Error analysis of variationally optimized upper‐ and lower‐bound wave functions for H

Abstract

AbstractWave functions which are a linear combination of H‐type elliptical orbitals are optimized to provide either an upper bound or a lower bound to the H ground state. For the latter, Temple's formula is used. Three criteria are considered to determine the relative accuracy of these wave functions: (i) energy (calculated versus exact eigenvalue); (ii) average error; and (iii) local energy. Although the lower‐bound optimized wave functions obtained are the most accurate available for H from approximate wave functions, they are still inferior to the corresponding upper‐bound wave functions by criteria (i) and (ii). In particular, using criterion (ii), it is shown numerically that the upper‐bound functions are “correct to second order,” while the lower‐bound functions are almost, but not quite, “correct to second order.” Despite this, the local energy analysis, criterion (iii), reveals that the lower‐bound wave functions can be more accurate than the upper‐bound functions in some regions of space, and hence give more accurate values for physical properties sensitive to these regions. Examples considered are the dipole–dipole and Fermi contact interactions.