Density and one-matrix functionals generated by constrained-search theory

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Abstract
In principle, it is possible to construct analytical energy functionals of the electron density and the one-particle density matrix by using the constrained-search theory of Levy [Proc. Natl. Acad. Sci. U.S 76, 6062 (1979)] to calculate the parameters in either model wave functions or two-particle density matrices. Illustrations are attempted through several specific examples. A very simple model set of two-particle matrices is shown to contain all forms of local exchange-correlation expressions for the electron-electron repulsion contribution to the total energy. A special feature of the model two-particle matrices is that they depend explicitly on the interelectronic distance. Constrained-search theory is used to derive values for the two parameters in the model two-particle matrices. The resulting functional shows promise for preserving charge distribution and bond directionality information absent in other functionals. A generalization of this model two-particle matrix allows derivation of a one-matrix functional, thereby skirting a common, implicit approximation in some local-density approximations to the total energy. Finally, outlines of constrained-search modifications to functionals due to Gunnarsson-Jones and to Colle and Salvetti illustrate the many possible variations of the central theme. For the Colle-Salvetti functional, a method is proposed that produces an N-representable total energy, rather than an electron-electron repulsion, functional, i.e., a functional based on a model wave function, and consequently obeys the variational principle for the true Hamiltonian system.