Ab initio molecular dynamics: Theory and Implementation

andE self = ∑ I1√2πZ 2 IR c I. (151)Here, the sums expand over all atoms in the simulation cell, all direct lattice vectorsL, and the prime in the first sum indicates that I < J is imposed for L = 0.3.2.3 Cluster Boundary ConditionsThe possibility to use fast Fourier transforms to calculate the electrostatic energyis one of the reasons for the high performance of plane wave calculations. However,plane wave based calculations imply periodic boundary conditions. This isappropriate for crystal calculations but very unnatural for molecule or slab calculations.For neutral systems this problem is circumvented by use of the supercellmethod. Namely, the molecule is periodically repeated but the distance betweeneach molecule and its periodic images is so large that their interaction is negligible.This procedure is somewhat wasteful but can lead to satisfactory results.Handling charged molecular systems is, however, considerably more difficult,due to the long range Coulomb forces. A charged periodic system has infiniteenergy and the interaction between images cannot really be completely eliminated.In order to circumvent this problem several solutions have been proposed. Thesimplest fix-up is to add to the system a neutralizing background charge. Thisis achieved trivially as the G = 0 term in Eq. (149) is already eliminated. Thisleads to finite energies but does not eliminate the interaction between the imagesand makes the calculation of absolute energies difficult. Other solutions involveperforming a set of different calculations on the system such that extrapolation tothe limit of infinitely separated images is possible. This procedure is lengthy andone cannot use it easily in molecular dynamics applications. It has been shown,that it is possible to estimate the correction to the total energy for the removalof the image charges 378 . Still it seems not easy to incorporate this scheme intothe frameworks of molecular dynamics. Another method 60,702,361 works with theseparation of the long and short range parts of the Coulomb forces. In this methodthe low–order multipole moments of the charge distribution are separated out andhandled analytically. This method was used in the context of coupling ab initioand classical molecular dynamics 76 .The long-range forces in Eq. (146) are contained in the first term. This termcan be written∫ ∫12dr dr ′ n tot (r)n tot (r ′ )|r − r ′ |= 1 2∫dr V H (r)n tot (r) , (152)where the electrostatic potential V H (r) is the solution of Poisson’s equation (seeEq. (80)). There are two approaches to solve Poisson’s equation subject to theboundary conditions V H (r) → 0 for r → ∞ implemented in CPMD. Both of themrely on fast Fourier transforms, thus keeping the same framework as for the periodiccase.The first method is due to Hockney 300 and was first applied to density functionalplane wave calculations in Ref. 36 . In the following outline, for the sake of simplicity,51