Molecular Schrödinger Equation. VII. Properties of the Energy Variance Function: The Estimation of Energy Eigenvalues

Abstract

The energy variance function U² is defined in terms of the eigenfunction expansion for a trial wavefunction ψ. The relationship of U² to certain upper‐ and lower‐bound formulas for the energy and purity functions associated with ψ is discussed. The concurrently allowed values of U² and the energy function ε fall within a bounded region, the shape of which depends partly upon the Schrödinger equation and partly upon those restrictions imposed by the incompleteness of the basis set used for the representation ψ. A number of exact relationships governing the behavior of ε, U², and their derivatives along the boundary curve are worked out. The concept of the derivative wavefunction along the boundary is introduced, and an explicit prescription for its computation is given. Ways and means for the estimation of the true energy eigenvalue from the properties of an imperfect wavefunction are presented and discussed.