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Density Functionals For Many-particle Systems: Mathematical Theory And Physical Applications Of Effective Equations [Kietas viršelis]

Edited by (Beijing Institute Of Technology, China & Centre For Quantum Technologies, Singapore), Edited by (Ludwig-maximilians-univ Munchen, Germany), Edited by (Nus, S'pore)
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Density Functionals For Many-particle Systems: Mathematical Theory And Physical Applications Of Effective EquationsDidesnis paveikslėlis
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Raktažodžiai:
"Density Functional Theory (DFT) first established it's theoretical footing in the 1960s from the framework of Hohenberg-Kohn theorems. DFT has since seen much development in evaluation techniques as well as application in solving problems in Physics, Mathematics and Chemistry. This review volume, part of the IMS Lecture Notes Series, is a collection of contributions from the September 2019 Workshop on the topic, held in the Institute for Mathematical Sciences, National University of Singapore. With contributions from prominent Mathematicians, Physicists, and Chemists, the volume is a blend of comprehensive review articles on the Mathematical and the Physicochemical aspects of DFT and shorter contributions on particular themes, including numerical implementations. The book will be a useful reference for advanced undergraduate and postgraduate students as well as researchers"--
Foreword vii
Preface ix
Participants xx
1 Mathematical elements of density functional theory
1(56)
Heinz Siedentop
1 Introduction
1(1)
2 Elements of quantum mechanics of electrons
2(7)
2.1 Electronic Hilbert spaces
2(1)
2.2 Electronic states
3(1)
2.3 Creation and annihilation operators
4(2)
2.4 Reduced densities and density matrices
6(1)
2.5 Observables
7(1)
2.5.1 The number operator
7(1)
2.5.2 The electronic Hamiltonian
8(1)
3 The Hohenberg-Kohn theorem
9(4)
4 Some results on concrete density functionals
13(28)
4.1 Thomas-Fermi theory
13(1)
4.1.1 Definition and basic properties
13(12)
4.1.2 Asymptotic exactness of Thomas-Fermi theory
25(6)
4.2 The Thomas-Fermi-Weizsacker functional
31(3)
4.3 The Engel-Dreizler functional
34(2)
4.4 Density functionals in phase space
36(1)
4.4.1 Marginal functionals: The position space
37(1)
4.4.2 Marginal functions: The momentum space
38(1)
4.4.3 Time-dependent equations
39(2)
5 Functionals of the one-particle density matrix
41(6)
5.1 The Hartree-Fock functional
41(4)
5.2 The Miiller functional
45(2)
Appendix: Maximal functions of powers and Thomas-Fermi energy of exchange holes
47(2)
Acknowledgments
49(1)
References
49(8)
2 A statistical theory of heavy atoms: Energy and excess charge
57(12)
Hongshuo Chen
Rupert L. Frank
Heinz Siedentop
1 Introduction
57(2)
2 Bound on the energy
59(3)
2.1 The domain of FWD
59(1)
2.2 Lower bound
59(1)
2.2.1 The Weizsacker energy
59(1)
2.2.2 The potential energy
59(1)
2.2.3 The Thomas-Fermi term
60(1)
2.2.4 The exchange energy
60(1)
2.2.5 The total energy
61(1)
3 Application to the excess charge problem
62(6)
Acknowledgments
68(1)
References
68(1)
3 Relativistic strong Scott conjecture: A short proof
69(12)
Rupert L. Frank
Konstantin Merz
Heinz Siedentop
1 Introduction
69(3)
2 Proof of the convergence
72(2)
3 The majorant
74(4)
Acknowledgments
78(1)
References
79(2)
4 Direct methods to Lieb-Thirring kinetic inequalities
81(36)
Phan Thanh Nam
1 Introduction
81(4)
2 Rumin method
85(12)
2.1 A simple proof of Sobolev inequality
85(2)
2.2 Lieb-Thirring kinetic inequality
87(3)
2.3 Eigenvalue bounds for Schrodinger operators
90(4)
2.4 Best known constant for kinetic inequality
94(3)
2.5 Further results
97(1)
3 Lundholm-Solovej method
97(14)
3.1 Kinetic inequality via local exclusion principle
97(7)
3.2 Kinetic inequality with semiclassical constant and error term
104(3)
3.3 Kinetic inequality for functions vanishing on diagonal set
107(2)
3.4 Lieb-Thirring inequality for interacting systems
109(2)
3.5 Further results
111(1)
Acknowledgments
111(1)
References
112(5)
5 Dynamics of interacting bosons: A compact review
117(38)
Marcin Napiorkowski
1 Introduction
117(6)
1.1 Setup
117(1)
1.2 Scaling regimes
118(2)
1.3 Types of approximation
120(3)
1.4 Outline
123(1)
2 Ground state properties
123(3)
2.1 Leading order approximation for the ground state
123(1)
2.2 Second order correction
124(2)
3 Leading order approximation
126(10)
3.1 Results for different scaling regimes
127(1)
3.1.1 Mean-field scaling
127(2)
3.1.2 NLS regime
129(1)
3.1.3 The GP regime
130(1)
3.2 Methods
131(1)
3.2.1 The BBGKY hierarchy
132(1)
3.2.2 Quantitative approaches
133(3)
4 Norm approximation
136(7)
5 Fock space approximation
143(3)
Acknowledgments
146(1)
References
146(9)
6 Corrections to the mean-field description for the dynamics of Bose gases
155(24)
Peter Pickl
1 Introduction
155(1)
2 The microscopic system and the effective system
156(1)
3 Types of convergence
157(3)
4 Proof of Theorem 1
160(8)
5 Corrections to the mean-field description
168(5)
6 Proof of Lemma 3
173(3)
Acknowledgments
176(1)
References
177(2)
7 Semiclassics: The hidden theory behind the success of DFT
179(72)
Pavel Okun
Kieron Burke
1 Introduction
180(5)
2 Basics
185(2)
3 Illustrations
187(7)
4 Scaling
194(5)
5 Box boundaries
199(4)
6 Real turning points
203(4)
7 Potential functionals
207(2)
8 Gradient expansions
209(2)
9 Three dimensions
211(5)
10 Atoms
216(2)
11 Exchange
218(4)
12 Correlation
222(2)
13 Ionization energies
224(4)
14 Practical functionals
228(1)
15 Summing up
229(7)
16 Conclusions
236(4)
Acknowledgments
240(1)
References
241(10)
8 Density-potential functional theory for fermions in one dimension
251(18)
Martin-Isbjorn Trappe
Jun Hao Hue
Berthold-Georg Englert
1 Introduction
251(2)
2 Density-potential functional theory in a nutshell
253(4)
3 Airy-averaged densities in ID
257(2)
4 DPFT densities for the Morse potential
259(3)
5 Airy-averaged energies
262(2)
6 Concluding remarks
264(1)
Acknowledgments
264(1)
References
265(4)
9 Remarks on the density functional theory of relativistic many-particle systems
269(18)
Reiner M. Dreizler
1 Introduction
269(1)
2 Nonrelativistic systems
270(3)
3 Relativistic systems on the basis of the Dirac equation
273(4)
4 Relativistic systems on the basis of field theory
277(7)
Acknowledgments
284(1)
References
284(3)
10 Energy functionals of single-particle densities: A unified view
287(22)
Berthold-Georg Englert
Jun Hao Hue
Zi Chao Huang
Mikolaj M. Paraniak
Martin-Isbjorn Trappe
1 Introduction
287(1)
2 Constrained search
288(5)
3 Configuration-space functionals
293(5)
4 Momentum-space functionals
298(2)
5 Thomas-Fermi atoms in configuration and momentum space
300(2)
6 Single-particle-exact functionals
302(1)
Appendix: Semiclassical eigenvalues
303(3)
Acknowledgments
306(1)
References
306(3)
11 Spin-density functional theory through spin-free wave functions
309(10)
Mel Levy
Federico Zahariev
Mark S. Gordon
1 Introduction
309(2)
2 The spin-free N-electron wave function
311(3)
2.1 The conventional definition of the universal functional by constrained search
312(1)
2.2 The spin-free definition of the universal functional by constrained search
312(2)
3 The spin-free constrained search from the spin-free iV-electron variational principle
314(1)
4 Spin-free coordinate-scaling constraints on the universal functional and the correlation-energy functional
315(1)
5 Concluding thoughts
316(1)
Acknowledgments
317(1)
References
317(2)
12 Advances in rectangular collocation for solution of the Schrodinger equation: From obviating integrals to machine learning
319(26)
Sergei Manzhos
Tucker Carrington
1 Introduction: The big picture
319(2)
2 Similarities and differences between the electronic and the nuclei Schrodinger equations
321(3)
3 Rectangular collocation method to solve the Schrodinger equation
324(2)
4 Rectangular collocation for the vibrational Schrodinger equation
326(6)
4.1 Treatment of the kinetic energy operator
329(1)
4.2 Use of non-integrable basis functions
330(1)
4.3 Advantage of rectangularity
331(1)
5 Rectangular collocation for the electronic Schrodinger equation
332(5)
6 Shaping the collocation point distribution
337(2)
6.1 Physical intuition based collocation point placement
337(1)
6.2 Machine learning guided collocation point placement
338(1)
7 Conclusions and perspectives
339(2)
References
341(4)
13 FLEIM: A stable, accurate and robust extrapolation method at infinity for computing the ground state of electronic Hamiltonian
345
Etienne Polack
Yvon Maday
Andreas Savin
1 Introduction
345(2)
1.1 Motivation
345(2)
1.2 Objective and structure of the paper
347(1)
2 Approach
347(5)
2.1 The model Schrodinger equation
347(2)
2.2 The correction to the model
349(1)
2.2.1 Using a basis set
349(1)
2.2.2 Approaching the Coulomb interaction
349(1)
2.2.3 Choice of the basis functions
350(1)
2.2.4 Reducing the basis set
350(2)
2.3 Computing other physical properties
352(1)
3 Numerical results
352(5)
3.1 Guidelines
352(1)
3.2 General behavior of errors
353(1)
3.3 Possibility of error estimates
353(3)
3.4 Expectation values with FLEIM: (r12) and (r2 12) for harmonium
356(1)
3.5 Comparison with DFAs
357(1)
4 Conclusion and perspectives
357(2)
Appendices
359(1)
A On density functional approximations
359(1)
B Basis functions
360(1)
C The empirical interpolation method
361(2)
C.1 Algorithm for EIM
362(1)
C.2 The forward looking empirical interpolation method (FLEIM)
363(1)
D Numerical details of the calculations
363(5)
D.1 Testing EIM and FLEIM with E(μ) = 1 + xj.(μ)
363(1)
D.2 Discretization for FLEIM
363(3)
D.3 Systems
366(1)
D.4 Obtaining the model energy
366(2)
E Change of coordinates in harmonium
368(1)
Funding
369(1)
References
369