Ab initio molecular dynamics: Theory and Implementation

where j l are spherical Bessel functions of the first kind. The local pseudopotentialand the projectors of the nonlocal part in Fourier space are given by∫ ∞∆V local (G) = 4π dr r 2 ∆V local (r)j 0 (Gr) (140)Ω 0P k (G) = √ 4π ∫ ∞(−i) l dr r 2 P k (r) j l (Gr) Y lm (˜θ, ˜φ) , (141)Ω0where lm are angular momentum quantum numbers associated with projector α.3.2 Electrostatic Energy3.2.1 General ConceptsThe electrostatic energy of a system of nuclear charges Z I at positions R I andan electronic charge distribution n(r) consists of three parts: the Hartree energyof the electrons, the interaction energy of the electrons with the nuclei and theinternuclear interactionsE ES = 1 ∫ ∫dr dr ′ n(r)n(r′ )2|r − r ′ |+ ∑ ∫dr Vcore(r)n(r) I + 1 ∑ Z I Z J2 |R I − R J |II≠J. (142)The Ewald method (see e.g. Ref. 12 ) can be used to avoid singularities in theindividual terms when the system size is infinite. In order to achieve this a Gaussiancore charge distribution associated with each nuclei is defined[n I c(r) = −Z ( ) ] 2I r − RI(R c π−3/2 exp −I )3R c I. (143)It is convenient at this point to make use of the arbitrariness in the definition of thecore potential and define it to be the potential of the Gaussian charge distributionof Eq. (143)∫Vcore(r) I =dr ′ nI c(r ′ )|r − r ′ | = − Z [ ]I |r −|r − R I | erf RI |R c I, (144)where erf is the error function. The interaction energy of this Gaussian charge49