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

Ground-state He and H— trial functions are scaled to satisfy the hypervirial relations generated by the family of hypervirial operators Wn=r1np1. These one-parameter hypervirial functions are used to calculate expectation values of the positive powers of the radial coordinate operators, rt, for t=1 through 20. Accurate values of 〈rt〉, 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 〈rt〉 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.