More about exchange

Let us define a new quantity

$$n_{xc}({\bf r},{\bf r}') = n({\bf r}')(g({\bf r},{\bf r}')-1).$$ (162)

We can easily see that the exchange correlation energy may be written

$$E_{xc}[n] = \frac{1}{2} \int d^3r_1 d^3r_2 n({\bf r}_1)n_{xc}({\bf r}_1,{\bf r}_2) \frac{1}{r_{12}}.$$ (163)

This is the Coulomb interaction of each electron with a charge distribution $n_{xc}$, whcih can be interpreted as a conditional density. This conditional density vanished as the distance between the particles goes to zero, and may be interpreted as a ``hole'' surrounding each particle, and it is named exchange correlation hole. We can see that this hole orrespond to a unit of charge by anotehr sum-rule:

$$\int d^3r n_{xc}({\bf r},{\bf r}') = -1.$$ (164)

We can consider $-n_{xc}$ as a normalizationo factor and define the radius of the exchange hole as:

$$\left \langle \frac{1}{R} \right \rangle = - \int d^3r \frac{n_{xc}({\bf r},{\bf R})}{\vert R\vert}.$$ (165)

Thsi leads to

$$E_{xc}[n] = -\frac{1}{2}\int d^3r n({\bf r}) \left \langle \frac{1}{R} \right \rangle$$ (166)

showing that, privided that the sum-rule is satisfied, the exchange-correlation energy depends only weakly on the details of $n_{xc}$. Thsi means that even if our approximation is nto able to describe the detailed spatial shape of the hole, as long as the sum-rule is fulfuilled, the errors are small, and LDA will produce good results.