# Coulomb potential in Kohn-Sham DFT

1. Nov 20, 2012

### molkee

In Kohn-Sham DFT, the Coulomb potential, which is a component of the Kohn-Sham potential, is given by:

$v_H(\mathbf{r}) = \int \frac{\rho(\mathbf{r'})}{|\mathbf{r}-\mathbf{r'}|}d\mathbf{r'}$

where $\rho(\mathbf{r'})$ is the electron density.

For molecular systems with exponential densities, this potential is known to be finite at any $\textbf{r}$.

How to prove it analytically just based on its definition?

Would a potential

$v(\mathbf{r}) = \int \frac{\rho(\mathbf{r'})}{|\mathbf{r}-\mathbf{r'}|^n}d\mathbf{r'}$

(where $n$ is some nonnegative integer)

be also finite at any $\mathbf{r}$?

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