Ab initio molecular dynamics: Theory and Implementation

a one-dimensional case is presented. The charge density is assumed to be non-zeroonly within an interval L and sampled on N equidistant points. These points aredenoted by x p . The potential can then be writtenV H (x p ) = L ∞∑G(x p − x p ′)n(x p ′) (153)N= L Np ′ =−∞N∑G(x p − x p ′)n(x p ′) (154)p ′ =0for p = 0, 1, 2, . . .N, where G(x p − x p ′) is the corresponding Green’s function. InHockney’s algorithm this equation is replaced by the cyclic convolutionṼ H (x p ) = L N2N+1∑p ′ =0˜G(x p − x p ′)ñ(x p ′) (155)where p = 0, 1, 2, . . .2N + 1, and{n(xp ) 0 ≤ p ≤ Nñ(x p ) =(156)0 N ≤ p ≤ 2N + 1˜G(x p ) = G(x p ) − (N + 1) ≤ p ≤ N (157)ñ(x p ) = ñ(x p + L) (158)˜G(x p ) = ˜G(x p + L) (159)The solution ṼH(x p ) can be obtained by a series of fast Fourier transforms and hasthe desired propertyṼ H (x p ) = V H (x p ) for 0 ≤ p ≤ N . (160)To remove the singularity of the Green’s function at x = 0, G(x) is modified forsmall x and the error is corrected by using the identityG(x) = 1 [ ] xx erf + 1 [ ] xr c x erfc , (161)r cwhere r c is chosen such, that the short-ranged part can be accurately described bya plane wave expansion with the density cutoff. In an optimized implementationHockney’s method requires the double amount of memory and two additional fastFourier transforms on the box of double size (see Fig. 6 for a flow chart). Hockney’smethod can be generalized to systems with periodicity in one (wires) and two (slabs)dimensions. It was pointed out 173 that Hockney’s method gives the exact solutionto Poisson’s equation for isolated systems if the boundary condition (zero densityat the edges of the box) are fulfilled.A different, fully reciprocal space based method, that can be seen as an approximationto Hockney’s method, was recently proposed 393 . The final expression forthe Hartree energy is also based on the splitting of the Green’s function in Eq. (161)E ES = 2π Ω ∑ GV MTH (G)n ⋆ tot(G) + E ovrl − E self . (162)52