Determination of a Wave Function Functional: The Constrained-Search Variational Method

The space of variations in standard variational calculations for the energy may be expanded by considering the wave function $\psi$ to be a functional of a set of functions $\chi: \psi = \psi[\chi]$, rather than a function. In this manner a greater flexibility to the structure of the wave function is achieved. Here we propose a constrained search within a \emph{subspace} over all functions $\chi$ such that the wave function functional $\psi[\chi]$ satisfies a constraint such as normalization or the Fermi-Coulomb hole charge sum rule, or the requirement that it lead to a physical observable such as the density, diamagnetic susceptibility, etc.. A rigorous upper bound to the energy is subsequently obtained by variational minimization with respect to the parameters in the approximate wave function functional. Hence, the terminology, the constrained-search variational method. The construction of such a constrained-search wave function functional is demonstrated by example of the ground state of the Helium atom.