Distribution of $r_{12} \cdot p_{12}$ in quantum systems

TL;DR

This paper introduces the Posmom intracule, a new two-particle probability density derived from the wavefunction, which provides insights into electron correlations in quantum systems and is contrasted with existing intracules.

Contribution

The paper presents the derivation and analysis of the Posmom intracule from many-particle wavefunctions, offering a novel tool for studying electron correlations.

Findings

Derived the Posmom intracule from wavefunctions

Compared Posmom intracule with Dot intracule and Wigner distribution

Applied the concept to various two-electron systems

Abstract

We introduce the two-particle probability density $X(x)$ of $x=\bm{r}_{12}\cdot\bm{p}_{12}=\left(\bm{r}_1-\bm{r}_2\right) \cdot \left(\bm{p}_1-\bm{p}_2\right)$. We show how to derive $X(x)$, which we call the Posmom intracule, from the many-particle wavefunction. We contrast it with the Dot intracule [Y. A. Bernard, D. L. Crittenden, P. M. W. Gill, Phys. Chem. Chem. Phys., 10, 3447 (2008)] which can be derived from the Wigner distribution and show the relationships between the Posmom intracule and the one-particle Posmom density [Y. A. Bernard, D. L. Crittenden, P. M. W .Gill, J.Phys. Chem.A, 114, 11984 (2010)]. To illustrate the usefulness of $X(x)$, we construct and discuss it for a number of two-electron systems.