quantum mechanics - Hartree Fock equations

I don't understand how the Hartree Fock equations define an iterative method!

For this discussion, I am referring to the HF equations as described here: click me!

Basically if you guess a bunch of initial wavefunctions, then you can plug them into the HF equation and get (by calculating the expectation value of the energy) an approximation for a single-electron energy, but I don't see how having this equation would define an iteration from which you can improve your wavefunctions?

My question is actually HOW you generate the new wavefunctions.

Imagine that we have $\Phi = \Pi_{i=1}^n \phi_i$ (so we neglect Pauli for simplicity) and $\phi_i = \sum_{k=1}^{n(i)} a_{i,k} \psi_{i,k}$. So you would start with some choice of the $a_{i,k}$ such that the wavefunction is normalized, but HOW do you get your new choice of the $a_{i,k}$ then?

If anything is unclear, please let me know.

Answer

You can think of Hartree-Fock as a self-consistent mean field method. The idea is that you start with each of the particles in their initial orbits. These particles generate a mean field, and you can solve for single-particle eigenfunctions of this mean field. This is done by solving the time-independent Schrodinger equation $$ -\frac{\hbar^2}{2m}\nabla^2\psi(x) + V(x)\psi(x) = E\psi(x)$$ where $V(x)$ is the mean field from the last iteration. The wave functions $\psi(x)$, which you solve for, are the single-particle eigenfunctions for this iteration.

You now place the particles in the lowest of these eigenfunctions, consistent with the Pauli principle. But now that the particles are in the new orbits, they generate a slightly different mean field. You now solve for the eigenfunctions of this new mean field, and place all the particles in these newest orbits. You iterate this process until the new eigenfunctions are the same as the one that generated the mean field. In a perhaps clearer way:

1. Place particles in initial guess orbits


2. Find the mean field potential generated by these orbits


3. Find the eigenfunctions of this mean field



4. If the eigenfunctions are different from the previous step, return to 2