Quantum Monte Carlo Formulation

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

<math>

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

</math>

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

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

<math>

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 

</math>

<math>

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 

</math>

The electron pair localization function is bounded in the <math>[-1,1]</math> interval, and is defined as

<math>

{\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}) }

</math>

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

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 <math>\min</math> function in the definition of <math>d_{\sigma \sigma}</math> and <math>d_{\sigma {\bar \sigma}}</math> 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 <math>|\Psi({\vec r}_1,\dots,{\vec r}_N)|^2</math>.


[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).