Convergence of a sequence of lower bounds for 〈1/ r 〉 for the noble gas, alkali, and alkaline earth atoms

This article considers 2 × 2, 3 × 3, 4 × 4, and 5 × 5 Gram determinantal inequalities among 〈 r n 〉, where r is the distance from the atomic nucleus. The elements of these determinants involve n = −1, 0, 1; n = −1, 0, 1, 2, 3; n = −1, 0, 1, 2, 3, 4, 5; and n = −1, 0, 1, 2, 3, 4, 5, 6, 7. The 2 × 2, 3 × 3, 4 × 4, and 5 × 5 determinantal inequalities are used to obtain lower bound estimates of 〈1/ r 〉 for atoms of spherically symmetric charge distributions. The 2 × 2, 3 × 3, and 4 × 4 inequalities have been applied to He, Ne, Ar, Kr, Xe; Li, Na, K, Rb; and Be, Mg, Ca, Sr. The 5 × 5 inequality has been applied only to He, Li, and Be. It is found that the lower bound values of 〈1/ r 〉, for all atoms considered, appear to converge to the quantum mechanical values of Boyd, who calculated them with the Roothaan–Hartree–Fock wavefunctions of Clementi and Roetti. © 1994 John Wiley & Sons, Inc.