A study of a lower bounding formula for the overlap between the exact and an approximate wavefunction

Abstract

Weinhold’s lower bounding formula for the overlap between the exact wavefunction and an approximate wavefunction is applied to the hydrogen atom. The lower bound is examined as a function of both the number of basis set terms used in the bounding inequality and the nonlinear variational parameter in this basis set. A method is then proposed and tested for approximating the integrals over H2 and H3 required by Weinhold’s technique as products of integrals involving only H. In the case studied, the resulting approximate overlap bounds converged to the true bound.

Theoretical Error Estimates in Atomic and Molecular Structure Calculations

“…On the other hand, if we employ more than a single member of a set of trial functions {<p.. ; m = 0, 1, ..., N} which satisfy the result is" where and which therefore' provide upper bounds to a set of eigenvalues provided that <P. is a sufficiently good approximation so that A multidimensional generalization of this result gives little quantitative improvement over (27) in calculations on the helium'S states. "…”

Approximating Ground States by Neural Network Quantum States

“…From these intermediate resolvents they obtained a new formula for the lower bound of the overlap between the approximate and exact wave functions of a quantum-mechanical system. Merkel [ 15 ] proposed a method and tested it for approximating the integrals over and required by Weinhold’s technique as products of integrals involving only H . Cioslowski [ 16 ] constructed a connected-moments expansion for the overlap between the approximate and the exact (but unknown) wave function of the ground state.…”