Long time ago, we talked about the mean-field calculation, an touched Hartree-Fock method. In that time, we explained excited-state approach. Now, we explain another approach by variation of the wave functions. This approach is inevitable in atomic physics, because the potential is fixed.

The Hamiltonian is

$H = H_1 + H_2 + V_{12}$

Since the spin component is anti-parallel, the space part of the total wave function is

$\Psi = \phi_1(r_1) \phi_2(r_2)$

The Schrodinger equation is

$H\Psi = E\Psi$

Integrate both side with $\phi_1(r_1)$

$\int dr_1 \phi_1(r_1) H \Psi = \int dr_1 \phi_1(r_1) E \Psi$

using the normalization and define $\left = \phi_1(r_1)$

$\left<1|H|1\right> \phi_2(r_2) = E \phi_2(r_2)$

similarly

$\left<2|H|2\right> \phi_1(r_1) = E \phi_1 (r_1)$

expand $H = H_1 + H_2 + V_{12}$

$(H_2 + \left<1|V_{12}|1\right>) \phi_2(r_2) = (E - \left<1|H_1|1\right>) \phi_2(r_2)$

$(H_1 + \left<2|V_{12}|2\right>) \phi_1(r_1) = (E - \left<2|H_2|2\right>) \phi_1(r_1)$

Now, we have 2 equations,  an initial guess of $\Psi = \phi_1(r_1) \phi_2(r_2)$,

The difficulty is that, the $\left<1|V_{12}|1\right>$ contains 2 variables.