Advances in Quantum Chemistry (eBook)

Contents 6
Contributors 10
Preface 12
Analytical Energy Gradients for Excited-State Coupled-Cluster Methods: Automated Algebraic Derivation of First Derivatives for Equation-of-Motion Coupled-Cluster and Similarity Transformed Equation-of-Motion Coupled-Cluster Theories 14
Introduction 15
The EOM-CCSD/PT and STEOM-CCSD/PT energy methods 25
The (ST)EOM-CCSD methods 26
The (ST)EOM-PT methods 41
The energy gradient and Lagrange multipliers (abstract expressions) 45
Lagrange's method of undetermined multipliers 45
The EOM-CC/PT intermediate and effective density matrices 48
The STEOM-CCSD/PT intermediate density matrix 55
Summary of steps in a STEOM gradient calculation 60
The SMART symbolic algebra package 61
Overview of SMART 62
Example derivations in SMART 64
Interconversion between H, H, and G2 amplitudes 68
Factorization approach 70
The (ST)EOM-CCSD/PT detailed algebraic gradient equations 71
The EOM-CCSD/PT and STEOM-CCSD/PT Lagrange multiplier energy functionals 72
Simplifications for the PT-based expressions 73
The (ST)EOM-CCSD/PT Z Lagrange multiplier equations 74
Homogeneous sides of the Z equations 74
Inhomogeneous sides of the Z equations 75
Three-body contributions to the Z equations and D 76
The EOM-CCSD/PT intermediate density matrix D 79
Inclusion of pure-excitation terms in the EOM-CCSD functional 79
The STEOM-CCSD/PT Z- and Z+ Lagrange multiplier equations 81
Decoupling of the Z± equations in the active external index 82
The STEOM-CCSD/PT intermediate density matrix D 83
Efficiency tweaks in implementation 83
The conversion of D to the effective density matrix D 84
Expression of the energy derivative in terms of elementary Hamiltonian derivative integrals 86
Tables of equations 88
Summary 109
Acknowledgements 110
References 110
Autoionizing States of Atoms Calculated Using Generalized Sturmians 116
The generalized Sturmian method for solving many-particle Schrödinger equations 116
The history of generalized Sturmians 117
Atomic calculations 118
Symmetry-adapted basis functions for the 2-electron isoelectronic series 121
The large-Z approximation 126
Range of validity of the large-Z approximation 127
References 131
Mathematical Elements of Quantum Electronic Density Functions 134
Introduction 136
Initial definitions 137
Definitions related to the construction of DF 138
Definitions leading to quantum similarity measures 141
Definitions related to the algebra of diagonal vector spaces and their applications 143
The structure of the generating N-dimensional VS 143
Expression of the DF and other problems 145
Inward matrix product: Definitions, properties and examples 146
Inward matrix product 146
IMP general features 147
IMP properties. 147
IMP unit element and inverse. 147
IMP powers and functions. 148
IMP applications 148
IMP and Taylor series expansions of multivariate functions 148
Scalar product of two hypermatrices 149
Sign separation in hypermatrix spaces and IMP 149
Sign separation in hypermatrix spaces and Boolean tagged sets: Hypermatrix signature 150
Quadratic form signature and strictly positive matrices 151
Normed vector semispaces: Minkowski norm 152
Minkowski norm 152
Matrix summation symbols on matrices 152
Shell structure in vector semispaces 152
alpha-shells 153
Homotheties and convex sets 153
Semispace partition and equivalence classes 153
Shell direct sums 154
Scalar products in vector semispaces 154
Minkowski scalar products 154
Inward matrix product structure and Minkowski scalar product of two vectors. 155
Minkowski scalar product main properties 155
Distributive law and root scalar products involving linear combinations 156
Angles subtended by two vectors 156
Minkowski metric properties 157
Minkowski product fundamental property involving unit shell vectors 157
A property of the elements of the unit shell vectors 158
Positive definite structure of Minkowski metric matrices involving two unit shell vectors 158
Linear independence of unit shell vectors 159
Positive definite nature of root metric matrices 159
Root distances in vector semispaces 159
Root distance properties 160
Generalized root scalar products and distances 161
Generalized root scalar products involving several vectors 161
Generalized root distances involving several vectors 161
Proving the fundamental property of generalized scalar products involving unit shell elements 162
Inward matrix structure of generalized root scalar products 163
Discrete DF forms and related questions 163
Quantum similarity measures involving two QO 163
Similarity matrices and discrete QOS 164
Stochastic transformations of QSM 165
Matrix representation of density functions 166
Diagonal elements of the density matrix 167
Matrix representation. 167
Mulliken approximation. 167
ASA approximation. 168
Atomic DMR. 168
Off-diagonal elements of the first order density matrix 169
The practical use of the DMR 170
Redundant solutions. 171
Averaged submatrices. 172
Optimization algorithm. 173
Simpler representation. 174
Approximate, generalized and average density functions 175
Convex conditions and ASA fitting 175
ASA and CASA 176
ASA coefficient optimization using elementary Jacobi rotations 177
Alternative approximate expression of density functions: Complete ASA (CASA) 179
Bader's analysis of ASA DF 180
ASA atomic and molecular density functions definition. 181
Gradient and Hessian of a molecular ASA density function. 181
(A) Gradient. 181
(B) Hessian. 182
(C) Further gradient and Hessian forms. 182
Bader analysis on ASA density functions. 183
Generalized density functions 184
Average density functions 184
Discrete conformational Boltzmann averages 185
Continuous conformational Boltzmann averages 186
Chiral R-S averages 187
General average DF 188
Frobenius average DF. 188
IMP average DF. 189
Concluding remarks. 190
Extended Hilbert spaces, Sobolev spaces and density functions 190
Expectation values 190
Problem setup 191
Extended Hilbert spaces 192
Considerations on EHS functions 193
Generating rules in EH spaces 194
Diagonal representations in EHS 195
Extended wave function projectors 195
Sobolev spaces 196
Generalized Sobolev spaces 196
Non-linear Schrödinger equation generated throughout extended Hilbert space wave functions 197
Classical Schrödinger energy expression from extended wave functions 197
Energy expression from generalized extended wave functions 198
Extended non-linear Schrödinger equation 198
General non-linear terms in Schrödinger equation 200
Density and shape functions semispaces 201
Hilbert semispaces and root products 201
Atomic shell approximation functions 201
Minkowski scalar products between ASA functions 201
ASA pseudo-wave functions 202
Minkowski scalar product between ASA functions belonging to different Hilbert semispaces 202
Density functions and generating wave functions 204
Structure of density functions 204
Convex conditions. 204
Generating wave functions. 205
Non-uniqueness of the generating wave functions. 207
Extended wave functions and the Schrödinger equation 208
Variational principle in density function formalism 209
Density functions difference, Fukui functions and quantum dissimilarity indices 209
The zero shell 209
Fukui functions 210
Vector spaces generated throughout semispace element subtraction 210
Quantum dissimilarity indices 210
Quantum similarity index: Carbó index 211
Convex sets of scalars. 211
Convex linear combinations within vector semispace shells. 212
Norm variation and Fukui functions 212
DFT variational theorem over shape functions 212
Conclusions 213
Used abbreviations 214
Acknowledgements 215
References 215
Quantum Monte Carlo: Theory and Application to Molecular Systems 222
Introduction 222
Numerical solution of the Schrödinger equation 223
Trial wave functions 224
Variational Monte Carlo 225
Diffusion Monte Carlo 226
Fixed-node approximation 229
Trial wave function optimization 229
Treatment of heavy elements 230
Applications 230
Singlet-triplet energy splitting in ethylene 231
Electronic excitations of free-base porphyrin 231
Characterization of CuSi4 and CuSi6 234
Acknowledgements 236
References 236
From Fischer Projections to QuantumMechanics of Tetrahedral Molecules: New Perspectives in Chirality 240
Introduction 240
Geometrical approach to central molecular chirality based on complex numbers 244
Fischer projections for tetrahedral molecules 246
Algebraic structure of central molecular chirality 247
Generalization to molecules with n stereogenic centres: A molecular Aufbau for tetrahedral chains 252
Quantum mechanical approach 254
Quantum chiral algebra and parity 256
Summary and conclusions 258
References 259
On the Canonical Formulation of Electrodynamics and Wave Mechanics 262
Introduction 263
Physical motivation 263
Historical and mathematical background 264
Gauge symmetry of electrodynamics 264
Gauge symmetry of electrodynamics and wave mechanics 265
Approaches to the solution of the Maxwell and Schrödinger equations 267
Canonical formulation of the coupled Maxwell-Schrödinger equations 269
Notation and units 271
Canonical structure 271
Lagrangian electrodynamics 272
Choosing a gauge 272
The Lorenz and Coulomb gauges 272
Hamiltonian electrodynamics 274
Hamiltonian formulation of the Lorenz gauge 275
Poisson bracket for electrodynamics 277
Hamiltonian electrodynamics and wave mechanics in complex phase space 277
Hamiltonian electrodynamics and wave mechanics in real phase space 280
The Coulomb reference by canonical transformation 281
Symplectic transformation to the Coulomb reference 282
The Coulomb reference by change of variable 286
Electron spin in the Pauli theory 287
Proton dynamics 288
Numerical implementation 289
Maxwell-Schrödinger theory in a complex basis 290
Maxwell-Schrödinger theory in a real basis 292
Overview of computer program 294
Stationary states: s- and p-waves 294
Nonstationary state: Mixture of s- and p-waves 294
Free electrodynamics 296
Analysis of solutions in numerical basis 297
Symplectic transformation to the Coulomb reference 300
Numerical implementation 302
Stationary states: s- and p-waves 302
Nonstationary state: Mixture of s- and p-waves 302
Free electrodynamics 303
Analysis of solutions in Coulomb basis 304
Asymptotic radiation 305
Proton dynamics in a real basis 306
Conclusion 307
References 309
Stopping Power-What Next? 312
Introduction 312
What is stopping power? 314
Methodology 315
Theory 315
Experiment 318
Other processes 319
Stopping at low projectile energy 320
Higher order Born terms 320
Charge fluctuation 321
Projectile excitation and ionization 323
Nuclear motion 324
Negative stopping 324
Orientation 325
Fragmentation 327
Relativistic considerations 329
Suggestions 329
Acknowledgements 330
References 330
Subject Index 334