Potential Energy of an Electron-Nuclei Interaction in DFT

[Dario56]

TL;DR

Potential Energy of Electron - Nuclei Interaction as a Functional of Electron Density

In density functional theory (DFT), electron density is a central quantity. Because of this, we want to calculate electron - nuclei potential energy as functional on electron density. If we know how potential energy varies across space, we can calculate this functional with plugging particular electron density into following equation:
$$ V[n] = \int V(r)n(r)d^3r $$
I am not sure where does this equation come from - it's derivation. Why does multiple $V(r)n(r)$ integrated over all space define this functional?

 

[anuttarasammyak]

Summary:: Potential Energy of Electron - Nuclei Interaction as a Functional of Electron Density

Why does multiple V(r)n(r) integrated over all space define this functional?

$$V(r)=-\frac{1}{4\pi\epsilon_0}\frac{Ze^2}{r}$$
and n(r) is density of electron cloud at r.
$$\int n(\mathbf{r}) d^3\mathbf{r} = Z$$
for neutral atom.