Can anyone help with bullet point one or two?
To set up the calculation of matrix elements, we express the Hamiltonian as
H = T + V_e-nuc(r) + V_e-e(r_i - r_j) + V_nuc-nuc
where m is the electron mass, e is the electron charge, V_e-nuc(r_i) = 2e^2/|r_i - R_B| describes the Coulomb repulsion between electrons i and j, and V_nuc-nuc = e^2/d_s accounts for the repulsion between the nuclei at a distance d_s.
The matrix elements involving states with 0 or 1 electrons are straightforward (note that V_e-e has no effect on these states). Using the definitions of the two-electron states, the following relations hold:
(0 0 1 1|H|11 0 0) = (o*o*|V_e-e|o o)
