Electronic energy functionals: Levy–Lieb principle within the ground state path integral quantum Monte Carlo

Abstract

We propose a theoretical/computational protocol based on the use of the Ground State Path Integral Quantum Monte Carlo for the calculation of the kinetic and Coulomb energy density for a system of N interacting electrons in an external potential. The idea is based on the derivation of the energy densities via the (N − 1)‐conditional probability density within the framework of the Levy–Lieb constrained search principle. The consequences for the development of energy functionals within the context of density functional theory are discussed. We propose also the possibility of going beyond the energy densities and extend this idea to a computational procedure where the (N − 1)‐conditional probability is an implicit functional of the electron density, independently from the external potential. In principle, such a procedure paves the way for an on‐the‐fly determination of the energy functional for any system. © 2012 Wiley Periodicals, Inc.

Density functional theory: Its origins, rise to prominence, and future

“…Condensed matter scientists have much experience with periodic boundary conditions and plane-wave orbital expansions, and this should aid the implementation of the method in extended systems. Another example is the reformulation of the constrained search approach in DF theory (Levy, 1979;Lieb, 1983) in terms of the density and the ðN − 1Þ-conditional probability density, which can be treated by ground state path integral QMC (Delle Site, Ghiringhelli, and Ceperley, 2013). It remains to be seen whether the computational demands usually associated with QMC can be reduced.…”

Shannon entropy and many‐electron correlations: Theoretical concepts, numerical results, and Collins conjecture

“…Its potentiality has never really been explored in full and it may turn extremely useful in connection with other electronic structure methods (see e.g., Ref. ) or in modern popular multiscale studies (see e.g., Ref. ).…”

“…and the revolutionary essence of the theorem is that in ρ(r) are contained (coded) all the properties of the ground state, even its wavefunction (see e.g. 13,48 ). Following the arguments above it becomes natural the following question: according to the idea of encoding/decoding of information theory, does the log ρ form of t c , v c and e c tell us that the correlation terms of the universal functional of Hohenberg and Kohn expresses the fact that the correlation energy corresponds (is proportional) to the average quantity of information needed to explicitly express the exact many-body behavior of the electrons?…”

Exact Density Functional Obtained via the Levy Constrained Search

“…There has been other recent work on the exact functional via a Lieb maximization [11], based on a search over potentials on which to carry out a many-body method such as full configuration interaction (FCI) [12][13][14][15][16][17] and in TDDFT [18,19], or Monte-Carlo based ideas to tackle the constrained search [20,21]. However, the Lieb maximization would fail to converge if the density is non-v-representable and the search over potentials becomes much harder as the density becomes more strongly correlated [15,22].…”

“…11,16 Also, there have been direct Monte Carlo proposals to tackle the constrained search. 17,18 In this Letter, we develop a method to explicitly carry out the Levy constrained search over many-body wave functions that integrate to the same density, Ψ → ρ as in eq 1 and, furthermore, to carry out the minimization over densities in eq 2, as an alternative to solving the Schrodinger equation. The importance of the exact functional is demonstrated for strongly correlated systems.…”