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A correlation function is a statistical correlation between random variables at two different points in space, time, or other parameter space, usually as a function of the variable distance between these points. In QFT, field autocorrelation functions are propagators, so use the "propagator" tag, instead.
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"Correlation energy" using the pair correlation function
In case someone else ends up here, here's what I think is going on:
As shown in the edit of the question the term containing $g(r)$ gives $E_{el-el}$, the Coulomb energy per particle of the electron- …
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"Correlation energy" using the pair correlation function
In this paper 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 …