Density Matrices and Density Functionals

A. JOHN COLEMAN

Density Matrices and

Density Functionals Proceedings o/the A. John Coleman Symposium

Edited by

Robert Erdahl and

Vedene H. Smith, Jr. Queen's University, Kingston, Canada

D. REIDEL PUBLISHING COMPANY

A MEMBER OF THE KLUWER ,. ACADEMIC PUBLISHERS GROUP

DORDRECHT/BOSTON/LANCASTER/TOKYO

Library of Congress Cataloging in Publication Data

Density matrices and density functionals.

Includes index. I. Density matrices-Congresses. 2. Density functionals-Congresses.

3. Coleman, A. John (Albert John), 1918- I. Coleman, A. John (Albert John), 1918- II. Erdahl, Robert. III. Smith, Vedene H. OCI74.17.D44D47 1987 530.1'529434 87-4718 ISBN-13: 978-94-010-8214-3 E-ISBN-13: 978-94-009-3855-7 DOl: 10.107/978-94-009-3855-7

Published by D. Reidel Publishing Company, P.O. Box 17,3300 AA Dordrecht, Holland.

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CONTENTS

PREFACE .......•.........................•....................•. i x

The publications of A. John Coleman ...•...•........•.•......... xiii

A Tribute to A. John Coleman - The "Tame" Mathematician ....... . P. O. Lowdi n

Reduced Density Matrices: 1929-1989 ......•........•........... 5 A.J. Coleman

Some Aspects on the Development of the Theory of Reduced Density Matrices and the Representability Problem .......•...... 21

P.O. Lowdin

Representability Conditions Robert Erdahl

51

On the Diagonal N-Representability Problem .............•....•.. 77 M.M. Mestechkin

Fermion N-Representability Conditions Generated by a Decomposition of the 1-Particle Identity Operator onto Mutually Orthogonal Proj ection Operators ..•........•........... 89

Hubert Grudzinski

The Unitarily Invariant Decomposition of Hermitian Operators ... 115 Zong-Hao Zeng and Chi a-Chung Sun

Building Up N-Electron States with Symplectic Symmetry......... 141 Z.H. Zeng, C.C. Sun and A.J. Coleman

Time Dependent Antisymmetrized Geminal Power Theory Using a Coherent State Formulation .....•......•...........•.... 167

E. Deumens, B. Weiner and Y. Ohrn

Griffiths Inequalities for Fermion Systems J.K. Percus

193

Entropy of Reduced Density Matrices ....................•....... 213 Mary Beth Ruskai

A Lower Bound to the Ground State Energy of a Boson System Wi th Fermion Source .•...•...•......•.........................•. 231

M. Rosina, B. Golli and R. M. Erdahl

vi CONTENTS

Reduced Density Operators, Their Related von Neumann Density Operators, Close Cousins of These, and Their Physical Interpretation •.•.•.••..•..•..•......•.....................••.. 249

Everett G. Larson

Theory and Practice of the Spin-Adapted Reduced Hamiltonians (SRHY .•....••..•.....•..•.........•..•...•........ 275

C. Valdemoro

Variational Principle with Built-In Pure State N-Representability Cgnditions. The N-Electron Case •..........• 289

Eduardo V. Ludena

Wigner Distributions as Representations of the Density Matrix .. 305 Leon Cohen

Inter-Relationships Between Various Representations of One-Matrices and Related Densities: A Road Map and an Exampl e •.•.....•.....•.......•..•.............................. 327

Ajit J. Thakkar, Anthony C. Tanner and Vedene H. Smith, Jr.

Current Problems in Density Functional Theory.................. 339 E. Zaremba

The Interface Between Reduced Density Matrices and Density Functional Theory...................................... 359

John E. Harriman

The Physics Underlying the Langreth-Mehl Scheme for Non-Uniform Systems .•.•..•.•.••...•......•...•..•.......•.• 375

David C. Langreth

Understanding Energy Differences in Density Functional Theory.. 391 S.H. Vosko and Jolanta B. Lagowski

Density Functional Calculations of Molecular Bond Energies ..•.. 443 A.D. Becke

Non-Local Effects on Atomic and Molecular Correlation Energies Studied with a Gradient-Corrected Density Functional ••..........•.•.•...........•..•.•.••..•..••..•...... 457

A. Savin, H. Preuss and H. Stoll

An Evaluation of Local Electron Correlation Corrections and Non-Local Exchange Corrections to the Hartree-Fock-Slater Method from Calculations on Bond Energies and Electronic Spectra of Molecular Systems ••.••.•...•......••...•.••.••...••• 467

Vincenzo Tschinke and Tom Ziegler

Correlation Energy Functionals of One-Matrices and Hartree-Fock Densi ties ..•..••.•...•...•..•..•...•.....••.••.••. 479

Mel Levy

CONTENTS vii

Some Remarks on Scaling Relations in Density Functional Theory .....••.•.........•...•..•........•....•................. 499

Wei tao Yang

Deduction of Semiempirical MO Methods from Density Functional Theory .............................................. 507

Einar Lindholm and Stig Lundqvist

Charge and Spin Densities in Molecular Solids: Local Density Functional Calculations Versus Experiment .•..•......... 517

Pierre J. Becker

A Functional of the Two-Particle Density Matrix for the Approximate Calculation of the Electronic Correlation Energy 545

Renato Colle and Oriano Salvetti

Extracules, Intracules, Correlation Holes, Potentials, Coefficients and All That •.......•............................. 553

Ajit J. Thakkar

The Exact Schr~dinger Equation for the Electron Density........ 583 Geoffrey Hunter

Adiabatic Separation, Broken Symmetries and Geometry Optimization................................................... 597

Osvaldo Goscinski and Vladimiro Mujica

Asymptotic Results for Density Matrices and Electron Density in Atoms and Nearly Spherical Molecules .......••....•.. 613

N.H. March and R. Pucci

An Algorithm for Calculating Isoelectronic Changes in Energies, Densi ties, and One-Matrices....................... 629

Jean E. Osburn and Mel Levy

Atoms and Ions in the Limit of Large Nuclear Charge ....•......• 643 Jose L. Gazquez, Marcelo Galvan, Elba Ortiz and Alberto Vela

Improved Thomas-Fermi Theory for Atoms •...•......•....•......•. 663 Swapan K. Ghosh and Robert G. Parr

A Bond Energy from Quantum Mechanics R.F.W. Bader and K.B. Wiberg

677

Measured Electron Densities and Band Structure Calculations ..•. 693 John Avery and Peter Sommer-Larsen

X-ray Orthonormal Orbital Model for Crystallography L.J. Massa

707

Index ....................•..•.........•.•.......••....•........ 717

PREFACE

THE COLEMAN SYMPOSIUM

This collection of papers is dedicated to Albert John Coleman for his enthusiastic devotion to teaching and research and his many

scientific accomplishments.

John was born in Toronto on May 20, 1918 and 21 years later

graduated from the University of Toronto in mathematics. Along the way he teamed up with Irving Kaplansky and Nathan Mendelson to win the first William Lowell Putnam Mathematical Competition in 1938. He earned his M.A. at Princeton in 1942 and then his Ph.D. at

Toronto in 1943 in relativistic quantum mechanics under the direction of Leopold Infeld. During this period he was secretary of

the Student Christian Movement in Toronto. Later, in 1945, he became traveling secretary of the World's Student Christian Federation in Geneva and in this capacity visited some 100 universities in 20 countries in the next four years. He spent the 50's as a member of the faculty at the University of Toronto and for 20 years, starting in 1960, he served as Dupuis Professor of Mathematics and Head of the Department at Queen's University. Since

1983 he has been Professor Emeritus at Queen's.

His 1963 paper in the Reviews of Modern Physics on the "Structure of Fermion Density Matrices" was a landmark and served as an inspiration to many. In that paper many of the difficult

mathematical and practical problems associated with density matrix

theory were isolated and the potential usefulness of density

ix

x PREFACE

matrices was brought before a wide audience. In the 60's John became a fixture at LOwdin's International Symposium at Sanibel and

at the Summer Institutes in Uppsala. During this period he lectured and wrote about reduced density matrix theory. He found a substantial audience, and as is evidenced by this volume, he had a profound effect on many careers. The AGP wave function has proved to be a flexible and useful tool in nuclear, molecular and solid state physics, and in this period, John was exploring its

fascinating mathematical and physical properties.

A great expositor of mathematics, John was thoroughly

appreciated by the theoretical chemists and physicists at Sanibel and Uppsala. In him they found a mathematician whom they could understand - even enjoy. It was the interdisciplinary flavor of these meetings - the chemists, physicists and mathematicians

learning from each other - which proved so fruitful. His articles "Induced and Subduced Representations" (1968) and "The Symmetric Group Made Easy" (1968) are remarkable in their crystal clarity and

accessibility. His treatment of the representations of the symmetric group is widely regarded as the best current one.

John is fond of quoting Alfred North Whitehead. The following

passage from Chapter 2 of "Science and the Modern World" vividly reflects - as all of his friends well know - his attitude that mathematics plays a far-reaching role in human affairs: "The paradox is now fully established that the utmost abstractions are the true weapons with which to control our thought of concrete fact". He had a passionate concern for his students, and tilis is the message that he brought them. An undergraduate once remarked

that "Regardless of what he taught, he got you thinking". Another

noted that only a portion of his lectures on Calculus were devoted to the subject. They contradicted every known text book on teaching

techniques. "I don't think he actually taught me calculus, but I

did learn it and I think he taught me how to learn it." He has

interacted with students in a wide variety of ways: in the classroom and in study groups at his home. He has spent hours with students on a one-to-one basis. He stimulated the creation of, and served for fifteen years as Chairman of, The International Student

Centre at Queen's.

PREFACE

from 1960 through 1964 he chaired the Ontario Mathematics Commission which brought the "new mathematics" to Canada and has recently been commended for bringing "an infectious and unrelenting enthusiasm" to this body. During this period he edited a widely

used series of mathematical texts for Grades 4 through 13 for the Canadian system, and co-authored several in the series. From 1972 to 1974 he served as president of the Canadian Mathematical Society. He served as chairman and then treasurer of the Development and

Exchange Commission of the International Mathematical Union in the period 1974-1982, taking special interest in improving mathematics teaching in Africa.

John is a man of many parts. He was active in the National

Committee of World University Service of Canada, 1949-1960, organizing a seminar in Mysore, India in 1952 which included many

future world leaders. He spoke in 1973 at a conference in Pont-a-Mousson, France, organized by the World Council of Churches on the Future of Technological Man. He was a member of the Science Council of Canada for four years and directed a study of the "Mathematical Sciences of Canada". His report, published in 1977,

contains a number of far-reaching recommendations for Canadian schools and was hailed by the "users" of mathematics. He was the sole layman from Canada invited to participate in the Lambeth

Conference in 1978, which brought together 350 bishops of the World

Anglican Communion. In 1980-1981 John worked for Marc Lalonde,

Minister cf Energy, Mines and Resources, acting as liaison between this department and the Parliamentary Committee on Alternative Energy and Oil Substitution.

Everyone who has visited John in Kingston has met Marie-Jeanne, whom he married in 1953. She is a graduate in theology of the University of Geneva. A much beloved and respected woman, her

powers extend far beyond theological matters. However we must remark that she was unable to stop John from entering federal

politics which he did in 1980. He was narrowly defeated by the

Conservative incumbent, Flora MacDonald, in the riding of Kingston and the Islands.

XI

Because of John's long association with the problem, five conferences and work shops on reduced density matrix theory were organized at Queen's University between 1967 and 1977. We all recall the informal atmosphere of these meetinga - taken to the

point of scheduling talks around sailing events on Lake Ontario. Noting John's belated interest in sailing and sport one of his colleagues composed the following lines:

"Now John loved to sail and ski and squash and jog around, Until his doctor told him that he really must slow down, John replied, that's nonsense, I'm really quite subdued,

The only exercise I get is when I tie my shoes.

And when I tie my shoes, myself I don't abuse, The only exercise I get is when I tie my shoes."

Read at John's retirement party, these lines will no doubt strike responsive chords in those who attended these earlier meetings on density matrix theory.

PREFACE

For the sixth meeting we decided that the density functional

theorists should be invited since there is so much common ground over which we can fight. The present volume contains the proceedings of a three day conference, held in August 1985 and hosted jointly by the Departments of Chemistry, and Mathematics and Statistics, of Queen's University. It is through the financial support of the Natural Sciences and Engineering Research Council of Canada that we were able to organize such a large international symposium.

We would like to take this opportunity to thank Linda Nuttall

and Betty McIntosh who assisted in organizing the Symposium and did

much of the work in preparing the final manuscript for publication. We are also indebted to our many colleagues who assisted in

reviewing the manuscripts contained in this volume.

Robert Erdahl

Vedene H. Smith, Jr. Kingston, Canada

THE PUBLICATIONS OF A. JOHN COLEMAN

1940 - 1986

1. 'Curves on a Surface', Am. Math. Monthly 42, 212-220 (1940).

2. 'Phase Space in Eddington's Theory', Phil. Mag. 36, 269 (1945).

3. 'A Simple Proof of Stirling's Formula', Am. Math. Monthly 58, 334 (1951).

4. 'The Betti Numbers of the Simple Lie Groups', Can. J. Math. 10, 349-356 (1958).

5. 'Density Matrices of n-Fermion Systems', Can. Math. Bull. 4, 209-212 (1961).

6. 'A Note on Nilpotent Operators, Jordan Canonical Form, Segre Characteristics and Minimal Polynomials', Quantum Chemistry Group, Uppsala University, Technical Note (1962).

7. 'The Structure of Fermion Density Matrices', Rev. Mod. Phys. 35, 668-687 (1963).

8. 'Induced Representations with Applications to the Symmetric and Full Linear Groups', Quantum Chemistry Group, Uppsala University, Technical Note No. 102 (1963).

9. 'The Onset of Superconductivity', Can. J. Phys. 42, 226 (1964).

10. 'Electron Pairs in the Quasichemical-Equilibrium Bardeen­Cooper-Schrieffer Theories', Phy. Rev. Letters 13, 406-407 (1964) •

xiii

xiv THE PUBLICATIONS OF A. JOHN COLEMAN

11. 'The Structure of Fermion Density Matrices.II. Antisymmetrized Geminal Powers', J. Math. Phy. 6, 1425-1431 (1965).

12. (with S. Pruski), 'Gorkov's Ansatz in Superconductivity Theory', Can. J. Phy. 43, 2142-2149 (1965).

13. 'Group Theory', Quantum Chemistry Group, Uppsala University, Technical Note, 92 pgs. (1965). Notes of Lectures at Abisko, prepared by J.L. Calais and S.O. Goscinski.

14. 'Induced Representations with Applications GL(n)', Queen's Papers in Pure and Applied 91 pgs., Queen's University (1966).

to Sand n

Mathematics, No.4,

15. 'Density Matrices in the Quantum Theory of Matter: Energy, Intracules and Extracules', Int. J. Quantum Chern. 1, 457-464 (1967).

16. 'Infinite Range Correlation and Large Eigenvalues of the 2-Matrix', Can. J. Phy. 45, 1271 (1967).

17. 'The Symmetric Group Made Easy', Advances in Quantum Chemistry 4, 83-108 (1968).

18. 'Induced and Subduced Representations', in Group Theory and its Applications, edited by E.M. Loebl, 57-118, Academic Press (1968) •

19. 'The Present Status of the N-Representability Problem', in Queen's Papers in Pure and Applied Mathematics, No. 11, edited by A.J. Coleman and R.M. Erdahl, 2-19, Queen's University (1968).

20. 'Interaction Between the Gravitational and Electromagnetic Fields', Proceedings of the 5th International Conference on Gravitation and the Theory of Relativity, held in Tbilisi,

Sept. 1-10, (1968).

21. 'Recent Results on Fermion N-Representability', Int. J. Quantum

Chem. 4, 355-358 (1971).

THE PUBLICATIONS OF A. JOHN COLEMAN

22. 'Necessary Conditions For N-Representability of Reduced Density Matrices', J. Math. Phys. 13, 214-222 (1973).

23. 'Two Necessary Conditions for Fermion N-Representability', Rep. Math. Phys. 4, 113-129 (1973).

24. 'N-Representability Circumvented', in Report of Boulder Conference, 1972: Energy, Structure and Reactivity, edited by D.W. Smith and W.R. McRae, 231-236, John Wiley (1973).

25. (with G.D. Edwards and K.P. Beltzner), Mathematical Sciences in Canada, Background Study No. 38 for the Science Council of Canada (1975).

26. 'A General Theory of Super Phenomena in Physics', in Quantum Physics and the Many-Body Problem, 239-248, Plenum Press (1975).

27. (with I. Absar), 'One Electron Orbitals Intrinsic to the Reduced Hamiltonian', Chern. Phys. Letters 39, 609-611 (1976).

28. (with I. Absar), 'Reduced Hamiltonian Orbitals I. A New Approach to the Many-Electron Problem', Int. J. Quantum Chern. 10, 319-330 (1976).

29. (with I. Absar and H. Kummer), 'Some Aspects of the N-Representability Problem in Finite Dimensions I. Operator Endomorphisms which Induce Necessary Conditions', J. Math. Phys. 18, 324-334 (1977).

30. 'The Convex Structure of Electrons', Int. J. Quantum Chern. 11, 907-916 (1977).

31. 'Reduced Density Operators and the N-Particle Problem', Int. J. Quantum Chem. 13, 67-82 (1978).

32. (with I. Absar), 'Reduced Hamiltonian Orbitals III. Unitarily Invariant Decomposition of Hermitian Operators', Int. J. Quantum Chern. 18, 1279-1307 (1980).

xv

xvi THE PUBLICATIONS OF A. JOHN COLEMAN

33. (with D. O'Shea), 'Local Characterization of Phase Diagrams, Phys. Rev. B 22, 3428-344?_(1980).

34. 'Calculation Of First And Second Order Reduced Density Matrices', in The Force Concept in Chemistry, edited by B.M. Deb, 418-448, van Nostrand (1981).

35. 'The Product of Unitary Reflections', Math. Reports of the Acad. Sc. of Canada 6, 371-373 (1984).

36. 'The State Labeling Problem - A Universal Solution', J. Math. Phys. 27, 1933-1943 (1986).

John contemplating one of his

favorite formulas