Variational Calculations of Correlation Effects in Light Atoms

The convergence of the "superposition of configurations" method is studied in a series of restricted calculations on the ground states of helium and lithium. A one-dimensional model with a singularity reminiscent of the cusp in the helium wave function is studied by the variational method, and the relative merits of a variety of different radial basis sets are compared. It is concluded that the basis ${e}^{\ensuremath{-}\ensuremath{\alpha}\mathrm{nr}}$ converges faster than the basis ${r}^{n\ensuremath{-}1}{e}^{\ensuremath{-}\ensuremath{\alpha}r}$ for the ground-state energies of the one-dimensional model and of helium. The two basis functions are also compared for the computed expectation values of $\ensuremath{\Sigma}{i}^{}{{r}_{i}}^{n},n=\ensuremath{-}2,\ensuremath{-}1,1,2$, and $\ensuremath{\Sigma}{i}^{}{\ensuremath{\delta}}^{3}({r}_{i})$ in the ground state of helium, and the difference between them is observed not to be as important as for the energy. The hyperfine-splitting parameter for the ground state of lithium is found to behave so erratically that attempts to calculate it with high precision for larger systems will prove to be very difficult by this method.