# The Electron Pair Localization Function

The Electron Pair Localization Function is a function defined in the three-dimensional space. It measures the degree of pairing of electrons in a molecule, with an increasing value as the electron pairing increases. Therefore chemical bonds, core domains and lone pairs can be visualized.

The EPLF [1] has been designed to describe local electron pairing in molecular systems. It is defined as a scalar function defined in the three-dimensional space and taking its values in the [-1,1] range. It is defined as follows:

$ {\rm EPLF}(\vec{r}) = \frac { d_{\sigma \sigma} (\vec{r}) - d_{\sigma {\bar \sigma}} (\vec{r}) } { d_{\sigma \sigma} (\vec{r}) + d_{\sigma {\bar \sigma}} (\vec{r}) } $

where the quantity $d_{\sigma \sigma} (\vec{r})$ [resp. $d_{\sigma {\bar \sigma}} (\vec{r})$] denotes the quantum-mechanical average of the distance between an electron of spin $\sigma$ located at $\vec{r}$ and the closest electron of same spin (resp., of opposite spin $\bar{\sigma}$).

The mathematical definition of these quantities can be written as

$d_{\sigma \sigma}(\vec{r}) = \int \Psi^2(\vec{r}_1,\dots,\vec{r_N}) \left[ \sum_{i=1}^N \delta(\vec{r}-\vec{r}_i) \min_{j\neq i;\sigma_j=\sigma_i}|\vec{r}_i - \vec{r}_j| \right] d\vec{r}_1 \dots d\vec{r}_N$

$d_{\sigma {\bar \sigma}}(\vec{r}) = \int \Psi^2(\vec{r}_1,\dots,\vec{r}_N) \left[ \sum_{i=1}^N \delta(\vec{r}-\vec{r}_i) \min_{j\ne i;\sigma_j\neq\sigma_i}|\vec{r}_i - \vec{r}_j| \right] d\vec{r}_1 \dots d\vec{r}_N$

where $\sigma$ is the spin ($\alpha$ or $\beta$), $\bar{\sigma}$ is the spin opposite to $\sigma$, $\Psi(\vec{r}_1,\dots,\vec{r}_N)$ is the wave function, and $N$ is the number of electrons.

In a region of space, if the shortest distance separating anti-parallel electrons is smaller than the shortest distance separating electrons of same spin, the EPLF takes positive values and indicates pairing of anti-parallel electrons. In contrast, if the shortest distance separating anti-parallel electrons is larger than the shortest distance separating electrons of same spin, the EPLF takes negative values and indicates pairing of parallel electrons (which in practice never happens). If the shortest distance separating anti-parallel electrons is equivalent to the shortest distance separating electrons of same spin, the EPLF takes values close to zero and indicates no electron pairing.

The original formulation of EPLF is extremely easy to compute in the quantum Monte Carlo framework. However, it is not possible to compute it analytically due to the presence of the min function in the definitions of $d_{\sigma \sigma}$ and $d_{\sigma {\bar \sigma}}$.

[1] Electron pair localization function, a practical tool to visualize electron localization in molecules from quantum Monte Carlo data
A. Scemama, P. Chaquin, M. Caffarel
J. Chem. Phys., vol 121, pp. 1725-1735 (2004)