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Many-Electron Densities and Reduced Density Matrices : Mathematical and Computational Chemistry - Jerzy Cioslowski

Many-Electron Densities and Reduced Density Matrices

Mathematical and Computational Chemistry

By: Jerzy Cioslowski (Editor)

Hardcover Published: 30th September 2000
ISBN: 9780306464546
Number Of Pages: 301

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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.

Properties of Reduced Density Matrices
RDMs: How Did We Get Here?
From Hylleraas to Coulsonp. 1
The Variational Approachp. 7
The Valdemoro-Nakatsuji-Mazziotti (VNM) Theoryp. 9
Next Stepsp. 15
Referencesp. 16
Some Theorems on Uniqueness and Reconstruction of Higher-Order Density Matrices
Introductionp. 19
The Unique Preimagep. 20
Some Definitionsp. 20
The Surface Pointsp. 22
The Reconstructionp. 25
The Antisymmetrized Geminal Power (AGP)p. 28
Summaryp. 30
Referencesp. 31
Cumulant Expansions of Reduced Densities, Reduced Density Matrices, and Green's Functions
Introductionp. 33
Reduced densitiesp. 36
One-Densityp. 36
Two-Densityp. 37
Motivation for the Cumulant Expansionp. 38
s-Particle Densities and Their Cumulant Expansionp. 39
Reduced Density Matricesp. 42
Green's Functionsp. 46
Equations of Motionp. 49
Particle-Number Distribution in Domainsp. 52
Higher-Order Fluctuationsp. 54
Referencesp. 55
On Calculating Approximate and Exact Density Matrices
Introductionp. 57
Approximate von Neumann Densitiesp. 60
Kth-Order Approximationsp. 60
Matrix Representationsp. 61
The Pauli Subspacep. 62
Additional Properties of Matrix Representationsp. 63
The Fundamental Optimization Theoremp. 64
Characterizing the Minimizerp. 65
A Symmetric Formulationp. 66
Second-Order Convergence for Algorithmsp. 66
Canonical Diagonalization of Operatorsp. 67
Minimizing the Energyp. 67
Interpreting the Representable Regionp. 70
Tracking the Correlations as [Lambda] [right arrow] [infinity]p. 72
Second-Order Estimatesp. 72
The Work of Garrod, Mihailovic, and Rosinap. 73
Dual Configuration Interaction and Correlation Representationsp. 74
Minimizing the Dispersionp. 76
Dispersion-Free Statesp. 79
Connection with the Work of Mazziotti, Nakatsuji, and Valdemorop. 80
The Prospects for Excited Statesp. 81
Fixing the Particle Numberp. 83
Referencesp. 84
The Contracted Schrodinger Equation
Density Equation Theory in Chemical Physics
Introduction and Definitionsp. 85
The Density Equationp. 89
The Hartree-Fock Theory as the Zeroth-Order DETp. 93
The Correlated Density Equationp. 94
Solving the DEp. 96
A Geminal Equation Derived from the DEp. 102
Application of DET to the Calculation of Potential Energy Surfacesp. 107
DET for Open-Shell Systemsp. 109
Conclusion and Future Prospectsp. 113
Referencesp. 114
Critical Questions Concerning Iterative Solution of the Contracted Schrodinger Equation
Introductionp. 117
Definitions, Notation, and Diagramsp. 119
The Reduced Density Matrices (RDMs)p. 120
The Hole RDMs and the Fermion Relationsp. 121
Brief Description of the it-CSE and the RDM Construction Proceduresp. 122
Construction Procedures for the 3- and 4-RDMsp. 123
The Correspondence between [superscript 2 Delta] and the Second-Order Correlation Matrix: A Generalizationp. 125
Higher-Order Correlation Matricesp. 128
Evaluation of [superscript 3 Delta]p. 129
New Approximation for [superscript 3 Delta]p. 130
The Role of the N-representability Conditions in the CSE Formalismp. 132
The Connection between the C-matrices and the N-representability G-conditionsp. 133
N-representability Tests at Convergence of it-CSEp. 135
Referencesp. 136
Cumulants and the Contracted Schrodinger Equation
Introductionp. 139
CSE Theoryp. 143
Derivation of TCSEp. 143
Nakatsuji's Theoremp. 144
Reconstruction of RDMsp. 145
Rosina's Theoremp. 145
Cumulant Theoryp. 146
Connected Reconstructionp. 149
Coupled Cluster Connectionsp. 152
CC via RTMsp. 152
CSE and CCp. 155
Ensemble Representabilityp. 156
An Applicationp. 158
Conclusionsp. 159
Referencesp. 162
Density Matrix Functional Theory
Natural Orbital Functional Theory
Introductionp. 165
Shortcomings of Kohn-Sham Schemesp. 166
Quantities Relevant to Natural Orbital Functional Theoryp. 168
Existence Proof of a Natural Orbital Functionalp. 170
Narrowing Down the Functional Form of a Natural Orbital Functionalp. 171
The Exact Natural Orbital Functional for the Two-Electron Casep. 172
General Properties of Natural Orbital Functionalsp. 173
Explicit Forms for Natural Orbital Functionalsp. 176
Shortcomings of the Present Natural Orbital Functionalsp. 177
Numerical Implementation of a Natural Orbital Functionalp. 178
Conclusionsp. 179
Referencesp. 179
The Pair Density in Approximate Density Functional Theory: The Hidden Agent
Introductionp. 183
Modeling the Pair Densityp. 183
Exact Density Functional Theory (DFT)p. 192
Old Faithful: The Local Density Approximationp. 197
Improving on The Local Density Approximationp. 200
Gradient Expansionsp. 200
Hybridsp. 203
Weighted Density Approximationp. 203
Self-Interaction Correction and Meta-GGAsp. 204
New Technologyp. 204
The Optimized Effective Potentialp. 204
Time-Dependent Density Functional Theoryp. 205
Conclusionsp. 206
Referencesp. 206
Functional N-representability in Density Matrix and Density Functional Theory: An Illustration for Hooke's Atom
Introductionp. 209
The Use of Energy Functionals in Quantum Mechanicsp. 211
N-representability and Functional N-representability of the 1- and 2-matricesp. 214
N-representable Functionals of the Two-Matrix: Hooke's Atomp. 218
Non-N-representable Functionals of the Two-Matrix: Hooke's Atomp. 219
N-representability of Functionals of the One-Particle Densityp. 220
N-representable Functionals of the One-Particle Density: Hooke's Atomp. 222
Non-N-representable Functionals of the One-Particle Density: Hooke's Atomp. 225
Conclusionsp. 227
Hooke's Atomp. 227
Referencesp. 228
Electron Intracule and Extracule Densities
Intracule and Extracule Densities: Historical Perspectives and Future Prospects
Introductionp. 231
Intracules and Extraculesp. 232
The Coulomb Holep. 232
The Fermi Hole and Hund's Rulep. 234
Intracule Densities and Hund Holesp. 236
Angular Aspects of Correlation Holesp. 237
Advances in the Calculation of Electron-Pair Functionsp. 237
Electron-Pair Functions as a Tool for Understanding Electron-Electron Interactionsp. 239
Accurate Electron-Pair Densities for Atomic Systemsp. 241
Neutral Atomsp. 241
Low-Lying Excited Statesp. 241
Charged Systemsp. 243
Electron-Pair Densities: Analysis in Position and Momentum Spacesp. 243
Intracule and Extracule Densitiesp. 243
Electron-Electron Coalescence and Counterbalance Densitiesp. 243
Electron-Pair Distances and Density Momentsp. 245
Referencesp. 246
Topology of Electron Correlation
Introductionp. 249
Topological Characteristics of Scalar Functions Defined in Cartesian Spacep. 251
The Correlation Cagep. 253
Correlation Cages in Simple Two-Electron Systemsp. 254
Evolution of the Correlation Cage in the Course of Bond Dissociationp. 255
Conclusionsp. 264
Referencesp. 264
Electron-Pair Densities of Atoms
Introduction and Definitionsp. 267
Mathematical Structure of Atomic Intracule and Extracule Densitiesp. 271
Intracule Densities and Momentsp. 272
Extracule Densities and Momentsp. 277
Electron-Electron Coalescence and Counterbalance Densitiesp. 280
Isomorphism between Intracule and Extracule Propertiesp. 281
Numerical Results for Atoms and Ionsp. 282
Intracule Propertiesp. 283
Extracule Propertiesp. 287
Approximate Isomorphic Relationsp. 290
Connection between One- and Two-Electron Momentsp. 292
Summaryp. 293
Recent Publications on Electron-Pair Densitiesp. 294
Referencesp. 296
Indexp. 299
Table of Contents provided by Syndetics. All Rights Reserved.

ISBN: 9780306464546
ISBN-10: 0306464543
Series: Mathematical and Computational Chemistry
Audience: General
Format: Hardcover
Language: English
Number Of Pages: 301
Published: 30th September 2000
Publisher: Springer Science+Business Media
Country of Publication: US
Dimensions (cm): 25.4 x 17.78  x 1.91
Weight (kg): 0.77

Earn 519 Qantas Points
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