3.4.3 Gradient for Nuclear PositionsThe derivative of the total energy with respect to nuclear positions is needed forstructure optimization <strong>and</strong> in <strong>molecular</strong> <strong>dynamics</strong>, 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 <strong>and</strong> the exchange <strong>and</strong> 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 <strong>and</strong> 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
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
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John von Neumann Institute for Comp
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The goal of this section is to deri
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and electrons r = {r i } can be wri
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The effective classical partition f
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up e.g. in Refs. 132,37,596,597,428
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The eigenvalues of A when multiplie
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frequency of the electronic degrees
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5.2 Solids, Polymers, and Materials
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the penetration of the oxidation la
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in terms of their electronic struct
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culations on very accurate global p
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AcknowledgmentsOur knowledge on ab
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57. M. Bernasconi, G. L. Chiarotti,
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Superiore di Studi Avanzati (SISSA)
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175. E. Ermakova, J. Solca, H. Hube
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244. H. Goldstein, Klassische Mecha
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313. T. Ikeda, M. Sprik, K. Terakur
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384. N. A. Marks, D. R. McKenzie, B
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442. S. Nosé and M. L. Klein, Mol.
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502. L. M. Ramaniah, M. Bernasconi,
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562. F. Shimojo, K. Hoshino, and Y.
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620. A. Tongraar, K. R. Liedl, and
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682. R. M. Wentzcovitch, Phys. Rev.