Ab initio molecular dynamics: Theory and Implementation

Table 3. Relative size of characteristic variables in a plane wave calculation. See text for details.silicon waterN at 1 3N p 1 1N b 2 4N PW 53 1000N D 429 8000N 1728 312503.8.1 CPMD Program: Data StructuresImportant quantities in the pseudopotential plane–wave method depend either notat all, linearly, or quadratically on the system size. Examples for the first kind ofdata are the unit cell matrix h and the cutoff E cut . Variables with a size that growslinearly with the system arer(3,N at ) nuclear positionsv(3,N at ) nuclear velocitiesf(3,N at ) nuclear forcesg(3,N PW ) plane–wave indicesipg(3,N PW ) mapping of G–vectors (positive part)img(3,N PW ) mapping of G–vectors (negative part)rhog(N PW ) densities (n, n c , n tot ) in Fourier–spacevpot(N PW ) potentials (V loc , V xc , V H ) in Fourier–spacen(N x ,N y ,N z ) densities (n, n c , n tot ) in real–spacev(N x ,N y ,N z ) potentials (V loc , V xc , V H ) in real–spacevps(N D ) local pseudopotentialrpc(N D ) core chargespro(N PW ) projectors of non-local pseudopotential.The pseudopotential related quantities vps, rpc, and pro are one–dimensional insystem size but also depend on the number of different atomic species. In thefollowing it is assumed that this is one. It is easy to generalize the pseudo codesgiven to more than one atomic species. For real quantities that depend on G–vectors only half of the values have to be stored. The other half can be recomputedwhen needed by using the symmetry relationA(G) = A ⋆ (−G) . (261)This saves a factor of two in memory. In addition G vectors are stored in a lineararray, instead of a three-dimensional structure. This allows to store only non–zerovariables. Because there is a spherical cutoff, another reduction of a factor of twois achieved for the memory. For the Fourier transforms the variables have to beprepared in a three-dimensional array. The mapping of the linear array to thisstructure is provided by the information stored in the arrays ipg and img.Most of the memory is needed for the storage of quantities that grow quadratically77