Exact Maps in Density Functional Theory for Lattice Models

TL;DR

This paper uses exact diagonalization on lattice models to explicitly construct the fundamental density-to-potential and density-to-wavefunction maps in density functional theory, revealing a new intra-system steepening feature related to correlation growth.

Contribution

It provides the first explicit construction of the exact density-to-wavefunction map in lattice models, uncovering the intra-system steepening phenomenon and its relation to the derivative discontinuity.

Findings

Identification of intra-system steepening as correlation increases

Connection between intra-system steepening and the inter-system derivative discontinuity

Explicit functional relations for excited states and entropy in terms of ground-state density

Abstract

In the present work, we employ exact diagonalization for model systems on a real-space lattice to explicitly construct the exact density-to-potential and for the first time the exact density-to-wavefunction map that underly the Hohenberg-Kohn theorem in density functional theory. Having the explicit wavefunction-to- density map at hand, we are able to construct arbitrary observables as functionals of the ground-state density. We analyze the density-to-potential map as the distance between the fragments of a system increases and the correlation in the system grows. We observe a feature that gradually develops in the density-to-potential map as well as in the density-to-wavefunction map. This feature is inherited by arbitrary expectation values as functional of the ground-state density. We explicitly show the excited-state energies, the excited-state densities, and the correlation entropy as functionals of the ground-state density. All of them show this exact feature that sharpens as the coupling of the fragments decreases and the correlation grows. We denominate this feature as intra-system steepening. We show that for fully decoupled subsystems the intra-system steepening transforms into the well-known inter-system derivative discontinuity. An important conclusion is that for e.g. charge transfer processes between localized fragments within the same system it is not the usual inter-system derivative discontinuity that is missing in common ground-state functionals, but rather the differentiable intra-system steepening that we illustrate in the present work.