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

TL;DR

This paper introduces a novel method using random numbers to invert density-to-potential mappings in density functional theory, eliminating the need for functional evaluations at each iteration.

Contribution

It presents a new approach that employs randomness to efficiently obtain the Kohn-Sham potential from a given density, simplifying the inversion process.

Findings

Successfully applied to atoms, clusters, and Hookium.

Avoids iterative functional evaluations in potential inversion.

Demonstrates computational efficiency and accuracy.

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}) + \Delta v_{xc}(\textbf{r})$, where $\Delta v_{xc}(\textbf{r}) = \frac{\delta S[\rho]}{\delta \rho(\textbf{r})} \Big |_{ \rho_i(\textbf{r})} - \frac{\delta S[\rho]}{\delta \rho(\textbf{r})}\Big|_{ \rho_0(\textbf{r})}$ and $S[\rho]$ 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 $S[\rho]$ in each iterative step. The method is demonstrated by calculating exchange-correlation potential of atoms, clusters and the Hookium.