Ab initio molecular dynamics: Theory and Implementation

correlation energyΩ ∑E xc = ε xc (n, ∇n)(R)n(R) = Ω ∑ N x N y N zRG˜ε xc (G)n(G) , (167)where ˜ε xc (G) is the finite Fourier transform of ε xc (R). This definition of E xcallows the calculation of all gradients analytically. In most applications the realspace grid used in the calculation of the density and the potentials is also usedfor the exchange and correlation energy. Grids with higher resolution can be usedeasily. The density is calculated on the new grid by use of Fourier transforms andthe resulting potential is transfered back to the original grid. With this procedurethe different grids do not have to be commensurate.The above redefinition has an undesired side effect. The new exchange andcorrelation energy is no longer translationally invariant. Only translations by amultiple of the grid spacing do not change the total energy. This introduces asmall modulation of the energy hyper surface 685 , known as ”ripples”. Highlyaccurate optimizations of structures and the calculation of harmonic frequenciescan be affected by the ripples. Using a denser grid for the calculation of E xc is theonly solution to avoid these problems.The calculation of a gradient corrected functional within the plane wave frameworkcan be conducted using Fourier transforms 685 . The flowchart of the calculationis presented in Fig. 7. With the use of Fourier transforms the calculation ofsecond derivatives of the charge density is avoided, leading to a numerically stablealgorithm. To this end the identity∂F xc∂(∂ s n) = ∂F xc∂|∇n|∂ s n|∇n|(168)is used.This is the place to say some words on functionals that include exact exchange.As mentioned in Sect. 2.7 this type of functional has been very popular recentlyand improvements of results over GGA–type density functionals for many systemsand properties have been reported. However, the calculation of the Hartree–Fockexchange causes a considerable performance problem in plane wave calculations.The Hartree–Fock exchange energy is defined as 604whereE HFX = ∑ ij∫ ∫drdr ′ ρ ij (r)ρ ij (r ′ )|r − r ′ |, (169)ρ ij (r) = φ i (r)φ j (r). (170)From this expression the wavefunction force is easily derived and can be calculatedin Fourier space1 ∂E HFXf i ∂c ⋆ i (G) = ∑ ∑V ijHFX (G − G′ )c j (G ′ ) . (171)j G ′The force calculation is best performed in real space, whereas the potential is calculatedin Fourier space. For a system with N b electronic states and N real space55