Ab initio molecular dynamics: Theory and Implementation

¢¢¢¢¢¢¢¢¢¢¢¢¢n(G)∂ s n(G) = iG s n(G)3 × inv FT∂ s n(R)ε xc|∇n| = ( ∑ s (∂ sn(R)) 2 ) 1/2∂F xc∂n∂F xc∂|∇n|A s (R) = ∂Fxc∂ sn∂|∇n| |∇n|3 × fw FTA s (G)∂A s (G) = iG s A s (G)3 × inv FT∂ s A s (R)E xc =ΩN xN yN z∑R ε xc(R)n(R)V xc (R) = ∂Fxc∂n (R) + ∑ s ∂ sA s (R)Figure 7. Flow chart for the calculation of the energy and potential of a gradient corrected exchangeand correlation functional.grid points, a total of 5Nb 2 three–dimensional transforms are needed, resulting inapproximately 25Nb 2 N log N operations needed to perform the calculation. Thishas to be compared to the 15N b N log N operations needed for the other Fouriertransforms of the charge density and the application of the local potential and the4Nb 2 N operations for the orthogonalization step. In calculations dominated by theFourier transforms an additional factor of at least N b is needed. If on the other handorthogonalization dominates an increase in computer time by a factor of 5 logN isexpected. Therefore, at least an order of magnitude more computer time is neededfor calculations including exact exchange compared to ordinary density functionalcalculations. Consequently, hybrid functionals will only be used in exceptional casestogether with plane waves 262,128 .56