Ab initio molecular dynamics: Theory and Implementation

The local part of the pseudopotential ∆Vlocal I (G) and the nonlocal projector functionsdepend on the cell matrix h through the volume, the Bessel transform integraland the spherical harmonics function. Their derivatives are lengthy but easy to calculatefrom their definitions Eqs. (140) and (141)∂∆V Ilocal (G)∂h uv= −∆Vlocal(G)(h I t ) −1uv∫ ∞+ 4π Ω 0∂FI,iα = √ 4π (−i) ∑ l∂h uv ΩG[(∂Y lm (˜θ, ˜φ)dr r 2 ∆V local (r)∂h uv∫ ∞+Y lm (˜θ, ˜φ)3.4.5 Non-linear Core Correction0c ⋆ i (G) S I(G)− 1 2 Y lm(˜θ, ˜φ)(h t ) −1uvdr r 2 P I α(r)( ) ∂j0 (Gr)Y lm (˜θ,∂h ˜φ) (198)uv) ∫ ∞dr r 2 Pα I (r) j l(Gr)0( )] ∂jl (Gr)∂h uv. (199)The success of pseudopotentials in density functional calculations relies on twoassumptions. The transferability of the core electrons to different environments andthe linearization of the exchange and correlation energy. The second assumption isonly valid if the frozen core electrons and the valence state do not overlap. However,if there is significant overlap between core and valence densities, the linearizationwill lead to reduced transferability and systematic errors. The most straightforwardremedy is to include “semi–core states” in addition to the valence shell, i.e. onemore inner shell (which is from a chemical viewpoint an inert “core level”) is treatedexplicitely. This approach, however, leads to quite hard pseudopotentials which callfor large plane wave cutoffs. Alternatively, it was proposed to treat the non–linearparts of the exchange and correlation energy E xc explicitely 374 . This idea doesnot lead to an increase of the cutoff but ameliorates the above–mentioned problemsquite a bit. To achieve this, E xc is calculated not from the valence density n(R)alone, but from a modified densityñ(R) = n(R) + ñ core (R) , (200)where ñ core (R) denotes a density that is equal to the core density of the atomicreference state in the region of overlap with the valence densityñ core (r) = n core (r) if r > r 0 ; (201)with the vanishing valence density inside r 0 . Close to the nuclei a model densityis chosen in order to reduce the cutoff for the plane wave expansion. Finally, thetwo densities and their derivatives are matched at r 0 . This procedure leads to amodified total energy in Eq. (176), where E xc is replace byE xc = E xc (n + ñ core ) , (202)61