Ab initio molecular dynamics: Theory and Implementation

density could give rise to large errors in the total energy 129 . The correction energycan be calculated from∆E tot = −2π Ω ∑ ( n in )(G)G 2 − nout (G)G 2 (n out (G)) ⋆G≠0−Ω ∑ G(Vinxc (G) − V outxc (G) ) (n out (G)) ⋆ , (180)where n in and n out are the input and output charge densities and Vxc in and Vxcout thecorresponding exchange and correlation potentials. This term leads to the so–called“non–self–consistency correction” of the force, introduced in Eq. (68).The use of an appropriate k–point mesh is the most efficient method to calculatethe total energy of a periodic system. Equivalent, although not as efficient, thecalculation can be performed using a supercell consisting of replications of theunit cell and a single integration point for the Brillouin zone. In systems wherethe translational symmetry is broken, e.g. disorder systems, liquids, or thermallyexcited crystals, periodic boundary conditions can still be used if combined witha supercell approach. Many systems investigated with the here described methodfall into these categories, and therfore most calculations using the Car-Parrinellomolecular dynamics approach are using supercells and a single k–point ”integrationscheme”. The only point calculated is the center of the Brillouin zone (Γ– point;k = 0). For the remainder of this chapter, all formulas are given for the Γ–pointapproximation.3.4.2 Wavefunction GradientAnalytic derivatives of the total energy with respect to the parameters of the calculationare needed for stable molecular dynamics calculations. All derivatives neededare easily accessible in the plane wave pseudopotential approach. In the followingFourier space formulas are presented1f i∂E total∂c ⋆ i (G) = 1 2 G2 c i (G)+ ∑ G ′ V ⋆loc(G − G ′ )c i (G ′ )∑ ( )Fα ⋆I,i hIαβ Pβ(G)S I I (G) , (181)+ ∑ Iα,βwhere V loc is the local potentialV loc (G) = ∑ I∆V Ilocal (G)S I(G) + V xc (G) + 4π n tot(G)G 2 . (182)Wavefunction gradients are needed in optimization calculations and in the Car-Parrinello molecular dynamics approach.58