# Electrostatic potential map and local ionization map

What's the difference between electrostatic potential map and local ionization map?

I think that the main difference is that the electrostatic potential map doesn't take into account resonance (delocalization), but only electronegativity, therefore nucleophilicity or electrophilicity of a species is much better estimated by using the local ionization map. Am I wrong?

• Can you provide a reference for the local ionization map? – pentavalentcarbon Sep 14 '16 at 16:36
• Huh? I've just started using Spartan and I couldn't find a neat explanation in the manual. – Marko Sep 14 '16 at 17:32
• Ok, it's from Spartan; that's what I meant by "reference". I've never heard of it before, but now I have a starting point. – pentavalentcarbon Sep 14 '16 at 18:20

## 1 Answer

The excellent MultiWFN is able to produce both functions and luckily, a reference for it has explanations for both as well:

# Electrostatic potential

According to the reference above,

$$V_{tot}(r) = V_{nu}(r) + V_{ele}(r) = \sum_A \frac{Z_A}{|r - R_A|} - \int \frac{\rho(r')}{|r - r'|} dr'$$ This function measures the electrostatic interaction between a unit point charge placed at $r$ and the system of interest. A positive (negative) value implies that current position is dominated by nuclear (electronic) charges. Molecular ESP has been widely used for prediction of nucleophilic and electrophilic sites for a long time. It is also valuable in studying hydrogen bonds, halogen bonds, molecular recognitions, and the inter-molecular interaction of aromatics.

# Average local ionisation energy

Again, according to the reference above,

$$\bar{I}(r) = \frac{\sum_i \rho_i(r) |\epsilon_i|}{\rho(r)}$$ where $\rho_i(r)$ and $\epsilon_i$ are the electron density function and orbital energy of the $i$th molecular orbital, respectively. This function has many uses, for example, reproducing atomic shell structure, measuring electronegativity, quantifying local polarizability and hardness, and predicting sites for electrophilic or radical attack.

Thus, while the electrostatic potential represents the hypothetical electrostatic interaction energy between the molecule and a positive unit point charge at $r$, the average local ionisation energy is the orbital density-weighted absolute orbital energy average.

Finally, resonance/delocalization/electronegativity are built in the method you use to produce the electron density $\rho(r)$ (edit: a nice discussion about it can be found here).