Is density functional theory a mean-field theory?

Exact statements

The Hohenberg-Kohn theorems, which are the theoretical foundation of DFT, essentially say that the ground state properties of a many-electron system are only a function of the electron density. Any quantity you want to calculate can be re-expressed in terms of the electron density $n(r)$, including the many-body ground state wavefunction, $\Psi(n)$. This is all exact.

You can then use this to write an energy functional $E[n]$, which will be minimized by the groundstate density $n_0(r)$. This is still all exact and is the core of DFT. But just as the Schrodinger Equation is "exact", it cannot actually be solved exactly for most of the case we would like to solve it for!

Approximations

If the electron density is slowly varying then the system can be mapped to a system of non-interacting particles in an effective potential (Kohn-Sham equations). This is an approximation but a very good one, as electron densities do not have singularities in real space. The Kohn-Sham orbitals (and therefore "particles") have no physical significance other than that they reproduce the true electron density of the system.

So far so good, but unfortunately, the form of the exchange correlation term $V_{XC}[n]$ as it appears in the Kohn-Sham equations is in general not know. This is what prevents us from actually solving the problem exactly. In order to solve the problem you have to make more severe approximations as to the form of the $V_{XC}[n]$.

There are several types of these approximations that are better suited to capturing different types of interactions.

• Local Density Approximation (LDA) says only the local electron density contributes to the the exchange interaction at that position.
• LDSA is an LDA which includes spin
• LDA+U includes the Hubbard model "U", which represents local electron repulsion
• GGA is like LDA but accounts for local changes in $n(r)$

So is DFT a mean field theory?

Well, the fact that you are solving a system of non-interacting particles which interact through a self-consistent effective potential sure sounds like it. But I would say the main difference lies in the fact that in mean field theory you are attempting to describe the actual particles as non-interacting particles in an effective potential, while in DFT you are mapping real particles to a set of fictitious particles and doing a "mean field theory" to reconstruct the exact results (in principle) of the fully interacting system.