  In a recent paper [Phys. Rev. Lett. \textbf{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 $\psi$ to be a
functional of functions $ \chi: \psi = \psi[\chi]$ rather than a function. The
space of variations is expanded because a search over the functions $\chi$ can
in principle lead to the true wave function. As the space of such variations is
large, we proposed the constrained-search-- variational method whereby a
constrained search is first performed over all functions $\chi$ such that the
wave function functional $\psi[\chi]$ satisfies a physical constraint such as
normalization or the Fermi-Coulomb hole sum rule, or leads to the known value
of an observable such as the diamagnetic susceptibility, nuclear magnetic
constant or Fermi contact term. A rigorous upper bound to the energy is then
obtained by application of the variational principle. A key attribute of the
method is that the wave function functional is accurate throughout space, in
contrast to the standard variational method for which the wave function is
accurate only in those regions of space contributing principally to the energy.
In this paper we generalize the equations of the method to the determination of
arbitrary Hermitian single-particle operators as applied to two-electron atomic
and ionic systems. The description is general and applicable to both ground and
excited states. A discussion on excited states in conjunction with the theorem
of Theophilou is provided.
