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