md2011-06-note_Part_2

md2011-06-note_Part_2 - D C M D I The Hamiltonian can be...

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Unformatted text preview: D C M D I The Hamiltonian can be written as H = - X I h 2 2 M I ∇ 2 I- X i h 2 2 m e ∇ 2 i + 1 4 πε X i < j e 2 | r i- r j |- X I , i e 2 Z I | R I- r i | + e 2 Z I Z J | R I- R J | = - X I h 2 2 M I ∇ 2 I + H e ( { r i } , { R I } ) (2) for the electronic { r i } and nuclear { R I } degrees of freedom. I Here, H e is assumed to be the Hamiltonian for the electronic system when the nuclei are stationary . Notes I Next, let’s assume that the exact solution of the corresponding time-independent electronic Schrödinger equation is H e ( { r i } ; { R I } ) Ψ k = E k ( { R I } ) Ψ k ( { r i } ; { R I } ) , (3) and is known for all possible positions of the nuclei. I Eigenfunctions are assumed to be normalized Z Ψ * k ( { r i } ; { R I } ) Ψ l ( { r i } ; { R I } ) d r = δ kl . (4) I Now it’s possible to expand the total wave function for time-dependent Schrödinger equation Φ ( { r i } ; { R I } ; t ) = ∞ X l = Ψ l ( { r i } ; { R I } ) χ l ( { R I } ; t ) . (5) Notes I...
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This note was uploaded on 02/14/2012 for the course CSE 6590 taught by Professor Kotakoski during the Winter '12 term at York University.

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md2011-06-note_Part_2 - D C M D I The Hamiltonian can be...

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