“Helium: energies for singlets and triplets” discusses rival explanations as to why helium's triplet states lie below corresponding singlets. Computations reported in the literature show that triplet states have more compact wave functions than do corresponding singlet states, even though this makes the electron–electron repulsion energy larger for triplets. A simplified variational method is applied to an attempted calculation of the energies, but is inadequate, showing how helium is numerically unfriendly.
Evaluating two-electron-repulsion integrals over arbitrary orbitals using zero variance Monte Carlo: Application to full configuration interaction calculations with Slater-type orbitals