in directing attention to the crucial points of a problem; for example, if we replace f “ by f in the matrix element above, we immediately obtain nonvanishing matrix elements. It follows that the representations listed in the matrix element cannot by themselves be sufficient to provide an explanation for the zeros. It is possible, of course, that some way of looking at the matrix elements has eluded us. The alternative is either that some group structure remains to be discovered or else that the zeros are purely accidental. This latter possibility should satisfy no one. The former is much more interesting; in fact, it may be said that the null entries in the tables of Nielson and Koster (1963) have stimulated considerable research over the years in atomic shell theory. Whether the zeros still outstanding will lead to some new grouptheoretical scheme remains to be seen; but the success of the methods described in this article have limited their number so severely that it seems rather unlikely that a major reorientation of our viewpoint will be required.

IX. Conclusion It would be wrong to suppose from the preceding remarks that the purpose of studying null matrix elements is simply to test the extent of our knowledge of shell theory, although this is certainly one of the more intriguing aspects of the analysis. Detailed calculations of the spin-spin and spin-other-orbit interactions are well under way in the f shell, and the additional checks that zero matrix elements provide are extremely welcome, particularly when much of the work is carried out by computers. Moreover, the possibility of being able to give a direct demonstration of a null matrix element frequently suggests a direct way of calculating the non-zero ones. As an example of this, we give the equation

for L“ even, which follows from an application of tensor algebra to the analysis of Section VII, B. It is perhaps worth remarking that the examples presented in this article have been drawn almost entirely from the f shell. This is partly a matter of convenience, and partly a response to the considerable interest in the properties of f electrons. As groups are used more extensively in such mixed configurations as (s + d)N,there seems little doubt that many of the methods described here will find new applications.

SELECTION RULES IN ATOMIC SHELLS

285

REFERENCES Armstrong, L., Jr. (1968) Phys. Rev. 170, 122. Armstrong, L., Jr. (1970). J . Phys. (Paris),31, Suppl. C 4 , 17; see also Phys. Rev. A 3, 1546 (1971). Armstrong, L., Jr., and Judd, B. R. (1970). Proc. Roy. SOC.,Ser. A 315, 27. Armstrong, L., Jr., and Taylor, L. H. (1969). J. Chem. Phys. 51, 3789. Bauche, J. (1970). Private communication. Cartan, E. (1894). “Sur la Structure des Groupes de Transformations Finis et Continus.” Nony, Paris. (Reprinted 1933.) Condon, E. U., and Shortley, G . H. (1935). “The Theory of Atomic Spectra.” Cambridge Univ. Press, London and New York. Crosswhite, H., and Judd, B. R. (1970). Atomic Data 1 , 329. Crosswhite, H. M., Crosswhite, H., and Judd, B. R. (1968). Phys. Rev. 174, 89. Donlan, V. L. (1970), US.Air Force Materials Lab. Tech. Rep. AFML-TR-70-249; see also J . Chem. Phys. 52, 3431 (1970). Edmonds, A. R. (1960). “Angular Momentum in Quantum Mechanics.” Princeton Univ. Press, Princeton, New Jersey. Elliott, R. J., and Stevens, K. W. H. (1953). Proc. Roy. Soc., Ser. A 218, 553. Flowers, B. H. (1952), Proc. Roy. SOC.,Ser. A 212, 248. Innes, F. R. (1953). Phys. Rev. 91, 31. Jahn, H. A. (1950), Proc. Roy. Soc., Ser. A 201, 516. Jucys, A., and Dagys, R. (1960). Lief. TSR Mokslu Akad. Darb., Ser B 1 , 59. Judd, B. R. (1963). “Operator Techniques in Atomic Spectroscopy.” McGraw-Hill, New York. Judd, B. R. (1966). Phys. Rev. 141, 4. Judd, B. R. (1967a). Phys. Rev. 162, 28. Judd, B. R. (1967b). “Second Quantization and Atomic Spectroscopy.” Johns Hopkins Press, Baltimore, Maryland. Judd, B. R. (1968a). In “Group Theory and Its Applications” (E. M. Loebl, ed), p. 183. Academic Press, New York. Judd, B. R. (1968b). J. Opt. SOC.Amer.58, 1311. Judd, B. R. (1970), J. P h j , ~(Paris), . Suppl. C4, 9. Judd, B. R., and Wadzinski, H. T. (1967), J . Math. Phys. 8 , 2125. Judd, B. R., Crosswhite, H. M., and Crosswhite, H. (1968), Phys. Rev. 169, 130. Karazija, R., VizbaraitC, J., Rudzikas, Z . , and Jucys, A. (1967). “Tables for the Calculation of Matrix Elements of Atomic Operators.” Academy of Sciences Computing Centre, Moscow. Littlewood, D. E. (1950). “The Theory of Group Characters.” Oxford Univ. Press, London and New York. McLellan, A. G. (1960). Proc. Phys. SOC.London 76. 419. Marvin, H. H. (1947). Phys. Rev. 71, 102. Murnaghan, F. D. (1938). “The Theory of Group Representations.” Johns Hopkins Press, Baltimore, Maryland. Nielson, C. W., and Koster, G. F. (1963). “Spectroscopic Coefficients for thep”, d”, a n d f ” Configurations.” M.I.T. Press, Cambridge, Massachusetts. Nutter, P. B. (1964). Raytheon Technical Memorandum T-544. O’Raifeartaigh, L. (1968). In “Group Theory and Its Applications” (E. M. Loebl, ed.) p. 480. Academic Press, New York. Pasternack, S., and Sternheimer, R. M. (1962). J. Math. Phys. 3 , 1280.

B. R. Judd Racah, G. (1942), Phys. Rev. 62,438. Racah, G. (1943). Phys. Rev. 63, 367. Racah, G. (1949). Phys. Rev. 76, 1352. Racah, G. (1551). “Group Theory and Spectroscopy,” Lecture Notes, Princeton Univ., Princeton, New Jersey. [Ergebnisse der Exakten Naturwissenschajien 37,28 (1965).] Racah, G. (1952). Phys. Rev. 85, 381. Shi Sheng-Ming (1965). Chin. Math. 6, 610 (Amer. Math. SOC.Trans. of Acta Math. Sinica, Peking). Shudeman, C. L. B. (1937). J . Franklin Inst. 224, 501. Slater, J . C. (1960). “Quantum Theory of Atomic Structure,” Vol. 11. McGraw-Hill, New York. Smith, P. R., and Wybourne, B. G . (1967). J . Math. Phys. 8, 2434. Smith, P. R., and Wybourne, B. G. (1968), J. Math. Phys. 9, 1040. Tan Wei-Han, Pi Wu-Chi, Tse Ting-Ting, Nieh Shun-Chuen, Shu Ping-Huo, Tsuei ChenChia, Chao Chen-Chou, Li Yung-Hwa, Tong Pong-Fu, and Fong Kuei-Chun (1966). Chin. J. Phvs. 22, 38 (Amer. Inst. Phys. Trans. of Acta Physica Sinica, Peking). Trees, R. E. (1951a). Phys. Rev. 82, 683. Trees, R. E. (1951b). Phys. Rev. 83, 756. Trees, R. E. (1952). Phys. Rev. 85, 382. Weyl, H. (1928). ‘‘ Gruppentheorie und Quantenmechanik.” Hirzel, Leipzig. [Trans]. by Robertson, H. P. (1931). “The Theory of Groups and Quantum Mechanics.” Dover, New York.] Wybourne, B. G. (1970). ‘‘ Symmetry Principles in Atomic Spectroscopy.” Wiley, New York.

GREEN'S FUNCTION TECHNIQUE IN ATOMIC AND MOLECULAR PHYSICS G Y. CSANAK' and H . S. TA YLOR' ' Department of Chemistry, University of Southern California Los Angeles, California

and

ROBERT YARIS2* Department of Chemistry, Washington University S t . Louis, Missouri

I. Introduction . . . . . . .

.............

11. Many-Particle Green A. The Many-Particl

290

One-Particle Green's Function. ................................. 290 B. The Two-Particle Green's Function. ................. 111. Coupled System of Equations for Green's Functions (The Method of Functional Differentiation; The Dyson Equation; The Bethe-Salpeter Equation) ...................................................... 305 A. A System of Equations for the Many-Particle Green's Function . . . . 305

the Method of Functional Differentiation IV. Scattering ..... ........... V. Nonperturbative ......................... VI. Perturbation Met A. G o Perturbat ...................................... B. Z Perturbation M e t h o d . . ...................................... C. Renormalized C Perturbation Method . . . . . . . . . . . .......... D. Perturbation Methods for R and .'A ............................

321 330 339 343 346 348

Work partially supported by the National Science Foundation under Grant No. GP-7861. * Alfred P. Sloan Foundation Fellow. Work partially supported by the National Science Foundation under Grant No. GP-9549, and partially by the Petroleum Research Fund of the American Chemical Society under Grant No. 4341-AC5. 287

288

Gy. Csanak, H . S. Taylor, and R. Yaris Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

354 358 359 360

I. Introduction The purpose of this review is t o familiarize the quantum chemist and molecular physicist with some of the ways one can apply the Green’s function technique to the problems of calculating excitation energies, ionization energies, ground state energies, transition matrix elements, electron absorption coefficients, frequency dependent polarizabilities for atomic and molecular systems, as well as electron-atom, electron-molecule elastic and inelastic scattering cross sections. The Green’s function technique was originally defined and applied in quantum field theory (Feynman, 1948; Schwinger, 1951)4 and extensively used in many-body physics (Galitski and Migdal, 1958) statistical mechanics (Matsubara, 1955; Martin and Schwinger, 1959) and nuclear physics (Migdal, 1967). The sources in these latter fields are difficult for the non-initiated t o read since there are different physical conditions, spatial homogeneities, terminologies, etc., which d o not apply in atomic and molecular physics. Another form of the field-theoretic or manybody method, the diagrammatic perturbation theory, has been used effectively in the last several years for atomic systems by Kelly (1968) for ground state energies, and by Kelly (1969), Dutta et al. (1969) for frequency dependent polarizabilities. There are also early attempts at using Green’s functions in quantum chemistry by Linderberg (1968), Reinhardt and Doll (1969), Hedin et al. (1969); and in atomic scattering theory by Schneider er al. (1970), Janev et a/. (1969), and Csanak et al. (1971). As will be seen, the advantages of this technique are: (a) Nonperturbatiue, self-consistent approximations (i.e., schemes similar t o the Hartree-Fock self-consistent theory) can be developed which go well beyond the Hartree-Fock ; e.g., a self-consistent theory of electron scattering cross section and the linear response (Schneider et al., 1970) which leads t o self-consistent excitation energies and transition probabilities. (b) The formalism works directly in terms of densities, transition amplitudes, and other quantities (e.g., linear response function) that are measurable, J. Schwinger (1951) has defined the many particle Green’s function (called renormalized -or real-or full Green’s function). The free particle Green’s function or propagator has been introduced by R. P. Feynman (1948, 1950). For further application, see any book on field theory, e.g., S. S . Schweber: An Introducrion to Relativistic Quantum Field Theory, Row Peterson and Co., Illinois (1960).

GREEN’S FUNCTIONI

289

and not in terms of wave functions. This is a great advantage of the Green’s function technique. These new functions lend themselves to known approximations that can now be more generally applied. For example, in the calculation of linear response functions (called also frequency dependent polarizabilities), coupled, time-dependent methods are known to be useful (Dalgarno and Victor, 1966; Janiieson, 1969). It will be seen here how such coupled time-dependent methods can be applied to the variety of problems listed in the first paragraph. (c) Density matrices and natural orbitals can be calculated directly without prior calculation of the wave functions (Reinhardt and Doll, 1969) and therefore the Green’s function technique is closely related to the density matrix methods already developed in quantum chemistry. (d) The optical potential (the effective one particle scattering potential) can be calculated using the experience gained in coupled time-dependent problems (Schneider et af., 1970). (e) Self-consistent perturbation theories can easily be formulated (i.e., we expand the optical potential in terms of the “ real ” Green’s function, then we choose a trial value for it and calculate an improved “real” Green’s function from the Dyson equation, which we substitute back to the perturbation series for the optical potential until self-consistency is achieved) and the derivation of the random phase approximation (R.P.A.) (Ehrenreich and Cohen, 1959; Thouless, 1961 ; Brout and Carruthers, 1963; Pines, 1961) (called also time-dependent Hartree-Fock theory) can be placed into the context of a hierarchy of approximations (Schneider et al., 1970). These approximations, in turn, can be made self-consistent at each truncation. As in simple Hartree-Fock theory, self-consistency will be seen to be a physically desirable feature of the approximation method. As will become evident, this review is formal, in that there exist at this time no self-consistent results. It is hoped that this exposition will encourage quantum chemists and molecular physicists with practical calculational experience to put the formalism to the test. The authors of this review believe the formalism is as practical as present-day configuration interaction, perturbation, close coupled and adiabatic methods that are presently being used. It is hoped that self-consistency will yield better results for equal effort. In Section 11 we shall introduce the formalism of second quantization, define the many-particle Green’s functions and outline some elementary properties of the one- and two-particle Green’s functions. It will be shown how physical information can be obtained from a knowledge of these functions. In Section 111 the hierarchy of equations for the Green’s functions is derived. As important are alternative forms of these hierarchies in terms of the spectral functions for the Green’s functions and in terms of optical (effective) one-, two-, and many-particle potentials. In Section IV the expressions

290

G y . Csanak, H . S . Taylor, and R . Yaris

for scattering cross sections are derived. In Sections V and VI, respectively, nonperturbative and perturbative self-consistent methods of solving for the Green's function are given. It is in Sections V and V1 that physical intuition comes into play. Throughout this review a familiarity with the methods of second quantization as well as the use of the Heisenberg and interaction representation will be assumed.'

11. Many-Particle Green's Functions and Physical Quantities6 A. THEMANY-PARTICLE GREEN'S FUNCTION-PHYSICAL QUANTITIES AND THE

ONE-PARTICLE GREEN'SFUNCTION The general n-particle Green's function (also called real or full or renormalized Green's function) is defined as G"(1, 2, ...) n ; l', 2', ...) n')

=(i>-"(yoI n" . . . $(n>$'(n') . . . $+U'>l I Yo)

(1)

Here, i = T i , ti;' I Yo) is the ground-state of an N-particle system; $(i) is the field operator in the Heisenberg representation, i.e., the operator which destroys an electron at position ri at time t i ; $'(i) is the analogous "creation" operator. As always in the Heisenberg representation, the operators are time dependent even when the corresponding operator is time independent in the Schroedinger representation, since ~

p

(

i

)= eiH"Op(ri, ~ , ~ O)e-'"'I ~ ~ ~

~ ~ e- i H r , = eiHriOp(ri)Schroedinger .

~

~

(2)

Here H is the Hamiltonian of the total system and h is set equal to I . It is well to remember that antisymmetry is automatically built into the second quantized formalism since all operators are written in terms of $ and Gt (e.g., the potential is written as

v =5

I

dr, drz $'cr,>$+crJ V ( I TI - rz I )$(rz)$(r,)

For elementary introduction to second quantization and many-body theory, see, e.g., Falkoff (1962), Roman (1965), Kirzhnits (1967), March e t a / . (1967) and Mattuck (1967). Matsubara (1955), Galitski and Migdal (1958), Martin and Schwinger (1959), Falkoff (1962), Roman (1965), Kirzhnits (1967), March e t a / . (1967), Mattuck (1967), and Migdal (1 967). 7 ' I = I , 2, . . . , n. ri means position and spin coordinate. Any integration for r means integration in position space and summation in spin space.

29 1

GREEN’S FUNCTIONS

in the Schroedinger representation, and as

v(t)= 9 j 4 dr2 V(r1 W t ( r 2 0 v I r1 - r2 I M(r2 0 $ @ 1 0

(3)

in the Heisenberg representation) and these in turn satisfy for equal times the anticommutation rules for Fermi-Dirac particles [$(rt),

$@‘01+= [$W$t(r’O1+ = 0

[$(rt), $+(r’t)]+ = 6(r - r’).

(4)

The symbol T in Eq. (1) is the Wick time-ordering operator, which, when applied to a product of operators, arranges them in chronological order of their time arguments with a multiplicative factor of + 1 depending on whether the chronological order is an even or odd permutation of the original order. As an example note

7-[$(1>$t(l’>l= $(l)$t(l’)@(tl

-

-$t(l’>$(l>@cfl’

tl’)

- tl)

(5)

where 0 is the Heaviside (unit step) function, hence Gi(1, 1’) = (;)-‘{@(fi -

-

tl’><$(l>$t(l’>>

W,’ - tl><$t(l’)$(l>>).

(6)*

From this formula flows a simple interpretation of the one-particle Green’s function. If 1,’ < t , then ( 7 1 , 1’) = ( l / i ) ($(1>$+(1‘)> = ( l / j ) p o ( t 1 r f 1 ’ ) ( y o i $ ( r ~ ) ~ - i ~ ( t i - ~$~t’(r1’)lYo) )

Now, $+(r,’)I Yo) represents a state where a particle was created at position r l ’ in the background of the N-particle ground-state and e-iH(fl-rI’)

t

$ (r,‘)I yo>

represents the state which was formed from the previous one (assuming it was created at time tl’) from time t , ’ to time t l . The second factor in the previous expression therefore represents the scalar product of this state with the state $t(rl)lYo) which is an extra particle at point r, with the groundstate background of N particles. The first factor is a phase factor. Therefore G(1, 1’) for t , ’ < t , is the probability amplitude that if an extra particle is Where it will not cause confusion we shall usually use angular brackets to denote an expectation value with respect to i.e.,


(Y-0

10, I Y.0)

Gy. Csanak, H . S . Taylor, and R . Yaris

292

created at time t,’ and point rl’, then it will be found at time f , at the point r,. G ( 1 , 1’) describes the propasation of a particle in an N-particle groundstate, therefore G( 1 , 1’) is also called the “ real ” or “ full ” or “ renormalized ” one-particle propagator, because it describes the actual propagation of an extra particle in the “medium” of the N-particle ground state. For t , < t,’ we destroy a particle at point r, at time t , and then measure that this destruction shows up at time t,‘ at the point rl’. We shall call this process the creation and propagation of a hole.” We can define the following related functions “

G’ (1, 1’)

= (l/i)<*(l)*t(l‘D

G‘ (1, 1’)

=

-(l/i)(*T(l’)*(l)).

These are called correlation functions, and GR(l, 1’)

= (l/i)@(tl - tl’)<[*(l),

*V’)I+)

the retarded one-particle Green’s function

@(I, 1’)

=

- ( l / i > @ ( f , ’- f1)([*+(1’)*(1)1+)

the advanced one-particle Green’s function. Since ($(l)+t(1’))

= (eiH‘l*(rl)e-iH‘leiHfl’

-

T =

i(H - Eo)r

t , - tl’, G, depends only on

G , ( l , 1‘)

rl’)e- iHr1’

7

*

t

(7)

(r,‘))

and can be written as

*

= (j)-l{~(T)(*(rl)e-i(H-Eo)r

- ~(-7)(*t(rl‘)ei(H-Eo)r Gi(ri9

)

iH(r 1 - f 1’) J ,, t (rl’)e-iEorl’)

(eiEof~,j(

= <*(rl)e-

where

$t(

ri’, 7)

(r1‘))

*(r1)))

(8)

which expresses time homogeneity for time-independent potentials. As will be seen later, similar, but more complicated formulas can be written for higher G, . To justify going further it will now be demonstrated how a knowledge of GI (or G‘) enables one to calculate the one-particle density matrix of the system (and thus all expectation values of one-particle operators over the ground state Y o )and also the ground-state energy. The density operator p,,(r), whose expectation value is the density at position r, is given in the Schroedinger representation as

GREEN'S FUNCTIONS

293

C d(r - ri)

(9)

pop(r> =

i= 1

and in the Heisenberg second quantized form as

= $t(rt)$(rt).

Therefore, the density at position r and time t is

P W

= ($+(~~bfw)) =

lim ($t(rt')$(rt)) r-z'-+-0

Since in taking the limit, t' approaches t from above (-0 means t' is always larger than t), we can insert T and write' p(rt) =

lim (T[$+(rt')$(rt)]) t-t'+-O

=

lim { - (~[$(rt)$+(rt')])} 2--1'-+

=

-0

I

lim {- id(r, r;

T)}

I-+-0

From Equation (1 1) one could alternatively have written p(rt) = - iG< (rt, rt)

(1 3)

and we see that G'(rt, rt) gives the density. In practice G(T)" will be determined as its time Fourier transform C(w) where ~ ( w =) J"

a,

dzG(z)ei"'

-00

for any complex w and the inverse transform

Similarly we can define G< (w), G' (w) as well as G"(o), GA(w). Then Eq. (12) simply means that if G(w) is known, then ip(r) is determined by integrating this function over a semicircular contour going along the real w axis from - 00 to + co by closing counterclockwise in the upper half plane (uhp). The closure is in the uhp because T is always negative, Where it will not cause confusion we shall sometimes drop the superscript 1 on GI, rlr t l , etc. Hence an unsubscripted G, r , t means GI, rl, tl, etc. i o Sometimes we waive writing out all coordinates. Here we omitted the space coordinates

of G.

294

Gy. Csanak, H. S. Taylor, and R. Yaris

hence eKiwrwill only vanish along the semicircle when Im o is positive. After fixing the path of integration we can take the limit behind the integral sign and we shall get a unit factor instead of the exponential. The sometimes confusing limit in Eq. (12) simply specifies a contour. Clearly the density is now known if G(7) or G < ( 7 ) or G ( o ) , or the poles and residues of G(o) are known in the uhp. In the practical problems which we are concerned with in this review, the residues and poles of G(w) shall be solved for and used to obtain the density. Such being the case, the so-called spectral representation of G ( o ) , which shows explicitly its poles and branchcuts, residues and discontinuities along the branchcuts, will soon be discussed. If for some reason the poles are more difficult to obtain than an expression for C(w), p(r) and (as we shall see shortly) E, , the ground state energy can be evaluated by integrating over the contour directly. This is accomplished most easily by rotating the contour counterclockwise by 4 2 . The spectral representation will show that the rotated contour encloses the same poles as the original one. On the infinite semicircle in the left half plane the form of G(w) will be obvious from the spectral representation and the integral can be performed analytically. On the part along the imaginary axis one can integrate numerically (there are n o singularities along the imaginary w axis) moving symmetrically (up and down) from the real axis until G(w) reaches its large w asymptotic form, at which point analytic integration again can be used. It will now be shown that the same residues and poles also determine the ground state energy. In this demonstration the equation qf motion for $(I), to be derived in the next section must be used. This equation is

1

i - - h(r) $(rt) K

t

= /dr’V(r -

r’>$’(r’t)$(r’t)$(rt)

(15)

where h is the one-particle part of the Hamiltonian and V is the two-particle potential. The key to obtaining the energy is to evaluate the derivative of G( rt; rt’) with respect to t in the special case o f t < t’ = 0 which specifies the time ordering, viz.

lim r”r t-r-0

dC(rt; r’0) at

GREEN'S FUNCTIONS

295

A comparison of (16) with the second quantized expressions for ( H , ) and ( V ) gives the value of the limit in (17) after integration for r as ( H , ) + 2( V ) . Since E, = ( H , ) + ( V ) one obtains

j dr lini

E, =

dG(r, r'; z)

r,+r

+ (H,)

r+-0

From Eq. (12) the second term on the right-hand side is ( H , ) = - i j d r lim h(r)G(r, r'; z) z+-O r'+r

hence the ground-state energy:

r'+r

1

j $ d w j d r lim [w + h(r)]G(r,r'; o) (1 9) 4rr r'+r Again the limit z -+ -0 simply means that one must close the contour in the w plane in the uhp, and also the knowledge of the spectral representation of G(w) will allow the calculation of E, . Note that both E, and p depend only on G' (z). From the interpretation of the one-particle Green's function it is obvious that GI will be related to the elastic scattering cross section (Bell and Squires, 1959; Kato et al., 1960; Namiki, 1960). Though the formula for the cross section is simple, the explanation is not straightforward and we should refer to the adiabatic principle and the formulation of field theory in the Heisenberg representation (Roman, 1965). We shall give only a simplified schematic description. The elastic scattering cross-section in the Green's function formalism was first formulated by Bell and Squires (1959) and by Namiki (1960). One can formally simplify the calculation using the so-called in-out or LSZ reduction formalism (Falkoff, 1962; Roman, 1965; Kirzhnits, 1967; March et al., 1967) in which adiabatic decoupling is applied to the operators. Let us denote by $(rt) the electron field operator and by 1 Y o )the ground state of the target (Roman, 1965). It can be shown that as t -+ T co with simultaneous adiabatic decoupling of the electron-atom interaction the field operator will converge to an asymptotic form which obeys the free equation of motion (Roman, 1965). Therefore, = -

in

Iim $(rt) i+Tm

=

Iim $""'(rt)= lim $'Iee(rt) t+Tm

i-Tm

Gy. Csanak, H . S. Taylor, and R . Yaris

296

where $’“(rt) and $““‘(rt) is the asymptotic form of $(rt) in the distant past and distant future. It is stressed that the adiabatic decoupling considered here refers to the electron-atom interaction. This process can be handled in a nontrivial manner in the field theoretic formalism. See, for example, Klein (1956)and Klein and Zemach (1957). t,hin(rt) and $ O U t ( r t ) can be expanded in terms of propagating plane waves qk(rt) of momentum k :

in

This defines ayt. Now, the scattering state corresponding to an { electron of momentum k can be written as

~

and from Eq. (20a) and (20b) in

alout= Iim akt(t) r+Tm

(20c)

where ak(t) is the expansion coefficient appearing in the expansion of $(rt)

Basically, Eq. (21) says that the scattering functions are those that connect adiabatically, when the electron-atom interaction is turned off infinitesimally slowly, to the state in which a free electron moving “ i n ” toward or “ o u t” from the target is created in the field of the ground state. This, of course, is an idealization of the experimental situation. The I Yk’) of Eq. (21) can be shown to satisfy the Lippman-Schwinger equation for electron-atom scattering (Bell and Squires, 1959; Namiki, 1960). (See also Klein, 1956; Klein and Zemach, 1957; Fetter and Watson, 1965.) The scattering matrix can be written out t i n

Sk’k = (y; I Yk+j = (YO la,, ak I y o ) = 1i m ( y o 1 ak‘(t‘)akt(t ) I Yo) f“W

I--m

=

lim ( y o1 ak’(t ’)a,t(t) I y o ) f‘-+ + m t--m

lim j d r d r ’ q z s ( r ’ t ’ ) ( y1o$(r’t’)$+(rr) 1 yo)(Pk(rt)

=

1--m 1’’

=i

+m

lim [dr’drq,*.(r’t’)G(r t+-m f’-‘+W

’f’,

rt)qk(rt).

~

GREEN’S FUNCTIONS

297

Note that since G contains only \ Y o ) ,as opposed to 1””) ( n # 0), it alone cannot supply inelastic scattering information. Equation (22) shows clearly how elastic scattering is related to a knowledge of G. A knowledge of G can be shown to give the natural orbitals and the ionization potential of the bound electrons of the target. The former statement is fairly obvious at this point, since a slight generalization of (12) shows that G gives the density matrix, and natural orbitals are those functions which diagonalize this matrix. Its general matrix element is p(r, r’) = N Jdr2 . . . dr, Y,*(r, r 2 , . . . , rN)yo(r’,rz , . . . , f N ) =

<++(W(r’))

or p(r, r’) = - i

lim G(r, r’; r ) = --iG<(l+, 1‘1. r+-0

(23)

The density matrix, since it is diagonal in the natural orbitals X,(r), can also be written as (Lowdin, 1955)

where n i are the occupation numbers. The X,(r) are orthonormal as p(r, r’) is an Hermitian matrix. The Green’s function method is clearly the way to calculate the density matrix and the natural orbitals directly. It might seem somewhat redundant to obtain natural orbitals once GI is known and E, can be calculated. This is really a question of practicality in that it has been shown that it is easier to calculate good approximate natural orbitals than p(r, r’) in that the occupation numbers in the method of Reinhardt and Doll (1969) are poor. It is then suggested that E, be calculated variationally with the natural orbitals as the basis set. Since the iterative methods, to be discussed later, were not tried by Reinhardt and Doll (1969), it is hoped that by using them a direct calculation of E, using Eq. (19) will be more successful. To further illuminate the relationship between the natural orbitals, the ionization energies and the density matrix it is well to write what is known as the spectral representation of GI (mentioned earlier in this section). Starting with Eq. (6) and inserting a complete set of N + 1 and N - 1 particle state functions in the first and second terms on the left-hand side, respectively gives

Gy. Csanuk, H . S. Taylor, and R. Yuris

298

The superscript now denotes the number of particles. Introducing the “ orbital ” definitions, called Feynman-Dyson amplitudes fn(1) = <

yolw)lY+l)

= <‘yolICl(rl>l~,N+l) exP[- i(E,N+’ - EoN)t1I E

=

fn(rl) exp[- i(Ef+ - EoN)tl]

<e-I W)IYoN)= <eI IC/(r,>I Y o N )exp[i(Ef-

= gn(rl)exp[i(Ef-’

(264

- EoN)tll

- EoN)t].

(26b)

Equation (25) becomes in terms of thefand g orbitals

= - if@(*)

Cf,(r>f.*(r’) exp[- ~ ( E , N1+- E , N ) ~ ] n

- @(-z)

gm(r)g2(r’)exp[-i(EoN -E;-’)T]]

(27)

m

From a comparison of (23), (24), and (27), it is tempting to equate the onehole ( N - N - 1) Feynman amplitudes with the natural orbitals, because we obtain ~ ( rr’) , = C gm(r’>gm*(r>

(284

m

Unfortunately, the y’s are not orthogonal, nor even linearly independent. The f ’s, the one-particle ( N o N + 1) Feynman amplitudes are also not linearly independent. In fact the only thing that can generally be proved aboutfand g is the completeness of the total set

C L(r)L*(r’) + C gm(r)gm*(r’) = d(r n

-

r’).

(28b)

m

Goscinski and Linder (1970) have shown how to use Lowdin’s (1956) method of canonical orthogonalization to transform the g’s into the x’s. This procedure shall not be reviewed here since it is readily available in the original reference and is clearly readable to quantum chemists. The ionization energies and electron attachment energies to various states of the N - 1 and N + 1 particle system clearly appear in Eq. (27). To see how the ionization energies are poles of GI, Eq. (27) is Fourier transformed using

299

GREEN'S FUNCTIONS

or 1

m

(29b)

The former formula is easily verified by performing an inverse transform of the right-hand side, i.e., multiply by e-iwi/271 and integrate from o = - a by completing the contour. For t < 0 the contour must be closed in the uhp resulting in a zero value for the integral. For t > 0 the contour must be closed in the Ihp and a single pole at a - iq, gives a residue which in the limit is e- i a r . Similarly,

1

m

-m

dt[O(- t)e-'"']eimr

=

lim

-i

tl++o o - a

- iq'

(30)

With this the Fourier transform of Eq. (27) is obtained using (14), i.e.,

(31) Clearly, in the limit, the poles of the second term are at the ionization energies which can take on both discrete and continuous values. One discrete value occurs for each bound state of the N - 1 particle system. For a neutral N electron system the physical poles of the second term appear in the second quadrant of the complex plane and fall on a line o = iy, q > 0, i.e., infinitesimally above the real axis. They are discrete for small negative real o,and merge into a branch cut for a larger negative o.The physical poles of the first term of (31) all lie in the lhp along a line o = -iq, q > 0. If there exist bound states of the negative ion discrete poles appear in the third quadrant. A cut always appears along this line in the fourth quadrant. The situation is schematized on Fig. 1. These analytic properties will be useful later when G(w) is found by solving for its spectrum. A knowledge of where these singularities lie narrows the region of the complex plane to be searched. Clearly our arguments must be modified if N does not represent a neutral system. The reader should also be reminded that if the definition of G(w) which is essentially given for real w , is extended into the whole complex plane, then the mentioned poles and branch cuts arise. Because G(w) has branch cuts on a part of the real axis we can define by analytic continuation a function on the double-sheet complex plane which has no branch cut. By this analytic continuation new poles will appear on the unphysical sheets which are reached when the path of continuation crosses the above-mentioned physical cuts. This shall be discussed more fully in the next section where the poles and residues are sought. One can prove that G(w) coincides with G A ( o )below the left-hand branch cut

300

Gy. Csanak, H . S. Taylor, and R . Yaris

Rotated integra tton \ p?h

\

\

\

I I

I I

Branch cut

x x

)ntegration

path

X I

/x x x

: x

I I I

/

t

Branch out

I I

/

/

1

/ ,

FIG.1. Poles and branch cuts of G ( w ) .

and G A ( o )is the analytic continuation of G(w) above this branch cut and G ( o ) coincides with G R ( o )above the right-hand branch cut and G R ( o )is the analytic continuation of G(w) into the lower half plane across this cut. GR(w) is regular in the upper half plane whereas G A ( o )is regular in the lower half plane. It will be seen that the ability to use finite discrete basis sets greatly simplifies this problem. B. THETWO-PARTICLE GREEN'SFUNCTION It is now useful to turn to a discussion of G,. In C, four times appear and as might be expected from the discussion of G,, different specific time orderings yield different information. An equation analogous to Eq. (6) could be written but each term would now require three Heaviside functions (four times three relative times). Fortunately for the purpose of this review, which is concerned with excitation energies and inelastic scattering of electrons from atomic and molecular targets, only a few specific time orderings shall be

301

GREEN’S FUNCTIONS

necessary to obtain such information. To illustrate this, several time orderings shall be considered. Case I. Set t,, t,’ > t,, tz‘ for arbitrary order o f t, and t,‘ and for arbitrary order o f t , and t,’. Then for this case GZ(1, 2; 1’, 2’)’ = (i)-’(YoN( T[$(1)$(2)$t(2’)~t(1’)I I yoN)

(YoNI ~[$(l)$t(l’)l~[$(2)$t(2’)lI YoN)

==

- 2 (YoNI T[$(l)$t(l’)l I ‘rnN>(Y,N I W(2W(2’)1 I YoN) n

=

-2 (YoNI wnfW fl)’

= -

I y.N>(YnNI mKWt(2’)1I YoN)

c xn(L 1’)ffl(2,2’).

(32)

n

Here an N particle-state closure has been inserted and the hole-particle (Bethe-Salpeter) amplitudes have been defined as xn(1, 1‘) = (YoNI w ( l ) ~ t ( l ’ ) l l ~ , N ) fn(l, 1‘) = (‘y,”I mw)$+(1’)1l y o ? .

(33)

It is well to note that it was the specific time ordering that resulted in two hole-particle products, i.e., the pairing of creation and annihilation operators. Other time orderings will give other pairings such as hole-hole and particleparticle. This point is important because only for hole-particle pairings do the intermediate states have to be N-particle states; it shall be seen that this results in poles related to the excitation energies of the N-particle system and not intermediate states representing double positive and negative ion states with poles representing double ionization and affinity energies, respectively. To see this explicitly, it is well to write out the right-hand side of Eq. (33) in detail using the Heaviside functions for the time of two time orderings and making the times explicit by using Eq. (2). If this is done, after some algebra the following result is obtained xn(l, 1’) = exp [(i/2)(EoN- ~ , ” ) ( + t ~tl‘)Ixn(rl, rl’; zl)

(34)

where xn(rl, rl’; zl) = @(z,) exp

[WON + EnN).rl/2]

x (Y o N I $(rl)e- iHrl$t(rl’)I Y n N )

+ EflN)rl/2]

- @( - zl)exp [ - i(EoN

I

I

x ( Y o $t(rl’)eiHrlwl) Y,N>

(35)

Using Eq. (32), this gives m

GAl, 2; l’, 2’)’ = -

2 exp

n=O

- E/)‘r1xn(rl, rl’: .rl)Mrz,rz‘;z2)

(36)

302

Gy. Csanak, H. S. Taylor, and R. Yaris

for t,, t,’ > t,, t,’ but arbitrary T ~ T,, . Here the following changes to relative time variables have been introduced. T = +(tl

+ tl’) - + ( t 2 + t,’)

zi = t i - ti’

(i = 1, 2).

The important point is that (36) has three relative times, the exponential argument of one of them being simultaneously completely factorable from the matrix elements and being an excitation energy. The only other time ordering giving this property is the other hole-particle case, t , , t,’ > t,, t,’. The result here is: Case II.

By an analysis similar to the one we have just gone through it can be verified that the other time orderings do not have factorable exponentials in which EON- EnNappear. In general, G 2 can be written

where the Heavisidefunctions satisfy the time orderings of Case I and I1 above. Let us define the first two terms as the hole-particle Green’sfunction,

The spectral representation of G 2 is obtained from (38b) by using (29b) and Fourier transforming the variable T, as in Eq. (14), to give

303

GREEN'S FUNCTIONS

where

x exp{-(i/2"

+ (E,N - EoN>ICI t 1 I + I t 2 I I>.

(40b)

Consequently G ; P ( ~=) G2(w)'

+ G:(w).

Let us define a new variable w, which runs through all EnN- EONand all = 1 and sgn (0,= ) - 1. Let us define

EON- EnNvalues. Correspondingly sgn (0,)

and

Xn(rl,rl'; r ) = Xn(rl, rl'; t)

for w, > 0

xn(rl, rl'; 7) = jn(rI, rl'; t)

for w, < o

-

Xn(rl,rl'; 7)

= X",,(rl,rl': T)

for w, > 0

2,,(rl, rl';

= Xn(rl, rl'; t)

for w,, < o

Let us define X,,(1, 1') where t'

= e-iuJ'X,(r,,

r,'; T

~ )

+ tl') and

= +(tl

Xn(l,1') = eiWnz'fn(rl,rl'; t l > then G:P(l, 2; l', 2') =

-

X,(1,1')X,(2,2')0(t-+lt1)

-31t21)

W,>O

=-

c X,(l,

on< 0

1')fn(2, 2')0(-t - 4/71I

- +1t21)

C Xn(19 1')2,,(2,2')0(sgn (0,)~ - $ 1 2 , I - +1t21> On

(404

304

Gy. Csanak, H . S. Taylor, and R . Yaris

and

and obviously (38) becomes G2(rl, r2, r,’, r2’; zl, z2, o)= Cip(r,, rz, rl’, r2’; zl, z2, w )

+ other terms.

(40e)

The “other terms” of G, have poles at real w values that differ from +(EflN- EON).Therefore, while (39) does not have Heaviside functions to distinguish its terms, they all have poles at different positions in the complex w plane. In this work we shall concentrate on G i p of Eq. (38b) and use it in the next section to obtain an equation for A’, . The information contained in Gip is clear, namely its poles give the target excitation (or deexcitation) energies. As interesting is that the residue of Gip at the nth pole gives X,,(r, r‘; z) which is, from (39, in the limit T ---t -0, just the matrix element ( Y o N I $t(r’)i+b(r)I Y n N )This . is a case which shall be, in practice, obtained in our approximate solutions (see Section IV). This residue then allows the calculation of all one-particle transition matrix elements between Y o Nand Y n NIf. Op“)

=

c Op(’)(ri) i

then in second quantized form 0p(1) = {cir$t(r)Op(l)(r)$(r) = {cir

Therefore

~r’~t(r’)~p(l)(r)i+b(r)6(r - r’).

1

(YoN1 Op(’)lY o N = ) dr dr’Op(‘)(r)(YON I i+bt(r’)$(r) I Y,”) 6(r - r ’)

GREEN'S FUNCTIONS

305

where the integral over the 6 function is taken only after the operator is applied. In particular, from (lo), (Yo I $+(r)$(r) I Y,) the diagonal element in the position representation is just (Yo I p(r) 1 Y,) the transition density. Moreover, if our Fourier transforms Eq. (9) to get

it is seen that the Fourier transform with respect to r of the element W O N

is just

I $+(MdI y " N ) N

1

(yoN I eik.rl1 Y n N ) i= 1

the generalized oscillator strength (Schneider, 1970). Therefore the pole and residues of the hole-particle part of the two-particle Green's function, in the form in which they shall be obtained by the approximation methods to be given later, shall yield the excitation energies, the generalized oscillator strengths, and enable one to calculate all one-particle transition matrix elements. To calculate expectation values of n-particle operators (n > 1) other than H , in the ground or in excited states takes a knowledge of higher Green's functions and the rest of G 2 . These are more difficult to calculate in the sense that the nonperturbative approximation to be discussed in Section IV will not yield this information. As such this review will not discuss these quantities though, in principle, extensions of the perturbative methods of Section V could formally, but at this time probably not practically, calculate them. It is now time to turn to the question of calculating G, and G, , in principle, exactly. Certainly their calculation is now amply justified.

111. Coupled System of Equations for Green's Functions (The Method of Functional Differentiation; The Dyson Equation; The Bethe-Salpeter Equation)' A. A SYSTEM OF EQUATIONS FOR

THE

MANY-PARTICLE GREEN'S FUNCTION

In this section various useful and equivalent forms of the equations of motion,for G, and G , shall be given. To determine the equation of motion for GI, the time-derivative of GI is " Galitski and Migdal (1958). For further references see Pines (1961), Abrikosov et al. (1963). Matsubara (1955), Martin and Schwinger (1959); for an introduction see Kadanoff and Baym (1962), Migdal (1967), Falkoff (1962), Roman (1965), Kirzhnits (1967), March et a1.(1967), Mattuck (1967).

306

Cy. Csanak, H . S. Taylor, and R. Yaris

needed. In the definition of GI the time only appears in the $ and $f (and in the time ordering) so it is evident that is needed. Since is a Heisenberg operator, its equation of motion is

+

@/W$(rO = [W), HI.

(41)

To evaluate the commutator in (41) one substitutes the second quantized form of H in the Heisenberg representation into the commutator bracket. The resulting commutator, now only between $ ( r t ) and the annihilation and creation operators corning from H can be evaluated, after a little algebra, by using the anticommutation relations between Fermi-Dirac particles, Eq. (4) and integrating over the resulting 6 functions, yielding

+

i(d/dt)$(rt) = (h(r) jdr’tjt(r’t)V( I r - r’ I)$(r’t)) $(rt)

(42)

where h will symbolize the one-electron parts of H . One should note that Eq. (42) is of the form of a time dependent one-particle Schroedinger equation for $(rt) (the Schroedinger matter field). A useful shorthand notation is obtained using the definitions 1 3 r,, t ,

V(1 - 2) = V ( Ir, - r21)6(tl - t , )

2 =- r,, t ,

and dl

= dr,

dt,

With this, (42) becomes [i(d/dt,)- h ( l ) ] $ ( l )= sd2V(1 - 2)$t(2)$(2>$(1). Now differentiating Eq. (6) with respect to t,’ (note d @ ( t ) / d t

(42‘)12

= 6(t)), gives

(43) Introducing the definition 6(1 - 1’) = 6(r, - rl’)6(tl - t , ’ )

(44)

using (4) and (42’) (for GI) gives i(a/dt,)c,(l, 1’)

= 6(1 -

1‘) + h(l)C,(l, 1‘)

- /d2V(I - 2)(T($+(2)+(2)+(1)$t(1‘))).

(45)

Different forms of the same equation shall be given the same number with primes to distinguish them.

307

GREEN’S FUNCTIONS

The matrix element in the last term on the right-hand side can be related to G,, if a positive infinitesimal time “ E ” is added to t 2 in $+(2)and the limit E -+ 0 is inserted inside the integral sign. Now that there are two different indices r, , t , + E(rZ,t , + E = 2 + ) and 2, permutations of time ordering under T can be made as discussed in the previous section. These permutations are made to give the ordering as in G, , with the result that (45) becomes [ i & - l7(l)]G1(l, 1’)

+ iSd2V(1 - 2)G,(1,

2 ; l’, 2’)

= d(1 -

l’),

(46)

where “2’ implies that the limit 2’ 2 ( E -+ 0 ) must be taken before integration. This equation is the.first equation in the coupled hierarchy relating G, to G,+land G,,-l. The rest of the hierarchy can be derived by analogous methods, i.e., differentiating the definition of G, with respect to t,, commuting, rearranging, etc., to express the ground state matrix elements on the righthand side in terms of G,+land the lower order G,’s. This coupled hierarchy is the new form of the Schroedinger equation in Green’s function theory. ”

--f

B. THEDYSON EQUATION AND

THE

SELF-ENERGY

Now, in preparation for the derivations and approximations in the next section it is useful to write some other exact forms qf’ the equation for G,. To do this it is efficient to define an unperturbed (i.e., V is temporarily set equal to zero in the Hamiltonian) one particle Green’s function Glo by [Glo(l, 1’)I-l

1

[

= i- al: - h(1) 6(1 - 1’).

Inserting (47a) into (46) gives

jdl”[GlO(l, l”)]-lGl(l”, 1’) + i S d 2 V ( 1 - 2)G,(1, 2 ; l’, 2’)

(474

= 6(1 -

1’). (46’)

The integral kernels in (46’) can be thought of as matrices with continuous indices 1, l’, etc., thus implying an operator equation

[Glo]-’Cl

+ iVG, = I.

(46“)

When (46”) is multiplied by GIo from the left G, = Glo

+ iGIoVG2

(46 ”)

which is an integral equation form of the equation of motion, i.e., G,(l, 1’) = GlO(l, 1’)

+ i l d 2 d2’G10(1,2)V(2 - 2’)G2(2,2‘; l’, 2+). (46’”)

Operating from the right on (46”) with G;’

[Gl0]-’

+ iVG,G;’

gives =

G 1- ’

(46”)

Gy. Csanak, H . S. Taylor, and R . Yaris

308

a form which lends itself to the introduction of a new function: C called the selfenergy, the optical potential, or the one-particle effective potential defined as

C = -iVG,G;’

(47b)

or C(1, 1’) = - i I d 2 d2‘1/(1 - 2)G2(l,2 ; 2‘, 2+)G1-’(2’, 1’).

(47b’)

With (47b), Eq. (46”) can be written as G;’

=

[Glo]-’

+ C.

Multiplying on the right by G, gives [Glo]-’Gl - CG,

=I

(48‘)

or

Kl 1 i-

- h(1) G I ( l , 1‘) - I d 2 C ( l , 2)G1(2,1’) = 6(1

-

1’).

(48”)

Multiplying Equation (47b) from the right by G, and inserting the results in (46”’) gives G , = G I o + GloCGl.

(48 “‘)

Equations (48”), (48”’),and (48) are, respectively, the differential, integral, and operator forms of the equation of motion for G, and are called the Dyson equation. In the Dyson equation G, is replaced, using (47b), by C and so the determination of G, and G, becoming equivalent to the determination of C and G,. Hence a change of functional variables has been made. I t will soon be seen that the reason for changing unknown functions is that our intuition as to approximate forms for G2 is not as well developed as it is for C. One sees from (48) that the Dyson equation, or the one-particle equation is closed for G, if C is known and that C plays the role of an effective potential, which is generally energy dependent, complex and nonlocal. Clearly, G, is not a true Green’s f ~ n c t i o n ’in~ that only if C is a local, energy-independent potential does one obtain the usual “operator times a Green’s function equal to a delta function” relation, but is rather a “driven” Green’s function equation. I t should be realized that (48) is a one-particle equation, which if I: is known is equivalent to the Schroedinger equation. The effective oneparticle potential has folded in it all the effects of the rest of the system; as such, it would indeed be surprising if it were not complex, nonlocal, and l 3 By “ t r u e ” Green’s function we mean the usual Green’s function discussed in mathematical physics in connection with differential equations.

309

GREEN’S FUNCTIONS

energy-dependent. Thus, an exact one-particle picture has been achieved at the price of a complicated potential. A detailed study of C will show it to be real when w is set to an energy value below the first inelastic threshold of the system. Above the threshold C is complex, the complex part can be shown (Mott and Massey, 1965) to represent the absorption of incident particles by the system. Since we are working with an effectively closed equation for GI, which in turn starts and ends in I Y o ) ,inelasticity must be taken into account by an absorption potential. Since such phenomenological potentials play a key role in optics, they are called optical potentials. At this point, it shall be assumed that C is known and the method of solving (48) shall be discussed, after which the discussion will return to the question of finding X. It will be useful to Fourier transform the time variable in (48”). I t should be noted that C(1, 1’) depends on t , - t,’ only, since from (48) it has the same time variable as G, and GIo which have this property. Using (14) and the Fourier convolution theorem, (48”) becomes [E

- /i(r)]G(r, r‘; E ) - /dr,C(r,

r,; E)G(rZ, r‘; E )

= 6(r

- r’)

(49)

As is well known, Green’s functions are more difficult to solve for than wave functions since they are basically nondiagonal matrix equations. As such it is reasonable to seek what is always sought in such cases, an eigenfunction expansion which diagonalizes Eq. (49). To do this, it is pedagogically useful to define an integral operator (Layzer, 1963)

L=h+C

and to write Eq. (49) symbolically as [E

- L(E)]G(E) = I .

(49‘)

Since L is generally non-Hermitian, G admits a biorthogonal expansion (Morse and Feshbach, 1953) in the assumed complete set of eigenfunctions v,(E)and (P.(E) of L(E)and its adjoint L + ( E ) respectively, , of the form, for a given E

I f (50) is put into (49) and its adjoint, respectively, and the limit taken, the set of equations obtained are

E

-+En(&)

310

Gy. Csanak, H . S. Taylor, and R . Yaris

h(r)(P,,(rs)

+ J dr’@,,(r‘;E)C*(r’, r; E ) = E,,*(e)(P,,(rE).

(5 1’)

For the nonperturbative methods of this review, (51’) shall be the form of the Dyson equation that will be used to determine the one-particle Green’s function via the En(&),(P,,(E), and (P,,(E). Equations (48) will be of formal use. While (51’) is a strange beast in theoretical chemistry and atomic and molecular physics, it is important to emphasize that the eigenvalue problem for non-Hermitian matrices can be solved by techniques which are standard in the applied mathematics literature (Wilkinson, 1965). In principle this non-Hermitian eigenvalue problem must be solved for each and every value of E in the complex plane. Each E gives a complete set of eigenfunctions and eigenvalues. Clearly, if all this were needed to represent Eq. (50) the method would be impractical. The saving grace is that as seen in Eq. (31) G(E)can be represented in a form where all the energy dependence comes into the denominator. In (31) only one set of functions { Lg } as well as a specific set of energies are needed for all real E . Of course for complex E the poles of G in (32) must be found so as to allow analytic continuation to achieve a complete complex plane representation. Hence, to obtain a form in the spirit of (31) it is desirable to use (51) to solve for the poles and residues of (50) and then to use the Mittag-Leffler theorem to represent G(E)in terms of its residues and poles [G(o)is analytic on the double-sheet except for poles]. Now a great deal is actually known about the poles of G(E).From the discussion of Eq. (31) it was seen that: (a) For an infinite system, branches exist parallel to, but iq above and below the real axis, in the second and fourth quadrants of the physical sheet. Therefore, all poles must lie on the nonphysical sheets, and the residues of these singularities are not simply related to the {f,g } set which are related to the branch cut. (b) For finite atomic and molecular systems, the branch cuts are replaced by a set of poles and adjoining branch cuts. Poles can exist on the physical sheet, as well as on the nonphysical sheets reached by continuing across the cut. The physical poles are clearly related to residues which are g,,g,,* from Eq. (31). (c) For a finite system described approximately by a finite basis set, both the branches are replaced by poles, since the effect of the finite basis set is

31 1

GREEN’S FUNCTIONS

to put the system into the Hilbert-space box” spanned by the set and hence to give only a discrete spectrum. In this latter case the poles are real (in the limit 9 + 0) and from Eq. (31) correspond t o ionization and electron-affinity energies. The residues are clearly related to gngn* andf , fn*. “

Now in atomic and molecular physics experience has shown that ground state problems, responsefunctions, and even effective scattering potential problems, can be represented by discrete basis sets which span (in a mean value sense) the short range region of real space (physical or configurational) in which the particles of the system are simultaneously interacting. It is in this region that the dynamics of the problem occur. The continuum relates the system to boundary conditions which describe the experiment being done on the system. Therefore, to solve for C and G,, which characterize the system, it will be physically reasonable and practical t o use finite basis sets; in particular, those bases known to give good energies, correlations, response functions (i.e., polarizabilities), etc. In the case (c) that concerns this review, a comparison of (31) and (8) shows that the poles occur near the real axis at solutions of En

(52)

= En(En).

The residues of the Green’s function at the poles are obtained by the standard procedure under the assumption that Eq. (50) has only simple poles. That is, it is assumed that there are no contributions from singularities in the energy dependence of (P,(E), @,(E), and En(&).Thus applying the standard formula for the residue of a simple pole lim ( E - E,)G(E) &+En

to the Green’s function in (50) the residue at the nth pole E,

= En(&,)is

rn V n ( E J C p n ( 4

(534

where

r,,, that is,

= [1

- (d/d~)En(~)Iz=&, ,

r, is the residue of I/[& - En(&)]taken at the E,

pole. Hence

Clearly in case (c) the (P, are related to the g’s andf’s. The E, with Re E, < 0 give residues g, gn* and with Re E, > 0 give residues r, (P, Cp,,. Thus to get G(E) for case (c) it is, in principle, necessary to solve Eq. (51’) for all real E and to choose from the complete spectrum those cp’s and E’s satisfying the eigenvalue

312

Cy. Csanak, H . S. Taylor, and R . Yaris

equation (52) (we shall discuss the finding of r, shortly). In practice one will solve at a finite set of real E values and extrapolate between them or for a finite basis set the determinant associated with (52) will be solved for. Since, as will be seen, Eq. (51‘) and the equation to be derived for C will be solved iteratively and self-consistently, the first approximation for G will be taken as that of the Hartree-Fock model. In this model the E, are the Hartree-Fock bond and virtual orbital energies. It should be noted that for the Hartree-Fock model (or indeed for any true single particle model) all r, = 1, since the model is a solution to an energy-independent effective potential. These Hartree-Fock poles can be used as a grid of E points on the next iterate. As in any iterative method a reasonable first guess, which the Hartree-Fock model is, is needed for convergence, thus it is hoped that the final poles will not differ greatly (in the mean) from the initial guess. This will simplify the search for E , . To find r, it is necessary to know En(&)in the neighborhood of the poles, but this is just what the extrapolation procedure will give us. Then knowing this one can find dE,/dE numerically. It is perhaps worth stressing that a method to solve for C , ( E )for a given C has now been sketched out. In doing this for the physical reasons discussed above it has been assumed that a finite discrete basis set will be adequate to represent C and GI. Mathematically this means that the branch cut has been replaced by a discrete set of poles on the real axis. This can be done if it is assumed that (a) the poles far from the real axis do not affect the physics and (b) that the (P,(E) and (P,,(E) do not give poles themselves. The former is justified (Goldberger and Watson, 1964) because poles far from the real axis describe effects that take place in extremely short times relative to the times for the total process, e.g., for a scattering process they describe events that occur before the particle fired from the gun can arrive at the target. The latter assumption follows from the idea that the (P,(E)’s are effective, one-particle orbitals for a given energy. If a singularity existed a small change in E would cause large changes in (P,(E) which would die out as E is changed again, infinitesimally. This is similar t o an extremely narrow resonance phenomenon which would not be observable in the finite resolution measuring processes used in experiments. In other words, if such poles existed they would not effect observables. Assumption (a) is further justified by realizing that a pole near the cut causes a heavy weighting of the continuum functions in the neighborhood of the pole. This weighting is analogous t o forming a wave packet out of the functions near the pole. The discrete basis essentially says that if only such poles exist, the spectral density result and packeting phenomenon can be anticipated by trying to represent the packets themsleves with a finite discrete function (calculated from a finite range basis). For poles far from the cut the wave packet is very broad containing contributions from a wide range of

313

GREEN'S FUNCTIONS

continuum functions. Such broad resonance type phenomena result in little change in the spectral density and are essentially unobservable. Once the poles and residues of GI are known the integrals for p(r, r') and E, are easily done.

C. HIERARCHIES FOR GREEN'S FUNCTIONS AND RELATEDQUANTITIES; THE METHODOF FUNCTIONAL DIFFERENTIATION Now that G, is known given C, equations for finding G , or C must be developed. In the spirit that the equations for G I , G,, . . . , or G , ,C, . . . , will be solved simultaneously, it will be assumed that GI is known and an equation for G , or C shall be the object of this subsection. Clearly the equation for G 2 or C will involve a knowledge of G, or its equivalent. As in the case of G I , physical insight and incisive approximation will be facilitated by writing the equation for G, in terms of a closed equation with an effective (twoparticle in the media) potential (the knowledge of which is equivalent to having G,) called E.Several sets of equivalent functional variables are in the process of being defined, viz., G I , G , , G , , . . .; G I , G , , =, . . .; G I , Z, E,. . .; G I , C , G,, . . . ; GI, C , 6C/fiU, ... ; { q n ,E"}, C, =, . . . : etc. The various hierarchies are all, of course, formally equivalent, in the sense that no new physical content will be contained in any of the alternative forms to the G,, G, , G,, . . . set of coupled equations. However, since the set of coupled equations must be truncated by an approximation at some stage, different forms of the hierarchy lead naturally to different approximations, and hence to different truncation procedures. In Section VI, perturbation expressions are derived which give G , or 2 in terms of V and G I . It will be pointed out in Section VI that the perturbation expansion for G2 (or C) in terms of GI, and the Dyson equation for GI form self-consistent perturbation theories for GI and G , . The perturbation expansion for G , involves G , and G,. The two closed equations for G I and G , and the expansion for G, again give a self-consistent perturbation theory of higher order. Of course, closing the set of equations at G, will be much more work than closing at G,. Only experience will show how high in the coupled equations the truncations must be made. Fortunately, since anything above G, will be inordinately difficult to calculate, diagrammatic analysis and experience (Kelly, 1968; Karplus and Caves, 1969) tends to indicate that G, will not be needed for most atomic and molecular problems. The completion of the G, truncation perturbative strategy does not require the derivation of the closed equation for G, in terms of E since it has already been indicated how the equation for G , in terms of G, is derived. The closed equation from the perturbative point of view is only an alternate equation in the sense that (48) is an alternate to (46). The real reason that this closed alternate form will be derived is to introduce the method of functional di'erentiation

-

-

314

Gy. Cdanak,H . S. Taylor, and R. Yaris

and the expression of G2 in terms of functional derivatives of lower order quantities. This method will lead to nonperturbative approximations to be discussed in the fifth ~ e c t i 0 n . lThe ~ fundamental idea of the functional derivative method of Schwinger is (Schwinger, 1951 ; Galitski and Migdal, 1958) to introduce into the problem an arbitrary nonlocal time dependent potential U(1, l’), that is turned on slowly (adiabatically) at t = -co and off at t = + co. This potential “probes” the system, and the physical quantities GI, G 2 , C, etc., are calculated in the limit U( 1, 1’) ---* 0. The idea is exactly that of studying generalized response properties of the system. The method is quite physical in that all experiments actually probe the system and measure its response in one of its several forms, e.g., absorption coefficient, dielectric constant, etc. It shall be shown that C, (for a given GI) can be replaced by a knowledge of 6Gl/6U1 o+o i.e., the variation of GI with respect to the small probe potential in the limit that the potential goes to zero. What will result is a sequence of coupled equations relating C (or G,) to S C j S U ; SCjSU to second derivatives, etc. It is worth repeating that the advantages of the new hierarchy over the original one are both physical and formal. Physical arguments will allow one to make approximations to variations that were not evident when G , , G,, etc., were used. The idea, as will be seen, is that approximations to C and G, that are inadequate to represent C and G, themselves may suffice to calculate small variations in C and G , . The formal advantage of the method is that closed equations for G, , etc., can be derived without invoking perturbational diagrammatic summation procedures. In the situation that a small arbitrary external nonlocal two time-dependent potential U(2’, 2) of the form

U(2’, 2)

=

U(r2’r,’, r, t,)O(t,’ - t 2 )

(where the Heaviside function maintains t = - m a n d o f f a t t = +co

t,

earlier than t2’) is turned on at

where the symbol U reminds us that the new Hamiltonian has U(2’, 2) added to the one of the previous section. Since the defining equation was for an arbitrary Hamiltonian, the derivation still holds and the equation can be written as G - ’ ( U ) = [G,O]-’ -

u - C.

(55)

l 4 Historically the equation for Gz i n terms of E, was first derived by Bethe and Salpeter, using formal complete diagrammatic perturbation summations. The method chosen here has the advantage of giving functional derivative expression for E and leads to approximations that might not be obvious in the diagrammatic formalism.

315

GREEN’S FUNCTIONS

dG,(l, 1’; U ) dU(2’, 2)

=

u=o

dG,(l, 1‘; U ) dU(2)

-G2(1, 2 ; l’, 2’) + Gl(l, 1’)G1(2, 2‘)

(56a)

+ Gl(l, 1’)G1(2, 2’).

(56b)

= -G2(1,

2 ; l’, 2’)

Lr=o

This is the desired functional relation that for a given G, replaces G2 by dC,/dUI “ = O . For a given G I , the derivation of the hierarchy for C in terms of dC/dU is easily carried through. Since

J d2G(1,2)G-’(2,

1’) = d(l - 1’)

taking the variational derivative with respect to U(4, 5) yields

which can be multiplied from left by G(l’, 2’) and integrated over d l ’ to give

6G( 1, 2‘) 6G-’(2, 1’) - - I d 2 dl’G(1,2) G(l‘, 2‘). dU(4, 5) dU(4, 5)

~-

(57)

To evaluate 6G-’/6U Eq. (55) is varied giving

6G-’(2, 1’) 6U(4,5)

=

-6(2 - 4) d(1’ - 5) -

dC(2, 1‘) SU(4, 5)

~

Substituting ( 5 8 ) into (57) gives

dC(2, 1‘) 2’) - G(1,4)G(5, 2’) + j d 2 dl’G(1, 2) G(l’, 2’) dU(4, 5) 6 U 4 , 5)

“(”

~-

~

(59)

316

Gy. Csanak, H . S . Taylor, and R . Yaris

which on using (56a) gives -GZ(l, 5 ; 2’, 4)

+ G l ( l , 2’)G1(5, 4) = G1(l, 4)G1(5, 2’) hC(21‘) + j d 2 dl’Gl(l,2) ___ Gl(l’, 2’). hU(4, 5 )

(604 Equation (60a) is the promised equation which replaces G, by SCjSU. Again the special case of a local potential is -Gz(l, 4; 2’, 4’)

+ Gl(1, 2’)G1(4, 4’)

=

G I ( ] , 4)G,(4, 2)

+ j d 2 dlrGl(l, 2) a q 2 , 1‘) Gl(l’, 2’). ~

6 U(4)

((job) We can now substitute 6C/6U for G , in the definition of C, Eq. (47b). It should be noted that in this definition Gz(1, 2; l’, 2’) the so-called “threepoint Green’s function appears rather than the more general four-point Green’s function G2 ( I , 2 ; l’, 2’) because the two-particle potential in the Hamiltonian is a two-point instantaneous potential (in the case of more general four-point retarded or advanced potentials the more general fourpoint Green’s function appears). Thus, in the hierarchy for C in terms of SCjSU only local potentials as in Eq. (60b) need be used. However, to develop a closed equation for the general four-point G , in terms of Z the more general functional derivative in terms of a nonlocal two-time potential will be needed. Hence, substituting (60b) into (47b) ”

“

C(1, 1‘) = - i j d l ” d 2 V ( I - 2)G,(1, 2; l”, 2+)G;’(l”, 1’) =

i jdl‘d2V(1 - 2) G l ( l , 2)G1(2, 1“) - G l ( I , 1“)G1(2,2’)

= -i

S(l - 1’) jd2V(1 - 2)G1(2, 2’)

+ iV(1 - l ’ ) G l ( l ,

I,+)

1‘) + i j d 2 d3V(1 - 2)G1(1, 3 ) 6C(3, SU(2) ~

”

317

GREEN’S FUNCTIONS

Equation (61) is the first equation of a new hierarchy which for a given G,, replaces the coupled equations for G , in terms of G , , G , in terms of G 4 , etc., with equations relating C to S C j S U , SCjSU to second variations, etc. Higher equations in the hierarchy are obtained by functionally differentiating (61). For example, the first variation yields

6G(1, 3’) 6C(3‘, 1’) + iV(1 - 1’) ”(”6 U(2)”+I + i Sd3 d3’1/(1 - 3) ____ 6U(2) 6U(3)

+i

d3 d3’1/( 1 , 3)G( 1 - 3’)

62C(3’, 1’) 6U(2) SU(3)’

D. THEBETHE-SALPETER EQUATION-ANOTHER HIERARCHY OF EQUATIONS For purposes of physical insight and ease of derivation of exact crosssection expressions, a third form of the hierarchy will now be derived. This third form, which is equivalent to the previous two forms stresses the ‘‘ optical ” potentials. Closed equations shall be given for each Green’s function. The unknown part of each equation will be an effective potential that requires a knowledge of higher Green’s functions or equivalently higher functional derivatives of C. The first equation of the hierarchy is again (48”) whose effective potential nature has already been discussed. The second equation is obtained by replacing G , by a more convenient functional variable called the generalized linear response function and defined as

R(121’2’)

=

%(I, 1’; U ) 6U(2’,

3-

0-0

It is a generalized linear response because it is the coefficient of the linear term in the expansion of G l ( l , 1’; U ) in a power series in U(2, 2‘). For small U , it is the most important term describing the effect U has on the system. The usual linear response is a special case of (63) and is (recalling that 1’+ =r,’, t , + E )

Therefore (64) gives the linear term in the expansion of the density matrix in a local potential. Comparing (56) and (63), it is clear that for a given G,, R and C, give equivalent information, i.e.,

318

Cy. Csanak, H . S. Taylor, and R. Yaris R(121’2’) = -Gz(l, 2; I’, 2’)

+ Cl(I, 1’)C1(2, 2‘)

(65) To obtain a closed equation for R, Eq. (59) is combined with (63) and SCjSU is replaced, using

=jd6

d7Z(2736)R(6574).

(66)

In the last step (63) has been used and the new definition

has been introduced. The result is, after some changes of dummy integration variables, the Bethe-Salpeter equation R(121’2‘) = Cl(l, 2’)G1(2, 1’)

+ j d 3 d3‘ d4 d4’G1(1, 3)G1(3’, lf>E(343’4’)R(4’242’).

(68)

Equation (68) is a closed equation for R (or G2), which when compared to Eq. (48”’) shows that it is an integral equation for R describing the motion of two Dyson ” or “dressed” particle^'^ [i.e., particles traveling in solutions of (48”)I-this is evident since C, and not G I o is the unperturbed part of (68)-interacting via the effective media potential which already has removed from it the purely single particle effects. Of course, 3 requires a knowledge of SCjSG,, which can be shown to be equivalent to a knowledge of G , or higher functional derivatives of C with respect to U.Similar closed equations for higher Green’s functions can be derived for dressed particles in the media. These shall not be needed here and shall be left out. The Dyson equation, (68) (called the Bethe-Salpeter equation) and the other effective potential equations form our last hierarchy.16 As in the case of the Dyson equation, solution of Eq. (68) is facilitated by deriving an equation for the spectral amplitude which has fewer variables. The property of (68) that t , and 2,‘ are parametric is very useful for such a derivation. Changing the parametric time variable to “

7, =

t , - t,’

2’ =

3(t2 + t2’)

dt2 dt2’ = dt2 dt,’.

Note that a Dyson particle can be a particle or a hole. The usual form of the Bethe-Salpeter equation that appears in the literature involves G , and not R and is derived in Appendix B. l5 l6

319

GREEN’S FUNCTIONS

Equation (68) can be written as ~ ( 1 1‘, , r 2 , r2’, t 2 , z2) = R O U TI’, r2, r2’, tZ, ~

2 )

+ j d 3 d3‘ d4 d4’R0(13’1’3)E(343’4’)R(4’, 4, r2, r,’,

t 2 , z2)

(694 where R0(121’2’)

=

Gl(I, 2’)G1(2, 1’).

(69b)

Now, Fourier transforming with respect to (- t 2 ) gives RU, 1’, r 2 , r2’, z2, 4

= Ro(L 1’,

r,, r2’, z2, E )

+ I d 3 d3’d4d4’R0(13’1’3)E(343’4’)R(4’, 4, r,, r2‘,z,

E).

(70) Since for the purposes of this review, only hole-particle amplitudes are of interest, further details will only be done for the hole-particle part of R, i.e., Rhp. The expression for RhP in terms of the amplitudes is from (65) exactly that for G;P given in (40d) except for an overall sign change (from “+” to “-”) and the n = 0 term is removed from the summations (it is cancelled by the GIGl term). Since the Fourier transform of G2 has been given in (40e), the Fourier transform of R can be obtained exactly as (40a) and (40b) were obtained. Simply noting from Eq. (38b) that ~ ~ ~ ( 1 2 1 ’=2 ’C ) ei”nr2X,(1, 1’)X,(r2, r,;

7),

““fO

and then proceeding exactly as in going from Eq. (38) to (40d) gives

xexP(-2sgn(o~)[E-o.l[lz,l i

+ Iz,ll).

(71)

Exactly as (38) compared to (40e), one obtains ~ ( 1 1’,, r 2 , r,’, z,

E) = Rhp(l,

If, r,, r2’, z 2 , E)

+ non-hole-particle terms with poles not at w, .

(72)

Substituting (72) into (70) and comparing the nth residues on both sides of and taking the the resulting equation and multiplying by E - o,- iq sgn (on), limit E o,gives, after the common factor xn(zZ) is canceled, --f

Xn(l, 1’)

=

J d3 d3‘ d4 d4’R0(13’1’3)2(343’4’)X,(4’,4).

(73)

Gy. Csanak, H . S. Taylor, and R . Yaris

320

Equation (73) is the desired equation for the Bethe-Salpeter amplitude. It has been assumed the R, has no poles at w, . This is clearly not true for scattering states and hence (73) is a bound-state equation. However, as was previously discussed in connection with solving the Feynman-Dyson equation, for atomic and molecular problems one can span the relevant region of space with a discrete basis set. Within such a discrete basis the assumption holds and (73) can be used for the X’s necessary to evaluate R. Equation (73), the Bethe-Salpeter amplitude equation for bound states, is clearly a closed equation for the two-particle, hole-particle amplitude. Since R, depends on GI and not G I 0 ,the hole and particle “move” in Dyson orbitals (the Dyson equation is assumed to be solved) and interact via Z which is the effective potential for hole-particle interactions of dressed particles and contains only true two-particle interactions with the singleparticle average effects removed. In order to obtain a Bethe-Salpeter equation valid for w, in the continuum, one must proceed slightly differently. Starting with the equation for R in terms of the X’s where the (continuous) index is now integrated over “

”

R(121’2’) = I d w m X m ( l ,I’)zm(2,2’) multiply both sides by X,(2,2’) and integrate over 2 and 2’. It is not, in general, true that the X’s are orthogonal, however, they have an exponential time dependence, Eq. (34), that goes as eiwnr. Since the times are integrated from -00 to + 00 one obtains a 6(wm- 0,) from the time integration. Hence, X,(l, 1’) = J d2 d2’R(121’2’)Xn(2,2’),

(74)

where the X’s are assumed to have already been normalized, and if there is any degeneracy that the X’s within the degenerate set have been orthogonalized (which can always be done). Notice that Eq. (75) says that R is the twoparticle kernel that propagates the two-particle amplitude X, from space time points {r, t , , r2’t2’} to { r l t l , rl’f,’). Now substituting (68) into (74) and then again using (74) in the resulting equation gives X,,(l, 1’)

=

J d 2 d2’G1(1,2’)G,(2, l‘)X,(2, 2’)

+ J’d3d3’ d4 d4’C,(I, 3)G1(3’,1’E(343’4’)Xn(4‘,4 )

(75)

which is the Bethe-Salpeter amplitude equation for continuum states, i.e., unbound hole-particle pairs. Note that it is of the same form as the bound state equation (73) except that it has an additional inhomogeneous term on the right-hand side. In Section IV we will elaborate this inhomogeneous term. A similar equation could have been derived for hole-hole or particle-

GREEN’S FUNCTIONS

321

particle amplitudes. The two-particle Green’s function G2 would have been used and the terms in Eq. (39) with the proper poles stressed. An equation exactly like (73) would result except that X , would be a hole-hole or particleparticle two-particle amplitude, and E would be replaced by W (see Appendix B). These equations will not be stressed in this review since they do not, as discussed in Section 11, lead to excitation energies. In summary, three equivalent forms of the hierarchy of equations have been derived in this section: (i) that for GI related to G,, G, to G, , etc.; (ii) that for GI related to C, C to 6C, 6C to h2C and ( c ~ C ) ~etc.; , and (iii) that for G, given by a closed equation with a potential C, which requires a knowledge of R (or G,) and a closed equation for R in terms of a potential Z, which depends on G, or higher variations in C. Much time has been spent in obtaining various useful equivalent forms of the hierarchy-equations.

IV. Scattering’’ The formulas and equations for scattering are here further developed using the exact relations of the previous section. First the case of elastic scattering is considered. The purpose is to give a simple prescription for calculating the T matrix when C is given in exact, or approximate, form. The S matrix has already been given in Eq. (22) in terms of G. Here it shall be expressed in terms of solutions of (51) with proper elastic outgoing boundary conditions. It is here assumed that C has already been solved for exactly, or, on a finite basis set, approximately. To do this consider the one-particle amplitude that corresponds to the boundary conditions of an “ in” or out ’’ elastic scattering experiment, i.e., the one related to y k * . This amplitude is, from Equation (26a), “

fi+) (rt) = (Yo“) I rG/(rt)I y k + ( N + 1)).

(76) Using (21), the inverse of (20), and inserting T since the limit already specifies the ordering, gives

fi+)(rt) =

i‘+

-m

lim

= 1”

=

lim ( Y oI rG/(rt)a,+(t’)IYo)

-m

j”nr~(~rG/(rt>rG/t(r‘t’>>yk(r~r~)

lim i /dr’Gl(rr, r’t‘)yk(r’t’) f’+-m

(77)

where it is recalled that yk is a free-particle function. ” SeeKlein (1956), Klein and Zemach (1957), Bell and Squires (1959), Namiki (1960), Fetter and Watson (1965), Janev et al. (1969), and Csanak el a / . (1971).

322

Gy. Csanak, H . S. Taylor, and R . Yaris

Putting (77) into (22) gives for the &k

=

lim i'+

m

s

"+"

case

dr'qi,(l')fi+)(l').

(78)

Therefore a knowledge offk+ (r' t ) gives S k , , k . Substituting (48'") into (77) and using (77) again the expression forf !+) becomes

dr'G,(rt, r't')qk(r't') r'+ - m

+ idr, dt, dr, dt, Go(rt, rltl)C(rltl, r2 t2)Gl(rz t , , r't)qk(dtr)] = ( P k k t>

+ [drl

dtl dr, dt, ~ , ( r t ,r l t l ) ~ ( r l t lr2 , t2)yi+) (r2 t,).

(79)

It is now necessary to Fourier transform Eq. (79). To do this, time homogeneity is invoked to give C(1, 1') = C(r, r'; t - t')

and the fact thatfb') (r, t ) can be rewritten as

f;+)(rt)

=

(yoN $ (/r t ) I ~ : ( ~ + ' ) ) = ( ~ ~ ~ ~ e ~ ~ ~ $ ( r ')) ) e - ~ ~ ~ l ~ q

= exp

[ - i(E:+

j $(r) 1 ~

-

q + ( '1)~ +

r e -ie qfq (+)(,.)

where cq = E:+' - E , ~

fi+)(r)

= (yoNl$(r)lyqi(N+l))

to give from Eq. (79) fs(+)(r)e-'~qt= qq(r)e-'Eq'

+ [dr,dl,dr,dt,G,(r,

r , : t - t,)C(r,, r,; t , - t 2 ) f ~ + ) ( r , ) e - ' E q ' z . (79')

Letting T =

t - t,

T , = t,

- t,

multiplying through by ei'qf, and integrating Fourier transform definitions, gives

fi"(r)

= q,(r)

+ lim n-t + O

1

dt, dt, T,

=

and

dr dr,

7,

, using the standard

Go(r, r , ; cq + iq)C(r, r,; cq)fi+)(r2) d r , dr,

(79")

where the iq is added because, since t + + GO, T is positive, which requires that the inverse transform of GO(&) be done by closing the contour in the lower

323

GREEN’S FUNCTIONS

half plane. The iq guarantees, by the residue theorem, that nonzero solutions only occur when the contour is closed properly, and that Eq. (79”) has the proper boundary conditions. Equation (79”) is the integral form of (51’) with the elastic “out-going scattering wave,” “ incoming free wave boundary conditions built in. It shows thatf,”) is a solution of Eq. (51) for a given energy &k = k2/2 and with outgoing boundary conditions. Hence, if X is solved for in any way, the solution of the elastic scattering problem is exactly the same as a nonlocal potential problem, for which many methods are known (Reinhardt and Szabo, 1970), the determinental method being especially appropriate. In deriving Eq. (79) the integral without C was replaced by q k ,the plane wave. To see the correctness of this procedure, note that “

”

”

lim i’+

-m

1

dr’Go(rt, r’t’)(Pk (r’t’) lim Jdr’(yoo 1 $o(rt)tjo+(r’t’)I‘Yoo)qk(r’t’)

=

i‘+ - m

lim (Y!ooI$o(rt)akt

= 1”

(~’)I‘Y~~)

= (Yool$o(rt)lYt+).

-m

Here the new “ 0 ” indicates “unperturbed” and ( Y o oI tjoI YE’) is clearly the free-particle one-particle amplitude, which by the Dyson equation, since C is here zero, is just an incoming plane wave. It is now convenient to have a T matrix form and then to convert from time variables to energy variables. To this end, if Eq. (79) is substituted into (81) and the analog for (27) is used for Go, 1

1

‘

j

Go(rf, r‘t’) = ; qj(rf)qj*(r’ r ’ )

the equation for

Sk,k= lim 1,- + m

&I

for

t

> t‘

becomes

( Jdr‘qk(r’t‘)q,*.(r‘t’)

Using the conservation of energy and momentum in the form

= d(Ep

- Ej)ijjp

Gy. Csanak, H . S. Taylor, and R. Yaris

324 gives 1Sk‘k

[

= d(&k - &k8) dkk,

2

+7

I

dr‘ df‘ dr” dt”cpz?(r’t’)C(r’t‘, ~“t”)f~+’(~”f”) . (8oa)

Now defining T by the standard relation S=l+T

gives 1

Tk’k

=y

1

]dl dl’V:f(l)x(l, l’)f$+)(l’).

(80b)

Note that T has the advantage over S that the limiting process which is difficult to perform is eliminated. To express the T matrix in terms of fk(r), (80b) is written as T~~= =

‘s

dl’ dl”cp,*(r) e(Ep”C(r’, r”; t ’ - t”)fA+)(r”)e-iW”

1

1 1

1”’ dr”cp:(r’) J^dd(r’,r”;

= 6(ep - Eq)

7) e i e p 7

s

dt”

f:

ei(Ep-Eq)f“

+

)(

r“)

27t

- Jdr dr’cp,*(r)x(r, r’; &,>f:+)(r’).

(81)

1

f6”

Equation (81) is what shall be used in practice since C(cp) and (r’) is what is usually available when time-independent methods (which are easier t o work with) are used. The next equation to be derived is for inelastic scattering. Analogous to the derivation of Eq. (21), the expression for the S matrix element for a scattering process that excites the target from state 0 to state n with electron initial and final momentum p and q, respectively, is = (~fl;;~+~)I\fln+d~+’= ))

lim (YI,Nlup(f’)aqt(f)IIflYnN> +

f‘m t+-m

It is evident that the two-particle, hole-particle amplitude is needed here, which requires a knowledge of E (or GJ. An alternative viewpoint comes from a comparison of the definitions of G, and x n , which shows that xn is reasonably considered an off diagonal one-particle Green’s function,” and should be related t o off diagonal optical potentials and responses. If the bound state Bethe-Salpeter equation for hole-particle amplitudes (73) is substituted into Eq. (82), another form, which is possibly more useful for making approximations, is obtained. “

325

GREEN’S FUNCTIONS

1

d2 d3 d4f~-’*(1)3(1423)~,(3,4)f,‘”(2)

= 7 [dl 1

(83)

where thef‘’s are special solutions of the ground state Dyson equation defined by fi”(2)

=

lim i [drz’qq(2’)G(2, 2’) 12’-

-m

f$-)*(l) = lim i t,‘+m

s

dr,’G(l’l)q~(l’).

The time independent form of Eq. (83) showing the proper energy conservation is obtained by substituting Eq. (79’) and (34) into (83), with the changes of variables

r

=

t 3 - t4

s = f(tl

t = $(t3

+ t,)

+ t4);

p = t,

a =s - t;

-

t,

W , = E,, -

Eo

to obtain =

i1’ sdc d t d p d r dr, dr, dr3 d r 4 f ~ - ) * ~ r ~ ) . ~ ~ ’ ) ( r Z ) e i ( ~ p - ~ q ) u i(Ep-Eq-iun)f

e

e

4

i(e + e q ) ~ / Z z

p

1 4

2 3 Wc)Xn(r3 7

r4>7).

Integration over t gives the energy conserving &functions. Integration over a and p Fourier transforms Z ( p , z,a) to Z[(cp + cq)/2, z,E~ - E ~ integration ] over r after substituting in the Fourier integral representations of Z(r) and ~ ( z )gives finally 1

= - 8(cp - E~ - w,)

i2

1

dr, dr, dr, dr4fj-)*(rl)ji+)(r2) sdc

If 3 is known (e.g., by solving for it perturbationally by the method to be discussed in Section VI, D) then the inelastic electron scattering cross section can be evaluated from Eq. (85). The equation for inelastic scattering of an electron off a target, since it only uses electron amplitudes for a ground state target [see, Eq. (84)], has a somewhat strange appearance. The more usual description of inelastic scattering as in Eq. (82) has the electron leaving the target with the target in its excited state. It is just this necessity of describing the system in terms of its excited state wavefunction that the Bethe-Salpeter amplitude equation does away with. That is, the excited state wavefunction is considered to have been

326

Gy. Csanak, H. S. Taylor, and R. Yaris

created adiabatically from an excitation of the ground state system, this process will be described in detail below for a somewhat different case. If one recalls the derivation of the Bethe-Salpeter amplitude equation in Section 111, where one starts with an equation for R in terms of an Xr? product, one sees that this is exactly analogous to the process of obtaining excited state information such as oscillator strengths by looking at the poles and residues of a ground state property, the frequency dependent polarizability. There, one obtains inelastic photon scattering information, the absorption oscillator strength, by using the polarizability, whose defining equation involves only ground state quantities. To complete the scattering picture, formulas for x,,are needed when onis greater than the ionization potential. Here x, is a continuum function and R, has poles (or rather a cut) at w , , so that the bound state Bethe-Salpeter equation no longer holds. Special continuum x,,are needed if exact R or G2 are to be calculated and the discrete basis approximation not made. A second reason for calculating the continuum xn is that it obviously contains information about the ionization continuum of the target. From this information it should be possible to obtain information about scattering from the ion. To do this it is necessary to study the effect of adiabatic decoupling on the electrons of the target atom itself. The ground state and some specific states of the ion can be described in the independent particle model as a hole in the grour,d state of the atom. where I@,) is the ground state in this approximation and m is a quantum number referring to a ground state occupied orbital. In the following the Hartree-Fock independent particle model is used and a", creates a HartreeFock hole. Also the field operator is

The Heisenberg operator is expressed as $(r

=

1a",(t>v,HF(r0 n

where qfF(r) and cpnHF(r t ) are the Hartree-Fock orbitals for the time independent and a freely propagating time dependent case, respectively. a",(t) is neither a pure Heisenberg nor interaction operator, but is a mixed representation. It is defined by the equation for $(r) and $(r t) which are related by the usual Heisenberg transform. The tilde on the a", simply distinguishes it from that for free (neutral targets) or Coulomb (charged target) waves. Now starting from the uncoupled state, by adiabatically turning on of the correlation potential, an exact ion state can be reached:

327

GREEN’S FUNCTIONS

I Y,)

= i$

IY o )

where

iz

=

Iim i,(t). i+-m

This I Y,) will be used as an ionic target state in the formalism used previously for electron-atom scattering. In this case, the scattering matrix assumes the form: SpmI,qm2=

Iyirnz)

(‘Yirnt

=(~mlIa~taJinIYm,)

lim ai (t1)crn,(t2)> +m

=

fti,--rn i,’f,‘+

=

Jdr, dr, dr,’ dr,’ G(rl’rl’, r2t2 ; r , t , , r2’tZ’)

lim tlt2-

f,“”

-m

+m

x cp,*(r,’ t 1‘)v3rz’t2’)cpq(r,

t i )cpZ *:(‘

f2)

(86)

where qp(rt), cp,(rt) are Coulomb waves whereas cpzy(rt), cpiy(rt) are HartreeFock one-particle stationary states (m,, m2 are Hartree-Fock quantum numbers). The change here from the electron-atom scattering case is that the indices p and q refer to Coulomb wavenumbers and cpp(rt) and cp,(rt) are Coulomb functions. The use of Coulomb functions is conditioned by the long range electron-ion interaction. The scattering matrix formula can be simplified by defining the Bethe-Salpeter amplitude referring to the state I ‘Pi,): xbil(1, 2) = (YOI T[$(l>$t(2>IIYim,) =

=

lim (Yo I T[IC/(~)$t(2)l~”,t(t)a:(t’)I Yo) -

f , i’-+m

lim f , f‘+

=

-m

J” dr dr’ (~[1~/(1)~+(2)$(rt)$t(r’t’)l) cp,(r’t’)cpH,f*(rt)

lim (- 1) /dr dr’ G2(l, rt; 2, r’t’)cp,(r’t’)cp:f*(rt)

,+-“. .-

(87)

f’+&+--m

Using this expression in the scattering matrix formula, it becomes

From the Bethe-Salpeter equation, an inhomogeneous (nonlinear) integral equation can be derived for the xbLi( I , 2 ) Bethe-Salpeter amplitude.

Cy. Csanak, H. S. Taylor, and R. Yaris

328

Inserting (65) and (68) into (88) and using the following identities lim

j d r dr’G(1, l’)qq(l’)q~F*(l)

f--m

-

I,+&- m

lim

=

j d r dr‘ ($(l)t,bt(l’))qq(l’)qm*(l)

f--m f’+&-+

=

-m

lim ( a m ( t ) a q t (t ’ ) )

=

(Y~IY~;))

=0

i+--00 f’+E+ -m

and

the following equation is obtained for the Bethe-Salpeter amplitude $)(1,

2) =f,“)( 1)gm*(2)+ Jd3 d3‘ d4 d4‘ R,(1423)3(34’43’)$,)(3’,

4’) (89)

wherefq(l) is the solution of the Dyson equation with incoming Coulomb wave of wave vector q boundary condition and gm(2) is the solution of the Dyson equation with Hartree-Fock boundary conditions

+ id3d4GHF(23 3)Ccorr(3, 4)9n(4) gn(2) = qj;’F(2)

(90)

where GH,(2, 3) is the Hartree-Fock one-particle Green’s function and = C - ZHF where CH, is the Hartree-Fock potential. This equation follows from the following form of the Dyson equation

C,,,,

= GHF

Now, the substitution of xb;!frorn given by Equation (88) gives %ml,qmz

= i llim ,-m

+ GHF ‘corr

G.

Eq. (89) to the scattering matrix expression

Jdr,’ q p * ( r l ’ t l ’ ) ~ ~ + ~ ( r , ’ flim l l ) idr2’q;F(ri fZ’+

ti )

m

x Iiin Jdr2r q;Y(r2’t2’)gm*(r2’t2’)i d 3 d3‘ d4 d4Y::’*(3) iz’+m

x .4,,(4)q34’43’)X;;;(3’,

4’).

(91)

329

GREEN’S FUNCTIONS

This formula can be simplified noticing that

The same expression occurred previously in the electron-atom scattering formula, however, p and p’ here refer to Coulombic functions. Similar expressions can be defined for the hole scattering ” “

lim

t’+m

s

dr’ cp,HFjr’t‘)g,.(r‘r’)

*

= S,,,,

.

(93)

Finally the following formula is obtained:

where

TpTm’, qm2 = j d 3 d3’ d4 d4‘”’’*(3>gm.(4)~(34’43’)~&~(3’, 4’).

(95)

The first term in Equation (94) describes the independent scattering of the particle and the hole, the second term expresses the interference of these scatterings. The energy-dependent form of the T matrix in (94) is

It is this form of the equation for the electron-ion T matrix which should prove most useful when combined with the perturbation expansion for E , as given in Section VI. The formula for the scattering matrix equation (94) can be made more symmetric. The equation for the Dyson orbital fi+)(l): fk(+)(l) = q k ( 1 ) f J”Go(1, 2)x(2, 3)&” (3) d 2 d3

(97)

can be solved in two steps. In the first step we construct the Hartree-Fock orbital q t F ( lwith ) outgoing Coulomb wave boundary condition :

(PFF(l)

%( 1)

1

Go( 1 , ~ ) C H F 3)rPkHF(3) (~, d2 d 3

and then solve the Dyson equation with C,,,, inhomogeneous term

= C - EHF using (pFF

(98) as the

Gy. Csanak, H . S. Taylor, and R . Yaris

330

S,,

=

lim tl’-m

.I‘dr,’ cP,*(rl’tl’)q~F(rl’tl’)+ lim 2 [dr,’ cp,HF(r,‘t,‘)

x ‘Pp*“l’t,’)

tl‘-*m r

1

~IHF*(2)~:,,,,(2,3)f;”(3> d 2 d3.

(100)

In the eigenphase representation the Hartree-Fock scattering matrix is diagonal, therefore

and consequently

where

and

s,, = 6,, + T,, .

The substitution of the form of Sp, into (94) gives

V. Nonperturbative Approximation Method’ Now that the formal equations are developed an “equation decoupling procedure” is needed to truncate any of the three equivalent sets of coupled equations of Section 111. This approximation will also simplify the scattering formulae involving Z. The idea behind the approximation is to guess a l 8 SeeDyson(1949); Feynman (1951); Matsubara (1955); Martin and Schwinger (1959); Baym and Kadanoff (1961); Schneider P t ul. (1970); Csanak ef a/. (1971),

331

GREEN’S FUNCTIONS

functional form for C (the reason that the G I , G, , G , , . . . hierarchy was replaced by the two other equivalent ones which stressed functional derivatives, was to make C more visible). A “functional form” means a specified dependance of C on G I , for unspecified GI . Since this guess defines the model and will be used to generate the physics of the problem, it must be physically well motivated. Once this is done the hierarchy can be closed in any one of an infinite number of ways each of which gives a higher order self-consistent set of equations, viz. : If C % CA(G,)is put into the Dyson equation, the system is truncated to one closed equation for GI requiring no other information. Since this equation is nonlinear it will be solved iteratively (or self-consistently). If, on the other hand, C z C”(G,) is used in 6C x 6CA(G,) with 6°C -+ 0, for n 2 2, a set of two coupled equations are obtained, namely the Dyson equation for G, in terms of X;a formula (not “eaqution”) for C in terms of G, , V , and R , and a closed equation for R in terms of GI . There two coupled equations must be solved iteratively and hence self-consistently. If the approximation is made for higher functional derivatives, i.e., SnC/8U”, n = 0, 1 is unspecified but a2C % 6’CA(G,) with S”Cj6U” = 0, n = 3,4, . . . larger and higher self-consistent sets are defined. Hence, it is seen how higher and different types of self-consistencies can be developed. It is hoped (and moreover physically reasonable) that higher self-consistencies are better, since the model C A is being used only to calculate smaller and smaller variations of C rather than C itself. This will be seen to be true explicitly in the context of perturbation theory in Section VI. The problem now is to choose the form o f C A ( G , ) ;from the discussion above, especially with reference to the single equation truncation, the choice of CHF(G1) as the Hartree-Fock functional form is suggested. In Section VI this will be shown to be the first-order perturbational approximation to C(G). Now x ~F ( 1 ,1’; GI)

= &(l,

1’)

= -i6(1

- 1’) /d2V(1

+ iV(1 - l‘)G,(l, 1”)

- 2)G1(2, 2’)

(106)

because if G, is taken as GHF (i.e., the Dyson orbitals and energies in the spectral representation of G, are taken as Hartree-Fock orbitals and energies) CHF(G1) becomes the Hartree-Fock potential. Note that CHF(G1)is generally not the Hartree-Fock potential since G, # GIHF. If this approximation is made in the Dyson equation, the total theory reduces to the Hartree-Fock model which is unsatisfactory for anything except the ground state density and expectation values of one particle operators. The next step, and the contribution by Schneider et al., (1970), is to do the second step, Le., leave X alone but use

6C + 6C”, .

( 107)

332

Gy. Csanak, H . S. Taylor, and R . Yaris

With this, using Equation (67), the approximate form of E(343‘4’) =

6C(3,3’) 6G1(4’, 4)

N

is obtained

dCHF(3,3’) = %,(343’4‘) 6G1(4‘, 4)

= i6(3 - 4‘)6(3’+ - 4)1/(3

- id(3 - 3’)6(4

- 3’)

- 4’+)1/(3 - 4’).

(108)

It will be shown in Section VI that Eq. (108) is the first-order term in the perturbation expansion of Z as a function of G. Now inserting (108) into (73) gives

X,,(l, 1’)

=

- i ~ d 2 d 2 ’ R 0 ( 1 2 ’ 1 ’ 2 ) V( 22‘)X,,(2’2’+)

+ i s d 2 d2’R0(12’1‘2)1/(2- 2‘)X,,(22’+).

(109)

Using the 6 ( t , - t,’) hidden in the definition of V ( 2 - 2’) shows that on the right-hand side of (109) the first and second terms contain, respectively, X,,(r2’,r,’; t 2 , -0) and X,,(r,, r,’; t Z , -0). Hence, in this approximation X,,(l, 1 ’ ) is obtained from a knowledge of X,,(r,, r,’; t 2 , -0) and G I . X,,(r,, r,’; t 2 , -0) is obtained from the closed equation obtained from (109) by choosing t,’ = t , E , so that { t l , t,’} + { t ’ = t , = tl’, z1 = -0}

+

X,,(rl, r,’; t ’ , -0)

= -i

s

dr, dr,’ dt2Ro(r,t‘,r, t 2 , rl,t’+,r, t 2 )

x V( I rz - r2‘ I >Xn(r;,r,’, tZ, - 0 )

+ -i

s

dr, dr,’ dt2Ro(r,t’,r,, t Z ,r , , t”, r,

x V ( I r2 - r2’ I )Jf&,>

r,’,

tZ)

tZ, - 0 )

(1 10)

where from Eqs. ( ~ O C ) , (34), and (35)

- e - i w ~ f ‘ ( Y o N ~ $ + ( r l ’ ) $ ( r ~ ) ~a,, ~ ~>) ,o a, < 0 - e - i o J ‘ ( y f l N l $t(r 1’14 v r I ) l ~ : h - -e-iwnf’Xfl(rl’, rl). (111)

X,,(rlrr,’, t‘, - 0 ) =

Thus Eq. (110) is now

X,,(r,‘, r,)eCion” = - i j d r , dr,’ dtZ{G,(r,t‘, r2 t2)Gl(r, t 2 , rl’t‘) x V( I r2

- r2’ I )Xn(r,’, r,’)

- Gl(rlt’, r, t2)Gl(r,’t2, r,’f’)V( J r , - r,’I)

x Xn(rZ’,r2)}e-ioJ2,

(112)

333

GREEN’S FUNCTIONS

Multiplying both sides of (112) by e’”““; replacing the four G,’s by their Fourier integral representations, and integrating over t2 to give &functions, which are then in turn integrated over, gives 1

Xn(rl’,r l ) = - Idr, dr2’ jde{G(rl, r,; &)G(r2’, rl’, E - 0,) 27c x x

V Ir2 - r2’ I )Xn(r2’,r2) - G(rl, r2, E)G(r2,

uI

r2

rl’, E - 0,)

- r2’ I )Xn(r2’,r2”.

In Appendix B the integrals over the definitions made that

E

(113)

are carried out. If these results are used

the resulting equation is

+ cw,, 99’

- N,)cp,(rl)cp;(r,’) Eq - Eq‘ - 0,

J dr3 dr4 (Pq*(r3)1/( Ir3 -

I

r 4 )‘Pqf(r4)Xn(r4

9

r3).

(115)

The ( N , - N,.) factor assures that only hole-particle X’s are solved for (Schneider et al., 1970). A closed set of equations has finally been achieved. Equation (115) can be solved by standard non-Hermitian matrix diagonalization techniques. If the Dyson orbitals and energies are known the knowledge of the Xn(rl,r2) and on values enables the complete construction of only a special case of the *general R(121’2’); namely, from (40d) the hole particle-Rhp(rl,r 2 , rl’, r2’; O f , O + ) ; i.e.,

R(121’+2’+)= Rhp(rlr2rl’r2’;O+o+s)

1

= -1

i

o

,

Xn(r1’9 ~

rl)-fn(r2

E~ - 0,

7

r2’)

sgn (0,)

+ irl sgn (0,)

(116)

334

Gy. Csanak, H . S. Taylor, and R. Yaris

This and G, are all that is needed to calculate C in this approximation consistent with (107). This result is then substituted into Eq. (71), which result is in turn substituted into (67) to give [recall U(2,2+) = U(2)] C(l, 1‘) = ZHF(l,1‘) - j d 2 d3V(1 - 2)G(1, 3)7/(1’ - 3)R(321’’2+)

+ G(l, 1’) I d 2 d3V(1 - 2)V(1’ - 3)R(32 3’2’).

(117)

Noting the d ( t ; - f,)in V(1’ - 3) indicates that the part of the general R needed in (1 17) is R(t,tzt,’tz’) = R(O’, O’, t‘ - t Z )which is the Fouriertransform of (116). In Fourier space equation (1 16) is

ZA(rl, rl’, z ) = &,(r,,

i

r,’) - - j d r , dr, dz’V(r, - r,) 2n

x Rhp(r, r, r, r 2 , O+O’z’)V(r, - r,’)G(r,, rl’, z

2n

- z’)

dr, dr, dz‘V(r, - r,)RA(r3r2rl‘r,, 0’0’~‘)

x V(r3 - rl’)G(rl, r 3 , z

- z’).

(118)

The final set of self-consistent approximate equations are, then, (1) Eq. (5 1) -the Dyson equation; (2) Eq. (1 15)-called the generalized R.P.A. equation; and (3) the formula in Eq. (1 18) after (1 15) is substituted in. The method of solution is then to start in Eq. (51) with ZA C,, and solve for the HartreeFock orbitals and energies which are the cp, and E, . These are then introduced into (115) which now becomes the R.P.A. equation (since the cp’s are cpHF’s).The X’s and 0,’s are found and combined with the cp,’~ and E,’S to form a new C. The procedure is repeated until selfconsistency is achieved. On higher iterates Eq. (115) is no longer the R.P.A. equation but the G.R.P.A. equation. Equation (115) need be solved only for w, > 0 as seen from (1 11). Once convergence is reached and G is known, Eq. (19) gives E,, , and Eq. (23) gives the one-particle density from which one-particle averages and natural orbitals can be found. As previously noted the w, are the excitation energies and the X , give the generalized oscillator strengths. Of course RhPis [compare Eqs. (64) and 1161 the linear response (not the general response) from a generalization of the R.P.A. or coupled time-dependent Hartree-Fock (which are just different names for the same equation) which is consistent with the one-particle orbital equation. The converged CA can be put in (51) with E = q2/2 set to the desired scattering energy to give an equation for thef;’) of Eq. (76). These scattering orbitals, with the boundary conditions given in Eq. (79), can be solved for by any, and all, methods used for solving potential scattering problems (Reinhardt

GREEN’S FUNCTIONS

335

and Szabo, 1970). When solving for fi’) the finite basis set used in solving Equations (51) and (115) is no longer used. Equation (80b) with C, substituted for C gives the approximate T matrix for elastic scattering. An approximate expression for inelastic scattering is obtained by substituting Eq. (108) into Eq. (83) which after Fourier transforming gives

The approximate equation for inelastic scattering (1 19) is expected to be less accurate than the other approximate equations given in this section. The reason for this is that the truncation approximation (107) is closer in the hierarchy to the desired quantity than in the case of quantities depending on GI. The philosophy of this approximation scheme is that a first-order approximation to E when integrated over gives a moderately good R (or X , ) , which when integrated over gives a good G I . Tn the case of inelastic scattering, however, one is not integrating over R (thus averaging out its deficiencies), but using the X , directly (and hence exposing them in their nakedness), so it would not be as surprising to see the approximation show deficiencies. If that proves to be the case, it will be necessary to either use higher order selfconsistent sets of coupled equations (e.g., S2C = S2C=,,) or use higher order truncations of the Bethe-Salpeter equation as discussed in Section VI. Clearly a generalization of Hartree-Fock theory has been achieved, with the advantage of overall self-consistency. The result is to give a theory that calculates with one basic approximation Eo , p(l’, l), s o k , &’, Sop, nq, X,, , o,, and the linear response. Hopefully, if the scheme is performed on a finite basis set, the calculations will be tractable. In solving these equations, the nonHermitian matrix diagonalization, the energy dependence of C,and the linear dependence of the set of Dyson orbitals, are all new but hopefully tractable problems.” The discrete basis set should not be any restriction since the cp’s and RhP represent phenomena that are localized in a small region of space. The linear dependance of the 31’s means that the set is overcomplete hence new roots of Eq. (1 16) may show up at w. = 0. Physically it is felt that these should be ignored and it is pleasing that the equation for Rhphas w. = 0 omitted from the summation. The word “may” is stressed since in Eq. (116) X is seen to be expanded in the product space qpx y h . So while the set q,,,9)h is overcomplete, and of course the product of two overcomplete sets in the product space, the partial, hole-particle, product is not necessarily overcomplete in the product space. In any case, such new roots, if they occur, will not cause any difficulty.

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Gy. Csanak, H. S. Taylor, and R . Yavis

The successful computation, using such bases, of orbitals and frequency dependent moments for atoms and molecules is well documented. The topic of the use of a discrete basis also brings up a point of caution. Equation (1 16) comes from Eq. (73) and is valid for bound-type functions only, i.e., functions that go to zero as rl or r2 go to infinity. In a finite basis this is true of all functions obtained. If a method of solution is used which includes continuum functions, the continuum hole-particle amplitude equation, (89), will have to be used for the continuum amplitudes. After Eq. (107) is substituted in and the usual Fourier transform (the changes are essentially the same as for the bound state problem) taken, an equation is derived that is exactly as Eq. (116) except that n - t p , m and an inhomogeneous term appears on the right-hand side which is simply the product of outgoing incoming wave particle and hole Dyson states with particle index p1 and standing hole index m, respectively. For the purpose of calculating R , the normalization of the Dyson orbitals does not matter since they differ by a phase factor which in turn causes the X’s to have different phase factors which cancels in the product Xx” appearing in (1 17). A convenient matrix form of X is easily derivable even though the q ’ s are linearly dependent. Simply noting that Eq. (116) can be written as

i i

with

Substituting Eq. (120) into (121) gives the desired matrix form

( 122a)

where

(ab( V(cd)

= Jdr

dr’cp,*(r)cp,*(r’)V(r - r’)qc(r)cpd(r’)

(122b)

GREEN’S FUNCTIONS

337

Similarly the matrix form of RhPis

x RhP(0 0 z >qp’ 4‘P +

+

(123a)

with (1 23b)

Now that the basic approximate equations are exposed, a further discussion of the physics of the approximation is in order. Up to now the approximation has been introduced by appealing to the attractiveness of having higher order self-consistent theories which generalize the Hartree-Fock and the coupled time-dependent Hartree-Fock approximations. Also the point has been stressed that even if C z C,, is not a very good approximation, 6C z 6C,, may be a much less damaging one. Further justification for expecting that the approximation will give good results comes from the experience that the coupled time-dependent Hartree-Fock equation gives in actual calculation good frequency dependent responses (Dalgarno et al., 1966), that the R.P.A. gives reasonable excitation energies (Dunning and McKoy, 1967) and that schemes which (i) use the R.P.A., (ii) solve for the ground state R.P.A. wave function, (iii) use the latter in place of i40) in (97) to solve for new orbitals, and (iv) use the orbitals in Eq. (1 IS) to get new w, and X , get generally improved agreement with experiment (Rowe, 1968; Gutfreund and Little, 1969) with only one iteration. Moreover, it has been shown that the types of correlation effects required in R, i.e., hole-particle effects (Kelly, 1968; Karplus and Caves, 1969) are all included in our iterated R. Note that the final linear response here is “nonlinear” in the sense that the zeroth order model has been greatly refined upon iteration. Perhaps the most persuasive argument for the approximation comes from the physical model of elastic scattering that is implied. Here one starts with the Hartree-Fock virtual continuum orbital as the scattering orbital. This is called the static exchange approximation. The target electrons are also taken as in Hartree-Fock orbitals with Hartree-Fock exclusion correlations. The scattered or “test electron in the virtual orbital now causes the target to respond, this response is calculated in the coupled time dependent HartreeFock (or R.P.A.) approximation. The response, which in this approximation is much better than perturbation theory applied to the Hartree-Fock Hamiltonian and which contains correlation and exchange effects and depends on the energy and position of the test particle, is then coupled with the HartreeFock orbitals (118) which “drive” the response to give a new effective potential. This potential is then used to calculate new target and scattering ”

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Gy. Csanak, H. S. Taylor, and R. Yaris

orbitals. Everything is then iterated until the scattering orbital, the target orbital, and the response are all self-consistent. Since the electrons are all indistinguishable, the correlations in the target are as well represented as the ones between the test particle and the target; this must be reasonably well done since as said above, even the R.P.A. gives good responses. To obtain the generalized R.P.A. equation for an electron scattering from an ion, in terms of quantities calculated for the neutral particle, (108) is substituted into (95) which after integrating over Lhe delta functions gives C.R.P.A. T q l m l ; qlrnZ=

i {dl d2fX:+)(l)g:,(2)V(l

- ~ ) x L I A ~2+) (~,

- i { d l d 2 f ~ , ' + ) ( l ) g ~ , ( l ) V-( l2)X&(2,2+).

(124)

This can further be approximated by its first iterate the RPA. Then C is replaced by CFIFin the generalized R.P.A. equation (1 15),fk+(l) by (pk+(l)HF and gm(2)by (pm(2)HF.These substitutions reduce Eq. (124) to

where the X's are the first iterate to Eq. (1 15). Also making the above substitution in (105) reduces the and s" matrices to their &function terms only. For the elastic R.P.A. case they become the unit matrix and the summations over p' and m' drop out. For the inelastic case the first term in (105) vanishes and in the second term there is only one nonzero term in the summation. We now remind the reader that the inelastic cross-section formula refer only to states that can be achieved by adiabatic coupling from a particle-hole independent particle state. For helium, there is only one particle-hole state, i.e., the lsq state and therefore for electron-helium scattering this formalism is not able to describe inelastic process at all. In Be one can think of two holeparticle states, i.e., (Is)'2sq and (ls)(2s)*q'; as such one can study elastic scattering on either of these states or inelastic scattering between them (though this inelastic process is not very interesting physically). We note also that even in the case where the formula can be applied for an inelastic process, the approximation is probably poor, because the R.P.A. is a refinement of the H F and the HF is not able to describe inelastic processes. The formula so obtained is well known as the R.P.A. elastic scattering formula (Dietrich and Hara, 1968; Lemmer and Veneroni, 1968). Dalgarno and Victor (1966) and Jamieson (1967) have used this equation by solving for ,'A using the R.P.A. equation for X with G taken as G,, , and with Hartree-Fock hole-particle

s

GREEN’S FUNCTIONS

339

boundary conditions. They solved this R.P.A. scattering elastic formula for the p-wave phase shift for the elastic scattering of an electron off a helium positive ion. Their agreement with the close coupled results of Burke and McVicar (1965) was quite good. Although obvious, it should be mentioned that in molecular systems, noting the facts that (1) for each geometry, excitation energies are calculated self-consistently and directly, and (2) Eq. (19) can be used to get the consistent absolute energy of the ground state (hopefully the errors i n this formula will not be very sensitive to the geometry changes), implies ,that all potential surfaces of the molecule can be calculated self-consistently in one calculation.

VI. Perturbation Methods2’ A. Go PERTURBATION METHOD One of the most straightforward ways (in principle) of solving for the one-particle Green’s function is as a perturbation expansion in terms of a “bare” or “free particle” (or unperturbed) Green’s function Go and the residual interaction potential V. Go satisfies the equation of motion (47a) where h(1) is some suitable single particle Hamiltonian (e.g., free particle, Hartree, or Hartree-Fock) leaving over an interaction potential V (1 - 2). While there are many ways to develop a perturbation series, we shall use the Dyson equation (48”’)and substitute the self-energy expressed in terms of the functional derivative of G, obtained from substituting (56b) into (47b’),

yielding

G(1, 1’) = GO(l,1‘) - i I d 2 d2’G0(1, 2)7.’(2- 2’) x

[G0(2’, 2’+)

]

- - G(2, 1’).

6 U(2’)

To develop a perturbation expansion, one just iterates on Eq. (127), that is, one first substitutes G0(2, 1’) for G(2, 1’) on the right-hand side and evaluates the resulting expression * O See Matsubara (1955), Martin and Schwinger (1959), Falkoff (1962), Roman (1965), March et al. (1967), Kirzhnits (1967), Mattuck (1967).

340

Cy. Csanak, H . S. Taylor, and R. Yaris G(1, 1’) = GO(1, 1’) - i J d 2 d2’GO(1, 2)1.’(2 - 2’)

]

G0(2’, 2’+) - -G0(2, 1’) + higher order terms, 6 U(2)

(128)

obtaining the lowest order corrections to Go. If one wishes to go further, one substitutes the lowest order terms back into (127) and repeats the process. Clearly in order to solve Eq. (128) we must be able to evaluate terms such as [6/SU(2)]Go(1, 1’). Varying Sd2[G0(1, 2)]-’Go(2, 1’) = 6(1 - 1’) yields 6G0(1, 1‘) 6 U(2)

3‘)I-l G0(3‘, 1‘) =-jd 3 d3’G0( 1, 3) [6G0(3, 6 U(2)

where [Go]-’ satisfies [in the presence of the external potential U(2)] [@(I, 1’)l-I

=

[i(d/dt,) - h(1) - U(1)]6(1 - 1‘)

hence dGO(1, 1‘)

=

6 U(2)

CO(l,2)G0(2, 1’).

This equation is the basic result necessary to develop a perturbation expansion. One should note that the right-hand side of Eq. (129) is an ordinary product and not an operator product. Substituting (129) into (128) yields the Green’s function correct to first order in the interaction potential G(1, 1’) = CO(l, 1‘) - i S d 2 d2’G0(1, 2)1/(2 - 2’) x [G0(2’, 2‘+)G0(2, 1’) - C0(2, 2’)C0(2’, l’)]

+ O(2).

(130) It is useful at this point to look at a diagrammatic interpretation of Eq. ( 130). G(I,I’)=

7 fl I

+L+a I’

+

0 (2)

I

(130)

GREEN’S FUNCTIONS

341

In (130‘) the diagrams correspond one to one with the three terms on the righthand side of Eq. (130). However, note that there is, as yet, no indication of the signs of the terms (we shall discuss how to determine the sign of a diagram presently). To interpret the diagrams one reads a straight line as a Go with the space time points at the two ends of the line as its arguments. A wavy line is a V interaction between the space time points at its two ends (remember that V has a &function of its time arguments denoting an instantaneous interaction). All internal points in parentheses are to be integrated over. Recall that since in the Green’s function both time orderings are present they represent both particle and hole propagators and all time orderings consistent with instantaneous interactions are implied. One can easily go over to the more common Goldstone diagrams (Goldstone, 1957) by taking account of the above orderings

(131a)

fi

=

+

(131c)

Hence (131a) depicts the propagation of a “free” particle or “free” hole depending on the time ordering; (131b) is the interaction of a particle (hole) with a passive particle, and (13 Ic) is the exchange term of (13 1b). Clearly these diagrams describe the Hartree-Fock interaction. By looking at Eq. (130) we see that (131b) and (131c) are multiplied by ( i ) and (131b) comes in with a negative sign. The general rule is to multiply each diagram by (i)”, where n is the number of V interactions (wavy lines, i.e., the order of perturbation theory) in the diagram and also multiply each diagram by ( - I)‘ where I is the number of closed loops composed solely of Go lines [notice (131b) has I = 1, (131c) has I = 01. To proceed to second order one merely replaces the G on the right-hand

342

Gy. Csanak, H . S. Taylor, and R . Yaris

side of the Dyson equation (127) by the first-order G, Eq. (130), and again uses (129) to evaluate the functional derivatives. Rather than write out the analytic forms for the second-order corrections to G we shall draw the diagrams

.

+ I'

(2)

.

(C)

I

+

I'

.

. (2) (dl

' I

(d

(3)*

(3)

+ >I'-&+:

1, (2) (1)

(2') (j)

I

(132)

As an example of going from the diagram to the analytic form of a perturbation term we shall write the formula for term (i). Reading from right to left on the diagram there is a GO(l,3); a V(3-3'); three Go lines: G0(3, 2 ) ; G0(3', 2'), and G0(2', 3'); then there is a V(2' - 2); followed by a G0(2, 1'). The integration is over space time points 2, 2', 3, and 3'. Since there are two V lines there is a factor ( i ) 2 , and one closed loop there is a factor (- 1). Thus,

GREEN’S FUNCTIONS

(132i) = (-i)’

343

j d 2 d2’ d3 d3’G0(1, 3)V(3 - 3’)G0(3,2)

x G0(3’, 2’)G0(2’, 3’)V(2 - 2’)G0(2, 1’).

The only point not illustrated in the above example is when a Go line begins and ends on the same vertex, such as in (132h), this is written as G0(3’, 3”). Thus the rules for obtaining a perturbation expansion of G in terms of Go and V can be summarized as follows: (1) C is the sum of all topologically distinct connected diagrams through the desired order in the interaction with two external solid lines. (Diagrams which differ only in their time ordering are not distinct, e.g., interchanging (2)4-+(3) and (2’)0(3’) in (131e) does not lead to a different diagram.) A diagram is connected if every vertex is connected to at least one other vertex by at least one Go or V line (i.e., it is not composed of disconnected pieces). (2) Interpret each straight line as a Go function of its end points reading from right to left. If there is any ambiguity in time interpret the furthest right point as having a + on it. (3) Interpret the wavy lines as V functions of its end points. (4) Integrate over the space time arguments of all internal points. (5) Give each diagram a multiplicative factor of (Q“,where n is the number of Vlines. (6) Give each diagram a multiplicative factor of (--l)’, where 1 is the number of closed loops made up solely of Go lines. In order to transform the perturbation series to a sum over states one first goes over to an energy dependent representation by introducing the time Fourier transformed Go’s in their spectral expansion form, Eq. (53b). One then integrates over the E’S instead of the times taking care to include the &functions coming from the V’s and the infinitesimal positive and negative imaginary contributions to the energy denominators coming from the particle and hole terms, respectively. The major difficulties with the Go perturbation method are concerned with rapidity of convergence. This becomes most evident from the fact that while G has poles off the real axis (on the nonphysical sheets) Go does not. Hence, one is building up the damping terms in G by a series expansion which does not converge rapidly for large times.

B. C PERTURBATION METHOD If we know the self-energy operator, then we can solve the non-Hermitian eigenvalue equation for G as described in Section 111. Hence as an alternative to obtaining C directly in a perturbation series one can develop a perturbation series for C and then solve for C. This can be done by resumming the G

Gy. Csanak, H . S. Taylor, and R. Yaris

344

perturbation series, or by again using the technique of functional differentiation. The procedure is much the same as that used in the preceding section. Starting with Eq. (61) for C as a function of V , G, and 6C/6U the lowest order terms in C are obtained by just replacing the G’s in the first two terms on the right-hand side of (61) by Go’s yielding c(1, 1’) = -i6(1 - 1’) jdZV(1 - 2)G0(2, 2’)

+ iV(1 - l‘)Go(l, 1’) + O(2)

(133)

or diagrammatically

(1 33’)

Clearly, this procedure can be continued by putting higher order terms in the perturbation expansion for G into the first two terms on the right-hand side of Eq. (61). There are also terms in the expansion for C which come from the third term on the right-hand side of (61). We shall evaluate the lowest order terms (second) coming from this third term by substituting Go for G and (133) for C,yielding SC(2‘ 1’) i d2 d2’V(1 - 2)G(l, 2’) A = Z(3)(1, 1’) 6 U(2)

s

=i

6

I d 2 d2’V(1 - 2)G0(1, 2’) 6 U(2)

x

[ -i6(2’

- 1’) sd3V(2’ - 3)G0(3, 3’)

+ W(2’ - 1’)G0(2’, 171 + O(3).

Using Eq. (129) for 6G0/6U and integrating the first term over 2‘ C(3)(l, 1’) = - (i)’ l d 2 d3V(1 - 2)G0(1, l’)V(l’

-

3)G0(3, 2)G0(2, 3 + )

+ i2 i d 2 d2’V(1 - 2)G0(1, 2’)V(2’ - 1’)G0(2’, 2)G0(2, 1‘) + O(3)

(134)

GREEN’S FUNCTIONS

345

or diagrammatically

q3)(1.1’)

=(3)

I’

I

(134‘) + 0 (3). One should notice that all of the diagrammatic rules given in Section V1,A for the perturbation expansion of G carry over unchanged for the perturbation expansion of C except for rule (1) which is modified as follows: (1’) C is the sum of all topologically distinct strongly connected diagrams through the desired order in the interaction with no external lines (however, there are two vertices which are not to be integrated over and have the space time designations of the two arguments of C). A connected diagram is strongly connected if it does not fall into two disconnected pieces when any single straight line, i.e., Green’s function line, is cut. For example, 132c,d,f,g,i,j are strongly connected second-order diagrams, the rest of (1 32) are not. The reason for the changes in rule ( I ) can be easily seen by expressing the Dyson equation (48”’) G = Go + GOCG

in diagrammatic form as

+-G=

- = + +

which on iteration becomes

(135)

y=--@-+ + t

From (136) we see that all weakly connected diagrams in G are composed of strongly connected C pieces connected by a single Go line. Hence, if we had included weakly connected diagrams in C we would have overcounted them when the equation of motion is solved for the resulting G. Hence, the C perturbation method is to solve for the self-energy perturbationally to the desired order and then to solve the equation of motion for the resulting Green’s function. While one has to do more work than in the Green’s function perturbation method, in that one must also solve the nonHermitian eigenvalue problem, one gets considerably more out of the method.

346

Gy. Csanak, H . S. Taylor, and R . Yaris

It is equivalent to iterating the diagrams considered in C to all orders in G [see (136)l. Naturally this implies that a smaller number of perturbational terms must actually be solved for in order to obtain good answers. Of course one can never show in any partially summed method that one is not losing some cancellation with diagrams that are not being considered. However, one hopes that by considering the most important diagrams in C (perhaps one should consider this as a model” self-interaction) and solving its equation of motion exactly will lead to useful results. Only future theoretical experimentation will tell. It is important to realize that it is the ability to solve the equation of motion for the Green’s function on a finite basis set for atoms and molecules that makes the C perturbation expansion a practical calculational method. “

C. RENORMALIZED C PERTURBATION METHOD One can also develop a renormalized or self-consistent perturbation expansion for C in terms of the interaction V and the full Green’s function G (not the “free particle” Go). One can obtain the expansion analytically using much the same functional differentiation method as that employed in the previous two sections; however, it is so easy to see diagrammatically that we shall proceed in that fashion. The diagrams in the renormalized perturbation expansion are again interpreted as in Section VI,A except that the straight lines will now denote the full one-particle Green’s function, and rule (1) which gives the class of diagrams t o be summed must again be modified as follows: (1”) C is the sum of all topologically distinct strongly connected basic diagrams with the desired number of interaction lines, full Green’s function lines, and no external lines (there are two free vertices). A diagram is a basic diagram if it cannot be constructed by inserting other diagrams into the Green’s function lines. For example,

3’

3

(137a)

and (137b)

GREEN’S FUNCTIONS

347

are basic, but (138a)

and

(138b)

are not. [Note that (137b and c) are the only second-order basic diagrams]. The reason for excluding insertion diagrams is that when the full Green’s function line is expanded as in (1 36) all insertion diagrams are included in the G line. Hence, putting them into the class of renormalized C diagrams will overcount them. Thus, there is a practical scheme for self-consistently solving for G using the renormalized C perturbation expansion : (i) Take a set of strongly connected basic C diagrams, correct to the desired order, and solve for C interpreting the Green’s function lines as “bare ” Go lines. (ii) Solve the equation of motion for the resulting G, which will be obtained in a spectral expansion form. (iii) Repeat step (i) now using the result of step (ii) for the Green’s functions. (iv) Repeat step (ii) now using the result of step (iii) for C, and repeat until self-consistency is achieved. Again we see that in the self-consistent perturbation method one must do more work, in the sense of repeatedly solving the same set of equations until self-consistency is achieved. However, again, one obtains more results for more work in that not only iterates of the basic diagrams considered in C are obtained to all orders in G, but one also obtains all diagrams which can be constructed by insertion of the basic diagrams into Green’s function lines (to all orders) with these multiple insertion diagrams also iterated to all orders in G. Thus, we would expect that starting with a rather small number of basic C diagrams solved for perturbationally would give good answers in the

Gy. Csunak, H . S. Taylor, and R . Yuris

348

self-consistent perturbation method. Again, it is our ability to solve the equation of motion for G in a finite basis set which makes this self-consistent perturbation approach practical for atomic and molecular calculations. D.

PERTURBATION

METHODS FOR R

AND

X,,

From the discussions in the previous sections it is evident that if one is interested in problems where there is a transition of the target system from one state to another, e.g., excitation energies, inelastic scattering, etc., then knowledge of GI is insufficient and one must look at the information contained in G, (or equivalently R ) . For this reason perturbation methods of solving for R , and X , , will now be derived. The hierarchy of perturbation methods will be derived in the opposite order from that of the previous subsections. That is, first an expansion for Z in terms of G, will be derived to be used in conjunction with the Bethe-Salpeter equation to find R (or X,,) as a function of G,, V, and R (or X,,) itself; this is analogous to the C perturbation method where C is used in conjunction with the Dyson equation to obtain G,. Then by expanding the Bethe-Salpeter equation a perturbation expression for R directly in terms of G , and V is derived (thus eliminating the need to solve the Bethe-Salpeter integral equation if one so desires). That such perturbation expansions can be derived is, of course, obvious, since their derivation just undoes the work of Bethe and Salpeter in summing the perturbation series to obtain an integral equation in the first place. The present philosophy is that it is preferable to derive perturbation expansions by expanding closed equations derived analytically rather than to sum perturbation series to obtain closed expressions. Since Z is defined as 6C/6G1 [Eq. (67)] a perturbation expansion for Z can be obtained by simply differentiating the expansion for X as a function of GI, obtained in Section VI, C , with respect to G,. Using the first-order correction to E, Equation (133) with the Glo’s replaced by GI’s (this is equivalent to functionally replacing X by the Hartree-Fock approximation to C) one obtains s l y 3 4 3’4’) = i

6 ( 4 ( 3 - 3’) p 5 q 3 - 5)G(5,5’) 6G(4’, 4)

~

+ V(3 - 3’)G(3, 3’)

i dG(5, 5 ’) 6G(4’, 4)

349

GREEN’S FUNCTIONS

= --i

6(3 - 3’) p V ( 3 - 5) 6(5 - 4’) 6(5+ - 4)

+ iV(3 - 3’) 6(3 - 4’) 6(3’ - 4) = - i6(3

- 3’) 6(4 - 4’) V(3 - 4’)

+ iV(3 - 3’) 6(3 - 4’) 4 3 ’ - 4).

(139)

Equation (139) is now inserted into the Bethe-Salpeter equation (68) and integrated over the &functions to give R(121’2’) = G(1, 2’)G(2, 1’) + i l d 4 d4’G(l, 4’)G(4, 1‘)1/(4 - 4‘) x R(4’242’) - i

d 3 d4’G(1, 3)G(3, 1’)V(3 - 4’)R(4’24’+2‘)

+ O(2)

( 140)

which is the same equation obtained in Section V by using C (106). Equation (140) can be expressed diagrammatically as

= CHF, Eq.

r2‘= 2‘

I

I

2

I‘

I

I‘

2

2‘

+ 0 (2).

4-

I’

(b)

(140‘)

2

where the heavy lines depict the R lines and the light lines are G, lines. The analytic expression corresponding to (140’a) carries a factor of (i) and that corresponding to (140’b) a factor of (- i). To obtain the second-order terms in Z one can just take the analytic expression for the second-order expression for C as a function of G,,which is given by the basic diagrams (137b) and (137c) and perform the differentiation

350

Gy. Csanak, H. S. Taylor, and R. Yaris

as above. However, it is more convenient, and much simpler, to perform this completely diagrammatically. It is clear looking at the C diagram, (137a), which leads to (140’a) that taking the derivative with respect to G(4’, 4) just deletes the Green’s function line G(3, 3‘) and gives two &function factors 6(3 - 4’) and 6(3‘ - 4)

w6 (3-4’)

6 (3’-4)

(140 bi

When this diagram is put into the Bethe-Salpeter equation and the &functions are integrated over the diagram is just linked up to the R diagram

with the &functions telling where the linkages go, the “free” vertices of the original C diagram (which in this case are also 3’ and 3) are linked up with the two Green’s function lines G(1, 3) and G(3‘, l’), and the intermediate points are integrated over leading to (140’a). Also since the original diagram carried a factor of (i) (one V line, no closed loops) so does the analytic expression corresponding to this diagram. Thus, there are a set of rules similar to those of the previous subsections for constructing the Bethe-Salpeter diagrams for R as a function of V, G,, and R itself to a given order in the interaction. R is the sum of all topologically distinct Bethe-Salpeter diagrams constructed by (1) Taking the basic strongly connected diagrams for C (in terms of GI and V ) to the desired order. (2) Erasing an internal G line in each of the C diagrams and connecting its exposed vertices to the free ends of the R diagram lines. (3) The free vertices of the original C diagram are each connected up to a G line. (4) Step (2) [followed by step (3)1 is repeated for each topologically distinct internal G line in each of the considered C diagrams.

The analytic formula corresponding to a given diagram is obtained in the same way as in the previous subsections, remembering that the G lines are now full G lines, with the following two exceptions: (i) The factor (- 1)’ must be obtained from the I: diagram before erasing the G line since one can open up a closed loop by erasing a line.

GREEN'S FUNCTIONS

351

(ii) If the G line being erased has g other topologically equivalent G lines the diagram carries a weight factor g, rather than repeating the diagram g times. As an example of the use of these rules, we shall now obtain the BetheSalpeter diagrams for the second-order corrections to R . The two secondorder basic diagrams are (137b) and (137c). Applying the above rules leads to the following four topologically distinct Bethe-Salpeter diagrams

-3-7-J-

x

(141a)

(141b)

(141c)

(141d)

Diagram (141a) comes from (137b) by erasing the 3'-3G line. Diagram (141b) is obtained by erasing either of the two 5'-5 lines in (137b) and hence carries a weight factor of 2. Diagram (141c) is obtained from (137c) by erasing either of the two topologically equivalent G lines 3'-5 or 5'-3, hence it also carries a weight factor of 2. Diagram (141d) comes from erasing the 5-5' G line in (137c). Diagrams (141a and b) have a factor of (-1) since (137b) has one closed loop, and all four have a factor of (i)' since there are two V lines. Since the Bethe-Salpeter amplitude equation (73) for X,, is of exactly the same form as that for R (except that X , has only two variables), one can immediately write down the perturbation expansion for X,,. All that is necessary is to define a symbol for X,, which has only two free lines corresponding to the two variables. Hence, X,, correct through second order is given by

Gy. Csanak, H . S. Taylor, and R . Yaris

352

x,

(I,

1’1

=

7 - J I /

I’

+

(142) where the terms have exactly the same interpretation (except for replacing the R piece in each diagram by an X , piece) as those in the Bethe-Salpeter expansion for R. Clearly, a self-consistent scheme of solution for GI and R is possible. That is, the Bethe-Salpeter perturbation series is truncated at some point. Then assuming a G (usually the Hartree-Fock G) one solves the truncated BetheSalpeter integral equation for R (or the X,,’s). Then knowing R one has the C for solving the Dyson equation for G, as was discussed in Section 111. The Dyson equation is then solved for G keeping C fixed. This G is then inserted back into the truncated Bethe-Salpeter equation and the process is repeated until self-consistency between G, and R is obtained. The above procedure is obviously a higher order (and more difficult) procedure than the self-consistent procedure described in Section VI, C, since it requires the self-consistent solution of a coupled pair of integral equations, and the Bethe-Salpeter equation now has an energy dependent kernel. The self-consistent method described in Section V as the generalized R.P.A. method is the lowest order case of the method described above, that is, the Bethe-Salpeter perturbation expansion is truncated at first order (a simple superposition approximation for G 2 is the truncation of the expansion at zeroth order which obviously eliminates the Bethe-Salpeter integral equation). Clearly higher order truncations will work better (especially for transition problems where one really needs R , or X,,) and also be correspondingly more work. As an alternative to the above Bethe-Salpeter perturbation series (which is really a perturbation series for B which is then inserted into the Bethe+

0 (3)

GREEN’S FUNCTIONS

353

Salpeter integral equation) the truncated expansion can be iterated thus eliminating the integral equation in favor of its perturbation expansion. This step undoes the original summation derivation of the integral equation. Diagrammatically this is just done by iterating the diagrams. That is, first replace the R part of a diagram by the zeroth order result (two G lines) thus obtaining the lowest order approximation to the original diagram. This is then used to replace R in the original diagram obtaining a higher order approximation, etc. For example, iterating one of the first order Bethe-Salpeter diagrams (140’a) gives

(143a) which is the “ ladder” diagram series. Similarly just iterating (140’b) gives the “ bubble” diagram series.

+

(143b)

There are also clearly combination diagrams of ladders and bubbles if one takes both (140’a) and (140’b) into account together by iterating all of (140’), such as (143c) to second order, and

354

Gy. Csanak, H . S. Taylor, and R. Yaris

to third order. The sum of all of the diagrams in (143) plus, of course, the diagram consisting of two G, lines gives the third-order perturbation approximation to the first order Bethe-Salpeter equation (140). This iteration can clearly be performed to any desired order starting with any set of Bethe-Salpeter diagrams, thus obtaining a truncated expression for R in terms of V and G, only. For a given set of Bethe-Salpeter diagrams one can thus eliminate having to solve the integral equation to find R . A price is again paid for this simplification in the amount of work, since the integral equation sums the diagrams to all orders and here one only sums to a finite order. Again, an iterative process can be carried through for self-consistently solving for R and C,. It is, however, exactly equivalent to the renormalized C self-consistent method described in Section VI, C, since knowing R (as a function of G and V ) to any given order is equivalent to knowing C (as a function of G and V ) to that order. Lower order approximations to R can be obtained by replacing the GI lines in diagrams like those of (143 a, b, c, d) by some perturbational approximation to G, as formulated in Section VI, A. R is then given by a perturbation expansion containing only GIo’s and V’s. Now it is no longer possible to solve for R and G, self-consistently, since G, does not appear in the truncated expression for R . Of course, R can be used together with the Dyson equation to solve for G,, but this is equivalent to the use of unrenormalized C perturbation theory as described in Section V1, B.

ACKNOWLEDGMENTS The authors acknowledge the kind help and interest of Dr. Barry Schneider, who initiated the application of Green’s functions in electron-atom scattering theory, and who has had numerous enlightening discussions with the present authors. They are also grateful to Mr. B.-S. Yarlagadda for several interesting remarks. The authors appreciate also the kindness of Mr. Lowell Thomas and Mr. J u Chu Ho for reading and correcting the manuscript.

Appendix A THEDERIVATION OF THE SCHWINGER RELATION It is useful to write G(1, 1’; U ) in terms of the quantities appearing in the case when U is zero, which is accomplished by going over to the interaction representation. To do this, the homogeneity of time is used to set the time at which the Heisenberg representation (in terms of which Equation (54) is written) and the interaction representation are equivalent at time t = -a.

355

GREEN’S FUNCTIONS

At this time I Y o ( U ) )= I Y o ) . To go over to the interaction representation one uses a generalization of Eq. (2) for a time-dependent Hamiltonian

The time ordering is necessary (Dyson, 1949; Feynman, 1951) because the Hamiltonian taken at two different times does not in general commute, hence the power series expansion of the exponential would be ambiguous as to the ordering of the operator. The time-ordering operator defines the ordering and hence removes the ambiguity. Thus

[

x exp{ - i H [ t - (- co)]}T exp - i

J‘

drHin,(r)]

-m

where use was made of the fact that the time-ordered exponential of a sum is equal to the time-ordered product of exponentials (this, of course, is not true without time ordering for noncommuting operators) and where J d s H,,,(T) = Jd2 d2‘U(2‘, 2)$+(2‘)$(2)

(A.2)

which since the time ordering is specified as t,’ > t , can be written as

J d r Hint(?)= - I d 2 d2’U(2’, 2)T[$(2)$+(2’)].

(‘4.3)

Defining S( t , t’)

=

[

T exp - i

I:

ds Elint(.)

1

Eq. (54) becomes 1

G(1, 1’; U ) = (Y,jTS(-co, t)$(l)S(t, t’)$+(l’)S(t’,- co)lYo). (A.5) 2

Using the fact that U

=0

at t

=

+a,Y o ( U ) is again Y o (to within an

Gy. Csanak, H. S. Taylor, and R. Yaris

356

unobservable infinite phase), the ( Y o / at t (‘POI at t = +co using

= - co

can be replaced by the

where the denominator accounts for the phase and preserves normalization. Equation (A.6) is justified by a series of steps quite similar to that gone through in (A.1). Substituting (A.6) into (AS), moving S( + 00, - co) to be thought of as lim t + m S(t, t’) under the T(which can be done since T2 = T ) t’+ - m

and combining the exponentials in the S’s gives

This is the desired expression for Gl(l, 1’; U ) in terms of the quantities that appear in GI for U = 0. To evaluate 6G1(U)/6U in the limit of small U , let U -+ U + 6U (where 6U is an arbitrary infinitesimal change in U ) and thus Hint-+ Hint+ 6Hint and G + G + 6G. The change in S arising from the infinitesimal change in U is given by

=

T S ( t , t ’ ) ( - i Jt:dz6HinI(z)]

(A.8)

yielding

G(1, 1’; u)

+ 6G(1, 1’; U )

- -i

(TS(

+ 00, - co)[l

- i J T d~z 6 ~ ~ , , ( z ) ] ~ ( l ) ~ ~ ( l ’ ) ) . (A.9)

( T S ( + w , -00)[l - l T m ~zGH~,~(z)])

If the denominator is expanded in powers of 6HinI,and only the lowest order term in the infinitesimal is retained then Eq. (A.9) gives, after (A.7) is used to subtract G,(U) from both sides dG(1, 1’; U ) = -i - G(1,

dz 6Hi,,I(z)$(l)$t(l‘) 1’; U)(TS(co,-co) j d z 6 H i n t ( z ) ) x) (S(co, -co))-’ (A.lO)

GREEN’S FUNCTIONS

357

which after substituting (A.3) into (A.lO) yields

dG‘(1, 1’; U ) =

(1

d2 d2’(TS(w, - ~)$(2)$~(2’)$(1)$~(1’))

x 6U(2’, 2)

+ 711 G(1, 1’; U i

I

s

d2 d2’(TS(w, - ~0)$(2)$~(2’))

x 6U(2’, 2) x (S(c0, -a))-’.

(A.11)

Permuting the creation and annihilation operators under T, with proper sign changes to give the order of the definitions of GI and G2 as given by Eq. (1) gives dG(1, 1’; U ) =

{

i d 2 d2’(TS(co, - ~ ) $ ~ ( 1 ’ ) $ ( 2 )

-

x $t(2’)$(l))6U(2’,

2)

1

+ 1 G(1, 1’; U)l d 2 d2‘

x (TS(W, - 03)$(2)$~(2‘))6U(2’, 2)

(A.12) The functional derivative is then

1

+ 7E G(1,1’; U)(TS(co, - m)$(2)$t(2’)))(S(m,

- “u))-l

(A.13) or 6Gl(1, 1’; U ) = dU(2‘2) u=o

-G2(1,

2; l’, 2’)

+ G l ( l , 1’)G1(2, 2’).

(A.14)

Several special cases of (A.14) which prove useful can be obtained. By letting 2‘ -+ 2‘+, i.e., 2 and 2‘, are at the same time with 2‘ infinitesimally earlier, one gets the case of the variation of G, with respect to a nonlocal one time potential U(2,2+) = U(r, r,, r2‘r2’)6(r2- rz’>

6G,(1,1‘; U ) = -G2(1, 2; 1’, 2”) + GI(1, 1’)G1(2, 2”). (A.15) 6U(2’+, 2) u = o The special case of the variation of G, with respect to a local potential U(2) is obtained from (A.15) by letting 2’ + 2

1

6G1(1, 1’; U ) SW2)

u=o

=

- G 2 ( l , 2; l’, 2’)

+ Gl(l, 1’)G1(2, 2’).

(A.16)

Cy. Csanak, H . S . Taylor, and R. Yaris

358

For the definition of the functional derivative we note that if F is a functional of U ( x ) and we change U ( x ) in the neighborhood of the point x by 6U(x), then F will change by SF. The limit of (l/A)[SF/6U(x)] is called the functional derivative (where A 4 0 and A is the length of the interval). If F can be written in the form

j

F [U ( x ) ]= K(x)U ( x )dx then

=I

XI

6F

K ( x ’ ) SU(x’) dx‘

xa

where [x,, x l ] is the interval around x where SU(x‘) is different from zero. If we write

6F = (XI- xo)K(X)GU(X) then

Appendix B Here Eq. (68) is converted to a closed equation for G,, with an effective potential W (sometimes called a vertex function) so as to give the usual form of the Bethe-Salpeter equation. Combining Eq. (68) with (65) gives GZ(1, 2; l’, 2’)

= Gl(1,

l’)G1(2, 2’) - G1(1,2’)G1(2, 1‘)

+ j d 3 d3’ d4 d4’C1(1, 3)C1(4, l’)E(34’ 43’)R(3‘2 4’2’).

(B.l)

Defining W by j d 3 ’ d4 d4’G(4, 1‘)=(34’43’)R(3’24’2’) =/d3’ d4 d4’G1(2, 4)W(343’4’)GZ(3’,4’, 1, ’2’) (B.2) (B. 1) becomes Gz(l,2; l’, 2’)

= GI(1,

l’)G1(2,2’) - Gl(1, 2’)G,(2, 1’)

+ j d 3 d3‘ d4 d4‘G1(1, 3)G1(2, 4)W(343’4’)G2(3’4’, 1’2’)

(B.3)

which is the usual form of the Bethe-Salpeter equation. Again the equation is for two dressed particles interacting through an effective potential W which

359

GREEN’S FUNCTIONS

has only the truly two-particle correlations. Experience shows that (B.3) is more convenient for hole-hole or particle-particle processes, while Eq. (68) is more useful for hole-particle processes of the type encountered in this work. Equation (B.2) can easily be converted into an integral equation for WG, which has Z as its kernel. Clearly a knowledge of Z and R is equivalent to a knowledge of Wand G, .

Appendix C Here the following integral is evaluated

If Eq. (53b) is inserted into (C.l) with the changes in notation

then Eq. (C.l) becomes

+ ?+o+ lim 1

11‘ ( E

gl(rl)sl(r2)gl’(r3>sl’(r4)

- el - ir])(e - E,

(C.3)

- el, - iv])

The first and fourth integrals vanish since both factors in each integral have poles only in the same half plane. If the integration contour is closed in the half plane which. does not contain any poles, the residue theorem assures us that the integrals are zero. For the other two integrals, since they have poles in both half planes the contours can be closed either way giving the same result. The second term can be closed in the lower half plane giving lim SdE ,,+o+

F(E) e - q +

iv]

=

-2rci lirn F ( E ~ iq) ,,+O+

=

-2ai

lim

,,-o+

Ek

- E, - el, - ir]

K.4) where the (- 1) comes in because the contour is traversed in a clockwise

Gy. Csanak, H . S. Taylor, and R . Yaris

360

manner. Similarly closing the contour for the third term in the upper half plane gives lim J d e

V‘Of

2ni

1 (& - &/

- iv])(&- E s - E k ‘

+ iv]) =

lim

v+O+

&I

- E,7 - Ek‘

+ iq

(C.5) ’

Hence the integral is

REFERENCES Abrikosov, A. A,, Gorkov, L . P., and Dzyaloshinski, I. E. (1963). “Methods of Quantum Field Theory in Statistical Physics.” Prentice-Hall, Englewood Cliffs, New Jersey. Baym, G., and Kadanoff, L. P. (1961). Phys. Rev. 124, 287. Bell, J. S., and Squires, E. J. (1 959). Phys. Rev. Letters 3, 96. Brout, R., and Carruthers, P. (1963). “Lectures on Many-Electron Problem.” Wiley, New York. (London) 86, 989. Burke, P. H., and McVicar, D. D. (1965). Proc. Phys. SOC. Csanak, Gy., Taylor, H . S., and Yaris, R. (1971). Phys. Rev. A3, 1322. Ser. A 291, 291. Dalgarno, A., and Victor, G. A., (1966). Proc. Roy. SOC. Dietrich, K., and Hara, K. (1968). Nucl. Phys. A 111, 392. Dunning, T. H., and McKoy, V. (1967). J . Chem. Phys. 47, 1735. Dutta, N. C., Ishibara, T., Matsubara, C., Pu, R. T., and Das, T. P. (1969). Phys. Rev. Letters 22, 8. Dyson, F. J. (1949). Phys. Rev. 75, 486, 1736. Ehrenreich, H., and Cohen, M. H. (1959). Phys. Rev. 115, 786. Falkoff, D. (1962). In “The Many-Body Problem” (Bergen School Lectures) (C. Fronsdal, ed.). Benjamin, New York. Fetter, A. L., and Watson, K. M. (1965). Advan. Theoret. Phys. 1, 115-194. Feynrnan, R. P. (1948). Rev. M o d . Phys. 20, 367. Feynrnan, R . P. (1950). Phys. Rev. 80, 440. Feynman, R. P. (1951). Phys. Rev. 84, 108. Galitski, V. M., and Migdal, A. (1958). Soviet Physics, JETP 34(7), 96. Cell-Mann, M., and Low, F. E. (1951). Phys. Rev. 84, 350. Goldberger, M. L., and Watson, K. M. (1964). “Collision Theory.” Wiley, New York. Goldstone, J. (1957). Proc. Roy. Soc. (London) A 239, 267. Goscinski, O., and Lindner, P. ( I 970). J . Math. Phys. 11, 13 13. Gutfreund, H., and Little, W. A. (1969). Phys. Rev. 183, 68. Hedin, L., Johansson, A,, Lundquist, B. I., Lundquist, S., and Samathiyakamit, V. (1969). Arkiv. Fysik 39, 97. Jamieson, M. J. (1969). Ph.D. Thesis, The Queen’s University of Belfast, Belfast, Northern Ireland.

GREEN’S FUNCTIONS

36 1

Janev, R. K., Zivanovic, D. J., and Maric, Z. D (1969). The Use of Green Functions in Electron-Atom Scattering Problem, In In/. Conf: Phys. Electron. At. Collisions. M.I.T. Press, Cambridge. Kadanoff, L. P., and Baym, G. (1962). “Quantum Statistical Mechanics.” Benjamin, New York. Karplus, M., and Caves, T. C. (1969). J . Chem. Phys. 50, 3549. Kato, T., Kobayashi, T., and Namiki, M., (1960). Pvogr. Theoret. Phys. Suppl. 15, 3. Kelly, H. P. (1968). Aduan. Theoret. Phys. 2, 75. Kelly, H . P. (1969). Phys. Rev. 182, 84. Kirzhnits, D . A. (1967). “Field Theoretical Methods in Many-Body Systems.” Pergamon, Oxford. Klein, A. (1956). Progr. Theoret. Phys. 14, 580. Klein, A,, and Zemach, C. (1957). Phys. Rec. 108, 126. Layzer, A. J. (1963). Phys. Reu. 129, 897. Lemmer, R. H . , and Veneroni, M. (1968). Phys. Rev. 170, 883. Linderberg, J. (1968). Mugy. Fiz. Fo‘oly. 16, 5. Lowdin, P. 0. (1955). Phys. Rev. 97, 1474. Lowdin, P. 0. (1956). Advan. Phys. 5, 1. March, N . H., Young, W. H., and Sampanthar, S. (1967). “The Many-Body Problem in Quantum Mechanics.” Cambridge Univ. Press, Cambridge, England. Martin, P. C., and Schwinger, J. (1959). Phj)s. Rec. 115, 1342. Matsubara, T. (1955). Pvogr. Theoret. Phys. (Kyoto) 14, 351. Mattuck, R. D. (1967). “A Guide to Feynman Diagram in the Many-Body Problem.” McGraw-Hill, New York. Migdal, A. B. (1967). “Theory of Finite Fermi Systems.” Wiley (Interscience), New York. Morse, P. M., and Feshbach, H. (1953). “ Methods of Theoretical Physics.” McGraw-Hill, New York. Mott, N. F., and Massey, H. S. W. (1965). “The Theory of Atomic Collisions.” Clarendon Press, Oxford, England. Namiki, M. (1960). Progv. Theovet. Phys. 23, 629. Pines, D., ed. (1961). “The Many-Body Problem.” Benjamin, New York. Pines, D . (1963). “The Many-Body Problem” (D. Pines, ed.). Benjamin, New York. Reinhardt, W. P., and Doll, J. D. (1969). J . Chem. Phys. 50, 2769. Reinhardt, W. P., and Szabo, A. (1970). Phys. Rev. Al, 1162. Roman, P. (1965). “Advanced Quantum Theory,” p. 308. Addison-Wesley, Reading, Massachusetts. Rowe, D. J. (1968). Rev. Mod. Phys. 40, 153. Schneider, B. (1970). Phys. Rev, A2, 1873. Schneider. B . , Taylor, H. S., Yaris, R. (1970). Phys. Reu. l A , 855. Schweber, S. S. (1961). “An Introduction to Relativistic Quantum Field Theory.” Harper, New York. Schwinger, J. (1951). Proc. Nut. Acud. Sci. U S . 37, 452. Thouless, D . J. (1961). “The Quantum Mechanics of Many-Body Systems.” Academic Press, New York. Wilkinson, J. H . (1965). “The Algebraic Eigenvalue Problem.” Clarendon, Oxford, England. Zmuidzinas, J. S. “ Self-consistent Green’s Function Approach to the Electron Gas Problem” (preprint).

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A REVIEW OF PSEUDOPOTENTIALS WITH EMPHASIS ON THEIR APPLICATION TO LIQUID METALS NATHAN WISER and A. J . GREENFIELD Department of Physics, Bar-llan University Ramat-Gan, Israel 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Underlying Ideas of the Pseudopotential ........................... 111. Simplification of the Form Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

IV. Formulations of v(y) Useful for Liquid Metals .....................

363 365 370 374

....................

384

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

386

VI. Comments

I. Introduction Over the past decade, the development of the pseudopotential into a practical computational tool has led to remarkable progress in our understanding of the properties of materials. Whereas previously the solid-state theoretician was limited in the study of metals, for example, to the analysis of highly idealized models, use of the pseudopotential has made it possible to treat explicitly the differences between one metal and another. Such advances have been made not only for solid metals, but also for liquid metals, semiconductors, semimetals, ionic crystals, molecular crystals, etc. However, in our discussion of the pseudopotential, we shall place particular emphasis on its applicability to liquid metals. We shall not enter into any discussion of the many other applications of the pseudopotential to the analysis of experimental data. That subject has been reviewed recently in a very thorough and comprehensive fashion by Cohen and Heine (1970). Of particular importance in their review is an exhaustive compilation for each metal of all experimental data available which relates to the fitting of pseudopotentials. 363

Nathan Wiser and A . J . Greenjeld

364

A key feature of the theory is that the same pseudopotential for a given element (say Al) is appropriate for this element either as a solid metal, as a liquid metal, as a semiconductor (e.g., AISb), or as an ionic crystal (e.g., AICl,). For this reason, it has been possible to carry over progress from one field to another. For the field of liquid metals, it was Ziman (1961) who first recognized that the pseudopotential method, developed for the solid phase, may be taken over bodily to the liquid phase. This recognition resulted in a dramatic advance in the field of liquid metals. It is only a slight exaggeration to say that within the last decade, the study of liquid metals was transmuted from a branch of metallurgy into a branch of physics. Ziman and co-workers (Ziman, 1961; Bradley et al., 1962; Faber and Ziman, 1965) derived formulas for the two independent transport coefficients, the electrical resistivity p and the thermoelectric power Q. They went on to apply these new ideas to a discussion of the Hall coefficient, optical properties, the Knight shift, and other properties of liquid metals. The existence for the first time of formulas for p and Q which could actually be evaluated and compared with experiment provided a tremendous stimulation, both theoretically (Edwards 1962; Baym, 1964; Springer, 1964; Sundstrom, 1965; Greene and Kohn, 1965; Enderby and March, 1965; Wiser, 1966; Ashcroft and Lekner, 1966; Ascarelli, 1966; Mott, 1966; Ballentine, 1966; Young et al., 1967) and experimentally (Cusack and Kendall, 1961 ; Hodgson, 1963; Endo, 1963; Johnson et al., 1964; Greenfield, 1964; Marwaha and Cusack, 1965; Busch et al., 1965; Thompson, 1965; Kaplow et al., 1965; Enderby and North, 1966; Adams and Leach, 1967; Cocking and Egelstaff, 1968; Halder et al., 1969; El-Hanany and Zamir, 1969) to the field of liquid metals. The current scope of work in liquid metals was already evident at the International Conference on the Properties of Liquid Metals held at Brookhaven National Laboratory in 1966. In addition to the conference proceedings,' a number of review articles have appeared over the last few years (Cusack, 1963; Ziman, 1964; Egelstaff, 1966; Faber, 1966; Harrison, 1966; Mott, 1967). The Ziman formulas for the electrical resistivity and thermoelectric power are

where

Published in three issues in Aduan. Phys. 16 (1967).

PSEUDOPOTENTIALS-EMPHASIS

ON LIQUID METALS

365

We use the standard notation: R, = atomic volume, k , = Fermi momentum, E , = Fermi energy, q = momentum transfer, v(q) = screened, electron-ion potential, and a(q) = static structure factor. In the equation for r , there appears the derivative of u2(q) with respect to k,. As we shall see in due course, v(q) is properly a function of k , as well as of q. A glance at Eqs. (1)-(3) shows that v(q) occupies a central position in the Ziman formulation. The only successful method developed to date for determining v(q) is based on the theory of the pseudopotential, and, in fact, is so closely related that v(q) is often referred to as the pseudopotential. It is worth emphasizing that at high temperatures the Ziman formulas for p and Q are equally applicable to a solid with a spherical Fermi surface, provided only that one insert for a(q) a structure factor appropriate to the solid. However, the pseudopotential v(q) is the same for the solid and liquid phase. This is because v(q) is a property of a single screened ion and is, hence, independent of the allotropic form of the metal. There are many seemingly different formulations of the pseudopotential, some of which are unfortunately disguised by the use of elegant terminology such as “ phase-shift analysis or t matrix.” Moreover, there are various approximations and differing models on which the formulations of the pseudopotential are based. This plethora of methods for determining v(q) has been responsible for a certain degree of confusion. It is the purpose of this article to review, clarify, and expose some of the subtleties and methods which have been used to generate pseudopotentials. We shall concentrate on simple ideas and basic concepts, taking pains to prevent beclouding these ideas by complicated mathematics. Indeed, it is our basic thesis that the best pseudopotentials have been obtained using practically no mathematics at all. In Section 11, we discuss the underlying ideas basic to the pseudopotential formulation resulting in the factorization of the relevant matrix element into structure factor and form factor. The form factor is discussed in detail in Section I11 and the various approximations involved are explained. In Section IV, five pseudopotentials which are particularly applicable to liquid metals are described. The important concept of screening is dealt with in Section V. Finally, in Section VI, we assess the strengths and weaknesses of the five potentials previously discussed. ”

“

11. Underlying Ideas of the Pseudopotential In this section, we shall concentrate attention on those aspects of the pseudopotential which are vital for their intelligent and effective use. As one might have surmised, for practical purposes it is not necessary to be familiar with the details of the basic derivation and mathematical justification of the

366

Nathan Wiser and A . J . Greenfield

general theory of pseudopotentials (Phillips and Kleinman, 1959; Cohen and Heine, 1961 ; Austin et al., 1962). Nevertheless, even the practical user of pseudopotentials should not disdain the benefits derived from the basic work on the theoretical underpinnings of pseudopotential theory. First, it gives vital insight into the construction of simple, but successful, phenomenological models for the pseudopotential. Second, it provides the practical user of pseudopotentials with a feeling of confidence far in excess of what would otherwise seem justified, in view of the simplicity of the mathematics involved. Still, it has been rather disconcerting that the many detailed and elegant calculations of very subtle and sophisticated improvements to the pseudopotential have generally turned out to be of little practical value. To be candid, the pseudopotentials which have proved most useful are those which are nothing more than a reasonable functional form for the pseudopotential containing a small number of parameters. These parameters are then determined by fitting to experimental data, and presto-one has a completely determined pseudopotential. Let us start with some simple, well-known definitions and concepts, applicable to a periodic solid. Practical calculations invariably require matrix elements of the Bloch potential VBloCh(r) between Bloch states

where uk(r) is periodic in the lattice vectors R , , and R is the volume of the crystal. A typical matrix element thus has the form

The practical difficulty in evaluating this matrix element is that both the function VBloch(r) and the function uk(r) are not really known very well. Indeed, it is this lack of knowledge that resulted in physicists' inventing the pseudopotential theory in the first place. Although one usually thinks of the pseudopotential as a tool for calculating energy levels, it is equally useful for calculating matrix elements. The basic result of this important aspect of the theory is contained in the scattering amplitude theorem of Austin, Heine, and Sham (1962). This theorem states that aside from a normalization factor, one can almost everywhere replace the above matrix element by the following matrix element: ((Pk(r)

I P(r) I (Pk'(r))

(6)

where qk(r) is the pseudo-wave function, and is also of the Bloch form where wk(r + R,)= wk(r). Here

P(r) is the pseudopotential, which is really

ON LIQUID METALS

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367

an operator rather than a simple function of r, as indicated by the caret over the I/. We hasten to emphasize that our new matrix element is just as difficult to calculate exactly as the previous matrix element. However, this substitution, called the pseudopotential transformation, is nevertheless extremely useful. Its usefulness lies in the fact that for many materials, a single plane wave, or a manageably small number of plane waves, constitutes a very reasonable approximation to qk(r). Moreover, calculations show that the matrix element is small, that is, of order several tenths of an electron volt, whereas the relevant energy differences entering perturbation theory or a variational calculation are typically several electron volts or more. Thus, one may use perturbation theory and other simple approximate methods to treat an otherwise extremely difficult problem. The cancellation theorem of Cohen and Heine (1961) explains why this smallness of the matrix element is to be expected on theoretical grounds. We should emphasize that whereas for calculating energy levels the pseudopotential formalism is very convenient, for calculating matrix elements such as those in Eq. ( 5 ) it is absolutely vital. The reason for this is that there is no means to date for obtaining the quantitatively reliable Bloch wave functions that are necessary for evaluating Eq. (5). This difficulty does not arise in calculating energy levels, and indeed there are a host of well-known, successful (if not easy) methods for obtaining the band structure of solids without the use of the pseudopotential formalism. It is generally recognized that all these methods (OPW, APW, KKR, k . p, etc.) fail to yield accurate wave functions, and hence, these methods are unreliable for calculating matrix elements. A key property shared by the matrix elements of both Eqs. ( 5 ) and (6) is that, for a periodic solid, the matrix element can be factorized in a particularly useful way. We begin by writing

P(r) =

c O(r

- R,)

I

where O(r - R,) is the pseudopotential due to a single atom situated at the lattice site R , . Recalling that qk(r) is of the Bloch form, we may write the matrix element as

x wk*(r - R$(r

- Rl)wk,(r - R,)eik"(r-Rf!

With the usual assumption of periodic boundary conditions, we can change variables from r to r - R, and thus eliminate R, entirely from the integrand.

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Nathan Wiser and A . J . Greenfield

Because the integral is independent of R,,we can pull the integral outside the summation sign. We may now write the matrix element as a product ((Pk(r)1 p(r) I (Pk‘(r)) = c(k, k’)S(k - k’)

where 1

(10)

1

u(k, k’) = - d3re-ik“wk*(r)D(r)wk.(r)eik”r 0, is called the form factor and

is called the structure factor. We have introduced R, as the atomic volume and N as the number of atoms in the crystal. For simplicity, we shall always pretend that there is only one atom per unit cell. Note that we nowhere need the fact that P(r) is periodic. Our restriction of Eq. (10) to periodic solids stems solely from the requirement that wk(r) be periodic. The traditional goal of the factorization obtained in Eq. (10) is to separate the matrix element into a product with one factor (the structure factor) depending only on the geometrical arrangement of the atoms and the second factor (the form factor) depending only on the properties of a single atom. We have not yet fully achieved this goal because c(k, k’) still depends on the arrangement of the atoms through wk(r) in Eq. (1 1). However, our goal can be fully achieved for those elements for which it is reasonable to make the approximation that wk(r) is a constant, that is, to assume that the pseudowave function (Pk(r) can be adequately represented by a single plane wave, fpk(r) = R g l / ’ e i k ‘ ’ = I k). The approximation simplifies Eq. (9) giving

= S(k - k’)u(k,

k’)

(13)

Though Eq. (13) appears formally the same as Eq. (lo), there is an enormous difference between the two. In Eq. (13), u(k, k’) depends only on the properties of a single atom and hence our separation of the matrix element is now complete. Therefore, if we determine tl(k, k’) from one experiment on a given metal (e.g., the de Haas-van Alphen experiment), we may apply the same u(k, k’) to interpret any other experiment on that metal or on any of its compounds. Moreover, this powerful method is not limited to perfect crystals, but can also be applied to a host of interesting situations where the atoms are not in their equilibrium position. These include studies of the phonon spectrum, transport properties involving electron-phonon scattering,

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disordered alloys, defects, and last but not least, properties of liquid metals. We reemphasize that the periodicity of the lattice was nowhere used in deriving Eq. (13). The vectors R, represent the positions of the array of atoms being treated. This array can be periodic, nonperiodic, or can even be timedependent, as occurs in the theoretical analysis of the lattice vibrations in terms of phonons and in the theory of electron-phonon scattering. A side benefit of our approximation, which the perceptive reader will not have failed to notice, is that the function w,(r), which we did not know anyway, has conveniently disappeared from the integral of v(k, k’). However, there is a price to pay. The treatment must now be restricted to those substances for which this single-plane-wave approximation to qk(r) is not unreasonable. The cancellation theorem (Cohen and Heine, 1961), indicates that, in practice, this approximation is quite suitable for simple metals (effectively, those metals for which the partially filled s and p bands are far from all d and f bands), semimetals, semiconductors, and their compounds and alloys. However, more enterprising physicists have not hesitated to apply this factorization without modification2 to the noble metals, transition metals, and even ionic crystals. The main conclusion to be drawn from our discussion regarding factorization is that the central problem is to determine the form factor tl(k, k’) for the atom of a given substance. Once one has found v(k, k‘), one is in a position t o calculate an impressively large number of properties of that substance. Before turning to a detailed discussion of how one in practice determines o(k, k’), we will first comment on the often misunderstood statement about the arbitrariness of the pseudopotential. This arbitrariness is generally dismissed by the experts with the offhand comment that it does not really matter, since when one goes to infinite-order perturbation theory, it can be shown rigorously that all valid pseudopotentials will give the same correct answer. For those not planning to carry their calculations to infinite order, this is not particularly helpful. A fruitful way of approaching this whole question is to note that there exists, in principle, a t-matrix formulation for the pseudopotential that properly incorporates all higher order corrections. Though no one knows the form of this function, the fact of its existence encourages us to approach the matter on a phenomenological basis. That is, assume that L)(k,k’) is not the simple integral appearing in Eq. (13) but rather that u(k, k’) already includes all higher order corrections, in the spirit of the t-matrix formulation. Any determination of v ( k , k’) is then visualized as an approximation to this exact form. With suitable modification, these ideas may indeed be extended to the noble and transition metals (e.g., see Hodges et a / . , 1966; Heine, 1967; Harrison, 1969). However, the necessary modifications are rather involved and we shall not discuss them here.

3 70

Nathan Wiser and A . J . Greenfield

There is one fly in this ointment. Once we visualize v(k, k’) as already containing higher order corrections, the form of v(k, k’) is no longer dependent only on the properties of a single atom. I t now depends as well on the geometrical arrangement of all the atoms, and our golden achievement, the factorization described in Eq. (1 3), is no longer rigorously true, Fortunately, there is reason to believe that this structural dependence of u(k, k’) is not that important. There have been many calculations (e.g., Cohen and Bergstresser, 1966) in recent years which have determined the form factor for one structure and applied it to a totally different structure, thereby ignoring entirely the structural dependence of v(k, k’). Nevertheless, these calculations have had a surprising measure of success in interpreting experimental data. We therefore conclude that the structural dependence of v(k, k’) can usually be safely ignored.

111. Simplification of the Form Factor As a first step toward the simplification of the form factor v(k, k’), we note that in practice, one invariably determines the form factor between plane waves, as in Eq. (13). Even when the pseudo-wave function is not a plane wave, as in Eq. ( I 1) for a periodic system, one proceeds by expanding wk(r) as a sum of plane waves

where G, is the nth reciprocal lattice vector. Inserting this expansion for wk(r) into Eq. (1 I), we obtain a sum of terms, each of which involves the pseudopotential between single plane waves. In all our subsequent discussion of evaluating v(k, k’), we shall always assume that we are dealing with one of the terms in such a sum and hence our v(k, k’) involves only single plane waves. However, even when considering a form factor involving only plane waves, we are still faced with the unpleasant task of evaluating u(k, k’), a function of six variables. Fortunately, for the cases of interest, namely the simple metals, semimetals, and semiconductors, the atoms are spherically symmetric, a fact which simplifies the problem immensely. By the expression “atom,” we mean what Ziman (1964) has aptly called the “neutral pseudoatom,” consisting of the closed-shell spherical ion together with its associated screening charge, which is also spherical in the usual approximation of linear screening. (The important concept of screening will be discussed in more detail presently.) For the case of spherical symmetry we shall designate the operator as D(r). From simple geometrical considerations one can show that for spherical symmetry r(k. k’) no longer depends on six variables but only on three, which are conveniently chosen to be q, k , and k‘, where q = I k - k‘ 1 .

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It should be clear that a spherically symmetric pseudopotential operator a(r) by no means implies a spherical Fermi surface. The symmetry of the operator 9 ( r ) is determined by the symmetry of a single atom (or screened ion), whereas the sphericity of the Fermi surface is dependent on the degree of validity of the nearly free electron model, which in turn depends on the magnitude of the matrix elements of D(r) near the Fermi surface. A very common approach to the evaluation of u(q, k , k ’ ) is to simplify it further by making what is called the momentum-independent or the local approximation. In this approximation, one replaces the pseudopotential operator D(r) by a simple function of position u(r). The effect of this approximation is to eliminate the dependence of the form factor on k and k‘, that is, u(q, k , k ’ ) --+ u(q). This can readily be seen from Eq. (13) for the form factor 1

u(q, k , k ’ ) = - jd3re-ik’rv(r)eik’‘r QO

When calculating the band structure of a perfect crystal, the local approximation leads to a form for the matrix elements which is extremely easy to treat. This is easily seen by rewriting Eq. (13) in the local approximation (kl V(r) I k’)

= S(k - k‘)u(q).

(16)

It is a straightforward matter to demonstrate that for a perfect lattice S(k - k’) is unity fork - k’ = G , , and zero otherwise. Thus, for this case, we only need to know u(q) = u(G,). Furthermore, we shall later see that u(q) vanishes when q exceeds several times G I , the smallest non-zero reciprocal lattice vector. Therefore, we need to know only a very few points on the u(q) curve for a complete band structure calculation. Recognizing this, Brust (1964) performed a complete variational band calculation for both Si and Ge using only three parameters, u(G,), u(G2), and u(C3).He determined these parameters by fitting to the three prominent energy gaps taken from optical data. Having determined these three parameters, he proceeded to a complete analysis of the reflectivity as a function of frequency. His overall agreement with experiment was quite good. A similar procedure, generalized to include spin-orbit coupling, was used by Falicov and Golin (1965), Falicov and Lin (1966), Lin and Falicov (1966), Golin (1968), and Weisz (1966) for computing the band structure of the semimetals As, Sb, and Bi, as well as metallic white Sn. Once again, after determining only four parameters, a complete band structure calculation was carried out including a determination of the Fermi surface. Comparison with Fermi surface experiments gave good agreement with this calculated Fermi

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Nathan Wiser and A . J . Greenfield

surface. It should be noted that such Fermi surface determinations provide a more stringent test of the band structure calculation than reflectivity calculations that involve averages over the band structure. In recent years, Cohen and his collaborators have used this approach for extensive calculations of the band structure of a large number of substances including semiconductors (Cohen and Bergstresser, 1966; Bergstresser and Cohen, 1967), insulators (Fong and Cohen, 1968), and ionic crystals (Cohen et al., 1967). Comparison with optical and other experimental data over a respectably wide range of energy was generally satisfactory. In view of all of this success, one might begin to wonder whether it is ever necessary to take into account the nonlocality of the pseudopotential. However, there are properties of materials for which the local approximation is clearly inadequate, for example, the cohesive energy, the elastic constants, and the phonon spectrum. The calculation of these properties involves taking sums over all of the electron states of the metal, rather than of select groups of electron states such as those on the Fermi surface. Moreover, for some metals, Zn and Cd, for example, even a band structure calculation does not give good agreement with experiment in the local approximation. However, Stark and Falicov (1 967) have shown that adding a carefully chosen parameterized nonlocal term to the pseudopotential achieves remarkably good agreement with a considerable number of experiments. These include de Haas-van Alphen experiments (Stark and Falicov, 1967), density of states (Allen et al., 1968) electron-phonon mass-enhancement factor (Allen et a/., 1968), temperature dependence of the Knight shift (Kasowski and Falicov, 1969; Kasowski, 1969b), radio-frequency size effect (Jones et al., 1968; Steenhaut and Goodrich, 1970), as well as the optical constants (Kasowski, 1969a). Harrison (1963a, 1964) has also made detailed calculations of the nonlocal character of the form factor. For a given q, he computed c(q, k , k’) for several values of k and k‘. If the local approximation were reasonable, the computed u(q, k , k’) would be nearly independent of k and k‘. In fact, the dependence of v(q, k , k’) on k and k’ is striking with differences of up to several electron volts in the important region q N G I (see Harrison, 1963a, Fig. 1 ; Harrison, 1964, Fig. I). This result occurred for each of the several metals that Harrison (1963b) examined. What is most significant about this result is that Harrison used a full pseudopotential calculation with no parameterization or phenomenology-in short, no fudging. In view of Harrison’s results, how is one to understand the success of the local approximation in many instances? We might as well admit straightaway that there is no good answer to this question. A partial answer sometimes given is that the integrals that enter the calculation tend to wash out the effects of the nonlocality, i.e., the k and k‘ dependence is less important for

PSEUDOPOTENTIALS-EMPHASIS

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373

the relevant integrals than might appear from examining the integrand. Still in all, it is a priori not clear when one can safely ignore the nonlocality of the pseudopotential. This is one aspect of the pseudopotential approach which remains an art, rather than a science. There have been a number of calculations that have explicitly taken into account the nonlocality of the pseudopotentials. The first serious effort to carry out detailed calculations of the form factor, including in full its nonlocal character, is the extensive work of Harrison (1963a,b, 1964) for eight simple metals. He calculated v(q, k k’) for various representative values of q, k , and k‘ and presented the results in tabular form. This work was a major step forward. However, we are here concerned primarily with providing a practical guide for obtaining form factors appropriate to liquid metals, without the need for extensive calculations. All of Harrison’s extensive calculations were carried out for the density corresponding to T = 0” K, and they cannot be adjusted to densities corresponding to other temperatures without repeating the entire detailed calculations. Therefore, further discussion of Harrison’s first-principles calculations is outside the scope of this review. For this same reason, we shall not discuss certain other proposed pseudopotentials, such as the GI model of Goddard (1968). The pseudopotential which has been most widely used is the HeineAbarenkov-Animalu model potential. In a series of papers, Abarenkov and Heine (1965), Heine and Abarenkov (1964), Animalu (1965, 1966), and Animalu and Heine (1965) developed a formulation of the pseudopotential which explicitly takes into account its nonlocal character. Its great advantage is that, for the first time, the nonexpert reader is enabled with relatively modest effort to obtain v(q, k , k’) for any desired values of these three variables. Moreover, suitable parameters for obtaining v(q, k , k’) are presented for no less than 25 elements. In view of these important advantages, the great popularity of the Heine-Abarenkov-Animalu potential is readily understood. It should be mentioned that Shaw and Harrison (1967; Shaw, 1968, 1969) have recently developed a method for optimizing some of the parameters of the Heine-Abarenkov-Animalu potential. There are many situations for which one is interested only in matrix elements for which both k and k’ lie on the Fermi surface. The calculation of the electrical resistivity, for example, or indeed of any transport coefficient, requires matrix elements of the pseudopotential only between electron states lying on the Fermi surface. In general, for a nonspherical Fermi surface, this restriction of both k and k’ to the Fermi surface provides no simplification whatsoever to u(q, k , k’). However, when discussing liquid metals, as well as solid Na and K, it is an excellent approximation to assume a spherical Fermi surface. For these cases, the Fermi wave number k , has the same value at each point on the Fermi surface, and therefore k = k‘ = k,. This leaves q

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Nathan Wiser and A . J . Greenfield

as the only remaining variable of the form factor v(q, k , , k,) = u(q). Since q is the change in wavenumber, and k and k' are restricted to the Fermi sphere, it is obvious that the maximum value of q is 2k,, the diameter of the Fermi sphere. It is well worth emphasizing that in spite of the fact that for a given k,, the form factor v(q) now depends only on a single variable q, nevertheless the nonlocality is fully included in v(q). It should be noted that were we to make a local approximation B(r) -,u(r), this would also lead to a function of a single variable q, but necessarily to a different function.

IV. Formulations of u(q) Useful for Liquid Metals We now turn to a detailed discussion of how to obtain the function v(q) in practice. There have been many suggested forms for v(q), leaving the nonexpert potential user in a quandary trying, often vainly, to assess the merits of the various pseudopotentials. It is our goal to provide for such users a handy guide to pseudopotentials and their intelligent use. We shall discuss the five forms for v(q) which are both applicable to a wide range of liquid metals, as well as being adjustable to the density changes which accompany temperature changes. From this point on, we shall follow the common ambiguous practice of referring to either v(q) or O(r) as the pseudopotential, rather than reserving the term pseudopotential for B(r) and designating v(q) as the form factor. A useful general form for a phenomenological pseudopotential due to an atom, or better, due to a Ziman (1964) neutral pseudoatom, can be obtained by separating v(q) into two parts 4 q ) = v i o n ( q ) + OeIec(q> (17) where vio,(q) is the contribution arising from the bare ion and uelec(q)is the contribution arising from the screening electrons. Cohen and Phillips (1961) have shown that, in the local approximation, the theory of linear screening permits one to express veleC(q)in terms of the dielectric function &(q),leading to the result

The simplest calculation (Lindhard, 1954) of E ( q ) is the self-consistent field approximation which gives the well-known formula

where x = q/2k,, and a, = h2/me2 is the Bohr radius. We shall defer a justification of Eq. (19) for E ( q ) until after a discussion of vion(q).

PSEUDOPOTENTIALS-EMPHASIS

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375

In a discussion of the various phenomenological forms of the pseudopotential, it is convenient to separate uiOn(q)into two parts

+ ucoul(4).

(20) The first term uCnre(4)arises from Ccore(r),by which we mean the nonlocal portion of the pseudopotential operator, Cohen and Heine (1961) have shown from quite general considerations that Ocore(r)vanishes outside the region of the core. We define the core region in terms of a core-radius parameter R,, which is generally taken to be somewhat larger than the actual core radius. The second term ucOul(q)arises from the Coulomb tail outside of the core region (see Fig. 1) u i o n ( 4 ) = ucnre(4)

_*_

Coulomb tail

region-

0

I

FIG.1. Graph of the Coulomb tail vcuul(r) vs. r . The core-radius parameter Re defines the extent of the core region within which the Coulomb tail is defined to be zero.

= 0,

r < R,

(21)

where z is the valence. Because ucou,(r) is a local potential, ucoul(q)is simply its Fourier transform,

I t is especially important to note that ucnre(4)should not be viewed as a Fourier transform of some core potential; because Ccore(r)is an operator, it has no Fourier transform in the usual sense. Rather, ucore(q)is given by

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Nathan Wiser and A . J . Greenfield

The importance of this distinction is that ucore(q)is really a function of k , , or in our earlier notation, ucore(q)= u,,,,(q, k , , k,). Therefore, when we shall later consider temperature changes, we must adjust the value of k , in ocore(q), as well as the more obvious adjustment of R,. By contrast, the only volume dependence in ucou,(q) is the normalization volume 52,. The various phenomenological potentials differ only in their choice of the core-radius parameter R, and the explicit form of the operator Dcore(r). The five potentials we shall discuss in turn are (1) the Harrison (1963b) (HAR) delta-function potential, (2) the Heine-Abarenkov-Animalu (Abarenkov and Heine, 1965; Heine and Abarenkov, 1964; Animalu, 1965, 1966; Animalu and Heine, 1965) (HAA) model potential, and its optimized version proposed by Shaw and Harrison (1967; Shaw, 1968, 1969), (3) the Cohen-Wiser (Cohen, 1962) (CW) potential, (4) the Ashcroft (1966, 1968; Ashcroft and Langreth, 1967) (ASH) potential, and (5) the Greenfield-Wiser (Greenfield and Wiser, 1971) (GW) potential. For the HAR, CW, and ASH potentials, Dcore(r)is written formally as a function of r, rather than as an operator. This function is to be viewed as the formal Fourier transform of ucore(q)= u,,,,(q, k , , k,), with ucore(q)formally continued for q > 2k,. Since this transforms depends on k , , we shall call it u:::(r, k,), where the k , dependence is commonly written as an E , dependence. Similarly, we shall define

uE;’(r, k F ) = ufcroa:,s(r,k F ) + ucoul(r). (24) Once again we emphasize that, viewed in this way, the nonlocality of these three potentials is fully included as long as we are dealing with matrix elements having initial and final electron states on the (spherical) Fermi surface. The HAR potential (see Fig. 2) is given by

u:zc(r, kF)

= P(kF)

6(r)

(25)

together with the condition that the core radius R, + 0. The physics of this potential is very easy to describe. It is assumed that since the ion core is reasonably small compared to the interatomic spacing, we can ignore its size altogether; hence, the &function. It then follows that all the effects of this point core can be lumped into a single strength parameter P(k,). The strength parameter is invariably positive since it represents primarily the fictitious repulsive potential which is central to pseudopotential theory. Typical values for p (Harrison, 1963b) are several tens of Rydberg-(Bohr radii)3. The Fourier transform is readily taken ucore(q)

= P(kF)/%

*

(26)

We see that ucore(q)is not a function of q at all, a result of the point-core approximation. It is fashionable to criticize the HAR potential on the grounds

PSEUDOPOTENTIALS-EMPHASIS

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377

that ucDre(q)does not approach zero as q co. However, Eq. (26) for ucDre(q) is defined only for q 5 2k,. Therefore, this criticism is misplaced. It is only when one interprets Eqs. (25) and (26) in an entirely different light that this criticism is in place. If one views Eq. (25) as a local approximation to Dcore(r), that is, u:,:(r, kF) ucDre(r)= pd(r), then ucore(q)is defined for all q and its large-q behavior is indeed a serious problem. Note, parenthetically, that another consequence of the local approximation is that the k, dependence of p disappears. Since we shall always view the HAR potential as a nonlocal potential, with q I 2k,, the large-q behavior is not relevant. -+

-+

fl potent la1

FIG.2. Graph of v ~ ~ " ' (k,) r , vs. Y for the HAR potential. The Coulomb tail extend all the way to the origin and the core is represented by a &function of strength P(kF)at the origin.

When considering volume changes, we must of course make some reasonable approximation for the kF dependence of P(kF). Happily, Harrison's (1963b) detailed calculation for A1 justifies ignoring this dependence entirely. Thus, we are left with a one-parameter, fully nonlocal potential. Harrison's (1963b) published values for p were determined by comparison with a firstprinciples calculation of u(q) at q = 2k,. It is now recognized that the state of the art for calculating u(q) is still no match for experiment. Therefore, for best results, one must determine p by matching to an experiment on the solid metal. The second potential on our list is the well-known HAA potential (Abarenkov and Heine, 1965; Heine and Abarenkov, 1964; Animalu, 1965, 1966; Animalu and Heine, 1965). This is the potential for which matrix elements have been calculated and published for the imposing number of over 25 elements. Furthermore, the general formula obtained by HAA is not limited to wave functions lying on the Fermi surface, but indeed, it is the only

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Nathan Wiser and A . J. GreenJieId

complete analytic expression for v(q, k, k ’ ) for arbitrary values of all three variables. For the HAA potential, I

=o

r > R,

(27)

where P,is the projection operator which picks out the lth angular-momentum component of the wave function and A , is an energy-dependent parameter having units of energy. Physically, this potential is a square well in the core region of depth A , , with the important proviso that each different angularmomentum component I of the wave function “feels” a well of different depth. A key point is of course the determination of the well depths A , ( E ) . HAA determine these parameters by fitting the energy levels arising from this potential to spectroscopic data observed for the isolated free ion. For simplicity, HAA set all A , = A , for 12 3. HAA have devoted considerable effort to the important question of determining the values of A , at the Fermi energy by extrapolation from the spectroscopic data which determine the A at a much lower energy. A second important point is the determination of the core-radius parameter R , . Unfortunately, they do not present any clear and unique way of determining this important parameter. All aspects of the HAA potential have been discussed in great detail in a series of technical reports, which also include tabulated values for v(q). These are available from the Solid State Theory Group at the Cavendish Laboratory, Cambridge, England. The most important of these tabulated values also appear in Harrison’s book (1966) on pseudopotentials. Important developments have recently occurred in the formulation of the HAA potential. Shaw and Harrison (1967) have corrected a conceptual inconsistency in the HAA potential. Subsequently, Shaw (1968, 1969) presented a somewhat improved version of the HAA potential, which we shall designate as the HAAS potential. The most important improvement was to propose a unique way of obtaining the optimum ” value of the core-radius parameter. The resulting pseudopotential is optimized in the sense that it produces the smoothest possible pseudo-wave function, in the sense discussed in detail by Cohen and Heine (1961). Shaw showed that optimization is achieved (i) by letting the core-radius parameter differ for each value of 1 (designated R , ) and (ii) by letting the values of R, satisfy

,

“

A , = ze2/R,.

(28)

This choice of R, makes each I-component of Dcore(r) join continuously to vcoul(r)at its core-radius R , . Because the HAAS potential is continuous for all r , its v(q) converges to zero for large q much more rapidly.

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Another change introduced by Shaw is to use the true potential, rather than the pseudopotential, for those higher I components of the wave function for which core states do not exist (e.g., only A , and A , are to be used for Mg which has only s andp core states). This eliminates the approximation of HAA that A , = A , for all 12 3. It can be shown that the mathematical content of Shaw's proposal is to replace each of the higher I constants A , by the function ze'jr. It is not hard to show on physical grounds that this replacement is very appropriate. The higher I components of the wave function do not extend very far into the core region. Therefore, their main contribution to the integral in Eq. (23) comes from the outer fringes of the core region where ze'jr 2: z e 2 / R , ,which in the Shaw optimization procedure is precisely equal to A , . Shaw (1968) gives the explicit form for the HAAS potential. For the matrix elements which we are considering ( k = k' = k,, E = E,), Shaw's expressions reduce to uion(q) =

-471(z - p)e2

Qo q2

477 10 R, 1 = 0

(

- - C ( 2 1 + 1)P, 1 - -

x A , kF /:R'dp(?.2

- k-, Ri Y ) ~ , ( Y )

(29)

where P l ( x ) and j , ( y ) are the Legendre polynomial and the spherical Bessel function, respectively. The parameters A , , R , , and the depletion hole p are tabulated for a number of metals in Tables I1 and 111 of Shaw (1968). The quantity I , is the highest I value for which a core state exists. The integral is easily performed for various values of k , as needed. The third potential is the CW potential (Cohen, 1962). Though it preceded the HAA potential, the former is actually a simplified form of the latter. The CW potential is also a square well type, nonlocal potential (see Fig. 3) and is given by trans

ucore

( r , kF) = - VdkF), = 0,

r < Rc r > Rc

(30)

leading to

As with the HAR potential, we ignore the k , dependence of the well depth V , , leaving two parameters Vo and R c . CW determined these two parameters for a number of metals by fitting to spectroscopic data for the isolated free ion. However, in the light of experience today, it is clear that a better procedure would be to compare to experimental data in the solid metal.

380

Nathan Wiser and A . J. Greenfield

I 0

potential Rc

r

FIG.3. Graph of v i y ( r , k F )vs. r for the CW potential. The core ( r < R,) is represented by a square well potential of depth Vo(kF).

Now we come to the fourth potential, the ASH potential (Ashcroft, 1966, 1968; Ashcroft and Langreth, 1967), which is a simplification of the CW potential. For the ASH potential, the parameter Vo is set equal to zero for all = 0 and ul::’(r, k F )= ucoul(r),as given in Fig. 1. metals, implying that This leaves only the single parameter R, in Eq. (22) to be determined. ASH wisely determines this parameter by fitting to experiments in the solid or liquid phase of the metal, rather than relying on free-ion data or a calculation. At first, the HAR potential with its u::z(r) equal to a delta function (Fig. 2) seems altogether different from the ASH potential with its equal to zero (Fig. I). But surprisingly enough, examination of v,,,(q) shows that these two potentials are almost identical. This can be seen by expanding the ASH potential as follows

u::z(r)

u::z(r)

where B A S H is q-independent. We immediately see that for sufficiently small q, where we can neglect the next term in the expansion of the cosine in Eq. (32), we get precisely the HAR potential. The ratio ofthe next term in theexpansion to the PAS”term is -(qR,)2/12 which has its maximum value when q = 2 k F . To take a few typical examples, the value of this ratio even at 2k, is only -0.21 for Na, -0.23 for K, and -0.36 for Al. Since the neglected terms are

PSEUDOPOTENTIALS-EMPHASIS ON LIQUID METALS

38 I

relatively small for all values of q, we can rewrite the last term in Eq. (32) to give BASH a modest q-dependence: BASH

-+

PASH(q)*

(33)

Thus, we see that the ASH potential is completely equivalent to the HAR potential except that BASH slowly decreases with increasing q, whereas PHAR is q-independent. One could equally view the matter from the opposite point of view. That is, one could say that the HAR potential is equivalent to the ASH potential except that the parameter R,,HAR has a modest q-dependence such that R,,HAR increases slowly with increasing q, whereas R,.ASH is qindependent. A point of general interest is that the well depth for all the square well type potentials is invariably negative or zero in the core region for all metals. This may seem strange in view of the repulsive nature of the fictitious potential central to pseudopotential theory. The explanation lies in the fact that Ocore(r) represents the sum of the very strong attractive potential and the almost equally strong fictitious repulsive potential. Numerically, it turns out that the effect is a slight net attraction, except for the ASH potential which artificially assumes perfect cancellation. Finally, we come to the fifth and last potential, the G W potential (Greenfield and Wiser, 1971). The GW potential is based on the recognition that it is not v::z(r) or Dcore(r), but rather ucore(q), which invariably enters practical calculations. Therefore, GW suggest the following simple empirical form

together with the condition that R, = 0 in uco,,(q). Thus,

Our discussion of the comparison between the HAR and ASH potentials shows that the GW potential is a generalization of these two. Note that two parameters, Po and are introduced in place of the single parameter P of HAR or R, of ASH. These two parameters are determined by comparison with experiments on the solid.

a2,

V. Screening Now that we have an understanding of the various phenomenological expressions in use for uion(q), we turn to the question of screening, contained in the term veleC(q) in Eq. (17). We have already pointed out that for local

382

Nathan Wiser and A . J. Greenjield

potentials, in the linear approximation the effect of uelec(q)is to divide ui,,(q) by the dielectric function e(q), as shown in Eq. (18). Even without knowing the true form of the dielectric function, it is possible to make some important comments regarding its general behavior. The exact, many-body e(q) must have the form

where x(q) approaches unity as q -+ 0 and approaches zero as q + 00. In the self-consistent field approximation for E ( q ) given in Eq. (19), we have

It is readily verified that this approximation to x ( q ) has the correct limiting form for both large and small q = 2k,x. Let us examine the small-q limit of u(q). In this limit, ucore(q)approaches a finite value and hence is negligible compared to ucoul(q)which diverges as q-', as seen from Eq. (22). This is because small q corresponds to large r , for which the core term is zero whereas the Coulomb term is long range, falling off only slowly as r - ' . Because the dominant Coulomb term is local, it is exact to use Eq. (18) for u(q). Thus, in the limit of small q, we have

-

-1E 3

F -

(38)

It is worth emphasizing that this well-known result involves no approximation whatsoever, either about the form of Dcore(r)or about the choice of the screening function. At the other extreme, for large q, there is no screening, since E ( q -+ 00) = 1. This can be understood by noting that q + co corresponds to r - 0 and screening out the potential down to r = 0 would involve piling a lot of electrons together at the origin. This is forbidden by the exclusion principle which demands that the electrons stay a respectable distance from each other. The relevance of the exclusion principle explains the appearance of EF in the expression for E ( q ) . It should be recognized that it is the exclusion principle and not the electron-electron interaction that prevents the electron from coming together at r = 0 to achieve perfect screening. The electron-electron interaction energy increases only as r-' for small r, whereas the exclusion principle produces an r-' dependence of the kinetic energy for small r

PSEUDOPOTENTIALS-EMPHASIS

ON LIQUID METALS

383

(easily seen from E , cs k,’ K RO-2/3cc r - ’ ) . Thus, the correct quantitative behavior is exhibited by the usual approximate self-consistent field expression for e(q), given in Eq. (19), even though this expression totally ignores electronelectron correlation and exchange. There are two popular approaches for improving the screening entering into the calculation of v(q). The first of these is to use the dielectric constant, that is, to assume local screening, but to improve e(q) by inserting correction factors to take into account such effects as correlation, exchange, orthogonalization, effective mass, etc. The most famous correction for correlation and exchange is due to Hubbard (1957, 1958). HAA have devoted considerable effort to including orthogonalization and effective-mass corrections to e(q). However, even after all this considerable labor on improving the calculation of e(q), it must be admitted that the true form is far from well established, as evidenced by the continuing stream of improvements (Sham and Ziman, 1963; Glick, 1963; Sham, 1965; Kleinman, 1968). The second approach for improving the treatment of the screening is to use nonlocal screening (Harrison, 1963a, 1966; Animalu, I966), rather than dividing ucore(q) by e(q). This procedure is indeed called for when dealing with a nonlocal potential. Unfortunately, with the available approximate treatment, nonlocal screening necessarily involves performing a number of nontrivial integrals for each desired value of q. The important practical drawback of nonlocal screening is the necessity of recalculating all these integrals each time the user wishes to apply such a pseudopotential to a different density, or to a nontabulated value of q. By contrast with local screening the user need merely insert the required value of k , and q into a given formula for e(q). This practical drawback of nonlocal screening is particularly annoying for work on liquid metals, where temperature dependences are generally very important. Moreover, tabulated values for v(q) are invariably calculated for the density at T = 0°K rather than the density of the liquid metal. All of the above corrections are quantitatively important when one is using a pseudopotential which requires determining velec(q) from a first-principles calculation. Hence, the uncertainty and/or difficulty of applying these corrections is a serious problem. However, if one determines the parameters of v(q) from an experiment on the solid, then it is no longer all that important to have a quantitatively correct screening function. Any error one makes by using an approximate local screening function is automatically compensated by the empirical fitting of the parameters of ucore(q).With such a fitting procedure, the screening function serves as merely an interpolation formula between measured points. Thus, any screening function having the proper qualitative behavior is quite adequate. In practice, the simplest local approximation to veleC(q),that is, use of e(q) given by Eq. (19), serves admirably.

384

Nathan Wiser and A . J . Greenjield

VI. Comments about the Various Pseudopotentials It is appropriate at this point to offer a few comments to aid in assessing the strengths and weaknesses of each of the five potentials discussed in Section IV. Each of these potentials contains one or more parameters. The most important feature of these potentials is the way in which their parameters were determined. For the HAA (Abarenkov and Heine, 1965; Heine and Abarenkov, 1964; Animalu, 1965, 1966; Animalu and Heine, 1965) (and HAAS, Shaw and Harrison, 1967; Shaw, 1968, 1969) and CW (Cohen, 1962) potentials, the parameters were determined by comparing to spectroscopic data of the free ion. As we have already pointed out, this corresponds to an experimental determination of ucore(q),leaving velec(q)to be determined by calculation. However, the theory of electron screening has not yet evolved to the point where accurate quantitative calculations can be carried out. Therefore, even if vcnre(q) were exact, the resulting v(q) would necessarily contain significant errors. This same criticism applies equally to the calculated values of p, the parameter characterizing the HAR (Harrison, 1963b) potential. In fact, the situation is even worse for these p values, since they correspond to a calculation of ucnre(q), as well as a calculation of uelec(q).It is esthetically very pleasing that, starting from first principles, Harrison has succeeded in calculating a u(q) which is reasonable. His calculation includes no input data other than the values of the fundamental constants e, ti, and m, as well as the valence and density of each metal he treats. However, for practical usage, where quantitatively accurate values of u(q) are necessary, the tabulated fl values, based on first-principles calculations, are not the optimum choice. The way to get around this problem is to determine u(q) itself from experiment, rather than ucnre(q),when evaluating the parameters in the potential. The advantage of this is obvious. At one stroke, one makes a best fit to both ucnre(q)and ueIec(q).In fact, with this procedure, the functional form of uelec(q),as well as of vcnre(q), serves essentially as an interpolation formula between experimentally determined points on the v(q) curve. For purposes of interpolation, even a qualitatively accurate choice for uelec(q)is quite sufficient to minimize any deviation from the true value of v(q) between determined points. For the ASH potential, Ashcroft (1966, 1968; Ashcroft and Langreth, 1967), wisely chose to determine the single parameter R, in ucore(q)by comparison with experimental data on the solid and/or liquid phase of the metal. However, it is rather disconcerting that among his various papers (Ashcroft, 1966, 1968; Ashcroft and Langreth, 1967), one can find quite different suggested values for the parameter R , for a given metal. The source of this difficulty is that Ashcroft unfortunately chose the electrical resistivity of the

PSEUDOPOTENTIALS-EMPHASIS

ON LIQUID METALS

385

liquid metal as the prime experiment for determining R,. However, the standard formula for the resistivity is not quantitatively reliable for liquid metals (Greenfield, 1966; Wiser and Greenfield, 1966; Greenfield and Wiser, 1967a,b, 1970; Adams and Ashcroft, 1967). Therefore, only those tabulated values of R, obtained from some other experiment should be used. The parameters Po and p2 of the GW potential were obtained by comparison with experiment on the solid, similar to the method of Ashcroft. The added flexibility of a second parameter in v(q) is shown to be necessary for achieving good agreement with experiment. Comparison with experiment is made at each reciprocal lattice vector G, < 2 k F , since one can essentially obtain v(G,) directly from experiment. For those metals having more than two such G,, Po and f12 are overdetermined and hence a measure of the accuracy of uGw(q)is readily obtained. The maximum error in uGw(q) for different metals ranges from 0.001 Ry to 0.01 Ry, being closer to the smaller value for most of the 11 metals treated.

VII. Conclusions There are two primary conclusions we have attempted to convey in this review. The first is that the pseudopotential has proved to be a major step forward in solid state physics, both conceptually and computationally. This is not to say that basic problems do not remain in the theoretical understanding of the underlying theory. Nevertheless, as in many other cases in the history of physics, overlooking such embarrassing difficulties and proceeding to use the pseudopotential has led to remarkable success in the interpretation of experimental data. The second conclusion is that the most successful procedure to date for obtaining accurate, reliable pseudopotentials is by empirical methods, utilizing direct comparison with experiment on the solid. By careful choice of an empirical method, for many metals one can achieve, with relatively little effort, a pseudopotential which is accurate to 0.01 Ry. It seems reasonable to expect that further efforts along these lines will produce pseudopotentials accurate to 0.001 Ry, that is, an order of magnitude greater accuracy. Such an achievement would lead to a much better understanding of the properties of solid and liquid metals.

ACKNOWLEDGMENTS It is a pleasure to thank Professor Marshall Luban for a critical reading of the manuscript. We also wish to thank Professor Marvin L. Cohen for sending us a copy of his review article prior to publication.

386

Nathan Wiser and A . J. Greenfield

REFERENCES Abarenkov, I. V., and Heine, V. (1965). Phil. Mag. 12, 529. Adams, P. D., and Ashcroft, N. W. (1967). Advan. Phys. 16, 597. Adams, P. D., and Leach, J. S. L. (1967). Phys. Rev. 156, 178. Allen, P. B., Cohen, M. L., Falicov, L. M., and Kasowski, R . V. (1968). Phys. Rev. Lett. 21, 1794. Animalu, A. 0. E. (1965). Phil. May. 11, 379. Animalu, A. 0. E.. and Heine, V. (1965). Phil. Mag. 12, 1249. Animalu, A. 0. E. (1966). Phil. Mag. 13, 53. Ascarelli, P. (1966). Phys. Rev. 143, 36. Ashcroft, N. W. (1966). Phys. Lett. 23, 48. Ashcroft, N. W. (1968). Proc. Phys. SOC. London (SolidStatePhys.) 1, 232. Ashcroft, N. W., and Langreth, D.C. (1967). Phys. Rev. 155, 682. Ashcroft, N. W., and Lekner, J. (1966). Phys. Rev. 145, 83. Austin, B. J., Heine, V., and Sham, 1.J. (1962). Phys. Rev. 127, 276. Ballentine, 1.E. (1966). Proc. Phys. SOC.London 87, 689. Baym, G . (1964). Phys. Rev. 135, A1691. Bergstresser, T. K., and Cohen, M. L. (1967). Phys. Rev. 164, 1069. Bradley, C. C., Faber, T. E., Wilson, E. G., and Ziman, J. M. (1962). Phil. Mag. 7, 865. Brust, D. (1964). Phys. Rev. 134, A1337. Busch, G., Guntherodt, H. J., and Tieche, Y . (1965). Phys. Verh. 16, 33. Cocking, S. J., and Egelstaff, P. (1968). Pvoc. Phys. SOC. London (Solid State Phys.) 1 , 561. Cohen, M. H. (1962). J . Phys. Radium 23, 643. Cohen, M. H., and Heine, V. (1961). Phys. Rev. 122, 1821. Cohen, M. H., and Phillips, J. C . (1961). Phys. Rev. 124, 1818. Cohen, M. L., and Bergstresser, T. K. (1966). Phys. Rev. 141, 789. Cohen, M. L., and Heine, V. (1970). SolidStatePhys. 24, 37. Cohen, M. L., Lin, P. J., Roessler, D. M., and Walker, W. C . (1967). Phys. Rev. 155, 992. Cusack, N. E. (1963). Rep. Progr. Phys. 26, 361. Cusack, N., and Kendall, P. (1961). Phil. Mag. 6, 419. Edwards, S. F. (1962). Proc. Roy. SOC.Ser. A 267, 518. Egelstaff, P. (1966). Rep. Pvogr. Phys. 29, 333. El-Hanany, U., and Zamir, D. (1969). Phys. Rev. 183, 809. Enderby, J. E., and March, N. (1965). Advan. Phys. 14, 453. Enderby, J. E., and North, D. M. (1966). Phil. Mag. 14, 961. Endo, H. (1963). Phil. Ma.9. 8, 1403. Faber, T. E. (1966). Advan. Phys. 15, 547. Faber, T. E., and Ziman, J. M. (1965). Phil. Mag. 11, 153. Falicov, L. M., and Golin, S. (1965). Phys. Rev. A 137, 871. Falicov, 1.M., and Lin, P. J. (1966). Phys. Rev. 141, 562. Fong, C. Y . ,and Cohen, M. L. (1968). Phys. Rev. Lett. 21, 22. Click, A. J. (1963). Phys. Rev. 129, 1399. Goddard, W. A. (1968). Phys. Rev. 174, 659. Golin, S. (1968). Phys. Rev. 166, 643. Greene, M. P., and Kohn, W. (1965). Phys. Rev. A 137, 513. Greenfield, A. J. (1964). Phys. Rev. A 135, 1589. Greenfield, A. J. (1966). Phys. Rev. Lett. 16, 6.

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Greenfield, A. J., and Wiser, N. (1967a). Achnn. Phys. 16, 591. Greenfield, A. J., and Wiser, N. (1967b). Adtan. Ph~.s.16, 601. Greenfield, A. J., and Wiser, N. (1970). Phys. Lett. A 32, 69. Greenfield, A. J., and Wiser, N. (1971). Proc. Phys. SOC.London (Solid State Phys.) to be published. Halder, N. C., North, D. M., and Wagner, C. N. J. (1969). Phys. Rev. 177, 47. Harrison, W. A. (1963a). Phys. Rev. 129,2512. Harrison, W. A. (1963b). Phys. Rev. 131, 2433. Harrison, W. A. (1964). Phys. Rev. 136, A1 107. Harrison, W. A. (1966). “ Pseudopotentials in the Theory of Metals.” Benjamin, New York. Harrison, W. A. (1969). Phys. Rev. 181, 1036. Heine, V. (1967). Phys. Rev. 153, 673. Heine, V., and Abarenkov, I. V. (1964). Phil. Mag. 9, 451. Hodges, L., Ehrenreich, H., and Lang, N. D. (1966). Phys. Reo. 152, 505. Hodgson, J. N. (1963). Phil. Mug, 8, 735. Hubbard, J. (1957). Proc. Roy. Soc., Ser. A 240, 539. Hubbard, J. (1958). Proc. Roy. SOC.,Ser. A 243, 336. Johnson, M. D., Hutchinson, P., and March, N. H. (1964). Proc. Roy. SOC.,Ser. A 282, 283. Jones, R. C., Goodrich, R. G., and Falicov, L. M. (1968). Phys. Rev. 174, 672. Kaplow, R . , Strong, S. L., and Averbach, B. L., (1965). Phys. Rev. A 138, 1336. Kasowski, R. V. (1969a) Phys. Rev. 187, 855. Kasowski, R . V. (1969b) Phys. Rev. 187, 891. Kasowski, R. V., and Falicov, L. M. (1969). Phys. Rev. Lett. 22, 1001. Kleinman, L. (1968). Phys. Rev. 172, 383. Lin, P. J., and Falicov, L. M. (1966). Phys. Rev. 142, 441. Lindhard, J. (1954). Kgl. Dan. Vidensk. Selsk. Mat.-Fys. Medd. 28, 8. Marwaha, A. S., and Cusack, N. E. (1965). Phys. Lett. 22, 556. Mott, N. F. (1966). Phil. Mug. 13, 989. Mott, N. F. (1967). Advan. Phys. 16, 49. Phillips, J. C., and Kleinman, L. (1959). Phys. Rev. 116, 287. Sham, L. J. (1965). Proc. Roy. SOC.,Ser. A 283, 33. Sham, L. J., and Ziman, J. M. (1963). Solid State Phys. 15, 221. Shaw, R. W. (1968). Phys. Rev. 174, 769. Shaw, R . W. (1969). Proc. Phys. SOC.London (Solid State Phys.) 2, 2335. Shaw, R. W., and Harrison, W. A. (1967). Phys. Reo. 163, 604. Springer, B. (1964). Phys. Rev. A 136, 115. Stark, R . W., and Falicov, L. M. (1967). Phys. Rev. Lett. 19, 795. Steenhaut, 0. L., and Goodrich, R. G. (1970). Phys. Rev. B 1, 451 I . Sundstrom, L. J . (1965). Phil. Mug. 11, 657. Thompson, J. C. (1965). Advan. Chem. Ser. SO, 96. Weisz, G. (1966). Phys. Rev. 149, 504. Wiser, N. (1966). Phys. Rev. 143, 393. Wiser, N., and Greenfield, A. J. (1966). Phys. Rev. Lett. 17, 586. Young, W. H., Meyer, A,, and Kilby, G. E. (1967). Phys. Rev. 160,482. Ziman, J. M. (1961). Phil. Mag. 6, 1013. Ziman, J. M. (1964). Advat7. Phys. 13, 89.

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Author Index Numbers in italics refer to the pages on which the complete references are listed. A Abarenkov, I. V., 373, 376, 377, 384, 386, 387 Aberth, W., 81, 82, 87, 94 Abragam, A,, 10, 40, 42 Abrikosov, A. A,, 305, 360 Adams, P. D., 364, 385, 386 Adams, W. H., 55, 60,87, 98, 107, 108, 116, 117, 118, 120, 138 Ahlrichs, R., 55, 60, 87, 91, 100, 138 Aiken, A. C . , 72, 87 Alder, B. J., 80, 94 Allan, D., 7, 8, 44 Allen, L. C., 59, 60, 65, 78, 87, 95 Allen, P. B., 372, 386 Allison, A. C . , 77, 84, 87, 89 Allison, D. C . , 85, 87 Altmann, S . L., 172, 219 Amdur, I., 80, 88, 90 Amemiya, A,, 68, 69, 70, 91, 145, 147, 150, 220 Amos, A. T., 137, 138, 182, 219 Anderson, P. W., 121, 138 Andresen, H. G . , 36, 43 Animalu, A. 0 . E., 373, 376, 377, 383, 384, 386 Arai, T., 162, 219 Arditi, M., 22, 24, 43 Armstrong, L., Jr., 257, 266, 270, 272, 275, 279, 282, 283, 285 Ascarelli, P., 364, 386 Ashcroft, N. W., 364, 376, 380, 384, 385, 386 Audoin, C., 4, 6, 12, 13, 17, 18, 19, 28, 32, 33, 41, 43, 44 Austin, B. J., 366, 386 Autler, S . H., 36, 43 Averbach, B. L., 364, 387 B Bagus, P., 54, 93 Baker, M., 3, 5, 7, 8, 45

Baht-Kurti, G. G., 73, 88 Ballentine, L. E., 364, 386 Balling, L. C., 10, 11, 32, 43 Bangham, M., 40, 43 Bardo, W. S., 17, 44 Bardsley, J. N., 238, 239, 242, 248 Barfield, M., 142, 219 Barnett, G., 80, 88 Barnett, M. P., 63, 88 Barr, T. L., 56, 88 Bartolotti, J., 57, 88 Basov, N. G., 2, 43 Bates, D. R., 49, 52, 53, 75, 77, 81, 84, 85, 88, 226, 232, 237, 248 Bauche, J., 282, 285 Baym, G., 305, 330, 360, 361, 364, 386 Bell, J. S . , 295, 296, 321, 360 Bender, C . F., 56, 60, 75, 77, 78, 80, 88, 92, 93, 112, 137, 138 Bender, P. L., 10, 11, 43 Berg, H. C . , 3, 11, 15, 31, 32, 43,44 Bergstresser, T. K., 370, 372, 386 Bernardini, O., 82, 87 Bernstein, R. B., 86, 92, 248 Berry, H. W., 81, 95 Berry, R. S . , 230, 248 Bersohn, M., 65, 95 Berstein, I., 2, 43 Bertoncini, P., 80, 88 Bethe, H. A., 75, 88 Bingel, W. A., 100, 138 Birnbaum, G., 86, 91 Birss, F. W., 66, 94, 185, 219 Bishop, D. M., 59, 88 Blaquiere, A., 2, 43 Bloom, S., 24, 43 Bogoliuboff, N., 14, 44 Bonaccorsi, R., 114, 115, 138, 139 Born, M., 100, 139, 144, 219 Bosomworth, D. R., 85, 88 Bottcher, C., 77, 88 Boys, S. F., 56, 57, 59, 61, 64, 88, 109, 111, 112, 113, 139

389

390

AUTHOR INDEX

Bradley, C. C., 364, 386 Brebner, M. A., 74, 90 Brenner, D., 4, 43 Brigman, G. H., 69, 88 Brink, D. M., 148, 202, 219 Brooker, R. H., 74, 88 Brout, R., 289, 360 Brown, R. E., 78, 88 Brown, R. T., 58, 88 Brown, W. B., 57, 86, 88, 92 Browne, J. C., 58, 59, 63, 75, 76, 77, 19, 80, 81, 84, 85, 86, 87, 88, 89, 90, 91, 92, 95 Bruch, L. W., 80, 88 Bruner, B. L., 56, 89 Brust, D . , 371, 386 Burhop, E. H . S . , 85, 88 Burke, E. A., 60, 91 Burke, P. H., 339, 360 Burnside, W., 172, 219 Busch, G., 364, 386

C

Cantu, A. A., 71, 73, 92 Carruthers, P., 289, 360 Carson, T. R., 49, 53, 85, 88 Cartan, E., 253, 285 Carver, T. R., 24, 43 Caves, T. C., 313, 337, 361 Certain, P. R., 135, 136, 139 Chan, A. C. H., 78,88 Chan, Y.M., 83, 89 Chao Chen-Chou, 253, 259, 286 Cheshire, I. M., 71, 89 Chiu, Y . N., 77, 89 Christoffersen, R. E., 62, 66, 89 Cizek, J., 55, 89 Claverie, P., 108, 137, I39 Clementi, E., 55, 65, 89, 167, 221 Cocking, S . J., 364, 386 Coffey, D. Jr., 82, 87, 89 Cohen, M. H., 118, 139, 289, 360, 366, 367, 369, 374,376,378, 379, 384,386 Cohen, M. L., 363, 370, 372, 386 Cohen-Tannoudji, C., 33, 34, 41, 42, 43, 44 Coleman, A. J., 145, 219 Colgate, S. O., 80, 88 Combrisson, J., 17, 43 Condon, E. U., 53, 89, 253, 260, 285

Conroy, H., 56, 64,89 Cook, D. B., 108, 139 Coolidge, A. S., 49, 90 Cooper, I. L . , 68, 69, 71, 74, 89 Corson, R. M., 147, 148, 219 Coulson, C. A,, 98, 139, 142, 219 Crampton, S. B., 3, 11, 15, 16, 30, 31, 43, 44 Crane, H. R., 31, 45 Crosswhite, H., 266, 270, 279, 285 Crosswhite, H. M., 266, 270, 285 Csanak, Gy., 288, 321, 330, 360 Cusachs, L. C., 126, 139, 140 Cusack, N., 364, 386 Cusack, N. E., 364, 386, 387 Cutler, L., 3, 5, 7, 8, 45

D Dagys, R., 265, 285 Daley, H. L., 82, 93 Dalgarno, A,, 57, 58, 59, 75, 77, 83, 84, 85, 87, 89, 95, 289, 337, 338, 360 Das, G . , 55, 56, 95, 167, 219 Das, T. P.,56, 91, 288, 360 Davidson, E. R., 50, 56, 60, 78, 80, 84, 88, 92, 93, 112, 137, 138, 175, 219 Davidson, W. D., 57, 83, 89 Debely, P. E., 4, 43, 45 Decious, D. R., 57, 91 Dehmelt, H. G., 10, 30, 43 Dellepiane, G., 64, 92 Del Re, G., 129, 139 de Mars, G., 8, 44 Demkov, Y . , 241, 248 Desaintfuscien, M., 6, 18, 19, 38, 43 Dewar, M. J. S., 190, 219 Dicke, R. H., 10, 11, 32, 45 Dietrich, K., 338, 360 Dimmock, J. O . , 195, 220 Diner, S . , 108, 137, 138, 139 Dirac, P. A. M., 102, 139 Doll, J . D., 228, 289, 297, 361 Donlan, V. L., 262, 285 Drake, G. W. F., 75, 89 Dunning, T. H., 133, 134, 139, 140, 337,360 Dutta, N. C., 56, 91, 288, 360 Dvoiatek, Z., 57, 89 Dworetsky, S., 82, 83, 89, 93

39 1

AUTHOR INDEX

Dyson, F. J., 330, 360 Dzyaloshinski, I. E., 305, 360

E Eckart, C., 54, 89, 142, 219 Edmiston, C., 60, 61, 89, 95, 102, 106, 120, 135, 139 Edmonds, A. R., 271, 274, 278, 285 Edwards, S. F., 364, 386 Egelstaff, P., 364, 386 Ehrenreich, H., 289, 360, 369, 387 El-Hanany, U., 364, 386 Eliezer, J., 248 Elliott, R. J., 260, 285 Ellis, D. E., 63, 64, 89 Empedocles, P., 74, 89 Empedocles, P. B., 143, 219 Enderby, J. E., 364, 386 Endo, H., 364, 386 Engeli, M., 66, 89 Epstein, I. R., 108, 140 Epstein, S. T., 56, 58, 59, 83, 89, 90 Everhart, E., 82, 90, 92, 232, 249 Eyring, H., 59, 68, 69, 89, 180, 219

F Faber, T. E., 364, 386 Falicov, L. M., 371, 372, 386, 387 Falkoff, D., 290, 295, 305, 339, 360 Fano, U., 10,43 Ferguson, A. F., 77, 89 Feshbach, H., 309, 361 Fetter, A. L., 296, 321, 360 Fettis, H. E., 74, 89 Feynman, R. D.,IO, 43, 288, 300,360 Field, G . B., 10, 44 Firsov, 0. B., 232, 248 Fischer, C. R., 81, 89 Fischer, I., 142, 219 Flannery, M. R., 62, 89 Fletcher, R., 58, 89 Flowers, B. H., 267, 269, 285 Fock, V., 101, 139 Foley, H. M., 82, 83, 93

Fong, C. Y . , 372, 386 Fong Kuei-Chun, 253, 259, 286 Ford, A. L., 75, 86, 89, 95 Forston, E. N . , 32, 43 Foster, J . M., 109, 111, 112, 139 Fraga, S., 185, 219 Francis, J. G . F., 74, 89, 90 Futrelle, R. P., 86, 90

G Gabriel, J. R., 64, 92, 150, 219 Galitski, V. M., 288, 290, 305, 314, 360 Gallup, G. A,, 55, 90, 175, 221 Garstang, R. H., 75, 90 Cell-Mann, M., 360 Gerratt, J., 71, 90, 144, 154, 220 Gershgorn, Z., 69, 90 Gilbert, M., 137, 138, 139 Gilbert, T. L . , 119, 120, 122, 139 Glaze, D., 7, 8, 44 Click, A. J., 383, 386 Gobeliewski, A,, 52, 95 Goddard, W. A,, 111, 55, 91, 93, 95, 143, 164, 182, 185,205,220, 373, 386 Goldberger, M. L., 312, 360 Goldenberg, H. M., 2, 3, 11, 12, 13, 20, 43, 44 Goldstone, J., 341, 360 Golebiewski, A., 78, 90 Colin, S., 371, 386 Goodisman, J., 57, 64, 88, 90 Goodrich, R. G., 372, 387 Gordon, J. P., 2, 43 Gorkov, L. P., 305, 360 Goscinski, O., 298, 360 Grad, J., 74, 90 Grant, D. M., 142, 219 Grasyuk, A. Z., 14, 17, 43 Green, T. A , , 81, 82, 90, 93 Greenawalt, E. M., 76, 90 Greene, M. P., 364, 386 Greenfield, A. J., 364, 376, 381, 385, 386, 38 7 Grivet, P., 4, 43 Grosset&te, F., 10, 43 Crotch, H., 30, 44 Guidotti, C., 64, 90

392

AUTHOR INDEX

Guntherodt, H. J., 364, 386 Gupta, B. K., 78, 90 Gush, H. P., 85, 88 Gutfreund, H., 337, 360

H Hagstrom, S., 72, 93 Halder, N. C., 364, 387 Hall, G. G., 54, 90, 172, 182, 219, 220 Hamermesh, M., 145, 147, 151, 220 Handler, G. S., 60, 91 Handy, N. C., 56, 57, 59, 64, 88, 90 Hansen, A. A., 77, 90 Hanson, R. J., 10, 11, 32, 43 Hara, K., 338, 360 Harkness, A. L., 80, 88 Haroche, S., 33, 34, 41, 43, 44 Harris, F. E., 55, 56, 58, 59, 60, 62, 63, 65, 68, 69, 70, 71, 77, 78, 79, 80, 90, 92, 94 Harrison, W. A., 364, 369, 372, 373, 376, 377, 378, 383, 384, 387 Hartmann, F., 12, 44 Hayes, E. F., 59, 90 Heastie, R., 86, 90 Hedin, L., 288, 360 Hegstrom, R., 30, 44 Hegyi, M. G., 56, 90 Heine, V., 118, 139,363, 366, 367, 369, 373, 376, 377, 378, 384, 386, 387 Heinz, O., 82, 94 Heitler, W., 180, 220 Helbig, H. F., 82, 90 Hellwarth, R. W., 10, 43 Hellwig, H., 7, 8, 33, 44 Henry, R. W., J., 59, 89 Herzberg, G., 48, 75, 84, 90, 147, 168, 197, 198, 220 Herzenberg, A., 238, 239, 242, 248 Hesser, J. E., 85, 90 Hinze, J., 58, 94, 167, 220 Hirschfelder, J. O., 49, 75, 93, 135, 136, 139 Hodges, L., 369, 387 Hodgson, J. N., 364,387 Hollis, P. C., 108, 139 Holm, L. M., 68, 69, 94 Hooke, R., 58, 90 Horak, Z . , 57, 89

Horie, H., 151, 220 Huang, K., 144, 219 Hubbard, J., 383, 387 Hughes, H. M., 31, 43 Huo, W. M., 78, 90, 142, 220 Hurley, A. C., 142, 143, 220 Hutchinson, P., 364, 387 Huzinaga, S., 59, 62, 65, 90, 95, 185, 220 Hylleraas, E. A., 54, 58, 90

1

Innes, F. R., 263, 285 Ishibara, T., 288, 360 Ishiguro, E., 68, 69, 70, 91, 145, 147, 150, 220 J Jahn, H. A., 143, 145, 147, 152, 157, 168, 220, 269, 285 James, H. M., 49, 90 James, T. C., 77, 90 Jamieson, M. J., 289, 360 Janev, R. K., 288, 321, 361 Jeeves, T. A., 58, 90 Johansson, A., 288,360 Johnson, B. R., 248 Johnson, E. H., 4, 5 , 7, 44 Johnson, M. D., 364, 387 Jones, L. L., 175, 219 Jones, R. C., 372, 387 Jordan, F., 137, 138, 139 Jordan, J. E., 80, 88, 90 Jousse, M., 24, 44 Joy, H. W., 60, 91 Jucys, A,, 265, 281, 285 Judd, B. R., 256, 257, 259, 266, 268, 269, 270,274,275, 277, 279, 281,282, 283, 285 Jungen, M., 55,91 Junker, B. R., 71, 81, 91 K Kadanoff, L. P., 305,330,360,361 Kahalas, S. L., 76, 91 Kaldor, U., 109, 139

AUTHOR INDEX

Kaplan, I. G., 143, 147, 149, 177, 220 Kaplow, R., 364, 387 Kapuy, E., 108, 137, 139 Karazija, R., 281, 285 Karl, G., 77, 91 Karplus, M., 65, 73, 88,94, 313, 337,361 Kasowski, R. V., 372, 386, 387 Kato, T., 295, 361 Kaufman, D. N., 56,94 Kay, K. G., 66,94 Kelly, H. P., 53,55, 56,77,91,288, 313, 337, 361 Kendall, P., 364, 386 Kennedy, M., 51, 52, 91 Kern, C. W., 76, 77, 92, 93 Kiang, H., 100, I39 Kilby, G. E., 364, 387 Kim, H., 60, 72, 91 Kimball, G . , 59, 68, 69, 89, 180, 219 Kimura, T., 68,69,70,91, 145, 147, 150,220 King, H. F., 60, 72, 91 Kirtman, B., 57, 91 Kirzhnits, D. A , , 290, 295, 305, 339, 361 Kiss, Z . J., 85, 91 Klein, A . , 296, 321, 361 Klein, D. J., 71, 80, 91 Kleinman, L., 118, 140, 366, 383, 387 Kleppner, D., 2, 3, 11, 12, 13, 15, 20, 30, 31, 32, 43, 44 Knudson, S. K., 52, 91 Kobayashi, T., 295, 361 Kohn, W., 232, 248, 364,386 Kolker, H. J., 78, 91 Kolos, W., 50, 57, 76, 84, 91 Koster, G. F., 195, 220, 259, 260, 262, 264, 273,278,280, 285 Kotani, M., 68, 69, 70, 91, 145, 147, 150, 198,220 Kramling, R. W., 55, 93 Krauss, M., 54, 60, 61, 65, 78, 79, 89, 91 Kryloff, N., 14, 44 Kunz, A. B., 119, 139 Kutzelnigg, W., 55, 60,87, 91, 100, 138, 139 L Ladner, R. C., 55, 91, 164, 220 Laine, D. C., 17, 44 Lamb, W. E., 12, 44

393

Lambe, E. B., 31, 44 Land, R. H., 63, 65, 95 Landau, L. D., 229, 232, 248 Lang, N. D., 369, 387 Langenberg, D. N., 29, 45 Langreth, D. C., 376, 380, 384, 386 La Paglia, S. R., 77, 91, 92 Larson, D. J., 30, 44 Larsson, S . , 60, 91 Layzer, A. J., 309, 361 Leach, J. S. L., 364, 386 Ledsham, K., 49, 53, 84, 88 Lee, T., 56, 91 Lehn, J. M., 138, 139 Lekner, J., 364, 386 Lemmer, R. H., 338, 361 Lennard-Jones, J. E., 99, 102, 103, 105, 139, 142,172,220 Lester, W. A,, Jr., 61, 91 Letcher, J. H., 133, 134, 139 Levine, H. B., 86, 91 Levine, M., 3, 4,5, 7, 8, 44, 45 Levine, R. D., 248 Levy, B., 138, 139 Levy, H., 62, 77, 89, 91 Levy, R., 138, 139 Lichten, W., 49, 51, 52, 82, 91, 233, 248 Lifschitz, E. M., 229, 248 Lin, P. J., 371, 372, 386, 387 Linderberg, J., 288, 361 Lindhard, J., 374, 387 Lindner, P., 298, 360 Linnett, J. W., 143, 219, 220 Lippmann, B. A . , 239,240,242,245,249 Lipscomb, W. N., 71, 76, 77, 90, 93, 108, 109, 110, 140, 144, 220 Lipsky, Z., 52, 92 Little, W. A , , 337, 360 Littlewood, D. E., 151, 177, 220, 272, 277, 285 Li Yung-Hwa, 253, 259, 286 Lockwood, G. J., 82, 92 Lowdin, P.-O., 53, 55, 60, 68, 69, 70, 71, 92, 99, 100, 101, 102, 112, 135, 139. 143, 182, 220, 297, 361 London, F., 230, 249 Lorents, D. C . , 81, 82, 87, 89, 94 Low, F. E., 360 Lundquist, B. I., 288, 360 Lundquist, S., 288, 360

394

AUTHOR INDEX

Lutz, B. L., 85, 90 Lykos, P. G., 130, 139 Lyness, J. N., 64, 92 Lyubarskii, G . Y . , 177, 220

M McCarroll, R., 75, 77, 88, 92 McCoubrey, A. O., 4, 44 MacDonald, J . K. L., 58, 92 McElwain, D. L. S., 74, 95 McGee, I. J., 80, 88 McGunigal, T. E., 4, 5, 7, 44 McKoy, V., 56, 66, 92, 93, 337, 360 Mackrodt, W. C., 77, 92 McLaughlin, D. R., 80, 94 McLean, A. D., 55, 63, 92,95 McLellan, A. G., 272, 285 McQuarrie, D. A,, 86, 92 McVicar, D. D., 339, 360 McWeeny, R., 55, 56, 68, 69, 70, 71, 77, 89, 92, 95, 108, 139 Magnasco, V., 64, 92, 108, 122, 123, 124, 125, 128, 129, 239 Magnusson, E. A., 64, 92 MaksiC, Z., 129, 140 Malli, G., 56, 89 Malrieu, J. P., 108, 137, 138, 139 Mandl, F., 238, 239, 242, 248 Manning, P. P., 167, 221 March, N., 364, 386 March, N. H., 290, 295, 305, 339,361, 364, 387 Marchetti, M. A., 77, 92 Marchi, R. P., 51, 52, 82, 92, 94 Maric, Z. D., 288, 321, 361 Marriot, R., 85, 88 Martin, D. H., 86, YO Martin, P. C., 288,290,305, 330, 339,361 Martin, R. S., 74, 92 Marvin, H. H., 263, 285 Marwaha, A. S., 364, 387 Massey, H. S. W., 75,88, 92, 232, 248, 309, 361 Matcha, R. L., 57, 77, 86, 92 Mathur, B. S., 30, 44 Matsen, F. A., 58, 67, 68, 69, 70, 71, 73, 78, 80, 88, YO, 91, 92, 147, 221

Matsubara, C., 288, 360 Matsubara, T., 288, 290, 305, 330, 339, 361 Matsumoto, G., 80,92 Mattheiss, L. F., 107, 139, 150, 151, 221 Mattuck, R. D., 290,305, 339, 361 Mehler, E. L., 5 5 , 63, 92 94 Melvin, M. A., 177, 221 Menoud, C., 7,44 Metiu, H. I., 138, 139 Meyer, A., 364, 387 Mezei, M., 56, 90 Michels, H. H., 58, 63, 65, 78, 79, 82, 90, 91, 92,93 Migdal, A. B., 288, 290, 305, 314, 360,361 Miller, J. M., 63, 92 Miller, K. J., 53, 55, 60, 92 Millie, P., 138, 139 Mizushima, M., 77, 92 Monkhorst, H. J., 65, 92 Morrison, R. C., 175, 221 Morse, P. M., 309, 361 Mott, N. F., 75, 76, 92, 309, 361, 364, 387 Mueller, C. R., 81, 93 Mueller, L., 3, 5 , 7, 8, 45 Mulliken, R. S., 123, 139 Munsch, B., 138, 139 Murnaghan, F. D., 268, 285 Murrell, J . M., 129, 139 Musher, J. I., 137, 138 Myint, T., 31, 44

N

Naniiki, M., 295, 296, 321, 361 Nazaroff, G. V., 52, 95 Nesbet, R. K., 53, 54, 55, 56, 61, 68, 69, 70, 72, 74, 86,92,93, 98, 112, 116, 140 Newton, M. D., 108, 109, 110, 140 Nicholls, R. W., 75, 84, 93 Nieh Shun-Chuen, 253, 259, 286 Nielson, C. W., 259,260,262,264,273,278, 280,285 Nikitin, A. I., 17, 44 North, D. M., 364, 386, 387 Novick, R., 82, 83, 89, 93 Nutter, P. B., 261, 263, 266, 272, 281, 283, 285

395

AUTHOR INDEX

0

Ohm, Y., 60, 93 Olive, J. P., 58, 93 Olson, R. E., 81, 82,87, 93 O’Malley, T. F., 49, 51, 52, 93, 239, 240, 241, 242, 243, 245, 249 0-ohata, K., 65, 95 Oppenheimer, R., 100, 139, 144, 219 O’Raifeartaigh, L., 258, 285 Orayevskiy, A. N., 14, 17, 43 Oskutz, I., 56, 94

Pople, J. A., 99, 102, 103, 105, 139, 142,220 Poshusta, R. D., 55, 59, 71, 88, 91, 92,93 Present, R. D., 49, 90, 93 Primas, H., 98, 140 Pritchard, H. O., 74, 95 Pritchard, R. H., 77, 93 Prokhorov, A. M., 2, 43 Prosser, F., 72, 93 Pu, R. T., 288, 360 Purcell, E. M., 10, 44

R P Pack, R. T., 49, 75, 93 Palke, W. E., 49, 75, 93 Parker, W. H., 29, 45 Parr, R. G., 59, 60, 61, 72, 90, 93, 95 Pasternack, S., 273,285 Pauling, L., 71, 93 Pauncz, R., 129,131,140,143, 182, 190,221 Peek, J. M., 53, 82, 93 Pekeris, C. L., 76, 94 Percus, J. K., 68, 93 Perel, J., 82, 93 Perico, A., 108, 122, 123, 124, 125, 128, 129, 139 Perkins, J. F., 60, 93 Peters, D., 126, 127, 128, 140 Peters, G., 74, 92, 93 Peters, H. E., 3, 4, 5, 7, 15, 16, 24, 44, 45 Petersson, G . A., 66, 93 Petrongelo, C., 114, 115, 138, 139 Phelps, A. V., 85, 93 Phillips, J. C . , 118, 140, 366, 374, 386, 387 Phillipson, P. E., 79, 80, 93 Piejus, P., 6, 38, 43 Pincelli, V., 137, 138, I39 Pines, D., 289, 305, 361 Pipano, A., 131, 140 Pipkin, F. M., 10, 11, 32, 43 Pitzer, R. M., 76, 77, 93, 109, 140 Pi Wu-Chi, 253, 259, 286 Polak, R., 129, 140 Politzer. P., 126, 139, 140 Poll, J. D., 77, 91

Racah,G., 143,155,221,252,253,254,257, 258, 260, 261, 262, 263, 266, 267, 268, 270, 272, 275, 278, 281, 286 Racine, J., 7, 44 Rajagopal, P., 64, 88, 93 Ramsey, N. F., 2, 3, 4, 7, 11, 12, 13, 15, 20, 30, 31, 32, 43, 44, 45 RandiC, M., 129, 140 Ransil, B. J., 131, 140 Reeves, C . M., 68, 69, 72, 93 Reid, C. E., 60, 93 Reinhardt, W. P., 288, 289, 297, 323, 335, 361 Richards, W. G., 77, 95 Roberts, H. G. F., 137, 138 Roberts, P. J., 66, 77, 93 Robinson, E. J., 83, 93 Robinson, H. G., 30, 31, 43, 44 Rodriguez, C . E., 80, 91 Roessler, D. M., 372, 386 Roman, P., 290, 295, 305, 339, 361 Roothaan,C. C. J., 54,93,98,140,142,166, 167, 168,220,221 Ros, P., 63, 89 Rose, M. E., 48, 93 Rosenthal, H., 82, 83, 93 Rotenberg, A,, 68, 93 Rothenberg, S., 50, 84, 93 Rowe, D. J., 337, 361 Rudzikas, Z., 281, 285 Ruedenberg, K., 53, 55, 60, 62, 63, 66, 89, 92, 94, 102, 106, 120, 121, 135, 139, I40 Rumer, G . , 151, 221 Russek, A., 52, 77, 92, 94, 233, 249

396

AUTHOR INDEX

S Sabelli, N., 58, 94 Sack, R. A., 66, 94 Sakai, Y., 59, 90 Salmon, L. S., 66,94 Salpeter, E. E., 75, 88 Salvetti, O., 64, 90 Samathiyakamit, V., 288, 360 Sampanthar, S., 290, 295, 305, 339, 361 Satchler, G. R., 148, 202, 219 Schaad, L. J., 60,91 Schaefer, H. F., 111, 55, 56, 64, 69, 78, 80, 94 Scherrnann, J. P., 6, 18, 19, 38, 41, 43, 44 Schiff, B., 76, 94 Schmeising, H. N., 130, 139 Schneider, B., 288, 289, 305, 330, 331, 333, 361 Schneiderman, S. B., 77, 94 Schroder, D. M., 77, 92 Schulman, J. D., 56, 94 Schwartz, C., 32, 44 Schweber, S. S., 361 Schwinger, J., 288, 290, 305, 314, 315, 330, 339,361 Scrocco, E., 114, 115, 138, 139 Secrest, D., 68, 69, 94 Serber, R., 151, 179, 197, 221 Sham, L. J., 366, 383, 386, 387 Sharrna, R. R., 66,94 Shavitt, I., 65, 69, 74, 75, 88, 90, 94 Shaw, R. W., 373,376,378,379,384,387 Shimoda, K., 2, 13, 44 Shirley, J. H., 8, 44 Shi Sheng-Ming, 263, 286 Shortley, G. H., 253, 260, 285 Shudeman, C. L. B., 256, 286 Shull, H., 71, 78, 88, 94 Shu Ping-Huo, 253, 259, 286 Siga, M., 198, 220 Silver, D. M., 5 5 , 62, 94 Silverstone, H. J., 61, 66, 94, 95 Sinanoglu, 0,,53, 5 5 , 56, 77, 91, 94, 99, 125, 140 Singer, K., 59, 61, 94 Skutnik, B., 99, 140 Slater, J. C., 54, 70, 72, 94, 253, 286 Slichter, C. P., 22, 44 Smith, F. J., 51, 52, 91

Smith, F. T., 49, 51, 52, 75, 76, 81, 82, 89 92, 93, 94, 224, 247, 249 Smith, P. R., 277, 286 Smith, V. H., Jr., 68, 94 Smith, W. H., 85, 95 Smith, W. W., 82, 89 Somorjai, R. L., 61, 95 Soshnikov, V. N., 84, 95 Spitzer, L., Jr., 85, 88 Springer, B., 364, 387 Squires, E. J., 295, 296, 321, 360 Stanton, R. E., 60, 72, 91 Stark, R. W., 372, 387 Statz, H., 8, 44, 195, 220 Stecher, T. P., 84, 95 Steenhaut, 0. L., 372, 387 Steinborn, O., 66, 95 Steiner, E., 56, 92 Stephens, T. L., 84, 89 Sternheimer, R. M., 273, 285 Stevens, K. W. H., 260, 285 Stevens, R. M., 108, 109, 110, 140 Stewart, A. L., 49, 53, 75, 84, 88, 93, 232, 248 Stewart, G. W., 74, 95 Stewart, R. F., 59, 95 Strakhovskiy, G. M., 17, 22, 44, 45 Strong, S. L., 364, 387 Stuckelberg, E. C., G., 232, 249 Sundstrom, L. J., 364,387 Sutcliffe, B. T., 56, 68, 69, 70, 72, 77, 95 Switkes, E., 108, 109, 110, 133, 140 Szabo, A,, 323, 335, 361 Szondy, E., 65,95 Szondy, T., 56, 90, 95

T Taketa, H., 65, 95 Tamir, I., 190, 221 Tang, K. C., 61, 95 Tan Wei-Han, 253, 259, 286 Taylor, B. N., 29, 45 Taylor, H. S., 52, 78, 90, 95, 242, 248, 249, 288, 289, 321, 330, 331, 333,360,361 Taylor, L. H., 266, 275, 279, 282, 285 Taylor, W. J., 66, 95, 105, 140 Teller, E., 168, 220 Thompson, J. C., 364, 387

397

AUTHOR INDEX

Thorson, W. R., 52, 76, 77, 80, 91, 95 Thouless, D. J., 289, 361 Tieche, Y., 364,386 Tolk, N., 82, 89 Tomassi, J., 114, 115, 138, 139 Tong Pong-Fu, 253,259,286 Tossel, J. A., 108, 140 Townes, C. H., 2, 13, 36, 43, 44 Trees, R. E., 264, 266, 272, 286 Trindle, C., 125, 140 Tse Ting-Ting, 253, 259, 286 Tsuei Chen-Chia, 253,259, 286

U

Ulrich, B., 86, 95 Undheim, B., 54, 58, 90 Unland, M. L., 134, 140 Uspenskiy, A. V., 22,45 Uzigiris, E. E., 4, 7, 45

V

Valberg, P. A., 30, 44 Vanier, J., 3, 7, 10, 11, 13, 15, 16, 24, 44, 45 Van Vleck, J. H., 148, 221 Van Wazer, J. R., 134, 140 van Wieringen, H., 145, 220 Veillard, A., 55, 89, 167, 221 Veneroni, M., 338, 361 Vernon, F. L., 10,43 Vessot, R. F. C., 3, 4, 5, 7, 8, 11, 15, 16,24, 44,45 Victor, G. A., 83,95,289, 337, 338,360 Viennet, J., 28, 43 Vizbaraite, J., 281, 285 Voge, H. H., 198, 221 von Neumann, J., 51, 95, 229, 249

W Wadzinski, H. T., 266, 270, 277, 281, 285 Wagner, C. N. J., 364, 387 Wahl, A. C., 55, 56, 63, 65, 80,88, 95, 142, 167,219,221 Walker, T. E. H., 77, 95

Walker, W. C., 372, 386 Wallis, A., 74, 95 Walter, J., 59, 68, 69, 89, 180, 219 Wang, T. C., 2, 13, 44 Watson, K. M., 296, 312, 321, 360 Weare, J. H., 61, 93, 95 Weber, T. A., 61, 95 Weinbaum, S., 142, 221 Weinstein, H., 129, 131, 140 Weiss, A. W., 76, 95 Weisz, G., 371, 387 Welsh, H. L., 85, 91 Weyl, H., 271, 286 Wheeler, R. G., 195, 220 Whisnant, D. M., 57, 86,88 Whitten, J. L., 59, 65, 95 Whittington, S. G., 65, 95 Wigner, E. P., 51,95,145,147,152,221,229, 249 Wilkinson, D. T., 31, 45 Wilkinson, J. H., 74, 92, 93, 95, 310, 361 Williams, D. A., 52, 77, 81, 83, 84, 88, 89, 95,226, 248 Williams, J. K., Jr., 248 Wilson, C. W., Jr., 55, 95 Wilson, E. G., 364,386 Wilson, G., 61, 95 Wind, H., 53, 95 Wiser, N., 364, 376, 381, 385, 387 Wittke, J. P., 10, 11, 32, 45 Wolniewicz, L., 50, 76, 84, 91, 95 Wright, J. P., 59, 95 Wyatt, R. E., 60, 72, 91 Wybourne, B. G., 261, 267,272,277,286

Y Yamanouchi, T., 147, 221 Yaris, R., 288,289, 321, 330, 331, 333,360, 361 Yoshimine, M., 55, 63, 92,95 Young, W. H., 290,295, 305, 339,361, 364, 387 Yue, C. P., 61, 95 Z Zamir, D., 364, 386 Zandomenghi, M., 64,90

398 Zaremba, S. K., 65, 95 Zauli, C., 64,92 Zehr, F. J., 81, 95 Zeiger, H. J., 2, 43 Zemach, C., 296, 321,361 Zener, C., 231, 232, 249

AUTHOR INDEX

Ziemba, F. P., 232,233, 249 Ziman, J. M., 364, 370, 374, 383, 386, 387 Zitzewitz, H., 7 , 8, 44 Zitzewitz, P. W., 4, 7 , 45 Zivanovic, D. J., 288, 321, 361 Zmuidzinas, J. S., 361

Subject Index B

A Adiabatic, see also Coupling operators for adiabatic states approximation, 226 electronic states, 223, 224, 228, 229, 233-236, 245 decoupling of, 296 fast passage, 6, 36 methods, 289 motion, 224 molecular states, 49-52, 75, 81, 82 principle, 295 Ammonia maser, 2, 12, 13. 17 molecule, 178 Amplitudes, Feynman, 298 Angular momentum coupling operator for, 75 electronic, 48, 67-69 electronic spin, 48, 67-69 nuclear rotation, 48 Analytic continuation, 245 Annihilation operator, 256 Ashcroft potential, 380, 381 At on1ic beam of cesium, 2 source of H atoms, 5 Atomic collisions, excitation in, 243 Atomic deuterium, hyperfine splitting of, 29 Atomic hydrogen, 5ff Bloch equation for, 13 double resonance in, 33-37 hyperfine splitting of, 8, 29 interaction with rf fields, 33, 34 Lamb shift in, 42 magnetic dipole moment, 9 multiple quantum transitions in, 40 Starck shift in, 32 Zeeman spectrum of, 41 Atomic nitrogen, 178 quadrupole coupling constant of, 31 Atomic tritium, quadrupole coupling constant of, 29 399

Banana bonds, 106 Band structure calculations, 371, 372 Basis standard, 149, 202 valence bond, I51 Young-Yamanouchi, 147 Basis sets, see also Orbital basis sets correlated, 56, 60, 61 of electronic states, 224, 225, 228, 230, 235, 239, 241, 243, 245 linear transformation of, 59, 61 of niolecular states, 47-50 orbital, 59-61 single center, 59 of spin functions, 157ff. 199, 201 Benzene molecule, 189 Bethe-Salpeter amplitude definition of, 300, 327 relation to generalized oscillator strength, 305 relation to transition density, 305 in the spectral representation for the holeparticle Green’s function, 302-304 time dependence of, 301 Bethe-Salpeter equation, 317-321, 324-328, 333, 348-354 Binding energies of molecules, 142, 206 Bi-orthogonal functions, 309, 310 Bloch equation for H-atoms, 13 Bloch potential, 366 Bonding in molecules, 194ff Born-Oppenheimer approximation, 144, 226ff, 246-247 Bulb storage averaging, 12 bulb coating, 4, 7 general, 2, 16 time constant, 12 C

Canonical orthogonalization, 298 Charge exchange, 232ff

400

SUBJECT INDEX

Charge distribution, 62, 63, 65 Chlorine molecule, 230-232, 234-236, 248 Clebsch-Gordan coefficient, 148, 185, 195, 21 1 Closed shell atoms, 192-194 Coefficients of fractional parentage, 143 Cofactors of determinants of overlap integrals, 71,72 of orbital transformation matrices, 73 Cohen-Wiser potential, 379 Cohesive energy in metals, 372 Collisions, see also Cross sections atomic, 243 heavy particle, 243 Commutators, 259, 274ff Configuration excited interaction, 54, 58, 109, 116, 170 Continuation, analytic, 245 Convergence, 116 Correlation angular, axial, and longitudinal, 175 dependence of correlation energy on internuclear separation, 55, 80 effect on transition matrix elements, 77 electronic, 112, 137, 383 between electrons with opposing spins, 142 energy, 292 functions, 292 formalism for computation of correlation energy, 53-56 horizontal, 175 interorbital, 106 pair, 137 radial and vertical, 190 treatment, 116 Coulomb interaction, 261, 270 Coupled equations for nuclear motion, 226, 23 1, 24 I , 246, 247 Coupling electronic, 226, 227, 230, 231, 235, 237, 238, 241 Hund rule, 180, 194, 197, 203 kinetic, 226, 227, 247 no coupling, 228, 229 rotational, 226, 247 of spin functions, 148, 152ff, 160, 165, 167, 177, 182 Coupling operator for adiabatic states electric dipole, 50, 75, 77, 78 due to nuclear motion, 49, 75-77

due to .relativistic term in Breit-Pauli Hamiltonian, 77 Covalent electronic states, 230ff, 234, 236 Creation operator, 256 for quasi-particles, 258 Cross sections for He+ on He, 81, 82 for He on He, 79-81 for Lit on He, 81 on Li, 82 oscillator structure of total, 82, 83 perturbation ofelastic differential,81, 82 for photon emission, 85 for photon absorption, 84-86 Crossing of potential energy surfaces, 229, 230, 233, 234, 236, 237, 243-245 D

Decoupling adiabatic, 296 procedure, 330 Density localized, 106 maximal overlap, 129 molecular charge, 128 operator, 125, 134 Density matrix, 71, 289, 297 diagonal elements of, 102 expression with the one-particle Green’s function, 293 natural expansion of, 101 one and two-particle reduced density matrix, 99, 101, 161 operator, 292 Deuterium, see Atomic deuterium Diabatic states, 223, 224, 232, 236, see also Electronic states of molecules Diabatic molecular states, 51, 52 Diagonal representation of electronic Hamiltonian, see Electronic states of molecules Diamagnetic shielding, 31 Dipole moments of He-H, 86 of He-Ar, 86 Dissociative attachment, 236-243 Dissociative recombination, 236-238, 243 Doppler effect first order, 3 second order, 12

401

SUBJECT INDEX

Double resonance, 34 Dynamical behavior, 14-19 Dyson equation, 307-311, 314, 318, 322, 329, 339, 342 Dyson orbitals, 320, 335

E Eigenfunction electronic, 144ff spin-valence theory of, 180 Eigenstates of electronic Hamiltonian, 224, 228, 234, 236, 245 Eigenvalue problems arising from molecular calculations non-Hermitian, 310 real-nonsymmetric, 74 real-symmetric, 73 Elastic scattering, 295, 321, 337, 338 Electron affinity, 311 attachment energy, 298 Electron-pair function, 142 Electronic eigenfunctions, 144ff spin-valence theory of, 180 Electronic Hamiltonian diagonal representation of, see Electronic states of molecules eigenstates of, 224, 228, 234, 236, 245 Electronic molecular wave functions, 47, 48 configuration interaction, 54 directly correlated, 55, 56 formalisms for computation of, 53-59 symmetry adaptation of, 67-69 valence bond, 58, 59 Electronic population, 123 Electronic states of molecules, see also Basis sets adiabatic (stationary, diagonal), 223, 224, 228, 229, 233-236, 245 covalent, 230ff, 234, 236 degenerate, 168 diabatic (nonstationary, nondiagonal) ionic covalent, 224, 230-232, 246 quasi-stationary (quasi-adiabatic), 224, 236ff, 240-246 resonance potential scattering, 239-249 single configuration molecular orbital, 224, 232-237, 243, 246 valence bond, 235, 243, 246

Energy localized functions, 106 Excitation energy, 304, 305 in atomic collisions, 243

F Factorization of matrix elements, 368, 369 Fermi surface, 373, 374 Feynman amplitudes, 298, 322, 329 Fictitious spin representation, 3, 22 Field operator, 290, 291, 295, 306 Fine structure constant, 29 Form factor, 365, 368-370 Fractional parentage, coefficients of, 143, 150, 157, 211 Franck-Condon overlap factor, 239 Franck-Condon principle, 23 1 Frequency dependent polarizability, 288, 289, 326, 327 Frequency stability, 2, 3, 8 Functional differentiation method, 313317, 344, 358

G Geminals, 137 Generalized linear response function definition of, 317 energy dependent form, 319 physical meaning of, 314, 317 relation to Green’s function, 318 spectral representation of, 319 oscillator strength, 305 random phase approximation (GRPA), 331-338, 349 Generators of groups, 255, 277fT Goldstone diagram, 341 Green’s function, 287-360 four particle, 316 hole-particle, 302-304 many particle, 290, 305 one particle, 290, 292-300, 324 three particle, 316 two particle, 300-305 Greenfield-Wiser potential, 381 Ground state energy, 294, 295

SUBJECT INDEX

Groups, general, 252ff G2,253,261,264 R(3), 253ff, 270 R(5), 253 R(7), 253, 263 R(21+ l), 253, 257, 280 R(41+ 2), 252 R(81+4), 267 Sp(lO), 257 Sp(14), 260, 267 Sp(41-t 2), 256, 267 SU(7), 253 U(7), 253 U(21+ I), 253, 257 U(4/+ 2), 255

Hydrogen atoms, see Atomic hy4rogen Hydrogen molecules, 243 Hydrogen fluoride molecule, 191 Hyperfine interaction frequency separation of deuterium, 30 hydrogen, 8, 30 Starck shift, 32 tritium, 30 frequency spectrum of line splitting, 35 Zeeman sublevels, 33, 34, 41 magnetic, 154, 261, 269 quadrupole, 261, 272

I

H Hamiltonian, 144 calculation of matrix elements of, 154 for diatomic molecules for adiabatic states, 48, 49 Breit-Pauli, 48 diagonalization of, 51, 52, 73-75 for diabatic states, 51, 52 for fixed nuclei, 48 for nuclear wave functions, 49 partitioning of, 49, 51, 52 eigenstates of electronic Hamiltonian, 224, 228, 234, 236, 245 Harrison delta function potential, 366, 367 Hartree-Fock function, 118, 141, 142, 166, 173, 180, 198 Hartree-Fock orbitals, 185, 288, 289, 312, 331, 334, 335 Hartree-Fock theory, 289, 337, 341 Heaviside function, Fourier representation of, 299 Heavy particle collisions, 243 Heine-Abarenkov-Heine potential, 377379 Heisenberg representation, 290, 306 Helium atom, interaction of ground state, 79-81 Helium ion, Hez+, 233ff, 243, 246 Hexapole magnet, 6 Hund rule, 180, 194, 197, 203 Hybrid orbitals, 126 Hybridization of orbitals, 107, 128

Inelastic scattering, 300, 324, 325, 335 Inner multiplicity problem, 254 Inner shells of electrons in molecules, 106 Integral evaluation methods, Slater type orbitals analytic methods, 66 bipoler coordinate methods, 66 direct numerical quadrature, 44, 45 elliptic coordinate methods, 65 Gaussian expansion methods, 65 integral transform method, 65 single-center expansion method, 63, 66 Integrals, two-electron Coulomb, two-center, 62 exchange, two-center, 63 general multi-center, 63-66 hybrid, two-center, 62 Interaction configuration, 109, 116 orbit-orbit, 261, 270 spin-orbit, 261, 271 spin-other orbit, 261, 270, 271, 279, 284 spin-spin, 261, 263, 270, 272, 275, 279, 284 Ionic states of mnlecules, 230ff, 234, 236, 248 Ionization potential of molecules, 166 Irreducible representations, 145, 168, 186 192, 258ff for symplectic groups, 260, 267 Isoscalar factors, 262, 273 Isospin, 269

SUBJECT INDEX

K

Kapur-Peierls resonant states, 238

L Landau-Zener formula, 232, 235 Land6 factor modification of, 42 nuclear-electronic, 9 of proton, 31 ratio of hydrogen-deuterium of, 30 Linewidth, 2, 3 Line broadening, 4, 37 Linear response function, 289, 31 1 Linear transfcrmation of basis sets, 59, 61 Liquid metals, 364, 369, 373 Localization, 119ff degree of, 122 functions, 123, 124 potential, 105 transformation, 102, 104 Localized molecular orbitals, 102 Lone pairs of electrons, 106, 179 localization function for, 124 separated, 137

M Magnetic dipole hyperfine interaction, 261, 269 radiation, 273 Magnetic shields, 7 Maser ammonia, 2, 12 hydrogen, 2-45 accuracy, 4, 7 as amplifier, 25 amplitude, 14, 24 dynamical behavior, 8, 16 frequency stability, 3, 7, 14 oscillations, 8, 15, 16, 19ff phase, 14, 15, 19, 20 proton, 17 relaxation time, 17-19, 23 synchronization, 26-28 rubidium, 14, 24 Matrix density, 161, 162,289, 297 non-Hermitian, 310, 333

403

overlap, 182 scattering, 296, 323-330 Matrix element formation computation of cofactors, Prosser-Hagstrom method, 72 determinantal methods, 70-72 nonorthogonality problem, 71-73 orbital transformation matrix method, 73 space-spin product method, 68, 70, 71 spin-free method, 68, 70, 71 symmetric group methods, 69-73 Matrix elements factorization of, 368, 369 null, 252ff, 275, 283 proportional to others, 274 of spin dependent operator, 211, 212 Measurement of atomic local population, 26, 39 of relaxation times longitudinal, 13, 19, 23 transverse, 16, 19, 23, 28 Methane molecule, 180, 194 Method of moments, 56, 57, 74 Microwave cavity electromagnetic field in, 13, 18 pulling, 3, 7, 16 Q-factor of, 7, 18 tuning, 16, 20 Molecular frequency standard, 2, 3 Molecular orbital configuration, see Electronic states of molecules Molecular orbitals, 97ff, see d s o Orbitals bond, 100 canonical, 103 distorted, 121 energy-localized, 104, 121 equivalent, 102, 107 exclusive, 111, 112, 138 invariant, 108, 111 localized, 102 maximum overlap, 130 natural, 101 natural spin, 136, oscillator, 112, 138 potential well localized, 121 quasi-invariant, 110 symmetry, 105 virtual, 112 Molecular ringing, 20, 22, 24 Momentum transfer, electronic, 77 Multiple quantum transitions, 40

404

SUBJECT INDEX

N Natrium, 230ff, 234-236, 246, 248 Natural orbitals, 60, 101, 136 Negative-ion molecular systems, 238, 244 Nitrogen, see Atomic nitrogen Noncrossing rules, 229, 230, 236, 237, 244 Nuclear kinetic energy operator, 225-227, 247 Nuclear motion equations, 48, 226, 231, 238, 241, 242, 246, 247

0

Occupation numbers, 100, 297 One-state problem, 228, 229 Operators annihilation, 256 classified by groups, 261 creation, 256 effective, 266, 270 projection, 143, 176 for quasi-particles, 258 spin-dependent, 163 time-ordering, 291 Young, 174 Optical pumping of Rb atoms, 13, 41 Optimization, 131 Orbit-orbit interaction, 261, 270 Orbital basis sets effect on integral evaluation, 61, 62 elliptic, 57 Gaussian, 59 Hulthen, 61 integral transform, 61 linear transformation of, 61 natural spin orbitals, 60 rational function, 61 Slater type, 59 Orbital exponents, choice of, 58 Orbital equations, 163ff correlation diagram of, 198-200 energy of, 166, 198, 199 operator, 187 Orbitals coupling schemes, 143 definitions, 218 degenerate, 180 Dyson, 320, 335 equivalent, 172

hybrid, 126 hybridization of, 107, 128 natural, 289, 297 normalization of, 171 one-particle, 3 12 orthogonality of, 182 symmetry properties of, 143, 185, 186 Oscillation level built up, 19 steady state, 15 transient, 16, 19 Oscillator strength effect of correlation error on, 77 examples of computation, 78, 84 Overlap, 72, 123, 129, 130, 133 Overlap matrix, 182

P Package programs for computation of molecular wave functions, 50 Pauling numbers, 71 Permutation representation, theory of, 146152, 171-180, 198 Perturbation general, 111, 137 many-body, 53-58 methods, 339-348 Rayleigh-Schrodinger, 57, 58 self-consistent, 313 theory, 289 Phonon spectrum, 368 Photodissociation, of Hz, 84 Plethysm, 277 Polarized hydrogen atoms, 2ff collisions of, 4, I 1 fictitious spin of, 9 magnetic state selection of, 6 recombination of, 12 sources of, 5 spin exchange between, 10, 32 storage of, 6, 12 Polyatoniic molecules, 78, 79 Population electronic, 123 partial overlap of, 133 Potential Ashcroft, 380, 381 Bloch, 366 Cohen-Wiser, 379

405

SUBJECT INDEX

energy excitation, 300, 305 Greenfield-Wiser, 381 Harrison delta function, 376, 377 Heine-Abarankov-Heine, 377-379 ionization, 297-301 optical, 289, 308, 309, 317 perturbing, 111 pseudo, see Pseudopotential scattering, 31 1 self, 307, 308 Potential curves adiabatic 49-52 avoided crossing and pseudocrossing of, 49-52 diabatic, 49-52 for H and H, 49 for H and H, 49, 50, 57 for He and He, 57, 78-81, 85 for He+ and Ne, 82 for H e + and He, 78, 82, 83 influence on cross sections, 81-83 for Li+ and He, 81 and Li, 82 for Li and H, 57, 78 for 0 and 0, 78 Projected atomic orbital method, 51, 52 Projection operator, 143, 176 Proton Land6 g-factor for, 31 maser, 17 Pseudopotential, general, 121, 363-385 of atomic core, 375 nonlocal, 372, 373 separation of, 375 t-matrix formulation of, 369 transformation of, 367 +

+

Q Quantum defect theory, 245 Quasi-adiabatic representation, see Electronic states of molecules Quasi-invariant molecular orbitals, 110 Quasi-particles, 258, 282 Quasi-spin, 256, 278 assignment of, 267ff Quasi-stationary electronic states, see Electronic states of molecules Quasi-stationary state formalism, 239-242, 244,246

R Radial integrals, 272 Radiative transitions electric dipole, 77-78, 83-87 electric quadruple, 77 Radio frequency field, 34 Random phase approximation (RPA), 289, 334-338 Relaxation magnetic, 11 spin exchange, 10, 32 wall collisions, 11 Representation of electronic Hamiltonian, see Electronic states of molecules Resonant scattering, 238-240 Resonant states, 237-244, 246 Resolvent operator computation of, 75 use of in computation of coupling matrix elements, 77 Ringing (molecular), 20ff Rotational coupling between electronic states, see Coupling Rydberg states, 237, 238, 244, 245 S

Scattering elastic, 295, 321, 337, 338 inelastic, 300, 324, 325, 335 matrix, 296, 329, 330 resonant, 238, 239, 240 potential, 31 1 Schwinger relation, 315, 318, 334-338 Selection rules within atomic shells, 251ff Self-consistent field approximation, 3748 method for computation of molecular wave function, 54, 55 Seniority numbers, 256, 272 Shell, half-filled, 277 Slater determinant, 67-69, 70-72, 101 Slater determinantal wave function, 143 Slater type orbitals, 44, 45, 63, 65, 66 Space-spin product wave functions, 68 Spin exchange coupling by, 30 cross section for, 19, 33 frequency shift of, 15, 16 relaxation time of, 32

406

SUBJECT INDEX

Spin functions, coupling of, 147, 148, 152, 160, 165, 167, 177, 182 Spin-free wave functions, 67, 68 Spin-orbit interaction, general, 261, 271, 279, 284 exceptional null matrix elements, 275,283 selection rule for mixed configurations, 274 Spin-spin interaction, general, 261, 270, 275, 279, 284 contact part, 261, 270 decomposed into irreducible parts, 263ff as quasi-spin scalar, 272 Stabilization method for computation of diabatic states, 52 State selector, 4, 6, 37, 39 States of molecules, see Electronic states of molecules Stationary electronic states, see Electronic states of molecules, adiabatic Structure factor (of atoms), 365, 368 Symmetries, conflicting, 276ff Symmetry adaptation of molecular wave functions, 67-69, 176ff, 205 molecular orbitals, 105 operators for diatomic molecules, 67 permutational, 146, 149 point, 168 Symplectic groups, 260, 267 Synchronization bandwidth, 28 level, 26

T Tensors, single electron, 253, 279 Transformation coefficients, 202 intrinisic and external, 121 Transition operators, see Coupling operators Translational absorption as molecular process, 85-87 Triangular conditions, 270ff, 280 stretched, 275 Tritium, see Atomic tritium Two-level approximation, 10 Two-particle effective potential in a coupled hierarchy of equations, 313 definition of, 318

GRPA form of, 332 physical interpretation of, 318, 320 Two-particle Green’s function energy-dependent form definition of, 302 poles and residues of, 304 time-dependent form definition of, 290 expression for special time orderings, 301, 302 Two-state problem, 224, 230, 235, 245 V

Valence bond electronic states, 235, 243, 246 Valence bond method, 142, 180 basis, 151, 213 Valence bond wave function, 58, 59 Valence state, 197 Variation principle, Rayleigh-Ritz, 54 Virtual molecular orbitals, 112 W Wave functions for atoms and molecules, 141ff configuration interaction, 54, 58 directly correlated, 56, 57, 60 electronic, 47-49 formalisms for computation configuration interaction, 54, 58 directly correlated wave functions, 56, 57 LCAO-SCF-MO, 54 many-body theory, 55, 56 niulticonfiguration method, 55 optimized value configuration method, 55 pair correlation method, 56 nuclear, 4 7 4 9 pair-correlated, 55, 56 single center, 59 spin coupled, 141-207 symmetry properties of, 168 transcorrelated, 56, 57 Wigner-Eckert theorem, 258, 262

Z Zeros intractable, 283 intriguing, 252 residual, 278ff