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 Schrodinger 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 Schrodinger Equation for a Molecular System H total ( r,Q ) t ( r,Q ) = E t 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 ) 2 summationdisplay 2 represents the electronic kinetic energy 2 m e r 2 i i T N ( Q ) 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 ) + U ( r,Q ) = H elec ( r,Q ) + U ( r,Q ) where U ( r,Q ) = U ( r,Q ) + 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 + + ... 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 Schrodinger Equations (i) The Dynamical equation for H elec ( r,Q ) { the Born representation } H elec ( r,Q ) i ( r,Q ) = i ( Q ) i ( r,Q ) [ T e ( r ) + U ( r,Q )] i ( r,Q ) = i ( Q ) i ( r,Q ) dynamical electronic wavefunctions: i ( r,Q ) Born Space (ii) The static equation for...
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33s_580ln_fa08 - MIT OpenCourseWare http://ocw.mit.edu 5.80...

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