Lower bound for the nonrelativistic ground state energy of the lithium atom

Abstract

A rigorous lower bound for the groundstate energy of the hydrogen atom Am.The upper and lower bounds of the ground state energies using the variational method Am.A Temple lower bound for the nonrelativistic ground state energy of the lithium atom is determined using large scale Hylleraas-type wave functions. Weinstein lower bounds for the three lowest nonrelativistic energy levels are also reported. All the component expectation values necessary to construct the lower bounds are given. Detailed information is given to assess the convergence of the calculations.

Bounds to electronic expectation values for atomic and molecular systems

“…when expectation values are calculated with the true wave functions. Thus we can replace the kinetic energy expectation value in inequality (12) with -E n to yield inequality (13).…”

“…The former is much more difficult to calculate than the latter. Notable efforts in the calculation of lower bounds to atomic and molecular energy levels are based on intermediate problems and Bazley's special choice [1][2][3][4][5][6][7]; the Temple bound and similar formulas [8][9][10][11][12][13][14][15]; effective fields [16][17][18][19][20]; and local energy [21][22][23].…”

Gegenbauer expansions for three‐electron integrals

“…For positive even values of , the expansion contains a finite number of terms (due to the cosine formula for r ij 2 ), but for positive odd it is an infinite series with the radial function R u (r i , r j ), becoming more complicated as increases. Use of this expansion for integer values of less than Ϫ1 is not needed for traditional nonrelativistic energy calculations, but the expansion of r ij Ϫ2 arises in lower-bound calculations [13,15] and in the treatment of relativistic effects [9,18,19]. Sack's formulation includes the case ϭ Ϫ2, but severe numerical and convergence problems have been encountered with its use for that value of , and several workers have been exploring alternatives or modifications [12,21].…”

Approximate lower bounds of the Weinstein and Temple variety

“…The requirement of the Temple bound for knowledge of E kϩ1 and the ambiguity of the Weinstein interval make rigorous application of these methods impossible in most cases. The most difficult chemical lower bound calculations to date have been on the electronic energy of the three-electron lithium atom by King [6] and by the collaboration of Lü chow and Kleindienst [5]. Using the Temple and Weinstein methods [6], or variations of these [5], both groups were unable to make their calculations rigorous, although their results are "trustworthy" (separate work by Russell and Greenlee [7], using the method of intermediate problems could be used to make their work rigorous).…”

“…The most difficult chemical lower bound calculations to date have been on the electronic energy of the three-electron lithium atom by King [6] and by the collaboration of Lü chow and Kleindienst [5]. Using the Temple and Weinstein methods [6], or variations of these [5], both groups were unable to make their calculations rigorous, although their results are "trustworthy" (separate work by Russell and Greenlee [7], using the method of intermediate problems could be used to make their work rigorous). Nevertheless, approximate bounds are still useful; after all, most error bars for experimental data are of a statistical nature and thus are not guaranteed to contain the true values.…”