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

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Abstract
For the Hartree–Fock ground state of atomic two‐electron systems, the variational function of Wilson and Silverstone, ϕ(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 ϕ(r) is indeed the outstanding two‐parameter function, at least for small and intermediate values of Z.

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