Ab initio molecular dynamics: Theory and Implementation

wavelet literature cited therein. Wavelet–based methods allow intrinsically to exploitmultiple length scales without introducing Pulay forces and can be efficientlyhandled by fast wavelet transforms. In addition, they are also a powerful route tolinear scaling or “order–N” methods 453,243 as first demonstrated in Ref. 241 withthe calculation of the Hartree potential for an all–electron uranium dimer.2.8.5 Mixed and Augmented Basis SetsLocalized Gaussian basis functions on the one hand and plane waves on the otherhand are certainly two extreme cases. There has been a tremendous effort tocombine such localized and originless basis functions in order to exploit their mutualstrengths. This resulted in a rich collection of mixed and augmented basis setswith very specific implementation requirements. This topic will not be coveredhere and the interested reader is referred to Refs. 75,654,498,370,371 and referencesgiven therein for some recent implementations used in conjunction with ab initiomolecular dynamics.2.8.6 Wannier FunctionsAn alternative to the plane wave basis set in the framework of periodic calculationsin solid–state theory are Wannier functions, see for instance Sect. 10 in Ref. 27 .These functions are formally obtained from a unitary transformation of the Blochorbitals Eq. (114) and have the advantage that they can be exponentially localizedunder certain circumstances. The so–called maximally localized generalized Wannierfunctions 413 are the periodic analogues of Boys’ localized orbitals defined forisolated systems. Recently the usefulness of Wannier functions for numerical purposeswas advocated by several groups, see Refs. 339,184,413,10 and references giventherein.2.8.7 Real Space GridsA quite different approach is to leave conventional basis set approaches altogetherand to resort to real–space methods where continuous space is replaced by a discretespace r → r p . This entails that the derivative operator or the entire energy expressionhas to be discretized in some way. The high–order central–finite differenceapproach leads to the expression− 1 2 ∇2 ψ i (r) h→0= − 1 2[∑Nn x=−N C n xψ i (r px + n x h, r py , r pz )+ ∑ Nn C y=−N n yψ i (r px , r py + n y h, r pz )]+ ∑ Nn z=−N C n zψ i (r px , r py , r pz + n z h)+ O ( h 2N+2) (103)for the Laplacian which is correct up to the order h 2N+2 . Here, h is the uniformgrid spacing and {C n } are known expansion coefficients that depend on the selectedorder 130 . Within this scheme, not only the grid spacing h but also the order are42