Lower Bounds to Expectation Values: Two-Electron Atoms

Abstract

Weinhold's lower bound formulas are used on rather accurate wavefunctions for the 1's ground states of H-, He, and Li+, and the 2IS and Z3S excited states of He. The results are found to be of very high quality, as judged by comparison with the best available variational calculations, and can be used to rule out many of the values calculated by Scherr and Knight from high-order perturbation theory. 260 (1973) Introduction The concept of attaching rigorous error limits to theoretical estimates of quantummechanical properties is attractive to many chemists and physicists. Such upper and lower bounds not only provide internal and a pviori measures of reliability, but also enable one to make more meaningful comparisons with experimental results. One of the recent developments in this direction is the application of Gramian determinantal ineq~alities.~ For the lower bounds to expectation values of positive operators, Weinhold (2-5) derived a wide variety of relatively simple formulas and illustrated their use with the familiar Is2 and lsls' trial functions for the 1'S ground state of the helium atom, with promising results. The purpose of this work is to test some of these procedures on better wavefunctions. On the basis of certain preliminary results, we were encouraged to employ some rather accurate wavefunctions which could be expected to yield bounds of a definitive quality for the simple 2-electron atomic systems.

Nonlinear programming approach to locally constrained variational calculations on the ground state of the helium atom

“…The expectation value <% 9 % > is required for the evaluation of the specific mass shift and the transition isotope shift, and is particularly sensitive to electron correlation effects. aThe exact values are taken from reference [19], except (r~), (r~) and (r~2) are from reference [21], (r~) is from reference [20], and (6(r;)) is from reference [31].…”

“…For the Chong-Weinhold [20] entry, the number of digits of precision is uncertain. Also reported in table 2 is the scale factor r/, defined by -89 (ll) r/ = (T) '…”

“…For the set {/, m, n}, the selection employed by Hart and Hertzberg [30] was utilized. This choice has 20 terms, and leads to a fairly accurate ground state energy and reasonably accurate values for a number of expectation values, based on comparison with the results of very elaborate calculations on the He atom [19][20][21]. In some respects, a smaller basis set would have been much more manageable in the nonlinear optimization phase of the project.…”

“…The small number of electrons involved allowed for a considerable amount of exploration with the nonlinear programming phase of the calculations described in section 2. For the Downloaded by ["Queen's University Libraries, Kingston"] at 18:16 03 February 2015 helium atom, there is an extensive literature available on the accurate evaluation of a range of properties [19][20][21]. These will provide an important comparison for the results of the present study.…”