Exchange-correlation energy and potential

Having calculated the electron density on the grid points, the exchange-correlation energy is obtained by summing over those grid points

$$E_{\mathrm {xc}} = \delta \omega \sum_{\bf r} n({\bf r}) \epsilon_{\mathrm {xc}}(n({\bf r}))$$ (7.12)

in the local density approximation. $\delta \omega$ is the volume of the supercell divided by the number of grid points. The exchange-correlation potential is similarly calculated at each grid point as

$$V_{\mathrm {xc}}({\bf r}) = \left[ \frac{\mathrm d}{{\mathrm... ...\epsilon_{\mathrm {xc}}(n) \right\} \right]_{n = n({\bf r})} .$$ (7.13)

In practice, the values of $\epsilon_{\mathrm {xc}}(n)$ and $\frac{\mathrm d}{{\mathrm d}n} \left[ n \epsilon_{\mathrm {xc}}(n) \right]$ are tabulated for various values of the electronic density $n$ and then interpolated during the calculation.