Phase Space, Density Matrices, Energy Densities, and Exchange Holes

Abstract

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 electronicstructure 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.

Exact-exchange energy density in the gauge of a semilocal density-functional approximation

“…The extent of locality of the exchange hole depends on the value of parameter ω. The maximal localization is achieved at ω = 1/2 [54,55]. We have numerically evaluated the transformed exact-exchange energy density e ex(ω) xσ (r) for various values of ω and found that ω = 0.92 leads to the best fit of the exact-exchange energy density to the TPSS meta-GGA.…”

Density functional theory on phase space

“…For instance, it suggests a description of the exchange hole close to being optimally localized [111]. Also for many atoms apparently t M obeys a local version of the Lieb-Oxford identities [112]. Now,…”

Correlation Energy Densities: E Pluribus Unum

“…For example, we can add ∇ 2 F (r) to either, where F (r) vanishes exponentially as |r| → ∞, without changing the integrated energy. We can also make coordinate transformations [16,17,18] which change the energy density but not the energy. Of course, the Kohn-Sham exchange-correlation potential is unique, and can be used to construct an "unam-biguous" energy density [19].…”

“…Unconventional energy densities can have holes which do not satisfy Eqs. (11) and 12at each r, but only in the system average over n(r) [16,17,18].…”