Bounds to electronic expectation values for atomic and molecular systems

Abstract

A generalization of a method to calculate lower bounds to expectation values of non-negative observables is presented. We consider bounds to three electronic expectation values hr 2 i, hri, and hr À1 i in the helium atom as an example. For both hr 2 i and hri, we are able to calculate improved lower bounds. The lower bound to hr À1 i does not improve, but we are able to calculate an upper bound which is much closer to the expectation value than the lower bound. Although our generalization allows for improved bounds and/or upper bounds, these bounds to general observables are much less precise than energy bounds and even the expectation values calculated from variational wave functions.

Comparison of an Improved Self-consistent Lower Bound Theory with Lehmann’s Method for Low-lying Eigenvalues

“…[34][35][36][37][38] Further lower bound calculation strategies are also available for He atoms. [39][40][41][42] A comprehensive discussion of these methods can be found in Ref. 43 and a historical review on lower bound theories can be found in Ref.…”

Optimization of the temple lower bound

“…For example, the variational upper bound provides the upper half of an error bar on the energy, but a lower bound is typically lacking. Furthermore, most bounds on properties other than energy rely on bounds to the latter (indirectly through the overlap of a trial function with the unknown exact wave function), making bounds to non-energy properties very difficult indeed [1][2][3][4][5]. Perhaps the best-known error bar for a system's property is the combination of the variational upper bound and the Temple lower bound to the ground-state energy, E 1 .…”