Quantum Monte Carlo
This is really simple. Just pick random positions in 3-D space, and evaluate the wavefunction and the hamiltonian.
Average over lots of positions, and that gives you an estimate for the energy of the atom. The wavefunction has parameters,
and we do a gradient descent to find the parameters that give the lowest energy.
This is still quite simple. We have two nuclei now and two electrons. The thing which makes this easy is that
we can assume that one electron is spin up and the other is spin down. That allows us to ignore spin and exchange symmetries
(although the wavefunctions we search for should be symmetric with respect to swapping the two electrons).