In [this paper](http://arxiv.org/abs/cond-mat/9806271) on the Quantum Hall effect the authors refer to something called the *correlation energy* of electrons. It is defined at the top of page 5 as

$E=\frac{n}{2}\int (g(r)-1)V(r)dA\ ,$

where $n$ is electron density, $g(r)$ is the pair correlation function, $V(r)$ is the potential and $dA$ is the area element (area because the system is 2D).

I'm trying to understand where this expression comes from, but searching around for "correlation energy" all I can find is connected to the Hartee-Fock approximation, which is unrelated I think. 

Testing the expression out it gives the same results as other ways of computing the energy per particle, with background charge included.

Does anyone have an explanation for this expression?

EDIT: prompted by an answer below I'll expand a bit on the mentioned "other ways of computing the energy." 

First we have the interaction between the electrons, which gives the following energy per particle:

$E\_{el}=\displaystyle\frac{\langle V\rangle}{N}=\frac{1}{N}\sum\_{i<j}^N\int\prod\_{k=1}^NdA\_k\psi^\*V(r\_{ij})\psi=\frac{N-1}{2}\int\prod\_{k=1}^NdA\_k|\psi|^2V(r)\ , \ \ \ \ \ \ \ \ \ \ (1)$

where $\psi(r)=\langle r|\psi\rangle$ is the wavefunction and we've used the fact that it is an eigenfunction of $r$ and antisymmetric in electron exchange.

At this point it is convenient to mention that all this takes place on a sphere; that is the electrons live on a spherical shell at the center of which we have a magnetic Dirac monopole. I didn't include this in the original post because I don't think it's relevant to the question of the expression for the correlation energy, but it explains the next term:

To also include the effect of the positive background charge we can place it all in the center. The energy per electron resulting from the interaction between the background chance and the electrons, and the background charge with itself, will then be: 

$E\_{bg}=-\frac{N}{2R}\ ,\ \ \ \ \ \ \ \ \ (2)$

where $R$ is the radius of the sphere (as explained better in e.g. Jain's book "Composite Fermions").

Returning to the expression for the correlation energy in the original post, we note that the definition of the pair correlation function for an isotropic system (so that we only need the relative coordinate, e.g. $r=r\_1-r\_2$), is:

$g(r)=\frac{N(N-1)}{n^2}\displaystyle\int\prod\_{k=3}^NdA\_k|\psi|^2$.

The term including $g(r)$ is therefore:

$\frac{n}{2}\displaystyle\int dA\ g(r)V(r)=\frac{N-1}{2}\int\prod\_{k=1}^NdA\_k|\psi|^2V(r)\ ,\ \ \ \ \ \ \ \ \ (3)$

where we've used the fact that $g(r)$ refers to *any* of the electrons to introduce an additional integration, over e.g. electron $1$, and cancel it's contribution with $1/A$, and the fact that $g(r)$ is isotropic to change an integration over a relative coordinate (the one in the original integral) to e.g. over electron $2$, bringing the integration to all electrons instead of just number $3$ to $N$.

Lastly we look at the last term of the original expression:

$-\frac{n}{2}\displaystyle\int dA\ V(r)=-\frac{n}{2}\int\_0^R\frac{4\pi r\ dr}{r}=-\frac{N}{2R}.\ \ \ \ \ \ \ (4)$

We see that (1) corresponds to (3) and (2) to (4), so that this way of calculating the energy gives the same answer as the "correlation energy." The reason I'm interested in finding out about the latter is that this expression seems to be valid in more general cases, or at least easier to calculate, than the first method above. One example of this is when in the second Landau level and using an effective interaction in the first Landau level, so that $V(r)\neq \frac{1}{r}$ and it is less clear how to handle the background charge.