Hypervirial Functions and the Positive Powers of the Radial Coordinate Operator in He and H—

Abstract

Ground‐state He and H^— trial functions are scaled to satisfy the hypervirial relations generated by the family of hypervirial operators W_n=r₁ⁿp₁. These one‐parameter hypervirial functions are used to calculate expectation values of the positive powers of the radial coordinate operators, r^t, for t=1 through 20. Accurate values of 〈r^t〉, for this range of t, are calculated using 20‐term Hylleraas functions. A comparison of results shows that simple hypervirial functions of the form exp[—K(r₁+r₂)] scaled to satisfy the hypervirial relations generated by W₂, give a good over‐all calculation of 〈r^t〉 for both He and H^—. The hypervirial theorem is used to develop a variety of expectation value relationships for one‐ and two‐electron central force problems in the Appendix.