Ab initio molecular dynamics: Theory and Implementation

used in calculations of different chemical environment resulting in calculations withcomparable accuracy.A first major step to achieve all this conflicting goals was the introduction of”norm–conservation” 622,593 . Norm–conserving pseudopotentials have to be angularmomentum dependent. In their most general form they are semi–localV PP (r,r ′ ) = ∑ lmY lm (r)V l (r)δ r,r ′Y lm (r ′ ) , (134)where Y lm are spherical harmonics. A minimal set of requirements and a constructionscheme for soft, semi–local pseudopotentials were developed 274,28 . Since thenmany variations of the original method have been proposed, concentrating eitheron an improvement in softness or in transferability. Analytic representations of thecore part of the potential 326,626,627,509 were used. Extended norm-conservation 564was introduced to enhance transferability and new concepts to increase the softnesswere presented 659,509,369 . More information on pseudopotentials and theirconstruction can be found in recent review articles 487,578,221 .Originally generated in a semi-local form, most applications use the fully separableform. Pseudopotentials can be transformed to the separable form using atomicwavefunctions 335,73,659 . Recently 239,288 a new type of separable, norm-conservingpseudopotentials was introduced. Local and non-local parts of these pseudopotentialshave a simple analytic form and only a few parameters are needed to characterizethe potential. These parameters are globally optimized in order to reproducemany properties of atoms and ensure a good transferability.A separable non-local pseudopotential can be put into general form (this includesall the above mentioned types)V PP (r,r ′ ) = (V core (r) + ∆V local (r)) δ r,r ′ + ∑ k,lP ⋆ k(r)h kl P l (r ′ ) . (135)The local part has been split into a core ( ∼ 1/r for r → ∞ ) and a short-rangedlocal part in order to facilitate the derivation of the final energy formula. Theactual form of the core potential will be defined later. The local potential ∆V localand the projectors P k are atom-centered functions of the formthat can be expanded in plane wavesϕ(r) = ϕ(|r − R I |) Y lm (θ, φ) , (136)ϕ(r) = ∑ Gϕ(G) exp[iG · r] S I (G) Y lm (˜θ, ˜φ) , (137)R I denote atomic positions and the so–called structure factors S I are defined asS I (G) = exp[−iG · R I ] . (138)The functions ϕ(G) are calculated from ϕ(r) by a Bessel transform∫ ∞ϕ(G) = 4π (−i) l dr r 2 ϕ(r) j l (Gr) , (139)048