We describe a practical algorithm for constructing the Kohn-Sham exchange-correlation potential corresponding to a given second-order reduced density matrix. Unlike conventional Kohn-Sham inversion methods in which such potentials are extracted from ground-state electron densities, the proposed technique delivers unambiguous results in finite basis sets. The approach can also be used to separate approximately the exchange and correlation potentials for a many-electron system for which the reduced density matrix is known. The algorithm is implemented for configuration-interaction wave functions and its performance is illustrated with numerical examples.

Several workers have deduced various exact expressions for the Kohn-Sham exchange-correlation potential in terms of quantities computable from the interacting and noninteracting wave functions of the system. We show that all these expressions can be obtained by one general method in which the interacting N -electron wave function is expanded in products of one-and (N − 1)-electron functions. Different expressions correspond to different choices of the latter functions. Our analysis unifies and clarifies the previously proposed exact treatments of the exchange-correlation potential, and suggests new ways of expressing this quantity.

Nonlocal exchange-correlation energy functionals are constructed using the accurate model exchange-correlation hole for the uniform electron gas developed by Gori-Giorgi and Perdew. The exchange-correlation hole is constrained to be symmetric and normalized, so the resulting functionals can be viewed as symmetrized versions of the weighted density approximation; we call them two-point weighted density approximations. Even without optimization of parameters or functional forms, the exchange-correlation energies for small molecules are competitive with those of the best generalized gradient approximation functionals. Two-point weighted density approximations seem to be an interesting new direction for functional development. A more general version of the conditions that define the energy for fractional electron number and fractional spin are presented. These "generalized flat-planes" conditions are closely linked to the normalization of the spin-resolved exchange-correlation hole at noninteger electron number. This and many other properties of the exact exchange-correlation functional can be imposed straightforwardly and directly in symmetrized weighted density approximation.

Developing a mathematical approach to the local hard/soft acid/base principle requires an unambiguous definition for the local hardness. One such quantity, which has aroused significant interest in recent years, is the unconstrained local hardness. Key identities are derived for the unconstrained local hardness, δμ/δρ(r). Several identities are presented which allow one to determine the unconstrained local hardness either explicitly using the hardness kernel and the inverse-linear response function, or implicitly by solving a system of linear equations. One result of this analysis is that the problem of determining the unconstrained local hardness is infinitely ill-conditioned because arbitrarily small changes in electron density can cause enormous changes in the chemical potential. This is manifest in the exponential divergence of the unconstrained local hardness as one moves away from the system. This suggests that one should be very careful when using the unconstrained local hardness for chemical interpretation.

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