Distribution of r₁₂ · p₁₂ in quantum systems

Abstract

We introduce the two-particle probability density X(x) of x= r ₁₂· p ₁₂=( r ₁ − r ₂)·( p ₁ − p ₂). 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.

Keywords:

• two-particle density distribution
• Wigner distribution
• intracule
• Posmom

PACS numbers:

• 31.15.X-
• 31.15.ve
• 31.15.vj
• 81.07.Ta

Acknowledgements

We thank Joshua Hollett for helpful comments. YAB thanks the ANU Research School of Chemistry for a PhD scholarship. PMWG thanks the Australian Research Council (Grants DP0984806, DP1094170, and DP120104740) for funding. PFL thanks the Australian Research Council for a Discovery Early Career Researcher Award (Grant DE130101441). PFL and PMWG thank the NCI National Facility for a generous grant of supercomputer time.

Notes

^a Reference [14]: 7 basis functions.

^bReference [46]: 11 basis functions.