Many-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 A Continuation Order Plan is available for this series. A continuation order will bring delivery of each new volume immediately upon publication. Volumes are billed only upon actual shipment. For further information please contact the publisher. 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