Ab initio molecular dynamics: Theory and Implementation
Ab initio molecular dynamics: Theory and Implementation
Ab initio molecular dynamics: Theory and Implementation
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¢££¤¤¢¢¢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 <strong>and</strong> 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 4<strong>and</strong> ¯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 <strong>and</strong> 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 <strong>and</strong> 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 <strong>and</strong> the boundary conditions are on asphere of radius R for the cluster. For a one-dimensional system we choose a torusof radius R <strong>and</strong> 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 y<strong>and</strong> J n <strong>and</strong> K n are the Bessel functions of the first <strong>and</strong> 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