Hartree--Fock Theory, Lieb's Variational Principle, and their Generalizations

Hartree–Fock theory in quantum mechanics is reviewed, from the proposal of the Hartree–Fock approximation right after quantum mechanics was formulated to its applications in modern physics. This includes the description of traditional Hartree–Fock theory in quantum chemistry, its generalizations of various kinds, and its importance for predicting the presence of symmetry breaking, or the absence thereof.