# How can density functional theory (DFT) be understood in many body perturbation theory (MBPT) language?

Many body interacting fermions problems are formulated in the many body perturbation theory language using Feynman diagrams and imaginary time formalism. To the best of my knowledge the kinetic energy should be the energy of the non interacting system and any correction coming from the interaction would come as self energy corrections. In practical we rarely take the dispersion of the free electrons., it is in general a better to take the DFT dispersion. What is very confusing is that DFT already takes some of the interaction into account, so if i am to use the DFT dispersion in a many body perturbative theory, what should i take for the interaction ?

I understand how can DFT get the dispersion, and i understand how the dispersion is modified by the self energy in MBPT. What i don't understand, is what does it mean to mix both approaches.

• I assume with dispersion you really mean London Dispersion not the whole van der Waals interactions. And when you say "I understand how can DFT get the dispersion" what method are you talking about there are different ways of getting dispersion interactions into account in DFT. – physicopath Jan 10 '18 at 19:15
• by dispersion i mean the band structure calculated from DFT, the one you obtain solving KS equations. – lakehal Jan 11 '18 at 16:46

\begin{align} G^{-1} &= G_0^{-1} - \Sigma \\ &= G_{DFT}^{-1} + V_{xc} - \Sigma \end{align}
Up to now, there is no advantage of using $G_{DFT}$, because you essentially go back to $G_0$. The advantage appears when you recall that the self-energy itself depends on the Green's function.
If you use some approximation based on skeleton diagrams, you need to perform self-consistent calculations, starting from a self-energy obtained from some initial Green's function. In this case, a good initial guess for $G$ (e. g., $G_{DFT}$) would speed up convergence. An example how this works within the $GW$ approximation can be found in [B. Holm and U. von Barth, "Fully self-consistent $GW$ self-energy of the electron gas"].
If you are not going to do self-consistent calculations of some skeleton diagrams and just want to construct a perturbation theory around $G_{DFT}$, you may simply substitute $G_0 = G_{DFT} + V_{xc}$ in the usual series in $G_0$ and $V$ — then both $V$ and $V_{xc}$ will be perturbations (though it does not look particularly useful).
• I am not very familiar with self-consistent calculations of the green's function, and don't know what are exactly skeleton diagrams. So if you have a good read on that, it would be very useful. I was thinking ( or hoping ) that one can replace $G_{0}$ by $G_{DFT}$ and renormalize the interaction and use MBPT. If i substitute $G_0=G_{DFT}+V_{xc}$ then is $G_0$ the free-electron propagator ? – lakehal Mar 2 '18 at 11:26