The constrained variational formulation of energy functionals

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Abstract
We present a generalization of the density functional formalism based upon constrained variations of the expectation value ℰ=〈Ψ‖Ĥ‖Ψ〉. The essential idea is to find the minimum value of E with respect to a set of state vectors Sa={‖Ψ〉a}, each member of which yields the specified expectation values ai≡a〈Ψ‖Ai‖Ψ〉a of set of observables Âi, i=1...M. a denotes the collection of M expectation values ai, i=1...M. If one or more N-electron vectors exist yielding the given M expectation values, a is said to be N representable. One thus obtains this minimum expectation value E(a)=min E over Sa as a function of the assigned expectation values. In terms of the function E(a), the exact ground state energy may be obtained by seeking its minimum value over all possible (N-representable) expectation values a. The traditional density functional formalism of Hohenberg and Kohn is seen to be a special case where the constraints are expressed by the expectation values of an indenumerable set of charge density operators, the members of which are indexed by the continuum of eigenvalues of the single particle position operator. The general ideas are illustrated by computing a function based upon constraining 〈1/r〉 and 〈1/r2〉 for a one-electron atom. In this example, the formalism is carried out by solving a modified Schrödinger equation which includes Lagrange multipliers in order to take into account the constraints. We conclude, however, with an outline of how the unitary transformation formalism can be used to obtain the function ℰ(a) in a systematic fashion.