The Hartree–Fock ground state of atomic two-electron systems and the Wilson–Silverstone 1s orbital

For the Hartree–Fock ground state of atomic two-electron systems, the variational function of Wilson and Silverstone, ?(r) = (a + kr)^?1 exp(-kr) / (4π)^1/2, can be optimized in two complementary ways. For small values of the atomic number Z, all intergrals have been calculated numerically and optimization can be performed accurately. However, as Z increases, loss of significant figures is increasingly detrimental to the optimization process. For sufficiently large values of Z, the integrals may be replaced by asymptotic expansions in terms of (2a)^?. As a result of optimization, the parameters and expectation values can be given as expansions in terms of (32Z)^?1/2. Both methods yield good results for Z ≈ 25, so that the whole range of Z can be treated accurately. The results have been compared with those derived from other analytical two-parameter functions. It is found that ?(r) is indeed the outstanding two-parameter function, at least for small and intermediate values of Z.