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I am quite a beginner in DFT and I am studying the theory behind the computational tools. I have some doubts about the Coulumb approximate functional term: $$U[n] = \frac{1}{2} \int d^3 r \int d^3 r^\prime \frac {n(\mathbf{r}) n(\mathbf{r}^\prime)}{|\mathbf{r} -\mathbf{r}^\prime|}$$ If I use the definition of the electronic density $$n[\mathbf(r)] = \sum_i^N \delta(\mathbf{r} -\mathbf{r}_i)$$ in the previous equation I obtain the square of the electronic charge over $\pu{0}$. In other words the energy diverges. Am I wrong?

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  • $\begingroup$ No, you are not wrong. DFT in almost all forms includes an interaction of the electron with itself as the approximate exchange functional doesn't exactly cancel this contribution to the Hartree term, unlike Hartree-Fock theory. For delta function charge distributions this self-interaction is infinite - try again with Gaussians for something that will work. $\endgroup$
    – Ian Bush
    Feb 26, 2021 at 9:55

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