Ab initio molecular dynamics: Theory and Implementation

f(G) are related by three–dimensional Fourier transformsThe Fourier transforms are defined by[inv FT [f(G)]] uvw=[fw FT [f(R)]] jkl=f(R) = inv FT [f(G)] (127)f(G) = fw FT [f(R)] . (128)N∑x−1j=0N∑ y−1k=0N∑z−1l=0f G jkl[exp i 2π ] [j u exp i 2π ] [k v exp i 2π ]l wN x N y N zN∑x−1u=0N∑ y−1v=0N∑z−1w=0f R uvw[exp −i 2π ] [j u exp −i 2π ] [k v exp −i 2π ]l wN x N y N zwhere the appropriate mappings of q and g to the indices(129), (130)[u, v, w] = q (131){j, k, l} = g s if g s ≥ 0 (132){j, k, l} = N s + g s if g s < 0 (133)have to be used. From Eqs. (129) and (130) it can be seen, that the calculationof the three–dimensional Fourier transforms can be performed by a series of onedimensional Fourier transforms. The number of transforms in each direction isN x N y , N x N z , and N y N z respectively. Assuming that the one-dimensional transformsare performed within the fast Fourier transform framework, the number ofoperations per transform of length n is approximately 5n logn. This leads to anestimate for the number of operations for the full three-dimensional transform of5N logN, where N = N x N y N z .3.1.5 PseudopotentialsIn order to minimize the size of the plane wave basis necessary for the calculation,core electrons are replaced by pseudopotentials. The pseudopotential approximationin the realm of solid–state theory goes back to the work on orthogonalizedplane waves 298 and core state projector methods 485 . Empirical pseudopotentialswere used in plane wave calculations 294,703 but new developments have considerablyincreased efficiency and reliability of the method. Pseudopotential are requiredto correctly represent the long range interactions of the core and to producepseudo–wavefunction solutions that approach the full wavefunction outside a coreradius r c . Inside this radius the pseudopotential and the wavefunction should be assmooth as possible, in order to allow for a small plane wave cutoff. For the pseudo–wavefunction this requires that the nodal structure of the valence wavefunctionsis replaced by a smooth function. In addition it is desired that a pseudopotentialis transferable 238,197 , this means that one and the same pseudopotential can be47