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Baroni_Structure_Dynamics - Molecular structure and...

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Molecular structure and dynamics with DFT 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 the plane-wave pseudo-potential way
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Ab initio simulations
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Ab initio simulations i ¯ h Φ ( r, R ; t ) t = - ¯ h 2 2 M 2 R 2 I - ¯ h 2 2 m 2 r 2 i + V ( r, R ) Φ ( r, R ; t )
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Ab initio simulations i ¯ h Φ ( r, R ; t ) t = - ¯ h 2 2 M 2 R 2 I - ¯ h 2 2 m 2 r 2 i + V ( r, R ) Φ ( r, R ; t ) The Born-Oppenheimer approximation (M m) M ¨ R I = - E ( R ) R I - ¯ h 2 2 m 2 r 2 i + V ( r, R ) Ψ ( r | R ) = E ( R ) Ψ ( r | R )
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V ( r, R ) = e 2 2 Z I Z J | R I - R J | - Z I e 2 | r i - R I | + e 2 2 1 | r i - r j | Density functional theory
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V ( r, R ) = e 2 2 Z I Z J | R I - R J | - Z I e 2 | r i - R I | + e 2 2 1 | r i - r j | Density functional theory
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V ( r, R ) = e 2 2 Z I Z J | R I - R J | - Z I e 2 | r i - R I | + e 2 2 1 | r i - r j | Density functional theory V ( r, R ) e 2 2 Z I Z J | R I - R J | + v [ ρ ( r )] ( r ) DFT
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ρ ( r ) = v | ψ v ( r ) | 2 - 2 2 m 2 r 2 + v [ ρ ( r )] ( r ) ψ v ( r ) = v ψ v ( r ) V ( r, R ) = e 2 2 Z I Z J | R I - R J | - Z I e 2 | r i - R I | + e 2 2 1 | r i - r j | Density functional theory V ( r, R ) e 2 2 Z I Z J | R I - R J | + v [ ρ ( r )] ( r ) DFT
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Kohn-Sham Hamiltonian ρ ( r ) = v | ψ v ( r ) | 2 - 2 2 m 2 r 2 + v [ ρ ( r )] ( r ) ψ v ( r ) = v ψ v ( r ) V ( r, R ) = e 2 2 Z I Z J | R I - R J | - Z I e 2 | r i - R I | + e 2 2 1 | r i - r j | Density functional theory V ( r, R ) e 2 2 Z I Z J | R I - R J | + v [ ρ ( r )] ( r ) DFT
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E ( R ) = min E [ { ψ } , R ] ψ * u ( r ) ψ v ( r ) dr = δ uv Kohn-Sham equations from functional minimization E [ { ψ } , R ] = - 2 2 m v ψ * v ( r ) 2 ψ v ( r ) r 2 dr + V ( r, R ) ρ ( r ) dr + e 2 2 ρ ( r ) ρ ( r ) | r - r | drdr + E xc [ ρ ( r )]
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H KS ψ v = v ψ v Kohn & Sham E ( R ) = min E [ { ψ } , R ] ψ * u ( r ) ψ v ( r ) dr = δ uv Kohn-Sham equations from functional minimization E [ { ψ } , R ] = - 2 2 m v ψ * v ( r ) 2 ψ v ( r ) r 2 dr + V ( r, R ) ρ ( r ) dr + e 2 2 ρ ( r ) ρ ( r ) | r - r | drdr + E xc [ ρ ( r )]
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H KS ψ v = v ψ v Kohn & Sham E ( R ) = min E [ { ψ } , R ] ψ * u ( r ) ψ v ( r ) dr = δ uv Kohn-Sham equations from functional minimization Helmann & Feynman E ( R ) R I = v ( r, R ) R I n ( r ) dr E [ { ψ } , R ] = - 2 2 m v ψ * v ( r ) 2 ψ v ( r ) r 2 dr + V ( r, R ) ρ ( r ) dr + e 2 2 ρ ( r ) ρ ( r ) | r - r | drdr + E xc [ ρ ( r )]
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Solving the Kohn-Sham equations - 2 2 m 2 + V ( r ) ψ n ( r ) = n ψ n ( r )
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ψ n ( r ) = i c ( i, n ) φ i ( r ) Solving the Kohn-Sham equations - 2 2 m 2 + V ( r ) ψ n ( r ) = n ψ n ( r )
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ψ n ( r ) = i c ( i, n ) φ i ( r ) Solving the Kohn-Sham equations - 2 2 m 2 + V ( r ) ψ n ( r ) = n ψ n ( r ) ψ n ( r ) c ( n, i )
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ψ n ( r ) = i c ( i, n ) φ i ( r ) Solving the Kohn-Sham equations - 2 2 m 2 + V ( r ) ψ n ( r ) = n ψ n ( r ) ψ n ( r ) c ( n, i ) representation & storage
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matrix eigenvalue problem j h ( i, j ) c ( j, n ) = n c ( i, n ) ψ n ( r ) = i c ( i, n ) φ i ( r ) Solving the Kohn-Sham equations - 2 2 m 2 + V ( r ) ψ n ( r ) = n ψ n ( r ) ψ n ( r ) c ( n, i ) representation & storage
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Requirements
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Requirements (effective) completeness easily checked and systematically improved
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Requirements (effective) completeness easily checked and systematically improved matrix elements easy to calculate and/or H ψ products easily calculated on the fly
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