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E-raamat: Quantal Density Functional Theory

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  • Ilmumisaeg: 12-Aug-2016
  • Kirjastus: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Keel: eng
  • ISBN-13: 9783662498422
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Quantal Density Functional TheorySuurem pilt
  • Formaat: PDF+DRM
  • Ilmumisaeg: 12-Aug-2016
  • Kirjastus: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Keel: eng
  • ISBN-13: 9783662498422
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This book deals with quantal density functional theory (QDFT) which is a time-dependent local effective potential theory of the electronic structure of matter. The treated time-independent QDFT constitutes a special case. In the 2nd edition, the theory is extended to include the presence of external magnetostatic fields. The theory is a description of matter based on the "quantal Newtonian" first and second laws which is in terms of "classical" fields that pervade all space, and their quantal sources. The fields, which are explicitly defined, are separately representative of electron correlations due to the Pauli exclusion principle, Coulomb repulsion, correlation-kinetic, correlation-current-density, and correlation-magnetic effects. The book further describes Schrödinger theory from the new physical perspective of fields and quantal sources. It also describes traditional Hohenberg-Kohn-Sham DFT, and explains via QDFT the physics underlying the various energy functionals and func

tional derivatives of the traditional approach to electronic structure. 

Introduction.- Schrödinger Theory from the Perspective of "Classical" Fields Derived from Quantal Sources.- Quantal Density Functional Theory.- The Hohenberg-Kohn Theorems and Kohn-Sham Density Functional Theory.- Physical Interpretation of Kohn-Sham Density Functional Theory.- Generalization of the Hohenberg-Kohn Theorem to the Presence of a Magnetostatic Field.- Quantal Density Functional Theory in the Presence of a Magnetostatic Field.- Generalization and Physical Interpretation of Runge-Gross Time-Dependent Density Functional Theory.- Hohenberg-Kohn and Percus-Levy-Lieb Proofs of Density Functional Theory.- Quantal Density Functional Theory of the Density Amplitude.- Quantal Density Functional Theory of the Discontinuity in the Electron-Interaction Potential Energy.- Further Insights Derived Via Quantal Density Functional Theory.- Epilogue.
1 Introduction
1(14)
References
12(3)
2 Schrodinger Theory from the `Newtonian' Perspective of `Classical' Fields Derived from Quantal Sources
15(52)
2.1 Time-Dependent Schrodinger Theory
16(1)
2.2 Definitions of Quantal Sources
17(5)
2.2.1 Electron Density ρ(rt)
18(1)
2.2.2 Spinless Single--Particle Density Matrix γ(Rr't)
18(1)
2.2.3 Pair--Correlation Density g(rr't), and Fermi--Coulomb Hole ρxc(rr't)
19(2)
2.2.4 Current Density j(rt)
21(1)
2.3 Definitions of `Classical' Fields
22(3)
2.3.1 Electron--Interaction Field Eee(rt)
22(1)
2.3.2 Differential Density Field D(rt)
23(1)
2.3.3 Kinetic Field Z(rt)
23(1)
2.3.4 Current Density Field J(rt)
24(1)
2.4 Energy Components in Terms of Quantal Sources and Fields
25(2)
2.4.1 Electron--Interaction Potential Energy Eee(t)
25(1)
2.4.2 Kinetic Energy T(t)
26(1)
2.4.3 External Potential Energy Eext(t)
27(1)
2.5 Schrodinger Theory and the `Quantal Newtonian' Second Law
27(2)
2.6 Integral Virial Theorem
29(2)
2.7 The Quantum--Mechanical `Hydrodynamical' Equations
31(1)
2.8 The Internal Field of the Electrons and Ehrenfest's Theorem
32(4)
2.9 The Harmonic Potential Theorem
36(2)
2.10 Time-Independent Schrodinger Theory: Ground and Bound Excited States
38(6)
2.10.1 The `Quantal Newtonian' First Law
38(2)
2.10.2 Coalescence Constraints
40(2)
2.10.3 Asymptotic Structure of Wavefunction and Density
42(2)
2.11 Examples of the `Newtonian' Perspective: The Ground and First Excited Singlet State of the Hooke's Atom
44(15)
2.11.1 The Hooke's Atom
44(2)
2.11.2 Wavefunction, Orbital Function, and Density
46(5)
2.11.3 Fermi--Coulomb Hole Charge Distribution ρxc(rr')
51(2)
2.11.4 Hartree, Pauli--Coulomb, and Electron--Interaction Fields EH(r), Exc(r), Eee(r) and Energies EH, Exc, Eee
53(3)
2.11.5 Kinetic Field Z(r) and Kinetic Energy T
56(1)
2.11.6 Differential Density Field D(r)
57(1)
2.11.7 Total Energy E and Ionization Potential I
57(2)
2.11.8 Expectations of Other Single--Particle Operators
59(1)
2.12 Schrodinger Theory and Quantum Fluid Dynamics
59(8)
2.12.1 Single--Electron Case
60(1)
2.12.2 Many--Electron Case
61(4)
References
65(2)
3 Quantal Density Functional Theory
67(68)
3.1 Time-Dependent Quantal Density Functional Theory: Part I
71(15)
3.1.1 Quantal Sources
72(3)
3.1.2 Fields
75(3)
3.1.3 Total Energy and Components in Terms of Quantal Sources and Fields
78(3)
3.1.4 The S System `Quantal Newtonian' Second Law
81(2)
3.1.5 Effective Field Feff(rt) and Electron-Interaction Potential Energy υee(rt)
83(3)
3.2 Sum Rules
86(3)
3.2.1 Integral Virial Theorem
86(1)
3.2.2 Ehrenfest's Theorem and the Zero Force Sum Rule
86(1)
3.2.3 Torque Sum Rule
87(2)
3.3 Time-Dependent Quantal Density Functional Theory: Part II
89(2)
3.4 Time-Independent Quantal Density Functional Theory
91(9)
3.4.1 The Interacting System and the `Quantal Newtonian' First Law
91(1)
3.4.2 The S System and Its `Quantal Newtonian' First Law
92(1)
3.4.3 Quantal Sources
93(1)
3.4.4 Fields
94(1)
3.4.5 Total Energy and Components
94(2)
3.4.6 Effective Field Feff(r) and Electron--Interaction Potential Energy υee(r)
96(1)
3.4.7 Sum Rules
97(1)
3.4.8 Highest Occupied Eigenvalue εm
98(1)
3.4.9 Proof that Nonuniqueness of Effective Potential Energy Is Solely Due to Correlation-Kinetic Effects
99(1)
3.5 Application of Q-DFT to the Ground and First Excited Singlet State of the Hooke's Atom
100(15)
3.5.1 S System Wavefunction, Spin--Orbitals, and Density
101(1)
3.5.2 Pair--Correlation Density; Fermi and Coulomb Hole Charge Distributions
102(4)
3.5.3 Hartree, Pauli, and Coulomb Fields EH(r), Ex(r), Ec(r) and Energies EH, Ex, Ec
106(2)
3.5.4 Hartree WH(r), Pauli Wx(r), and Coulomb Wc(r) Potential Energies
108(2)
3.5.5 Correlation--Kinetic Field Ztc(r), Energy Tc, and Potential Energy Wtc(r)
110(4)
3.5.6 Total Energy and Ionization Potential
114(1)
3.5.7 Endnote on the Multiplicity of Potentials
114(1)
3.6 Quantal Density Functional Theory of Degenerate States
115(1)
3.7 Application of Q-DFT to the Wigner High-Electron-Correlation Regime of Nonuniform Density Systems
116(2)
3.8 Quantal Density Functional Theory of Hartree--Fock and Hartree Theories
118(17)
3.8.1 Hartree--Fock Theory
119(3)
3.8.2 The Slater--Bardeen Interpretation of Hartree--Fock Theory
122(2)
3.8.3 Theorems in Hartree--Fock Theory
124(1)
3.8.4 Q--DFT of Hartree--Fock Theory
124(3)
3.8.5 Hartree Theory
127(3)
3.8.6 Q--DFT of Hartree Theory
130(2)
References
132(3)
4 Hohenberg--Kohn, Kohn--Sham, and Runge--Gross Density Functional Theories
135(50)
4.1 The Hohenberg--Kohn Theorems
140(8)
4.1.1 The First Hohenberg-Kohn Theorem
141(2)
4.1.2 Implications of the First Hohenberg-Kohn Theorem
143(2)
4.1.3 The Second Hohenberg-Kohn Theorem
145(1)
4.1.4 The Primacy of the Electron Number in Hohenberg-Kohn Theory
146(2)
4.2 Generalization of the Fundamental Theorem of Hohenberg-Kohn
148(6)
4.2.1 The Unitary Transformation
149(2)
4.2.2 New Insights as a Consequence of the Generalization
151(3)
4.3 Inverse Maps
154(1)
4.4 The Percus-Levy-Lieb Constrained-Search Path
155(3)
4.5 Kohn--Sham Density Functional Theory
158(6)
4.6 Runge-Gross Time-Dependent Density Functional Theory
164(3)
4.7 Generalization of the Runge-Gross Theorem
167(3)
4.8 Corollary to the Hohenberg--Kohn and Runge-Gross Theorems
170(15)
4.8.1 Corrollary to the Hohenberg-Kohn Theorem
172(6)
4.8.2 Corollary to the Runge-Gross Theorem
178(2)
4.8.3 Endnote
180(2)
References
182(3)
5 Physical Interpretation of Kohn--Sham Density Functional Theory via Quantal Density Functional Theory
185(30)
5.1 Interpretation of the Kohn--Sham Electron--Interaction Energy Functional EKSee[ ρ] and Its Derivative υee(r)
187(4)
5.2 Adiabatic Coupling Constant Scheme
191(7)
5.2.1 Q--DFT Within Adiabatic Coupling Constant Framework
192(2)
5.2.2 KS--DFT Within Adiabatic Coupling Constant Framework
194(2)
5.2.3 Q--DFT and KS--DFT in Terms of the Adiabatic Coupling Constant Perturbation Expansion
196(2)
5.3 Interpretation of the Kohn--Sham `Exchange' Energy Functional EKSx[ ρ] and Its Derivative υx(r)
198(1)
5.4 Interpretation of the Kohn--Sham `Correlation' Energy Functional EKSc[ ρ] and Its Derivative υc(r)
199(1)
5.5 Interpretation of the KS--DFT of Hartree--Fock Theory
200(1)
5.6 Interpretation of the KS--DFT of Hartree Theory
201(1)
5.7 The Optimized Potential Method
202(6)
5.7.1 The `Exchange--Only' Optimized Potential Method
203(5)
5.8 Physical Interpretation of the Optimized Potential Method
208(7)
5.8.1 Interpretation of `Exchange--Only' OPM
208(1)
5.8.2 A. Derivation via Q--DFT
208(3)
5.8.3 B. Derivation via the XO--OPM Integral Equation
211(1)
References
212(3)
6 Quantal Density Functional Theory of the Density Amplitude
215(16)
6.1 Density Functional Theory of the B System
217(4)
6.1.1 DFT Definitions of the Pauli Kinetic and Potential Energies
220(1)
6.2 Derivation of the Differential Equation for the Density Amplitude from the Schrodinger Equation
221(3)
6.3 Quantal Density Functional Theory of the B System
224(5)
6.3.1 Q--DFT Definitions of the Pauli Kinetic and Potential Energy
228(1)
6.4 Endnote
229(2)
References
230(1)
7 Quantal Density Functional Theory of the Discontinuity in the Electron--Interaction Potential Energy
231(22)
7.1 Origin of the Discontinuity of the Electron--Interaction Potential Energy
232(4)
7.2 Expression for Discontinuity Δ in Terms of S System Eigenvalues
236(3)
7.3 Correlations Contributing to the Discontinuity According To Kohn--Sham Theory
239(1)
7.4 Quantal Density Functional Theory of the Discontinuity
239(11)
7.4.1 Correlations Contributing to the Discontinuity According To Q--DFT: Analytical Proof
242(2)
7.4.2 Numerical Examples
244(6)
7.5 Endnote
250(3)
References
251(2)
8 Generalized Hohenberg-Kohn Theorems in Electrostatic and Magnetostatic Fields
253(30)
8.1 The Classical Hamiltonian and Properties
256(3)
8.1.1 Classical Physics
256(3)
8.2 The Quantum-Mechanical Hamiltonian and Properties
259(6)
8.3 Generalized Hohenberg-Kohn Theorems
265(12)
8.3.1 Proof of Generalized Hohenberg-Kohn Theorems: Case I: Spinless Electrons
266(6)
8.3.2 Proof of Generalized Hohenberg-Kohn Theorems: Case II: Electrons with Spin
272(5)
8.4 Remarks on Spin and Current Density Functional Theories
277(4)
8.4.1 Remarks on Spin Density Functional Theory
277(2)
8.4.2 Remarks on Paramagnetic Current Density Functional Theory
279(2)
8.5 Endnote
281(2)
References
281(2)
9 Quantal-Density Functional Theory in the Presence of a Magnetostatic Field
283(30)
9.1 Schrodinger Theory and the `Quantal Newtonian' First Law
285(6)
9.2 Quantal Density Functional Theory
291(4)
9.3 Application of Quantal Density Functional Theory to a Quantum Dot
295(18)
9.3.1 Quantal Sources
296(5)
9.3.2 Fields and Energies
301(4)
9.3.3 Potentials
305(5)
9.3.4 Eigenvalue
310(1)
9.3.5 Single-Particle Expectations
310(1)
9.3.6 Concluding Remarks
310(1)
References
311(2)
10 Physical Interpretation of the Local Density Approximation and Slater Theory via Quantal Density Functional Theory
313(28)
10.1 The Local Density Approximation in Kohn--Sham Theory
316(14)
10.1.1 Derivation and Interpretation of Electron Correlations via Kohn--Sham Theory
316(3)
10.1.2 Derivation and Interpretation of Electron Correlations via Quantal Density Functional Theory
319(5)
10.1.3 Structure of the Fermi Hole in the Local Density Approximation
324(5)
10.1.4 Endnote
329(1)
10.2 Slater Theory
330(11)
10.2.1 Derivation of the Exact `Slater Potential'
330(3)
10.2.2 Why the `Slater Exchange Potential' Does Not Represent the Potential Energy of an Electron
333(3)
10.2.3 Correctly Accounting for the Dynamic Nature of the Fermi Hole
336(2)
10.2.4 The Local Density Approximation in Slater Theory
338(1)
References
338(3)
11 Epilogue
341(8)
Curriculum Vitae
347(2)
Appendix A A Derivation of the `Quantal Newtonian' Second Law 349(6)
Appendix B Derivation of the Harmonic Potential Theorem 355(10)
Appendix C Analytical Expressions for the Properties of the Ground and First Excited Singlet States of the Hooke's Atom 365(10)
Appendix D Derivation of the Kinetic--Energy--Density Tensor for Hooke's Atom in Its Ground State 375(4)
Appendix E Derivation of the S System `Quantal Newtonian' Second Law 379(4)
Appendix F Derivation of the `Quantal Newtonian' First Law in the Presence of a Magnetic Field 383(8)
Appendix G Analytical Expressions for the Ground State Properties of the Hooke's Atom in a Magnetic Field 391(6)
Appendix H Derivation of the Kinetic-Energy-Density Tensor for the Ground State of Hooke's Atom in a Magnetic Field 397(4)
Appendix I Derivation of the Pair--Correlation Density in the Local Density Approximation for Exchange 401(6)
Index 407
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