Exchange and Correlation Functionals

Gradient corrected exchange and correlation functionals and their potentials are defined as

$$E_{xc} = \int F_{xc}(n,\nabla n) d{\bf r}$$ (239)

$$V_{xc} = {\delta E_{xc} \over \delta n} = {\partial F_{xc} \over \partial ... ...r_{\alpha}} \left[ {\partial F_{xc} \over \partial (\nabla_{\alpha} n)} \right]$$

$$= {\partial F_{xc} \over \partial n} - \sum_{\alpha = 1}^3 {\partia... ...ver \partial \vert\nabla n\vert} {\partial n \over \partial r_{\alpha}} \right]$$ (240)

The function F and its derivatives with respect to n and $\vert\nabla n\vert$ have to be calculated in real space. The derivatives with respect to r can be calculated most easily in reciprocal space. The following scheme outlines the steps necessary to perform the calculation

1. $n(R) \stackrel{FFT}{\longrightarrow} n(G)$
2. ${\partial n \over \partial r_{\alpha}} = i G_{\alpha} n(G)$
3. $i G_{\alpha} n(G) \stackrel{INVFFT}{\longrightarrow} {\partial n \over \partial r_{\alpha}} (R)$
4. $\vert\nabla n(R)\vert$
5. $F(n(R),\nabla n(R))$
6. ${\partial F(R) \over \partial n}$
7. ${1 \over \vert\nabla n(R)\vert}{ \partial F(R) \over \partial \vert\nabla n(R)\vert}$
8. $H^1(R) = {\partial F(R) \over \partial n} \stackrel{FFT}{\longrightarrow} H^1(G)$
9. $H^2_{\alpha}(R) = {1 \over \vert\nabla n(R)\vert}{ \partial F(R) \over \partia... ... \over \partial r_{\alpha}}(R) \stackrel{FFT}{\longrightarrow} H^2_{\alpha}(G)$
10. $V_{xc}(G) = H^1(G) - \sum_{\alpha = 1}^3 i G_{\alpha} H^2_{\alpha}(G)$
11. $V_{xc}(G) \stackrel{INVFFT}{\longrightarrow} V_{xc}(R)$