Ab initio molecular dynamics: Theory and Implementation

distributions is now added and subtracted from the total electrostatic energyE ES = 1 ∫ ∫dr dr ′ n(r)n(r′ )2|r − r ′ |+ 1 ∫ ∫dr dr ′ n c(r)n c (r ′ )2|r − r ′ |∫ ∫+ dr dr ′ n c(r)n(r ′ )|r − r ′ |+ ∑ ∫dr Vcore I (r)n(r) + 1 ∑ Z I Z J2 |R I − R J |II≠J− 1 ∫ ∫dr dr ′ n c(r)n c (r ′ )2|r − r ′ , (145)|where n c (r) = ∑ I nI c (r). The first four terms can be combined to the electrostaticenergy of a total charge distribution n tot (r) = n(r) + n c (r). The remaining termsare rewritten as a double sum over nuclei and a sum over self–energy terms of theGaussian charge distributionsE ES = 1 ∫ ∫dr dr ′ n tot(r)n tot (r ′ )2|r − r ′ |⎡ ⎤+ 1 ∑ Z I Z J2 |R I − R J | erfc ⎣ |R I − R J |√ ⎦ − ∑2 + R c J2II≠Jwhere erfc denotes the complementary error function.3.2.2 Periodic SystemsR c I1√2πZ 2 IR c I, (146)For a periodically repeated system the total energy per unit cell is derived fromthe above expression by using the solution to Poisson’s equation in Fourier spacefor the first term and make use of the quick convergence of the second term in realspace. The total charge is expanded in plane waves with expansion coefficientsn tot (G) = n(G) + ∑ n I c(G)S I (G) (147)I= n(G) − 1 ∑ Z I√ exp[− 1 ]Ω 4π 2 G2 R c I 2 S I (G) . (148)This leads to the electrostatic energy for a periodic systemE ES = 2π Ω ∑ G≠0I|n tot (G)| 2G 2 + E ovrl − E self , (149)where⎡ ⎤E ovrl = ∑ ′ ∑ Z I Z J|R I − R J − L| erfc ⎣ |R I − R J − L|√ ⎦ (150)2 + R c J2I,JLR c I50