# Electrostatic Potential (ESP)¶

The electrostatic potential (ESP) is used to analyze the positively and negatively charged regions of a molecule. In atomic units, it is defined as the interaction energy of the molecule with an infinitesimal positive point charge, per unit charge:

$\Phi \left(\mathbf{r}\right) = \sum_{A=1}^{N_\text{atoms}} \frac{Z_A}{\rvert \mathbf{R}_A - \mathbf{r} \lvert} - \int \frac{\rho \left(\mathbf{r^\prime}\right)}{\rvert \mathbf{r} - \mathbf{r^\prime} \lvert} d\mathbf{r^\prime}$

Here, $$Z_A$$ is the charge on an atomic nucleus, $$\mathbf{R}_A$$ is the coordinates of an atomic nucleus, and $$\rho(\mathbf{r})$$ is the electron density. The ESP can also be written in terms of the external potential, $$\mathit{v}(\mathbf{r})$$, used in DFT:

$\Phi \left(\mathbf{r}\right) = - \left(\mathit{v}(\mathbf{r}) + \int \frac{\rho \left(\mathbf{r^\prime}\right)}{\rvert \mathbf{r} - \mathbf{r^\prime} \lvert} d\mathbf{r^\prime}\right)$

Typically, hard acids/electrophiles attack a molecule where the electrostatic potential is most negative, and the hard bases/nucleophiles attack a molecule where it is most positive. To predict how a molecule would interact with an approaching reagent, one commonly plots the electrostatic potential on a van der Waals surface of the molecule or an iso-electron-density surface of about $$\rho(\mathbf{r})=0.002$$.

Todo

$\begin{split}E(Z_A \pm 1) - E(Z_A) \stackrel{\text{1st order}}{\approx} \pm \frac{\partial E}{\partial Z_A} = \pm \left( \sum_{B=1 \\ B \neq A}^{N_\text{atoms}} \frac{Z_B}{\rvert \mathbf{R}_B - \mathbf{r} \lvert} - \int \frac{\rho \left(\mathbf{r^\prime}\right)}{\rvert \mathbf{r} - \mathbf{r^\prime} \lvert} d\mathbf{r^\prime}\right)\end{split}$