Determination of wave-function functionals: The constrained-search variational method

In a recent paper [Phys. Rev. Lett. 93, 130401 (2004)], we proposed the idea of expanding the space of variations in variational calculations of the energy by considering the approximate wave function ψ to be a functional of functions χ, ψ=ψ[χ], rather than a function. A constrained search is first performed over all functions χ such that the wave-function functional ψ[χ] satisfies a physical constraint or leads to the known value of an observable. A rigorous upper bound to the energy is then obtained via the variational principle. In this paper we generalize the constrained-search variational method, applicable to both ground and excited states, to the determination of arbitrary Hermitian single-particle operators as applied to two-electron atomic and ionic systems. We construct analytical three-parameter ground-state functionals for the H^- ion and the He atom through the constraint of normalization. We present the results for the total energy E, the expectations of the single-particle operators W=∑ᵢrᵢⁿ, n=-2,-1,1,2, W=∑ᵢδ(rᵢ), and W=∑ᵢδ(rᵢ-r), the structure of the nonlocal Coulomb hole charge ρc(rr^'), and the expectations of the two particle operators u²,u,1∕u,1∕u², where u=rᵢ-rⱼ. The results for all the expectation values are remarkably accurate when compared with the 1078-parameter wave function of Pekeris, and other wave functions that are not functionals. We conclude by describing our current work on how the constrained-search variational method in conjunction with quantal density-functional theory is being applied to the many-electron case.