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

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.