md2011-06-note_Part_4

md2011-06-note_Part_4 - If the ground state potential UE is...

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I If the ground state potential U E e is approximated by an analytical function of the form U E e U AP e = I U 1 ( R I ) + X I < J U 2 ( R I , R J ) + X I < J < K U 3 ( R I , R J , R K ) + . . . , (16) we are left with the “standard” MD equation of motion M I ¨ R I ( t ) = - I U AP e ( { R I ( t ) } ) . (17) Notes
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I This highlights the motivation behind using analytical potential functions U AP e in the classical MD approach as parametrizations of the averaged ground state electronic potential. I In contrast, in AIMD methods the electronic potential is always evaluated for the positions of the nuclei either at ground state (Born-Oppenheimer) or as a dynamical system (Ehrenfest). Notes
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E B -O MD Ehrenfest MD I In the case of Ehrenfest MD, the forces are calculated “on-the-fly” as the nuclei are propagated using classical mechanics. I They can be numerically solved simultaneously from the coupled set of quantum/classical equations M I ¨ R I ( t ) = - I Z Ψ * H e Ψ d r = - I h H e i (18) i h ∂Ψ t = " - X i h 2 2 m e 2 i + U n - e ( { r i } , { R I ( t ) } ) # Ψ = H e Ψ . (19) Notes
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