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In a recent paper [Phys. Rev. Lett. \textbf{93}, 130401 (2004)], we proposedthe idea of expanding the space of variations in variational calculations ofthe energy by considering the approximate wave function $\psi$ to be afunctional of functions $ \chi: \psi = \psi[\chi]$ rather than a function. Thespace of variations is expanded because a search over the functions $\chi$ canin principle lead to the true wave function. As the space of such variations islarge, we proposed the constrained-search-- variational method whereby aconstrained search is first performed over all functions $\chi$ such that thewave function functional $\psi[\chi]$ satisfies a physical constraint such asnormalization or the Fermi-Coulomb hole sum rule, or leads to the known valueof an observable such as the diamagnetic susceptibility, nuclear magneticconstant or Fermi contact term. A rigorous upper bound to the energy is thenobtained by application of the variational principle. A key attribute of themethod is that the wave function functional is accurate throughout space, incontrast to the standard variational method for which the wave function isaccurate only in those regions of space contributing principally to the energy.In this paper we generalize the equations of the method to the determination ofarbitrary Hermitian single-particle operators as applied to two-electron atomicand ionic systems. The description is general and applicable to both ground andexcited states. A discussion on excited states in conjunction with the theoremof Theophilou is provided.Comment: 26 pages, 4 figures, 5 table

Determination of wave-function functionals: The constrained-search variational method