Exact Density Functional Obtained via the Levy Constrained Search

Abstract

A stochastic minimization method for a real-space wave function, Ψ(r, r...r ), constrained to a chosen density, ρ(r), is developed. It enables the explicit calculation of the Levy constrained search, F[ρ] = min⟨Ψ| T̂ + V̂ |Ψ⟩, which gives the exact functional of density functional theory. This general method is illustrated in the evaluation of F[ρ] for densities in one dimension with a soft-Coulomb interaction. Additionally, procedures are given to determine the first and second functional derivatives, δ F/δρ(r) and δ F/[δρ(r)δρ(r')]. For a chosen external potential, v(r), the functional and its derivatives are used in minimizations over densities to give the exact energy, E , without needing to solve the Schrödinger equation.

Fermionic Statistics in the Strongly Correlated Limit of Density Functional Theory

“…In previous work, 39 the Levy constrained search was found for the exact density-matrix functional of the two-site Hubbard model using an analytic formula. However, in this work, the constrained search is carried out via a stochastic minimization of the wave function as in ref ( 40 ) to give the exact density functional of eq 32 .…”

“…With these initial wave functions that integrate to ρ( x ), the key to the procedure is to define moves of the spatial part of the wave function that maintain the density. When the density is represented on a grid (we generally use 200 grid points), this can be done based on a move of four points of the wave function at once as outlined in ref ( 40 ). These moves are attempted and accepted if they lower the energy of eq 32 .…”

Functional derivative of the zero-point-energy functional from the strong-interaction limit of density-functional theory

“…In the appendix, we show that, using (33) and (34) in (31) and inserting the result in (30), δF ZPE /δρ(x) can be expressed as (see Appendix for details)…”

“…For this reason, our investigation starts from a simple, yet non trivial, case: two electrons confined in one dimension (1D). Similar 1D models have been widely used to investigate features in exact KS DFT, proving to provide a good qualitative description of the relevant features of their 3D counterparts [27,[31][32][33][34][35].…”

Application of neural network for exchange-correlation functional interpolation

“…Calculations made by post-Hartree-Fock methods were also used to improve the quality of the XC functionals for molecular systems [4,11]. The search for advanced XC functional is ongoing, and still a very active direction of research [12][13][14][15].…”