Why is Hartree-Fock considered a mean-field approach?

Question

In studying the Hartree-Fock method for solving systems of interacting particles, I have often found that the method is referred to as a mean-field approach. Wikipedia's page for instance says that the mean-field approximation is implied. I don't see why this is the case though.

In the HF method, one considers a single Slater determinant and minimizes the energy with respect to the orbitals that compose the determinant. The equations that follow this procedure are the HF equations and while it's true that they resemble the Schrodinger equations of independent electrons in a mean-field, they also have an exchange term, which makes those equations different from the Hartree equations, which actually do look like a mean-field approximation. In any case, I don't see where a mean-field approximation is implied.

Answer 1

Hartree-Fock reduces the multi-particle problem to a one-particle one, representing its interaction with the rest of the world as an interaction between the particle and some average (mean) field. This is perhaps less obvious when treating a two-particle system, where both particles appear on an equal footing, but it becomes rather obvious when one is treating a system of N-particles.

Slater determinant is a manifestation of that: it gives an exact wave function for $N$ non-interacting particles, but once the interaction is on, the factorization into single-particle orbitals becomes an approximation.

Remarks: The exchange term may appear unusual in comparison to the usual form of the Coulomb interaction, but it remains a one-particle interaction with the rest. When applying more formal mean field frameworks, e.g., via the Habbard-Stratonovich transformation, both terms appear naturally, on equal footing.

Example: This is not to give a summary of all the mean-field approaches, but just to give an idea why it is natural to call HF mean field. Let us consider a second-quantized electron Hamiltonian with Coulomb interaction: $$ \mathcal{H} = \sum_{k,\sigma}\epsilon_{k,\sigma}c_{k,\sigma}^\dagger c_{k,\sigma} + \sum_{k,k',\sigma,\sigma'}V_{k,\sigma;k',\sigma'}c_{k,\sigma}^\dagger c_{k',\sigma'} + \\ \frac{1}{2}\sum_{k,k', k'', k''', \sigma, \sigma'}U_{k, k', k''', k''}c_{k,\sigma}^\dagger c_{k',\sigma'}^\dagger c_{k'',\sigma'} c_{k''',\sigma} $$ Here the second term is an interaction with an external field, which potentially can be spin-dependent (hence the spin indices). The last term is the usual Coulomb interaction, which we will factorize in a HF manner as $$ \frac{1}{2}\sum_{k,k', k'', k''', \sigma, \sigma'}U_{k, k', k''', k''}c_{k,\sigma}^\dagger c_{k',\sigma'}^\dagger c_{k'',\sigma'} c_{k''',\sigma} \approx\\ \frac{1}{4}\sum_{k,k', k'', k''', \sigma, \sigma'}U_{k, k', k''', k''}c_{k,\sigma}^\dagger c_{k''',\sigma}\langle c_{k',\sigma'}^\dagger c_{k'',\sigma'}\rangle - \frac{1}{4}\sum_{k,k', k'', k''', \sigma, \sigma'}U_{k, k', k''', k''}c_{k,\sigma}^\dagger c_{k'',\sigma'} \langle c_{k',\sigma'}^\dagger c_{k''',\sigma}\rangle $$ here the first term in the last expression is the Hartree term, whereas the last term is the Fock term - they differ just by the order of pairing operators that will be replaced by their mean, i.e. $$ c_{k',\sigma'}^\dagger c_{k'',\sigma'}\rightarrow \langle c_{k',\sigma'}^\dagger c_{k'',\sigma'}\rangle $$ We can further rewrite the Coulomb term as $$ \frac{1}{2}\sum_{k,k', k'', k''', \sigma, \sigma'}U_{k, k', k''', k''}c_{k,\sigma}^\dagger c_{k',\sigma'}^\dagger c_{k'',\sigma'} c_{k''',\sigma} \approx \sum_{k,k',\sigma,\sigma'}U^{eff}_{k,\sigma;k',\sigma'}c_{k,\sigma}^\dagger c_{k',\sigma'}, $$ which has the same form as the one-particle potential (external field term). The use of mean and field comes rather naturally in this context and has a clear meaning. Note also that in this formalism there is no really apparent difference between Hartree and Fock terms - they differ by the spin indices, which are necessarily the same in the Hartree term, but even these could be symmetrized, as is done in the classical reference on the field theory methods in condensed matter, although for more detailed derivations of this type and introduction to the second quantization I would recommend more pedestrian Fetter & Walecka. These books however may not call it mean field, which is a term used mainly by the people raised on path integrals, for which a good condensed matter reference is Negele & Orland.

Answer 2

In HF we assume that we can separate the wavefunction into a product ansatz $$ \Psi(r_1, r_2, ...)\Psi^*(r_1, r_2, ...)=|\Psi(r_1, r_2, ...)|^2=\rho(r_1, r_2, ...) $$

$$ \rho(r_1, r_2, ...)\approx\rho_1(r_1)\rho_2(r_2)\cdots $$

$$ \rho_1(r_1)=\psi_1(r_1)\psi^*_1(r_1) $$

This allows us to define single particle potentials that are summed to form the mean field potential $V_H$, $$ \begin{aligned} V_H(r_i) &=\frac{-e}{4\pi\varepsilon_0}\sum_{j=1(\neq i)}^N \int dr_j \frac{\rho_j(r_j) }{|r_j-r_i|}\\[1.5em] V_H(r_i) &=\frac{-e}{4\pi\varepsilon_0}\sum_{j=1(\neq i)}^N \int dr_j \frac{\psi_j(r_j)\psi^*_j(r_j) }{|r_j-r_i|} \end{aligned} $$

The potential energy of an particle is then calculated by interaction with this mean field potential of the remaining particles, $$ \begin{aligned} E_{pot,i} &= q_e\int_V dr_i \psi_i(r_i)\psi^*_i(r_i)\frac{e^2}{4\pi\varepsilon_0}\sum_{j=1(\neq i)}^N \int dr_j \frac{\psi_j(r_j)\psi^*_j(r_j) }{|r_j-r_i|}\\ &=q_e\int_V dr_i \psi_i(r_i)\psi^*_i(r_i)V_{H}(r_i) \end{aligned} $$

I would consider this the reason why HF is called a mean field method.