Ab initio molecular dynamics: Theory and Implementation

¢££¤¤¢¢¢n tot (G)inv FTn tot (R)ñ tot (G)fw FTñ tot (R)Ṽ H (G) = Φ(G)ñ tot (G)inv FTṼ H (R)V H (G)fw FTV H (R)E H =ΩN xN yN z∑R V H(R)n tot (R)Figure 6. Flow chart for the calculation of long-ranged part of the electrostatic energy using themethod by Hockney 300 . The part inside the dashed box is calculated most efficiently with theprocedure outlined by Eastwood and Brownrigg 173 .The potential function is calculated from two parts,V MTH (G) = ¯V H (G) + Ṽ H (G) , (163)where ṼH(G) is the analytic part, calculated from a Fourier transform of erfcṼ H (G) = 4π ( [1 − exp − G2 r 2 ])cn(G) (164)G 2 4and ¯V H (G) is calculated from a discrete Fourier transform of the Green’s functionon an appropriate grid. The calculation of the Green’s function can be done at thebeginning of the calculation and has not to be repeated again. It is reported 393that a cutoff of ten to twenty percent higher than the one employed for the chargedensity gives converged results. The same technique can also be applied for systemsthat are periodic in one and two dimensions 394 .If the boundary conditions are appropriately chosen, the discrete Fourier transformsfor the calculation of ¯VH (G) can be performed analytically 437 . This ispossible for the limiting case where r c = 0 and the boundary conditions are on asphere of radius R for the cluster. For a one-dimensional system we choose a torusof radius R and for the two-dimensional system a slab of thickness Z. The electrostaticpotential for these systems are listed in Table 2, where G xy = [ gx 2 + ] 1/2g2 yand J n and K n are the Bessel functions of the first and second kind of integer ordern.Hockney’s method requires a computational box such that the charge density isnegligible at the edges. This is equivalent to the supercell approach 510 . Practicalexperience tells that a minimum distance of about 3 Å of all atoms to the edges of53