# 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.

• Just to have an idea of your experience, how comfortable are you with using either Newton's method or the secant method for minimizing 1D functions? Mar 2, 2015 at 23:08
• I know Newton's method Mar 2, 2015 at 23:16

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.