The Exchange Correlation Hole

The last term of equation (2.27) can be separated into the classical, $J[\rho]$, and non-classical parts by defining
$$\begin{align}\rho_{2}({\bf r}_{1},{\bf r}_{2}) = \frac{1}{2}\rho({\bf r}_{1})\rho({\bf r}_{2})[1+h({\bf r}_{1},{\bf r}_{2})] \end{align}$$
where $h({\bf r}_{1}{\bf r}_{2})$ is the pair correlation function. Slater [76] looked at this in a slightly different way, defining the exchange-correlation hole by
$$\begin{align}\rho_{xc}({\bf r}_{1},{\bf r}_{2}) = \rho({\bf r}_{2})h({\bf r}_{1},{\bf r}_{2}). \end{align}$$
Using the spinless equivalent of equation (2.23) we find the condition
 $$\begin{align} \int \rho_{xc}({\bf r}_{1},{\bf r}_{2}) \, d{\bf r}_{2} = -1 \end{align}$$
which must hold for all values of ${\bf r}_{1}$. The electron repulsion term can then be written
$$\begin{align}V_{ee} = J[\rho] + \frac{1}{2} \iint \frac{1}{r_{12}}\rho({\bf r}_{1})\rho_{xc}({\bf r}_{1},{\bf r}_{2})\,d{\bf r}_{1}d{\bf r}_{2}, \end{align}$$
where the non-classical part has been expressed as a repulsion between the density and the exchange correlation hole, a distribution of unit positive charge centered around ${\bf r}_{1}$. The coulomb potential due to the non-classical part has been shown to have the asymptotic behaviour [77]
$$\begin{align}\lim_{r_{1}\to\infty} v_{xc}({\bf r}_{1}) = \lim_{r_{1}\to\infty} \... ...{xc}({\bf r}_{1},{\bf r}_{2})\,d{\bf r}_{2} = -\frac{1}{{\bf r}_{1}} \end{align}$$