# The "Hartree-Fock energy" in the Feynman formalism vs the Hartree-Fock method

This question has been previously asked, but I do not understand the answer.

When calculating the ground state energy of an interacting system by a perturbative expansion in terms of Feynman diagrams, say for the interacting electron gas, the contribution of the first-order diagrams is often referred to as the "Hartree-Fock energy" (see Nolting, just under equation 5.139, or Section 43.2 in Lancaster & Blundell).

I am conceptually familiar with the Hartree-Fock method for finding ground states of interacting systems by assuming that the quantum state can be written as a single Slater determinant, then iteratively solving for the right set of basis wavefunctions.

Am I missing some kind of deeper connection here? The answer to the linked questions seems to suggest that one can rederive the Hartree-Fock method by summing diagrams at all orders (via the self-energy), which of course involves going beyond first order. But then I don't understand why the first-order contribution is called the Hartree-Fock energy, if the connection involves going beyond first order. Is this just an unfortunate naming convention?

For context: I am interested in computing the ground state energy for a system of fermions interacting by s-wave scattering (so the pair interaction is a delta function). I computed the two first-order Feynman diagrams, and found that they cancelled out. Obviously, I know that there should be some energy change to the ground state associated with adding an interaction, so I don't see how the contributions of first-order Feynman diagrams could possibly be equivalent to Hartree-Fock.

The self-energy is a functional of $$G_0$$, $$\Sigma[G_0]$$, which is not self-consistent since $$G_0$$ is the free propagator, which is approximated in the HF approximation by two diagrams (that we give a contribution that we call $$\Sigma_{HF}[G_0]$$.
However, one can reconciliate the two approach by making the equation for the HF self-energy self-consistent, i.e. $$\Sigma_{HF}[G_0]\to\Sigma_{HF}[G]$$. This is now a more complicated equation, since $$G=G_0-G_0\Sigma_{HF}[G]G$$ cannot be solved easily. However, writing $$G(x,y)$$ in a matrix form in terms of eigenvalues (corresponding roughly to the inverse of the energy of each orbitals) and eigenvectors (corresponding to the orbitals), one should recover the standard HF equation of quantum chemistry. In the jargon of QFT, this method corresponds to work to the lowest order approximation of the Luttinger-Ward functional in the 2 Particle Irreducible formalism.