The Hartree-Fock (HF) approach relies on assuming that the ground state
of a many-fermion system can be uniquely characterized by the
one-body density matrix (59). There are many ways of deriving
the HF equations; the simplest one is to use the variational principle
together with the following approximation of the two-body density
matrix (60):
From Eq. (75) it is clear that not the real interaction , but the effective interaction , must be used in the HF method. Indeed, when the density-matrix (75) is inserted in the expression for the HF energy (70), one recovers the action of the effective interaction on the two-body product wave functions (61). It is now obvious that the determination of the effective interaction must be coupled to the solution of the HF equations, and performed self-consistently. Namely, for a given effective interaction one solves the HF equations, and the obtained HF orbitals (74) are in turn used in the Bethe-Goldstone equation to find effective interaction. Such a doubly self-consistent procedure is called the Brueckner-Hartree-Fock method.
Modern understanding of the HF approximation is not directly based on the variational method applied to Slater determinants. Certainly, the basic approximation for the two-body density matrix (69) is an exact result for a Slater determinant, but the key element of the approach is expression (70), which states that the ground-state energy can be approximated by a functional of the one-body density matrix.