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Michael Springborg, J. P. Perdew, K. Schmidt
Phase Space, Density Matrices, Energy Densities, and Exchange Holes
In the general case, quantum-mechanical quantities are represented by operators in position- or momentum-space representations, but in phase space they are represented by functions. The correspondence between classical mechanics and quantum mechanics is non-unique as a consequence of [x^,p^] ≠ 0, and therefore the phase-space representation of quantum mechanics is also non-unique. We explain how different correspondence rules lead to different phase-space functions and how the latter are related to first-order reduced density matrices. As a special example, we discuss the phase-space representation of different terms of the total energy within the Hartree-Fock approximation for electronic-structure calculations. In particular we discuss how one may use the phase-space representation to define energy densities in position space, putting special emphasis on kinetic and exchange energy densities which are not unique. While the standard exchange energy density has an exchange hole which is normalized, the Weyl and other exchange energy densities have exchange holes which are more localized. We also make a numerical study of statistical correlations among variables commonly used in density functional theory and several energy densities, including the standard and Weyl exchange energy densities. Finally, we examine the spherical average of the standard exchange hole for the molecule LiH, finding that it reflects both the ionic and the covalent character of the bond.
Zeitschrift für Physikalische Chemie, Oldenbourg Wissenschaftsverlag
Print ISSN: 0942-9352
Volume: 215, 10/2001
Pages: 1243