Using random numbers to obtain Kohn-Sham potential for a given density

Abstract

Most of the density-to-potential inversion methods developed over the years follow a general algorithm $ v_{xc}^{i+1}(\textbf{r}) = v_{xc}^{i}(\textbf{r}) + Δv_{xc}(\textbf{r})$, where $Δv_{xc}(\textbf{r}) = \frac{δS[ρ]}{δρ(\textbf{r})} \Big |_{ ρ_i(\textbf{r})} - \frac{δS[ρ]}{δρ(\textbf{r})}\Big|_{ ρ_0(\textbf{r})}$ and $S[ρ]$ is an appropriately chosen density functional. In this work we show that this algorithm can be used with random numbers to obtain the exchange-correlation potential for a given density. This obviates the need to evaluate the functional $S[ρ]$ in each iterative step. The method is demonstrated by calculating exchange-correlation potential of atoms, clusters and the Hookium.