We extend the idea of the constrained-search variational method for the construction of wave-function functionals psi[chi] of functions chi. The search is constrained to those functions chi such that psi[chi] reproduces the density rho(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 psi[chi] 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 psi[chi] that reproduce the density as given by the Kinoshita correlated wave function. The expectation of single-particle operators W = Sigma(i) r(i)(n), n = -2,-1,1,2, W = Sigma(i) delta(r(i)) are exact, as must be the case. The expectations of the kinetic energy operator W = -1/2 Sigma(i) del(2)(i), the two-particle operators W = Sigma(n) u(n), n = -2,-1,1,2, where u = vertical bar r(i) - r(j)vertical bar, and the energy are accurate. We note that the construction of such functionals psi[chi] is an application of the Levy-Lieb constrained-search definition of density functional theory. It is thereby possible to rigorously determine which functional psi[chi] is closer to the true wave function.
Slamet, Marlina; Pan, Xiao-Yin; and Sahni, Viraht, "Wave-function functionals for the density" (2011). CUNY Academic Works.