Building Kohn-Sham potentials for ground and excited states

TL;DR

This paper investigates the inverse problem in Density Functional Theory, demonstrating the existence of potentials for specific states and densities, and introduces an improved inversion algorithm that handles degeneracies.

Contribution

It provides new theoretical results on the existence of potentials for given densities and states, and proposes an enhanced inversion algorithm accounting for degeneracies.

Findings

Existence of potentials with specific bound states close to target densities

Pure-state representability conjecture for 2D and non-density for 3D

An inversion algorithm that manages degeneracies effectively

Abstract

We analyze the inverse problem of Density Functional Theory using a regularized variational method. First, we show that given $k$ and a target density $\rho$, there exist potentials having $k^{\text{th}}$ bound mixed states which densities are arbitrarily close to $\rho$. The state can be chosen pure in dimension $d=1$ and without interactions, and we provide numerical and theoretical evidence consistently leading us to conjecture that the same pure representability result holds for $d=2$, but that the set of pure-state $v$-representable densities is not dense for $d=3$. Finally, we present an inversion algorithm taking into account degeneracies, removing the generic blocking behavior of standard ones.