Distribution of r₁₂ · p₁₂ in quantum systems

We introduce the two-particle probability density X(x) of x=r ₁₂· p₁₂=(r₁ - r₂) ·(p₁ - p₂). The fundamental equations involved in the derivation of this new intracule X(x), which we call the Posmom intracule, are derived and we show how to derive X(x) from the many-particle wave-function. We contrast it with the Dot intracule [Y.A. Bernard, D.L. Crittenden, and 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, and P.M.W. Gill, J. Phys. Chem. A 114, 11984 (2010)]. To illustrate the information provided by the Posmom intracule, we apply this new formalism to various two-electron systems: the three-dimensional parabolic quantum dot, the helium-like ions and the ground and excited states of the helium atom.