After long preparation, I am ready to do this problem.

The two electron in the helium ground state occupy same spacial orbital but difference spin. Thus, the total wavefunction is

Since the Coulomb potential is spin-independent, the Hartree-Fock method reduce to Hartree method. The Hartree operator is

where the single-particle Hamiltonian and mutual interaction are

In the last step, we use atomic unit, such that . And the energy is in unit of Hartree, .

We are going to use Hydrogen-like orbital as a basis set.

I like the left the , because in the integration , the can be cancelled. Also, the is a compact index of the orbital.

Using basis set expansion, we need to calculate the matrix elements of

Now, we will concentrate on evaluate the mutual interaction integral.

Using the well-known expansion,

The angular integral

where the integral .

From this post, the triplet integral of spherical harmonic is easy to compute.

The Clebsch-Gordon coefficient imposed a restriction on .

The radial part,

The algebraic calculation of the integral is complicated, but after the restriction of from the Clebsch-Gordon coefficient, only few terms need to be calculated.

The general consideration is done. now, we use the first 2 even states as a basis set.

These are both s-state orbital. Thus, the Clebsch-Gordon coefficient

The radial sum only has 1 term. And the mutual interaction becomes

The angular part

Thus, the mutual interaction energy is

The radial part

We can easy to see that . Thus, if we flatten the matrix of matrix, it is Hermitian, or symmetric.

Now, we can start doing the Hartree method.

The general solution of the wave function is

The Hartree matrix is

The first trial wave function are the Hydrogen-like orbital,

Solve for eigen system, we have the energy after 1st trial,

After 13th trial,

Thus, the mixing of the 2s state is only 3.7%.

Since the eigen energy contains the 1-body energy and 2-body energy. So, the total energy for 2 electrons is

In which ,

So the energies for

From He to He++.

From He+ to He++, .

From He to He+, is

The experimental 1 electron ionization energy for Helium atom is

The difference with experimental value is 2.175 eV. The following plot shows the Coulomb potential, the screening due to the existence of the other electron, the resultant mean field, the energy, and

Usually, the Hartree method will under estimate the energy, because it neglected the correlation, for example, pairing and spin dependence. In our calculation, the energy is under estimated.

From the , we can see, the mutual interaction between 1s and 2s state is attractive. While the interaction between 1s-1s and 2s-2s states are repulsive. The repulsive can be easily understood. But I am not sure how to explain the attractive between 1s-2s state.

Since the mass correction and the fine structure correction is in order of , so the missing 0.2 eV should be due to something else, for example, the incomplete basis set.

If the basis set only contain the 1s orbit, the mutual interaction is 1.25 Hartree = 34.014 eV. Thus, the mixing reduce the interaction by 5.07 eV, just for 3.7% mixing

I included the 3s state,

The mutual energy is further reduced to 1.05415 Hartree = 28.6848 eV. The . If 4s orbital included, the . We can expect, if more orbital in included, the will approach to .