Ab initio molecular dynamics: Theory and Implementation

Table 2. Fourier space formulas for the Hartree energy, see text for definitions.Dim. periodic (G 2 /4π)V H (G) V H (0)0 – (1 − cos [R G]) n(G) 2πR 2 n(0)1 z (1 + R (G xy J 1 (RG xy ) K 0 (Rg z )−g z J 0 (RG xy ) K 1 (Rg z ))) n(G) 02 x, y (1 − (−1) gz exp [−GZ/2]) n(G) 03 x, y, z n(G) 0the box is sufficient for most systems. The Green’s function is then applied to thecharge density in a box double this size. The Green’s function has to be calculatedonly once at the beginning of the calculation. The other methods presented in thischapter require a computational box of double the size of the Hockney method asthey are applying the artificially periodic Green’s function within the computationalbox. This can only be equivalent to the exact Hockney method if the box is enlargedto double the size. In plane wave calculations computational costs grow linearlywith the volume of the box. Therefore Hockney’s method will prevail over the othersin accuracy, speed, and memory requirements in the limit of large systems. Thedirect Fourier space methods have advantages through their easy implementationand for small systems, if not full accuracy is required, i.e. if they are used withsmaller computational boxes. In addition, they can be of great use in calculationswith classical potentials.3.3 Exchange and Correlation EnergyExchange and correlation functionals implemented in the CPMD code are of the localtype with gradient corrections. These type of functionals can be written as (seealso Eqs. (88) and (84))∫E xc = dr ε xc (n, ∇n) n(r) = Ω ∑ ε xc (G)n ⋆ (G) (165)Gwith the corresponding potentialV xc (r) = ∂F xc∂n − ∑ s[ ]∂ ∂Fxc∂r s ∂(∂ s n), (166)where F xc = ε xc (n, ∇n)n and ∂ s n is the s-component of the density gradient.Exchange and correlation functionals have complicated analytical forms thatgive rise to high frequency components in ε xc (G). Although these high frequencycomponents do not enter the sum in Eq. (165) due to the filter effect of the density,they affect the calculation of ε xc . As the functionals are only local in real space, notin Fourier space, they have to be evaluated on a real space grid. The function ε xc (G)can then be calculated by a Fourier transform. Therefore the exact calculation ofE xc would require a grid with infinite resolution. However, the high frequencycomponents 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 and54