# Quantum Monte Carlo Formulation

The EPLF [1] has been designed to describe local electron pairing in molecular systems. An electron $i$ located at $\vec{r}_i$ is said to be paired to an electron $j$ located at $\vec{r}_j$ if electron $j$ is the closest electron to $i$. The amount of electron pairing at point $\vec{r}$ is therefore proportional to the inverse of

$d({\vec r}) = \sum_{i=1}^N \langle \Psi | \delta(\vec{r}-\vec{r}_i) \min_{j\ne i} r_{ij} | \Psi \rangle $

where $d({\vec r})$ is the shortest electron-electron distance at ${\vec r}$, $\Psi({\vec r}_1,\dots,{\vec r}_N)$ is the N-electron wave function and $r_{ij} = |{\vec r}_j - {\vec r}_i|$.

There are two different types of electron pairs: pairs of electrons with the same spin $\sigma$, and pairs of electrons with opposite spins $\sigma$ and $\bar \sigma$. Hence, two quantities are introduced:

$d_{\sigma \sigma}({\vec r}) = \sum_{i=1,N} \langle \Psi | \delta(\vec{r}-\vec{r}_i) \min_{j\ne i ; \sigma_i = \sigma_j} r_{ij} | \Psi \rangle $

$d_{\sigma {\bar \sigma}}({\vec r}) = \sum_{i=1,N} \langle \Psi | \delta(\vec{r}-\vec{r}_i) \min_{j ; \sigma_i \ne \sigma_j} r_{ij} | \Psi \rangle $

The electron pair localization function is bounded in the $[-1,1]$ interval, and is defined as

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

When the pairing of spin-unlike electrons is predominant, $d_{\sigma \sigma} > d_{\sigma {\bar \sigma}}$ and ${\rm EPLF}({\vec r}) > 0$. When the pairing of spin-like electrons is predominant, $d_{\sigma \sigma} < d_{\sigma {\bar \sigma}}$ and ${\rm EPLF}({\vec r}) < 0$. When the electron pairing of spin-like and spin-unlike electrons is equivalent, ${\rm EPLF}({\vec r}) \sim 0$.

This localization function does not depend on the type of wave function, as opposed to the Electron Localization Function[2] (ELF) where the wave function has to be expressed on a single-electron basis set. The EPLF can therefore measure electron pairing using any kind of wave function: Hartree-Fock (HF), configuration interaction (CI), multi-configurational self consistent field (MCSCF) as well as Slater-Jastrow, Diffusion Monte Carlo (DMC), Hylleraas wave functions, etc. Due to the presence of the $\min$ function in the definition of $d_{\sigma \sigma}$ and $d_{\sigma {\bar \sigma}}$ these quantities cannot be evaluated analytically, so quantum Monte Carlo (QMC) methods have been used to compute three-dimensional EPLF grids via a statistical sampling of $|\Psi({\vec r}_1,\dots,{\vec r}_N)|^2$.

[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., 121, pp. 1725-1735 (2004)

[2] A. D. Becke and K. E. Edgecombe
J. Chem. Phys. 92, 5397 (1990).