Quantal Density Functional Theory II (eBook)

Preface 7
Contents 9
1 Introduction 17
2 Schrödinger Theory from a ``Newtonian'' Perspective 30
2.1 Time-Independent Schrödinger Theory 30
2.2 Schrödinger Theory from a ``Newtonian'' Perspective: The Pure State Differential Virial Theorem 32
2.3 Definitions of Quantal Sources 33
2.4 Definitions of ``Classical'' Fields 37
2.5 Energy Components in Terms of Quantal Sources and Fields 39
2.6 Integral Virial, Force, and Torque Sum Rules 41
2.7 Coalescence Constraints 44
3 Quantal Density Functional Theory 49
3.1 Quantal Density Functional Theoryfrom a ``Newtonian'' Perspective 50
3.2 Definitions of Quantal Sources Within Quantal Density Functional Theory 51
3.3 Definitions of ``Classical'' Fields Within Quantal Density Functional Theory 55
3.4 Total Energy and Its Components in Terms of Quantal Sources and Fields 57
3.5 Effective Field Feff (r) and Electron-Interaction Potential Energy vee (r) 61
3.6 Integral Virial, Force, and Torque Sum Rules 62
3.7 Highest Occupied Eigenvalue em 63
3.8 Quantal Density Functional Theory of Degenerate States 65
4 New Perspectives on Hohenberg–Kohn–Sham Density Functional Theory 66
4.1 The Hohenberg–Kohn Theorems and Corollary 67
4.2 Kohn–Sham Density Functional Theory 72
4.3 Generalization of the Fundamental Theorem of Hohenberg–Kohn 80
5 Nonuniqueness of the Effective Potential Energy and Wave Function in Quantal Density Functional Theory 86
5.1 The Interacting System: Hooke's Atom in a Ground State 89
5.2 Mapping to the S system in Its 11S Ground State 89
5.3 Mapping to an S system in Its 21S Singlet Excited State 95
5.4 Nonuniqueness of the Wave Function of the S system in an Excited State 98
5.5 Proof that Nonuniqueness of Effective Potential Energy Is Solely Due to Correlation-Kinetic Effects 107
6 Ad Hoc Approximations Within Quantal Density Functional Theory 111
6.1 The Q-DFT of Hartree Theory 115
6.2 The Q-DFT of Hartree–Fock Theory 119
6.3 Time-independent Quantal-Density Functional Theory 125
6.4 The Case of Nonconservative Fields 131
7 Analytical Asymptotic Structure in the Classically Forbidden Region of Atoms 137
7.1 The Wave Function 139
7.2 The Single-Particle Density Matrix and Density 142
7.3 The Pair-Correlation Density 144
7.4 The Work Done in the Electron-Interaction Field 145
7.5 The Correlation-Kinetic Potential Energy 149
7.6 Endnotes 150
8 Analytical Asymptotic Structure At and Near the Nucleus of Atoms 153
8.1 Proof of Finiteness of Potential Energies vee (r and vBee (r) at the Nucleus 155
8.2 Criticality of the Electron–Nucleus Coalescence Condition to Local Effective Potential Energy Theories 157
8.3 General Structure of vee (r) Near the Nucleus of Spherically Symmetric and Sphericalized Systems 160
8.4 Exact Structure of vee(r) Near the Nucleus of Spherically Symmetric and Sphericalized Systems 165
8.5 Endnote 177
9 Application of the Q-DFT Hartree Uncorrelated Approximation to Atoms 179
9.1 Electronic Structure of the Neon Atom 180
9.2 Atomic Shell Structure and Core–Valence Separation 187
9.3 Total Ground State Energies 191
9.4 Highest Occupied Eigenvalues 191
10 Application of the Q-DFT Pauli Correlated Approximation to Atoms and Negative Ions 199
10.1 Ground State Properties of Atoms 200
10.2 Ground State Properties of Mononegative Ions 226
10.3 Static Polarizabilities of the Neon Isoelectronic Sequence 229
11 Quantal Density Functional Theory of the Density Amplitude: Application to Atoms 233
11.1 Quantal Density Functional Theoryof the Density Amplitude 234
11.2 Application to Atoms 238
11.3 Conclusions and Endnotes 243
11.4 Consequences for Traditional Density Functional Theory 244
12 Application of the Irrotational Component Approximation to Nonspherical Density Atoms 246
12.1 Scalar Effective Fermi Hole Source pxeff(r) 247
12.2 Vector Vortex Fermi Hole Source Jx (r) 250
12.3 Irrotational ExI (r) and Solenoidal ExS (r) Components of the Pauli Field Ex (r) 252
12.4 Path-Independent Pauli Potential Energy WxI(r) 256
12.5 Endnotes on the Approximation 257
13 Application of Q-DFT to Atoms in Excited States 260
13.1 The Triplet 2 3S State Isoelectronic Sequence of He 262
13.2 One-electron Excited States of Li 263
13.3 One-electron Excited States of Na 265
13.4 Multiplet Structure of C and Si 267
13.5 Doubly Excited Autoionizing States of He 269
13.6 Endnote 272
14 Application of the Multi-ComponentQ-DFT Pauli Approximation to the Anion–Positron Complex: Energies, Positron and Positronium Affinities 274
14.1 Equations of the Multi-Component Q-DFT Pauli Approximation 275
14.2 Brief Remarks on Hartree–Fock Theory of Positron Binding to Anions 278
14.3 Total Energy of the Anion–Positron Complex and Positron Affinities 279
14.4 Positronium Affinities 283
15 Application of the Q-DFT Fully Correlated Approximation to the Helium Atom 286
15.1 The Interacting System: Helium Atom in Its Ground State 287
15.2 Mapping to an S System in Its 11S Ground State 288
15.3 Endnotes 298
16 Application of the Q-DFT Fully Correlated Approximation to the Hydrogen Molecule 300
16.1 The Interacting System: Hydrogen Molecule in Its Ground State 300
16.2 Mapping to an S System in Its (g1s)2 GroundState Configuration 301
16.3 Endnotes 312
17 Application of Q-DFT to the Metal–Vacuum Interface 314
17.1 Jellium Model of a Metal Surface 317
17.2 Surface Model Effective Potential Energies and Orbitals 322
17.3 Accuracy of the Model Potentials 325
17.4 Structure of the Fermi Hole at a Metal Surface 327
17.5 General Expression for the Pauli Field Ex(x) and Potential Energy Wx (x) 339
17.6 Structure of the Pauli Field Ex(x) and PotentialEnergy Wx(x) 343
17.7 Analytical Structure of the Pauli Potential Energy Wx(x) 346
17.8 Analytical Structure of the Lowest Order Correlation-Kinetic Potential Energy W(1)tc (x) in the Classically Forbidden Region 349
17.9 Analytical Structure of the Coulomb Wc (x) and Second-and Higher-Order CorrelationKinetic W2tc (x), W3tc (x) … Potential Energiesin the Classically Forbidden Region 355
17.10 Analytical Asymptotic Structure of the Effective Potential Energy vs (x) in the Classically Forbidden Region 361
17.11 Endnote on Image-Potential-Bound Surface States 364
18 Many-Body and Pseudo Møller-Plesset Perturbation Theory within Quantal Density Functional Theory 365
18.1 Many-Body Perturbation Theory within Q-DFT 366
18.2 Pseudo Møller–Plesset Perturbation TheoryWithin Q-DFT 377
19 Epilogue 382
A Quantal Density Functional Theory of Degenerate States 384
B Generalization of the Runge–Gross Theorem of Time-Dependent Density Functional Theory 391
C Analytical Asymptotic Structure of the Correlation-Kinetic Potential Energy in the Classically Forbidden Region of Atoms 395
D The Pauli Field Ex (r) and Potential Energy Wx (r) in the Central Field Approximation 401
E Equations of the Irrotational Component Approximation as Applied to the Carbon Atom 404
F Ground State Properties of the Helium Atom as Determined by the Kinoshita Wave Function 409
G Approximate Wave Function for the Hydrogen Molecule 413
References 415
Index 429