We extend the idea of the constrained-search variational method for the construction of wave-function functionals ψ[χ] of functions χ. The search is constrained to those functions χ such that ψ[χ] reproduces the density ρ(r) while simultaneously leading to an upper bound to the energy. The functionals are thereby normalized and automatically satisfy the electron-nucleus coalescence condition. The functionals ψ[χ] are also constructed to satisfy the electron-electron coalescence condition. The method is applied to the ground state of the helium atom to construct functionals ψ[χ] that reproduce the density as given by the Kinoshita correlated wave function. The expectation of single-particle operators W = ∑i r ni , n = −2,−1,1,2, W = ∑i δ(ri ) are exact, as must be the case. The expectations of the kinetic energy operator W = −½ ∑i ∇2i; , the two-particle operators W = ∑n un, n = −2,−1,1,2, where u = |ri − rj|, and the energy are accurate. We note that the construction of such functionals ψ[χ] is an application of the Levy-Lieb constrained-search definition of density functional theory. It is thereby possible to rigorously determine which functional ψ[χ] is closer to the true wave function.
Slamet, Marlina, "Wave-Function Functionals for the Density" (2011). Physics Faculty Publications. 2.