In a nutshell, the only approximation in Hartree Fock is the assumption that the electronic wave function has a very specific form. Namely, that it is a Slater determinant of orbitals, and that each orbital is a linear combination of atomic orbitals from a fixed basis set. The linear coefficients of the orbitals are then solved for via the (exact) variational method.

Of course, the true wave function is generally not a Slater determinant. In particular, electrons in a Slater determinant with different spins are uncorrelated.

The standard approach to resolving this is density functional theory. In that model, the main approximation is the choice of an “exchange correlation functional” which approximates the electron exchange and correlation energy. The choice of a functional is unfortunately a dark art in the sense that they can only be evaluated empirically rather than from first principles.

The classic reference for Hartree Fock is Modern Quantum Chemistry by Szabo and Ostland: https://books.google.com/books/about/Modern_Quantum_Chemistr...

It is very well written and I highly recommend it.

I also wrote up some notes here: https://www.daniellowengrub.com/blog/2025/07/26/scf