Ab initio molecular dynamics: Theory and Implementation

Distances in scaled coordinates are related to distances in real coordinates by themetric tensor G = h t h(r i − r j ) 2 = (s i − s j ) t G(s i − s j ) . (106)Periodic boundary conditions can be enforced by usingr pbc = r − h [ h −1 r ] NINT , (107)where [· · ·] NINT denotes the nearest integer value. The coordinates r pbc will bealways within the box centered around the origin of the coordinate system. Reciprocallattice vectors b i are defined asand can also be arranged to a three by three matrixb i · a j = 2π δ ij (108)[b 1 ,b 2 ,b 3 ] = 2π(h t ) −1 . (109)Plane waves build a complete and orthonormal basis with the above periodicity(see also the section on plane waves in Sect. 2.8)with the reciprocal space vectorsf PWG (r) = 1 √Ωexp[iG · r] = 1 √Ωexp[2π ig · s] , (110)G = 2π(h t ) −1 g , (111)where g = [i, j, k] is a triple of integer values. A periodic function can be expandedin this basisψ(r) = ψ(r + L) = √ 1 ∑ψ(G) exp[iG · r] , (112)Ωwhere ψ(r) and ψ(G) are related by a three-dimensional Fourier transform. Thedirect lattice vectors L connect equivalent points in different cells.3.1.2 Plane Wave ExpansionsThe Kohn–Sham potential (see Eq. (82)) of a periodic system exhibits the sameperiodicity as the direct latticeGV KS (r) = V KS (r + L) , (113)and the Kohn–Sham orbitals can be written in Bloch form (see e.g. Ref. 27 )Ψ(r) = Ψ i (r,k) = exp[ik · r] u i (r,k) , (114)where k is a vector in the first Brillouin zone. The functions u i (r,k) have theperiodicity of the direct latticeu i (r,k) = u i (r + L,k) . (115)44