Ab initio molecular dynamics: Theory and Implementation

2.8.3 Generalized Plane WavesAn extremely appealing and elegant generalization of the plane wave concept 263,264consists in defining them in curved ξ–spacef GPWG (ξ) = Ndet 1/2 J exp [iG r(ξ)] (101)∣ det J =∂r i ∣∣∣∣∂ξ j ,where det J is the Jacobian of the transformation from Cartesian to curvilinearcoordinates r → ξ(r) with ξ = (ξ 1 , ξ 2 , ξ 3 ) and N = 1/ √ Ω as for regular planewaves. These functions are orthonormal, form a complete basis set, can be usedfor k–point sampling after replacing G by G + k in Eq. (101), are originless (butnevertheless localized) so that Pulay forces are absent, can be manipulated viaefficient FFT techniques, and reduce to standard plane waves in the special case ofan Euclidean space ξ(r) = r. Thus, they can be used equally well like plane wavesin linear expansions of the sort Eq. (65) underlying most of electronic structurecalculations. The Jacobian of the transformation is related to the Riemannianmetric tensorg ij =3∑k=1∂ξ k ∂ξ k∂r i ∂r jdet J = det −1/2 {g ij } (102)which defines the metric of the ξ–space. The metric and thus the curvilinear coordinatesystem itself is considered as a variational parameter in the original fullyadaptive–coordinate approach 263,264 , see also Refs. 159,275,276,277,278 . Thus, a uniformgrid in curved Riemannian space is non–uniform or distorted when viewed inflat Euclidean space (where g ij = δ ij ) such that the density of grid points (or the“local” cutoff energy of the expansion in terms of G–vectors) is highest in regionsclose to the nuclei and lowest in vacuum regions, see Fig. 2 in Ref. 275 .Concerning actual calculations, this means that a lower number of generalizedplane waves than standard plane waves are needed in order to achieve a given accuracy263 , see Fig. 1 in Ref. 275 . This allows even for all–electron approaches toelectronic structure calculations where plane waves fail 431,497 . More recently, thedistortion of the metric was frozen spherically around atoms by introducing deformationfunctions 265,266 , which leads to a concept closely connected to non–uniformatom–centered meshes in real–space methods 431 , see below. In such non–fully–adaptive approaches using predefined coordinate transformations attention has tobe given to Pulay force contributions which have to be evaluated explicitely 265,431 .2.8.4 WaveletsSimilar to using generalized plane waves is the idea to exploit the powerfulmultiscale–properties of wavelets. Since this approach requires an extensive introductorydiscussion (see e.g. Ref. 242 for a gentle introduction) and since it seemsstill quite far from being used in large–scale electronic structure calculations theinterested reader is referred to original papers 134,674,699,652,241,25 and the general41