Ab initio molecular dynamics: Theory and Implementation

The number of plane waves for a given cutoff depends on the unit cell and thek–point. An estimate for the size of the basis at the center of the Brillouin zone isN PW = 12π 2 Ω E3/2 cut , (122)where E cut is in Hartree units. The basis set needed to describe the density calculatedfrom the Kohn-Sham orbitals has a corresponding cutoff that is four timesthe cutoff of the orbitals. The number of plane waves needed at a given densitycutoff is therefore eight times the number of plane waves needed for the orbitals.It is a common approximation in density functional theory calculations 536,169to use approximate electronic densities. Instead of using the full description, thedensity is expanded in an auxiliary basis. An incomplete plane wave basis can beconsidered as an auxiliary basis with special properties 371 . Because of the filterproperty of plane waves the new density is an optimal approximation to the truedensity. No additional difficulties in calculations of the energy or forces appear.The only point to control is, if the accuracy of the calculation is still sufficient.Finally, sums over all unit cells in real space have to be truncated. The onlyterm in the final energy expression with such a sum is the real space part of theEwald sum (see Sect. 3.2). This term is not a major contribution to the workloadin a density functional calculation, that is the cutoff can be set rather generously.3.1.4 Real Space GridA function given as a finite linear combination of plane waves can also be definedas a set of functional values on a equally spaced grid in real space. The samplingtheorem (see e.g. Ref. 492 ) gives the maximal grid spacing that still allows to holdthe same information as the expansion coefficients of the plane waves. The realspace sampling points R are definedR = h Nq , (123)where N is a diagonal matrix with the entries 1/N s and q is a vector of integersranging from 0 to N s − 1 (s = x, y, z). To fulfill the sampling theorem N s has tobe bigger than 2 max(g s ) + 1. To be able to use fast Fourier techniques, N s mustbe decomposable into small prime numbers (typically 2, 3, and 5). In applicationsthe smallest number N s that fulfills the above requirements is chosen.A periodic function can be calculated at the real space grid pointsf(R) = ∑ G= ∑ g= ∑ gf(G) exp[iG · R] (124)f(G) exp [ 2π i ( (h t ) −1 g ) · (hNq) ] (125)[ ] [ ] [ ]2π 2π 2πf(G) exp ig x q x exp ig y q y exp ig z q zN x N y N z. (126)The function f(G) is zero outside the cutoff region and the sum over g can beextended over all indices in the cube −gsmax . . .gsmax . The functions f(R) and46