33s_580ln_fa08

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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 5.80 Small-Molecule Spectroscopy and Dynamics Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms . Lecture # 3 Supplement Based on a lecture written by Professor Patrick H. Vaccaro. Outline (i) “true” Eigenstates: A long, hard climb; (ii) the “total” molecular Hamiltonian and its Schr¨ odinger Equation; (iii) the electronic Schr¨ odinger Equation; (iv) transformation of the molecular Schr¨odinger Equation; (v) the Adiabatic Approximation; (vi) Adiabatic corrections; (vii) Non-Adiabatic corrections; (viii) the transition moment of the A tildewide 1 A 2 ← X tildewide 1 A 1 absorption in H 2 CO: a vibronic coupling model. Image removed due to copyright restrictions. Figure 1: Various routes to approach the exact non-adiabatic wavefunction. From “What Does the Term ‘Vibronic Coupling’ Mean” by T. Azumi and K. Matsuzaki, Photochemistry and Photobiology 25 , 315-326 (1977). 1 3 5.80 Lecture # 3 Supplement Time-Independent Schr¨odinger Equation for a Molecular System H total ( r,Q )Ψ t ( r,Q ) = E t Ψ t ( r,Q ) where H total ( r,Q ) = T e ( r ) + T N ( Q ) + U ( r,Q ) + V ( Q ) “ r ” represents electronic coordinates “ Q ” represents mass-weighted nuclear coordinates describing displacements from a reference configuration “ Q 0 ” T e ( r ) ≈ − ℏ 2 summationdisplay ∂ 2 represents the electronic kinetic energy 2 m e ∂r 2 i i T N ( Q ) ≈ − ℏ 2 summationdisplay ∂ 2 represents the nuclear kinetic energy 2 ∂Q 2 n n U ( r,Q ) represents the Coulombic potential energy V ( Q ) represents the potential energy of the nuclei PROBLEM: Hamiltonian does not permit separation of variables. Therefore, exact solution is not possible. Consider only the terms depending on the electronic coordinates (i.e. the so-called Electronic Hamilto- nian) H elec ( r,Q ) = T e ( r ) + U ( r,Q ) = T e ( r ) + U ( R,Q ) + Δ U ( r,Q ) = H elec ( r,Q ) + Δ U ( r,Q ) where U ( r,Q ) = U ( r,Q ) + Δ U ( r,Q ) H elec ( r,Q ) = T e ( r ) + U ( r,Q ) . Note that: U ( r,Q ) = U ( r,Q ) + summationdisplay bracketleftbigg ∂U ( r,Q ) bracketrightbigg Q n ∂Q n n 1 summationdisplay bracketleftbigg ∂ 2 U ( r,Q ) bracketrightbigg Q n Q m + ... + 2 ∂Q n ∂Q m nm 3 summationdisplay bracketrightbigg summationdisplay bracketrightbigg summationdisplay summationdisplay summationdisplay summationdisplay summationdisplay bracketleftBigg summationdisplay bracketrightBigg summationdisplay Page 3 Consequently: bracketleftbigg ∂U ( r,Q ) bracketleftbigg ∂ 2 U ( r,Q ) 1 + + ... Δ U ( r,Q ) ≈ Q n Q n Q m ∂Q n 0 2 ∂Q n ∂Q m n n,m Define two types of Electronic Schr¨odinger Equations (i) The Dynamical equation for H elec ( r,Q ) { the “Born” representation } H elec ( r,Q ) ψ i ( r,Q ) = ǫ i ( Q ) ψ i ( r,Q ) [ T e ( r ) + U ( r,Q )] ψ i ( r,Q ) = ǫ i ( Q ) ψ i ( r,Q ) dynamical electronic wavefunctions: ψ i ( r,Q ) ⇒ Born Space (ii) The static equation for...
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This note was uploaded on 11/28/2011 for the course CHEM 5.74 taught by Professor Robertfield during the Spring '04 term at MIT.

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33s_580ln_fa08 - MIT OpenCourseWare http/ocw.mit.edu 5.80...

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