The constrained-search--variational method: application to the ground state of Helium atom

n a recent paper we proposed the expansion of the space of variations in energy calculations by considering the approximate wave function $\psi$ to be a functional of functions $\chi: \psi = \psi[\chi]$ rather than a function. For the determination of such a wave function functional, a constrained search is first performed over the subspace of all functions $\chi$ such that $\psi[\chi]$ satisfies a physical constraint or leads to the known value of an observable. A rigorous upper bound to the energy is then obtained by application of the variational principle. To demonstrate the advantages of the expansion of variational space, we apply the constrained-search--variational method to the ground state of the negative ion of atomic Hydrogen, the Helium atom, and its isoelectronic sequence. The method is equally applicable to excited states, and its extension to such states in conjunction with the theorem of Theophilou is also described.