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Unformatted text preview: Densityfunctional 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, MCCUIUC, 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 HellmannFeynman theorem E ( λ ) = min Ψ  H ( λ )  Ψ Ψ  Ψ = 1 The HellmannFeynman theorem g ( λ ) = min x G [ x, λ ] E ( λ ) = min Ψ  H ( λ )  Ψ Ψ  Ψ = 1 The HellmannFeynman 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 HellmannFeynman 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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This note was uploaded on 12/07/2011 for the course CHEM 350 taught by Professor Duanejohnson during the Summer '06 term at University of Illinois, Urbana Champaign.
 Summer '06
 DuaneJohnson
 Mole, pH

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