Many-Electron Densities and Reduced Density Matrices Book

I.	Properties of Reduced Density Matrices
Chapter 1.	RDMs: How Did We Get Here?
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 Reconstruction of Higher-Order Density Matrices
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, Reduced Density Matrices, and Green's Functions
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
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
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 |[Lambda]| [right arrow] [infinity]	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
1.	Introduction and Definitions	85
2.	The Density Equation	89
3.	The Hartree-Fock Theory as the Zeroth-Order DET	93
4.	The Correlated Density Equation	94
5.	Solving the DE	96
6.	A Geminal Equation Derived from the DE	102
7.	Application of DET to the Calculation of Potential Energy Surfaces	107
8.	DET for Open-Shell Systems	109
9.	Conclusion and Future Prospects	113
	References	114
Chapter 6.	Critical Questions Concerning Iterative Solution of the Contracted Schrodinger Equation
1.	Introduction	117
2.	Definitions, Notation, and Diagrams	119
2.1.	The Reduced Density Matrices (RDMs)	120
2.2.	The Hole RDMs and the Fermion Relations	121
2.3.	Brief Description of the it-CSE and the RDM Construction Procedures	122
2.4.	Construction Procedures for the 3- and 4-RDMs	123
3.	The Correspondence between [superscript 2 Delta] and the Second-Order Correlation Matrix: A Generalization	125
3.1.	Higher-Order Correlation Matrices	128
3.2.	Evaluation of [superscript 3 Delta]	129
3.3.	New Approximation for [superscript 3 Delta]	130
4.	The Role of the N-representability Conditions in the CSE Formalism	132
4.1.	The Connection between the C-matrices and the N-representability G-conditions	133
4.2.	N-representability Tests at Convergence of it-CSE	135
	References	136
Chapter 7.	Cumulants and the Contracted Schrodinger Equation
1.	Introduction	139
2.	CSE Theory	143
2.1.	Derivation of TCSE	143
2.2.	Nakatsuji's Theorem	144
3.	Reconstruction of RDMs	145
3.1.	Rosina's Theorem	145
3.2.	Cumulant Theory	146
3.3.	Connected Reconstruction	149
4.	Coupled Cluster Connections	152
4.1.	CC via RTMs	152
4.2.	CSE and CC	155
5.	Ensemble Representability	156
6.	An Application	158
7.	Conclusions	159
	References	162
III.	Density Matrix Functional Theory
Chapter 8.	Natural Orbital Functional Theory
1.	Introduction	165
2.	Shortcomings of Kohn-Sham Schemes	166
3.	Quantities Relevant to Natural Orbital Functional Theory	168
4.	Existence Proof of a Natural Orbital Functional	170
5.	Narrowing Down the Functional Form of a Natural Orbital Functional	171
6.	The Exact Natural Orbital Functional for the Two-Electron Case	172
7.	General Properties of Natural Orbital Functionals	173
8.	Explicit Forms for Natural Orbital Functionals	176
9.	Shortcomings of the Present Natural Orbital Functionals	177
10.	Numerical Implementation of a Natural Orbital Functional	178
11.	Conclusions	179
	References	179
Chapter 9.	The Pair Density in Approximate Density Functional Theory: The Hidden Agent
1.	Introduction	183
2.	Modeling the Pair Density	183
3.	Exact Density Functional Theory (DFT)	192
4.	Old Faithful: The Local Density Approximation	197
5.	Improving on The Local Density Approximation	200
5.1.	Gradient Expansions	200
5.2.	Hybrids	203
5.3.	Weighted Density Approximation	203
5.4.	Self-Interaction Correction and Meta-GGAs	204
6.	New Technology	204
6.1.	The Optimized Effective Potential	204
6.2.	Time-Dependent Density Functional Theory	205
7.	Conclusions	206
	References	206
Chapter 10.	Functional N-representability in Density Matrix and Density Functional Theory: An Illustration for Hooke's Atom
1.	Introduction	209
2.	The Use of Energy Functionals in Quantum Mechanics	211
3.	N-representability and Functional N-representability of the 1- and 2-matrices	214
3.1.	N-representable Functionals of the Two-Matrix: Hooke's Atom	218
3.2.	Non-N-representable Functionals of the Two-Matrix: Hooke's Atom	219
4.	N-representability of Functionals of the One-Particle Density	220
4.1.	N-representable Functionals of the One-Particle Density: Hooke's Atom	222
4.2.	Non-N-representable Functionals of the One-Particle Density: Hooke's Atom	225
5.	Conclusions	227
Appendix	Hooke's Atom	227
	References	228
IV.	Electron Intracule and Extracule Densities
Chapter 11.	Intracule and Extracule Densities: Historical Perspectives and Future Prospects
1.	Introduction	231
2.	Intracules and Extracules	232
2.1.	The Coulomb Hole	232
2.2.	The Fermi Hole and Hund's Rule	234
2.3.	Intracule Densities and Hund Holes	236
2.4.	Angular Aspects of Correlation Holes	237
3.	Advances in the Calculation of Electron-Pair Functions	237
4.	Electron-Pair Functions as a Tool for Understanding Electron-Electron Interactions	239
5.	Accurate Electron-Pair Densities for Atomic Systems	241
5.1.	Neutral Atoms	241
5.2.	Low-Lying Excited States	241
5.3.	Charged Systems	243
6.	Electron-Pair Densities: Analysis in Position and Momentum Spaces	243
6.1.	Intracule and Extracule Densities	243
6.2.	Electron-Electron Coalescence and Counterbalance Densities	243
6.3.	Electron-Pair Distances and Density Moments	245
	References	246
Chapter 12.	Topology of Electron Correlation
1.	Introduction	249
2.	Topological Characteristics of Scalar Functions Defined in Cartesian Space	251
3.	The Correlation Cage	253
4.	Correlation Cages in Simple Two-Electron Systems	254
5.	Evolution of the Correlation Cage in the Course of Bond Dissociation	255
6.	Conclusions	264
	References	264
Chapter 13.	Electron-Pair Densities of Atoms
1.	Introduction and Definitions	267
2.	Mathematical Structure of Atomic Intracule and Extracule Densities	271
2.1.	Intracule Densities and Moments	272
2.2.	Extracule Densities and Moments	277
2.3.	Electron-Electron Coalescence and Counterbalance Densities	280
2.4.	Isomorphism between Intracule and Extracule Properties	281
3.	Numerical Results for Atoms and Ions	282
3.1.	Intracule Properties	283
3.2.	Extracule Properties	287
3.3.	Approximate Isomorphic Relations	290
3.4.	Connection between One- and Two-Electron Moments	292
4.	Summary	293
Appendix	Recent Publications on Electron-Pair Densities	294
	References	296
	Index	299