Ab initio molecular dynamics: Theory and Implementation

3.4.3 Gradient for Nuclear PositionsThe derivative of the total energy with respect to nuclear positions is needed forstructure optimization and in molecular dynamics, that is∂E total= ∂EPP local+ ∂EPP∂R I,s ∂R I,snonlocal∂R I,s+ ∂E ES∂R I,s, (183)as the kinetic energy E kin and the exchange and correlation energy E xc do notdepend directly on the atomic positions, the relevant parts are∂E PPlocal∂R I,s= −Ω ∑ G∂E PPnonlocal∂R I,s= ∑ if i∑iG s ∆V Ilocal(G) S I (G) n ⋆ (G) (184)α,β∈I{ ( )(F )α ⋆I,i hI ∂Fβ (I,i ∂Fα ) ⋆}I,iαβ + h I∂R I,s ∂Rα,βF β I,i(185)I,s∂E ES= − Ω ∑ n ⋆iGtot(G)s∂R I,s G 2 n I c(G) S I (G) + ∂E ovrl. (186)∂R I,sG≠0The contribution of the projectors of the non-local pseudopotentials is calculatedfrom∂FI,iα = − 1 ∑√ iG s P∂Rα(G) I S I (G) c ⋆ i (G,k) . (187)I,s ΩR c IGFinally, the real space part contribution of the Ewald sum is⎧⎡ ⎤∂E ovrl= ∑ ′ ∑ ⎨Z I Z J∂R I,s ⎩|R I − R J − L| 3erfc ⎣ |R I − R J − L|√ ⎦J LR c2 I+ R c J2⎡⎤⎫+ √ 2 1 Z I Z J√ π 2 + R c |R I − R J − L| exp ⎣− |R I − R J − L| 2 ⎬√ ⎦J22 2 + R c ⎭J2×(R I,s − R J,s − L s ) . (188)The self energy E self is independent of the atomic positions and does not contributeto the forces.R c I3.4.4 Internal Stress TensorFor calculations where the supercell is changed (e.g. the combination of the Car–Parrinello method with the Parrinello–Rahman approach 201,55 ) the electronic internalstress tensor is required. The electronic part of the internal stress tensor isdefined as 440,441 (see also Sect. 4.2.3)Π uv = − 1 Ω∑s∂E total∂h ush t sv . (189)59