Table Of ContentMany-Electron
Densities and Reduced
Density Matrices
MATHEMATICAL AND COMPUTATIONAL CHEMISTRY
Series Editor: PAUL G. MEZEY
University of Saskatchewan
Saskatoon, Saskatchewan
MANY-ELECTRON DENSITIES AND REDUCED DENSITY MATRICES
Jerzy Cioslowski
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Many-Electron
Densities and Reduced
Density Matrices
Edited by
Jerzy Cioslowski
Florida State University
Tallahassee, Florida
Springer Science+Business Media, LLC
Library of Congress Cataloging-in-Publication Data
Many-electron densities and reduced density matrices/edited by Jerzy Cioslowski.
p. cm. — (Mathematical and computational chemistry; 1)
Includes bibliographical references and index.
ISBN 978-1-4613-6890-8 ISBN 978-1-4615-4211-7 (eBook)
DOI 10.1007/978-1-4615-4211-7
1, Density matrices. 2. Chemistry, Physical and theoretical—Mathematics. I.
Cioslowski, Jerzy. II. Series.
QD462.6.D46 .M36 2000
541.2'8—dc2t
00-042336
ISBN 978-1-4613-6890-8
©2000 Springer Science+Business Media New York
Originally published by Kluwer Academic / Plenum Publishers, New York in 2000
Softcover reprint of the hardcover 1st edition 2000
All rights reserved
No part of this book may be reproduced, stored in a retrieval system, or transmitted in any form or by any
means, electronic, mechanical, photocopying, microfilming, recording, or otherwise, without written
permission from the Publisher
Contributors
A. Artemiev, Centro de Quimica Instituto Venezolano de Investiga
ciones Cientificas (IVIC), Apartado 21827,-Caracas 1020-A, Venezuela
Kieron Burke, Departments of Chemistry and Physics, Rutgers Uni
versity, 610 Taylor Road, Piscataway NJ 08854
Russell J. Boyd, Department of Chemistry, Dalhousie University, Hal
ifax, Nova Scotia, Canada B3H 4J3
Jerzy Cioslowski, Department of Chemistry and CSIT, Florida State
University, Tallahassee, Florida 3230G, USA
A. John Coleman, Department of l\lathematics and Statistics, Queen's
University, Kingston ON, Canada K7L 3NG
Robert Erdahl, Department of Mathematics and Statistics, Queen's
University, Kingston ON, Canada K7L 3NG
S. Goedecker, Departement de recherche fondamentale sur la mati 'ere
condensee, SP2M/NM, CEA-Grenoble, 38054 Grenoble cedex 9, France
D. Gomez, Centro de Quimica Instituto Vellezolano de Investigaciones
Cientfficas (IVIC), Apartado 21827, Caracas 1020-A, Venezuela
Beiyan Jin, Department of lVIathematics and Statistics, Queen's Uni
versity, Kingston ON, Canada K7L 3NG
V. Karasiev, Centro de Quimica Instituto Venezolano de Investiga
ciones Cientificas (IVIC), Apartado 21827, Caracas 1020-A, Venezuela
Toshikatsu Koga, Department of Applied Chemistry, Muroran Insti
tute of Technology, Muroran, Hokkaido 050-8585, Japan
E. V. Ludeiia, Centro de Quimica Institut.o Vellezolano de Investiga
ciones Cientificas (IVIC), Apart.ado 21827, Caracas 1020-A, Venezuela
v
vi Contributors
Neepa T. Maitra, Departments of Chemistry and Physics, Rutgers
University, 610 Taylor Road, Piscataway NJ 08854, USA
David A. Mazziotti, Department of Chemistry, Duke University, Dur
ham, NC 27708, USA
Hiroshi Nakatsuji, Department of Synthetic Chemistry and Biologi
cal Chemistry, Graduate School of Engineering, Kyoto University, Kyoto
606, Japan
E. Perez-Romero, Departamento de QUlmica-Fisica, Universidad de
Salamanca, 37008 Salamanca, Spain
Mitja Rosina, Department of Physics, University of Ljubljana and J.
Stefan Institute, P.O.B. 2964, SI-1001 Ljubljana, Slovenia
L. M. Tel, Departamento de Quimica-Fisica, Universidad de Sala
manca, 37008 Salamanca, Spain
C. J. Umrigar, Cornell Theory Center, Ithaca NY 14850, USA
Jesus M. Ugalde, Kimika Fakultatea, Euskal Herriko Unibertsitatea,
Posta Kutxa 1072, 20080 Donostia, Euskadi, Spain
Carmela Valdemoro, Instituto de Matem aticas y Fisica Fundamental,
CSIC, Serrano 123, 28006-Madrid, Spain
Elmer Valderrama, Kimika Fakultatea, Euskal Herriko Ullibertsitatea,
Posta Kutxa 1072, 20080 Donostia, Euskadi, Spain
Paul Ziesche, Max Planck Institute for the Physics of Complex Sys
tems, Nothnitzer Str. 38, D-01187 Dresden, Germany
Preface
Science advances by leaps and bounds rather than linearly in time.
I t is not uncommon for a new concept or approach to generate a lot of
initial interest, only to enter a quiet period of years or decades and then
suddenly reemerge as the focus of new exciting investigations. This is
certainly the case of the reduced density matrices (a k a N-matrices or
RDMs), whose promise of a great simplification of quantum-chemical
approaches faded away when the prospects of formulating the auxil
iary yet essential N-representability conditions turned quite bleak. How
ever, even during the period that followed this initial disappointment,
the 2-matrices and their one-particle counterparts have been ubiquitous
in the formalisms of modern electronic structure theory, entering the
correlated-level expressions for the first-order response properties, giv
ing rise to natural spinorbitals employed in the configuration interaction
method and in rigorous analysis of electronic wavefunctions, and al
lowing direct calculations of ionization potentials through the extended
Koopmans'theorem.
The recent research of Nakatsuji, Valdemoro, and Mazziotti her
alds a renaissance of the concept of RDlvls that promotes them from
the role of interpretive tools and auxiliary quantities to that of central
variables of new electron correlation formalisms. Thanks to the economy
of information offered by RDMs, these formalisms surpass the conven
tional approaches in conciseness and elegance of formulation. As such,
they hold the promise of opening an entirely new chapter of quantum
chemistry.
This book has the ambitious aim of raising awareness of these ex
citing developments among the practitioners of electronic structure the
ory and calculations, while showcasing other applications of RDl\Is and
their sister quantities known as many-electron densities. To this end, we
present here a compilation of cutting-edge contributions from the lead
ing experts in these fields. Three major areas are covered. First, the
properties of RDMs and many-electron densities are reviewed in detail.
Following these preliminaries, the new approaches to direct determina
tion of RDMs are put forward. Next, the density matrix functional
vii
viii Pr'eface
theory, which employs the l-mat.rix as the cent.ral quant.ity, is discussed
and the role played by the t.wo-electron density in density functional the
ory is exposed. Finally, applications of electron int.racule and ext.racule
in analysis of electronic structures of atoms and molecules are presented.
Being intended for a wide audience of readers with diverse interests
that range from solid st.ate physics to comput.er simulations of chemical
species, t.his book is eclectic by design. Chapt.ers devoted to formal
mat.hematical theorems and t.heir rigorous proofs are presented side-by
side with t.hose concerned wit.h historical overviews and compilations of
practical calculations. With t.his somewhat unconventional structure,
we hope for this book t.o be equally attractive to the pundits and their
apprentices.
Usefulness of even the most interesting monograph can be greatly
diminished by poor formatting and/or lack of uniformit.y in not.ation.
Thus, I find it appropriat.e to conclude this preface by acknowledging
the excellent edit.ing carried out by my assistant Dr. Agnieszka Szarecka
who spent long hours meticulously checking and revising the chapters. I
also thank Ms. Katarzyna Pernal for t.echnical help with wordprocessing.
Jerzy Cioslowski
Contents
I. Properties of Reduced Density Matrices
Chapter 1. RDMs: How Did We Get Here?
A. John Coleman
1. From Hylleraas to Coulson .................................... 1
2. The Variational Approach ..................................... 7
3. The Valdemoro-Nakatsuji-Mazziotti (VNM) Theory ........... 9
4. Next Steps ................................................... 15
References ....................................................... 16
Chapter 2. Some Theorems on Uniqueness and Reconstruc
tion of Higher-Order Density Matrices
Mitja Rosina
1. Introduction ................................................. 19
2. The Unique Preimage ........................................ 20
2.1. Some Definitions ........................................ 20
3. The Surface Points ........................................... 22
4. The Reconstruction .......................................... 25
5. The Antisymmetrized Geminal Power (AGP) ................. 28
6. Summary .................................................... 30
References ....................................................... 31
Chapter 3. Cumulant Expansions of Reduced Densities, Re-
duced Density Matrices, and Green's Functions
Paul Ziesche
1. Introduction ................................................. 33
2. Reduced densities ............................................ 36
2.1. One-Density ............................................. 36
2.2. Two-Density ............................................ 37
2.3. Motivation for the Cumulant Expansion ................. 38
2.4. s-Particle Densities and Their Cumulant Expansion ...... 39
3. Reduced Density Matrices .................................... 42
ix
x Content.s
4. Green's Functions .................................... , ....... 46
5. Equations of Motion ......................................... 49
Appendix A: Particle-Number Distribution in Domains ........ , .. 52
Appendix B: Higher-Order Fluctuations .......................... 54
References ....................................................... 55
Chapter 4. On Calculating Approximate and Exact Density
Matrices
Robert Erdahl and Beiyan Jin
1. Introduction ................................................. 57
2. Approximate von Neumann Densities ........................ 60
2.1. Kth-Order Approximations ....................... '" .... 60
2.2. Matrix Representations .................................. 61
2.3. The Pauli Subspace ..................................... 62
2.4. Additional Properties of Matrix Representations ......... 63
3. The Fundamental Optimization Theorem ..................... 64
3.1. Characterizing the Minimizer ................... " ....... 65
3.2. A Symmetric Formulation ............................... 66
3.3. Second-Order Convergence for Algorithms ............... 66
3.4. Canonical Diagonalization of Operators .................. 67
4. Minimizing the Energy ....................................... 67
4.1. Interpreting the Representable Region ................... 70
4.2. Tracking the Correlations as IAI -; 00 ................... 72
4.3. Second-Order Estimates ................................. 72
4.4. The Work of Garrod, Mihailovic, and Rosina ............ 73
4.5. Dual Configuration Interaction and Correlation
Representations ......................................... 74
5. Minimizing the Dispersion ................................... 76
5.1. Dispersion-Free States ................................... 79
5.2. Connection with the Work of Mazziotti, Nakatsuji,
and Valdemoro .......................................... 80
5.3. The Prospects for Excited States ........................ 81
5.4. Fixing the Particle Number .............................. 83
References ....................................................... 84
II. The Contracted Schrodinger Equation
Chapter 5. Density Equation Theory in Chemical Physics
Hiroshi Nakatsuji
1. Introduction and Definitions ................................. 85
2. The Density Equation ........................................ 89
3. The Hartree-Fock Theory as the Zeroth-Order DET .......... 93
