Ab initio molecular dynamics: Theory and Implementation

The index i runs over all states and the states have an occupation f i (k) associatedwith them. The periodic functions u i (r,k) are now expanded in the plane wavebasisu i (r,k) = √ 1 ∑c i (G,k) exp[iG · r] , (116)ΩGand the Kohn–Sham orbitals areφ i (r,k) = √ 1 ∑c i (G,k) exp[i(G + k) · r] , (117)ΩGwhere c i (G,k) are complex numbers. With this expansion the density can also beexpanded into a plane wave basisn(r) = 1 ∑∫dk f i (k) ∑ c ⋆ iΩ(G′ ,k)c i (G,k) exp[i(G + k) · r] (118)G,G ′= ∑ Gin(G) exp[iG · r] , (119)where the sum over G vectors in Eq. (119) expands over double the range givenby the wavefunction expansion. This is one of the main advantages of the planewave basis. Whereas for atomic orbital basis sets the number of functions neededto describe the density grows quadratically with the size of the system, there isonly a linear dependence for plane waves.3.1.3 K–Points and CutoffsIn actual calculations the infinite sums over G vectors and cells has to be truncated.Furthermore, we have to approximate the integral over the Brillouin zone by a finitesum over special k–points∫dk → ∑ w k , (120)kwhere w k are the weights of the integration points. Schemes on how to choose theintegration points efficiently are available in the literature 30,123,435 where also anoverview 179 on the use of k–points in the calculation of the electronic structure ofsolids can be found.The truncation of the plane wave basis rests on the fact that the Kohn–Shampotential V KS (G) converges rapidly with increasing modulus of G. For this reason,at each k–point, only G vectors with a kinetic energy lower than a given maximumcutoff12 |k + G|2 ≤ E cut (121)are included in the basis. With this choice of the basis the precision of the calculationwithin the approximations of density functional theory is controlled by oneparameter E cut only.45