We report a series of calculations of error bounds to the Hartree-Fock approximation for the 2s-2p and 3s-3p transitions in lithium and sodium, and the singly ionized ions ${\mathrm{Be}}^{+}$ and ${\mathrm{Mg}}^{+}$. Our purpose is to test the efficacy of various modifications of Weinhold’s effective-bounds formula for several simple but realistic examples of multielectron systems. We therefore assume the overlap error ε to be known, adopting overlap values from extensive variational calculations. We find angular momentum projection of the transition operator to be effective in tightening the bounds, while the variational optimization of a mixture of length and velocity operators is not. We also found a wave-function projection based on Brillouin’s theorem to be especially effective. When used in conjunction with angular momentum projection, this last procedure has yielded bounds for lithium very close to the known Hartree-Fock error.

• Received 7 March 1988

DOI:https://doi.org/10.1103/PhysRevA.38.1760