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

Add links to example gallery.

The value of the electrostatic potential at a nucleus is often interesting, because it shows how the energy changes when the atomic number of the nucleus changes (to first order). Therefore it is relevant for alchemical changes as an atom changes to an adjacent atom in the periodic table,

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

For example, at a hydrogen nucleus, the electrostatic potential correlates tightly with the proton affinity (ergo pKa).