Hartree-Fock approximation

Question

The Hartree-Fock approximation is used in solving many-body quantum mechanical systems. The problems of these type of systems are the e-e repulsion term in the Hamiltonian and the single electron wavefunction which is unkown.

I understand that:

1. The e-e repusion term contains two positions variable and is not separable, therefore to solve such problems we need an approximated expression of such term.
2. By using Fock operator, we can determined the single electron wavefunction and solve the problem.

My questions are: how this Fock operator is related with e-e repulsion term and how to proceed from this term to Fock operator? what is the physical interpretation of Fock-operator? Also in particular, which type of systems can be solved from this approximation.

Answer 1

The Hartree-Fock approximation makes an assumption on the state, namely that your $N$-electron wave function is the antisymmetric product of single-electron states; if you drop the antisymmetry (i. e. you forget that electrons are fermions), you get the Hartree approximation. You can implement the antisymmetrization procedure by plugging the single-particle wave functions into a Slater determinant.

To find an approximate ground state of the system, you can use a variational principle under the constraint that all the single-particle wave functions are orthogonal (that's the Pauli principle again). The Hartree-Fock equations now come out as the Euler-Lagrange equations to that variational principle. Note that the ground state is approximate, because instead of minimizing over all antisymmetric wave functions, you are only admitting antisymmetrized product states.

In summary, you approximate the true ground state of the system by an antisymmetrized product state of single-particle wave that solve a non-linear equation. The true ground state will not be a product state, but in many situations it is energetically close to a product state.