Ab initio molecular dynamics: Theory and Implementation
Ab initio molecular dynamics: Theory and Implementation
Ab initio molecular dynamics: Theory and Implementation
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distributions is now added <strong>and</strong> 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 <strong>and</strong> 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 <strong>and</strong> make use of the quick convergence of the second term in realspace. The total charge is exp<strong>and</strong>ed 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