Variational Principles for Expectation Values Applied to the Ground State of the Helium Isoelectronic Sequence

Analytic calculations of the expectation values of $r^n$, $n=-2,-1,1,2\mathrm{}$, the kinetic energy, and the charge density at the origin, have been performed for two-electron atoms with nuclear charge $Z$ using a variational principle first proposed by Delves. The trial wave function ${\psi }_{0T}$ was taken as a product of single-particle hydrogenic states with a single variational parameter $\overline{Z}$. In each case the complementary function ${\psi }_{1T}$ was obtained by exactly solving a differential equation relating ${\psi }_{1T}$ to both ${\psi }_{0T}$ and the operator whose expectation value was to be calculated. For $Z\ge 2$, the least accurate results obtained are for $〈r^2〉$ which are 1.9 and 0.7% lower than the highly accurate values obtained by Pekeris for He and ${\mathrm{Li}}^{+}$, respectively. In all cases the reduction in the error obtained using a one-parameter wave function given by the variational principle for minimizing the energy increases as $Z$ increases, and the results are, in many cases, comparable to those given by the Hartree-Fock treatment. A condition proposed by Aranoff and Percus involving the minimization of an auxiliary expression was used to renormalize ${\psi }_{1T}$. This method left unchanged or improved the results for all cases considered.