Ab initio molecular dynamics: Theory and Implementation

An important identity for the derivation of the stress tensor is∂Ω∂h uv= Ω(h t ) −1uv . (190)The derivatives of the total energy with respect to the components of the cell matrixh can be performed on every part of the total energy individually,∂E total∂h uvlocal∂h uv= ∂E kin+ ∂EPP∂h uv+ ∂EPP nonlocal+ ∂E xc+ ∂E ES. (191)∂h uv ∂h uv ∂h uvUsing Eq. (190) extensively, the derivatives can be calculated for the case of a planewave basis in Fourier space 202 ,∂E kin∂h uv= − ∑ i∂E PPlocal∂h uv∂E PPnonlocal∂h uv= Ω ∑ I= ∑ i∂E xc∂h uv= − ∑ G∑ ∑f i G u G s (h t ) −1sv |c i (G)| 2 (192)∑Gf i∑IGs( ) ∂∆VIlocal(G)S I (G)n ⋆ (G) (193)∂h uv∑α,β∈I{ ( )(F )α ⋆I,i hI ∂FβI,iαβ +∂h uvn ⋆ (G) [V xc (G) − ε xc (G)](h t ) −1uv+ ∑ ∑iG u n ⋆ (G)s G∂E ES= 2π Ω ∑ ∑{− |n tot(G)| 2∂h uv G 2G≠0 s(+ n⋆ tot(G) ntot (G)G 2 G 2I( ∂FαI,i∂h uv) ⋆h I α,βF β I,i}(194)( )∂Fxc (G)(h t ) −1sv (195)∂(∂ s n)δ us) }+ 1 ∑n I2c(G)(R I c) 2 G u G s G u G s (h t ) −1sv+ ∂E ovrl∂h uv. (196)Finally, the derivative of the overlap contribution to the electrostatic energy is⎧⎡ ⎤∂E ovrl= − ∑ ′ ∑ ⎨Z I Z J∂h uv ⎩|R I − R J − L| 3erfc ⎣ |R I − R J − L|√ ⎦I,J LR c2 I+ R c J2⎡⎤⎫2 Z I Z J+√√π R c2 I+ R c |R I − R J − L| exp ⎣− |R I − R J − L| 2 ⎬√ ⎦J22 R c2 I+ R c ⎭J2× ∑ s(R I,u − R J,u − L u )(R I,s − R J,s − L s )(h t ) −1sv . (197)60