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Quantum fluctuations and kinetic correlation in the strongly Interacting Limit of
Density Functional Theory
Grossi, J.
2020
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Grossi, J. (2020). Quantum fluctuations and kinetic correlation in the strongly Interacting Limit of Density
Functional Theory.
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C O N T E N T S
1 introduction 1
Overview of the Thesis and main contributions 2
2 fundamentals of density functional theory 5
2.1 Preliminaries 5
2.2 The Hohenberg-Kohn theorem and the Levy-Lieb con-strained formulation 6
2.3 The Lieb density matrix constrained-search functional 8
2.4 The Kohn-Sham self-consistent scheme 9
2.5 Adiabatic connection formalism 11
3 density functional theory for strongly corre-lated systems 15
3.1 Strongly Interacting Limit in Density Functional The-ory 15
3.2 Strongly Interacting Limit as an Optimal Transport problem 20
3.3 Zero Point Oscillations 21
4 fermionic statistics in the strongly interacting limit of density functional theory 31
4.1 Introduction 31
4.2 Wrap-up: again, SCE+ZPE in a nutshell 32
4.3 Constrained search method for two electrons in 1D 38
4.4 Adiabatic connection at large l: numerical and asymp-totic results 39
4.5 The effects of the spin state at large l 41
4.5.1 Explicit antisymmetrization of the ZPO wave-function 42
4.6 Alternative strategies to include the statistics in the
l 1 regime 43
4.6.1 Results for the singlet-triplet splitting 48
4.7 Conclusions and Perspectives 48
5 functional derivative of the zero point energy functional 51
5.1 Introduction 51
5.1.1 Cliff Notes on SCE + ZPE for N=2 electrons in
1D 52
5.2 Functional derivative of FZPE[$]for N=2, D=1 53
5.2.1 Explicit expression 53
5.2.2 Numerical results for selected densities 55
5.2.3 Divergences of dFZPE/d$(x)in 1D 58
5.3 Exchange-correlation potential for a 1D dimer 59
5.4 Conclusions 63
vi contents
6 kinetic correlation functionals from the en-tropic regularization of the sce problem 65
6.1 Introduction 65
6.2 The entropic regularization of the SCE functional 66
6.2.1 Interpretation of the parameter t and of the en-tropy S[g] 69
6.3 Analytic and numerical examples of the entropic regu-larization problem 71
6.3.1 Harmonic interactions case 71
6.3.2 Regularized Coulomb interaction case 73
6.4 Comparison with the Hohenberg-Kohn functional 74
6.4.1 Inequalities and approximations 75
6.5 Analytical and numerical investigation 77
6.5.1 Repulsive Harmonic interaction 77
6.5.2 Effective Coulomb interaction 79
6.6 Conclusions and Outlook 80
7 ks equations with functionals from the strictly-correlated regime: investigation with a spec-tral renormalisation method 83
7.1 Introduction 83
7.2 Theoretical background 85
7.2.1 Density Functional Theory 85
7.2.2 Kohn-Sham Equations 86
7.2.3 Approximations for the xc functional 87
7.3 Spectral Renormalization Algorithm 91
7.3.1 Bosons 93
7.3.2 Fermions 93
7.4 Numerical Implementation 94
7.5 Results 96
7.5.1 Convergence of the LDA and SCE approxima-tion 97
7.5.2 Convergence of the SCE+ZPE approximation 98
7.6 Conclusions and Outlook 102
8 zero point oscillations in hartree-fock theory 105
8.1 Introduction 105
8.2 Bird’s eye survey on adiabatic connection in Hartree-Fock theory 106
8.2.1 The strong interacting limit in HF 107
8.3 The Jellium model from the adiabatic connection in Hartree-Fock theory 109
8.3.1 Hartree-Fock approximation for the rs 1 regime 111
8.4 Variational computation of the Zero Point Hartree-Fock energy 114
contents vii Appendix
a kinetic energy operator at different orders 123
b co-motion functions for the analytical densi-ties 127
c calculation details for dFZPE/d$(x) 129
d lda for electrons in 1d 133
e code to produce N comotion functions 135
f asymptotic expansion of the exchange opera-tor 137
g electrostatic energy of a hartree product of gaussian orbitals 141
$2(x) 145
