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

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

The Schrodinger equation is

Integrate both side with

using the normalization and define

similarly

expand

Now, we have 2 equations, an initial guess of ,

The difficulty is that, the contains 2 variables.