Baroni_DFPT - Density-functional perturbation theory...

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Unformatted text preview: Density-functional perturbation theory Stefano Baroni Scuola Internazionale Superiore di Studi Avanzati & DEMOCRITOS National Simulation Center Trieste - Italy Summer school on Ab initio molecular dynamics methods in chemistry, MCC-UIUC, 2006 forces, response functions, phonons, and all that Energy derivatives H = H + i λ i v i E [ λ ] = E- i f i λ i + 1 2 ij h ij λ i λ j + · · · Energy derivatives H = H + i λ i v i E [ λ ] = E- i f i λ i + 1 2 ij h ij λ i λ j + · · · Energy derivatives H = H + i λ i v i • structural optimization & molecular dynamics E [ λ ] = E- i f i λ i + 1 2 ij h ij λ i λ j + · · · Energy derivatives • (static) response functions elastic constants dielectric tensor piezoelectric tensor Born effective charges . . . H = H + i λ i v i • structural optimization & molecular dynamics E [ λ ] = E- i f i λ i + 1 2 ij h ij λ i λ j + · · · Energy derivatives • (static) response functions elastic constants dielectric tensor piezoelectric tensor Born effective charges . . . H = H + i λ i v i • vibrational modes in the adiabatic approximaton • structural optimization & molecular dynamics Lattice dynamics R’ R V ( r ) = V ( r ) = R v ( r- R ) E = E Lattice dynamics V ( r ) = V ( r ) + R u ( R ) · ∂ v ( r- R ) ∂ R E = E + 1 2 R , R u ( R ) · ∂ 2 E ∂ u ( R ) ∂ u ( R ) · u ( R ) + · · · R’ R u(R) u(R’) V ( r ) = V ( r ) + R u ( R ) · ∂ v ( r- R ) ∂ R Energy derivatives & perturbation theory E [ λ ] = E- i f i λ i + 1 2 ij h ij λ i λ j + · · · H = H + i λ i v i Energy derivatives & perturbation theory E [ λ ] = E- i f i λ i + 1 2 ij h ij λ i λ j + · · · H = H + i λ i v i f i =- ∂ E ∂λ i λ =0 =- Ψ | v i | Ψ Energy derivatives & perturbation theory E [ λ ] = E- i f i λ i + 1 2 ij h ij λ i λ j + · · · H = H + i λ i v i f i =- ∂ E ∂λ i λ =0 =- Ψ | v i | Ψ h ij = ∂ 2 E ∂λ i ∂λ j λ =0 = 2 n Ψ | v i | Ψ n Ψ n | v j | Ψ- n h ij = ∂ 2 E ∂λ i ∂λ j λ =0 = 2 n Ψ | v i | Ψ n Ψ n | v j | Ψ- n = 2 Ψ | v i | Ψ j Energy derivatives & perturbation theory E [ λ ] = E- i f i λ i + 1 2 ij h ij λ i λ j + · · · H = H + i λ i v i f i =- ∂ E ∂λ i λ =0 =- Ψ | v i | Ψ h ij = ∂ 2 E ∂λ i ∂λ j λ =0 = 2 n Ψ | v i | Ψ n Ψ n | v j | Ψ- n = 2 Ψ | v i | Ψ j = 2 Ψ i | v j | Ψ Energy derivatives & perturbation theory E [ λ ] = E- i f i λ i + 1 2 ij h ij λ i λ j + · · · H = H + i λ i v i f i =- ∂ E ∂λ i λ =0 =- Ψ | v i | Ψ E ( λ ) = min Ψ | H ( λ ) | Ψ Ψ | Ψ = 1 The Hellmann-Feynman theorem E ( λ ) = min Ψ | H ( λ ) | Ψ Ψ | Ψ = 1 The Hellmann-Feynman theorem g ( λ ) = min x G [ x, λ ] E ( λ ) = min Ψ | H ( λ ) | Ψ Ψ | Ψ = 1 The Hellmann-Feynman theorem g ( λ ) = min x G [ x, λ ] ∂ G ∂ x x = x ( λ ) = 0 E ( λ ) = min Ψ | H ( λ ) | Ψ Ψ | Ψ = 1 The Hellmann-Feynman theorem g ( λ ) = min x G [ x, λ ] g ( λ ) = G [ x ( λ ) , λ ] ∂ G ∂ x x = x ( λ ) = 0 E ( λ...
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Baroni_DFPT - Density-functional perturbation theory...

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