Exchange and Correlation Energy

Exchange and correlation functionals almost exclusively used in plane wave calculations are of the local type with gradient corrections. These type of functionals can be written as

$$E_{\rm xc} = \int \! d{\bf r} F_{\rm xc}(n,\nabla n) = \int \!... ...n({\bf r}) = \Omega \sum_{\bf G} \varepsilon_{\rm xc}({\bf G}) n^\star({\bf G})$$ (93)

with the corresponding potential

$$V_{\rm xc} ({\bf r}) = {\partial F_{\rm xc} \over \partial n} - \... ...eft[ {\partial F_{\rm xc} \over \partial ( \partial_{s} n )} \right] \enspace ,$$ (94)

where $F_{\rm xc} = \varepsilon_{\rm xc}(n,\nabla n) n$ and $\partial_{s} n$ is the s-component of the density gradient. Exchange and correlation functionals have complicated analytical forms that can give rise to high frequency components in $\varepsilon_{\rm xc}({\bf G})$ . Although these high frequency components do not enter the sum in Eq. (93) due to the filter effect of the density, they affect the calculation of $\varepsilon_{\rm xc}$ . As the functionals are only local in real space, not in Fourier space, they have to be evaluated on a real space grid. The function $\varepsilon_{\rm xc}({\bf G})$ can then be calculated by a Fourier transform. Therefore, the exact calculation of $E_{\rm xc}$ would require a grid with a very high resolution. However, the high frequency components are usually very small and even a moderate grid gives accurate results. The use of a finite grid results in an effective redefinition of the exchange and correlation energy

$$E_{\rm xc} = {\Omega \over N_{\rm x} N_{\rm y} N_{\rm z}} \sum_{\... ... \Omega \sum_{\bf G} \tilde \varepsilon_{\rm xc}({\bf G}) n({\bf G}) \enspace ,$$ (95)

where $\tilde \varepsilon_{\rm xc}({\bf G})$ is the finite Fourier transform of $\varepsilon_{\rm xc}({\bf R})$ . This definition of $E_{\rm xc}$ allows the calculation of all gradients analytically. In most applications the real space grid used in the calculation of the density and the potentials is also used for the exchange and correlation energy.

The above redefinition has an undesired side effect. The new exchange and correlation energy is no longer translationally invariant. Only translations by a multiple of the grid spacing do not change the total energy. This introduces a small modulation of the energy hypersurface, known as ``ripples''. Highly accurate optimizations of structures and the calculation of harmonic frequencies can be affected by the ripples. Using a denser grid for the calculation of $E_{\rm xc}$ is the only solution to avoid these problems.