Variational Principles and Methods in Theoretical Physics and Chemistry

Half-title......Page 3
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
Preface......Page 15
I Classical mathematics and physics......Page 17
1 History of variational theory......Page 19
1.1 The principle of least time......Page 20
1.2 The variational calculus......Page 21
1.2.1 Elementary examples......Page 23
1.3 The principle of least action......Page 24
2.1 Lagrangian formalism......Page 27
2.1.2 Dissipative forces......Page 28
2.1.3 Lagrange multiplier method for constraints......Page 29
2.2.1 The Legendre transformation......Page 30
2.2.2 Transformation from Lagrangian to Hamiltonian......Page 31
2.2.3 Example: the central force problem......Page 32
2.3 Conservation laws......Page 33
2.4 Jacobi\'s principle......Page 34
2.5 Special relativity......Page 36
2.5.1 Relativistic mechanic of a particle......Page 37
2.5.2 Relativistic motion in an electromagnetic field......Page 38
3.1 Linear systems......Page 41
3.2 Simplex interpolation......Page 42
3.2.1 Extremum in n dimensions......Page 43
3.3 Iterative update of the Hessian matrix......Page 44
3.3.1 The BFGS algorithm......Page 45
3.4 Geometry optimization for molecules......Page 46
3.4.2 The BERNY algorithm......Page 47
II Bound states in quantum mechanics......Page 49
4 Time-independent quantum mechanics......Page 51
4.1.1 Sturm–Liouville theory......Page 52
4.1.2 Idiosyncracies of the Schrödinger equation......Page 54
4.1.3 Variational principles for the Schrödinger equation......Page 56
4.1.4 Basis set expansions......Page 57
4.2.2 The hypervirial theorem......Page 59
4.2.3 The virial theorem......Page 60
4.3.1 The N-electron Hamiltonian......Page 61
4.3.2 Expansion in a basis of orbital wave functions......Page 62
4.3.3 The interelectronic Coulomb cusp condition......Page 64
4.4 Symmetry-adapted functions......Page 65
4.4.1 Algorithm for constructing symmetry-adapted functions......Page 66
4.4.2 Example of the method......Page 67
5 Independent-electron models......Page 69
5.1 N-electron formalism using a reference state......Page 70
5.1.1 Fractional occupation numbers......Page 71
5.1.2 Janak’s theorem......Page 72
5.2.1 Explicit components of the energy functional......Page 73
5.2.2 Orbital Euler–Lagrange equations......Page 74
5.2.3 Exact correlation energy......Page 75
5.3.1 Closed shells – unrestricted Hartree–Fock (UHF)......Page 77
5.3.3 Open-shell Hartree–Fock theory (RHF)......Page 78
5.3.5 Multiconfiguration SCF (MCSCF)......Page 80
5.4 The optimized effective potential (OEP)......Page 81
5.4.1 Variational formulation of OEP......Page 83
5.5.1 The Hohenberg–Kohn theorems......Page 84
5.5.2 Kohn–Sham equations......Page 86
5.5.3 Functional derivatives and local potentials......Page 87
5.5.4 Thomas–Fermi theory......Page 88
5.5.5 The Kohn–Sham construction......Page 90
6 Time-dependent theory and linear response......Page 93
6.1 The time-dependent Schrödinger equation for one electron......Page 94
6.2 The independent-electron model as a quantum field theory......Page 95
6.3.2 The screening response......Page 97
6.4.1 Remarks on time-dependent theory......Page 99
6.4.3 Definition of the response kernel......Page 100
6.5 Reconciliation of N-electron theory and orbital models......Page 101
6.6 Time-dependent density functional theory (TDFT)......Page 102
6.7 Excitation energies and energy gaps......Page 105
III Continuum states and scattering theory......Page 107
7 Multiple scattering theory for molecules and solids......Page 109
7.1 Full-potential multiple scattering theory......Page 111
7.1.2 Two-center expansion......Page 112
7.1.3 Angular momentum representation......Page 113
7.1.4 The surface matching theorem......Page 115
7.1.5 Surface integral formalism......Page 116
7.1.6  Muffin-tin orbitals and atomic-cell orbitals......Page 117
7.1.7 Tail cancellation and the global matching function......Page 118
7.1.8 Implementation of the theory......Page 119
7.2.1 Kohn–Rostoker variational principle......Page 120
7.2.2 Convergence of internal sums......Page 122
7.2.3 Schlosser–Marcus variational principle......Page 124
7.2.4 Elimination of false solutions......Page 127
7.3.1 The LMTO method......Page 129
7.3.2 The LACO method......Page 131
7.3.3 Variational theory of linearized methods......Page 132
7.4 The Poisson equation......Page 134
7.5 Green functions......Page 136
7.5.1 Definitions......Page 137
7.5.2 Properties of the Green function......Page 140
7.5.3 Construction of the Green function......Page 141
8.1 Scattering by an N-electron target system......Page 145
8.1.1 Cross sections......Page 148
8.1.2 Close-coupling expansion......Page 149
8.2 Kohn variational theory......Page 150
8.2.1 The matrix variational method......Page 151
8.2.2 The Hulthén–Kohn variational principle......Page 153
8.2.3 The complex Kohn method......Page 155
8.3 Schwinger variational theory......Page 156
8.3.1 Multichannel Schwinger theory......Page 159
8.3.2 Orthogonalization and transfer invariance......Page 161
8.4 Variational R-matrix theory......Page 163
8.4.1 Variational theory of the R-operator......Page 170
8.4.2 The R-operator in generalized geometry......Page 172
8.4.3 Orbital functional theory of the R-matrix......Page 173
Bound–free correlation......Page 174
9 Electron-impact rovibrational excitation of molecules......Page 177
9.1 The local complex-potential (LCP) model......Page 179
9.1.1 The projection-operator method......Page 180
9.2 Adiabatic approximations......Page 182
9.2.1 The energy-modified adiabatic approximation (EMA)......Page 184
9.3 Vibronic R-matrix theory......Page 185
9.3.1 Phase-matrix theory......Page 188
9.3.2 Separation of the phase matrix......Page 189
9.3.3 Phase-matrix formalism: EMAP......Page 190
9.3.4 Nonadiabatic theory: NADP......Page 191
IV Field theories......Page 195
10 Relativistic Lagrangian theories......Page 197
10.1 Classical relativistic electrodynamics......Page 198
10.1.1 Classical dynamical mass......Page 200
10.1.2 Classical renormalization and the Dirac equation......Page 201
10.2 Symmetry and Noether’s theorem......Page 202
10.2.1 Examples of conservation laws......Page 203
10.3 Gauge invariance......Page 205
10.3.1 Classical electrodynamics as a gauge theory......Page 206
10.3.2 Noether’s theorem for gauge symmetry......Page 207
10.3.3 Nonabelian gauge symmetries......Page 208
10.3.4 Gauge invariance of the SU(2) field theory......Page 211
10.4.1 Energy and momentum in classical electrodynamics......Page 213
10.4.2 Energy and momentum in SU(2) gauge theory......Page 215
10.5 The Standard Model......Page 217
10.5.1 Electroweak theory (EWT)......Page 218
10.5.2 Quantum chromodynamics (QCD)......Page 219
References and bibliography......Page 221
Index......Page 241