Table Of ContentDensity 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.
Sold and distributed in the U.S.A. and Canada
by Kluwer Academic Publishers,
101 Philip Drive, Assinippi Park, Norwell, MA 02061, U.S.A.
In all other countries, sold and distributed
by Kluwer Academic Publishers Group,
P.O. Box 322, 3300 AH Dordrecht, Holland.
All Rights Reserved
© 1987 by D. Reidel Publishing Company, Dordrecht, Holland
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner
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 51
Robert Erdahl
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 193
J.K. Percus
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 677
R.F.W. Bader and K.B. Wiberg
Measured Electron Densities and Band Structure Calculations ..•. 693
John Avery and Peter Sommer-Larsen
X-ray Orthonormal Orbital Model for Crystallography 707
L.J. Massa
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 XI
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.
