C566lecture18_000

C566lecture18_000 - C566 Master Lecture Notes Lecture 18...

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1 C566 Master Lecture Notes Lecture 18 Electronic spectroscopy SELECTION RULES… continued Wavefunction, assuming BO approximation is valid, rotations and vibrations are not coupled mol trans ( R com ) vib ( R )Y J,M ( , ) el ( r 1 , r 2 ,… r N ; R) The translational bit- Free particle wavefunction (Ae ikx + Be -ikx ) which we’ll neglect from now on. The vibrational wavefunction is our harmonic oscillator (function of R ) The rotational wavefunction is the spherical harmonics ( , ) The electronic bit is an N-electron wavefunction, based on a particular bondlength, R Interaction between light and matter is strongest via dipole interaction with the field, and the dipole of the molecule is: = Z 1 e R 1 + Z 2 e R 2 e r j = n + el Transition dipole integral: M =  mol | n | mol  +  mol | el | mol  =  vib Y J ,M el | n | vib Y J ,M el  +  vib Y J ,M el | el | vib Y J ,M el  1. First term:  mol | n | mol  The n operator only operates on nuclear coordinates and therefore equals:  el | el  vib Y J ,M | n | vib Y J ,M = 0 (why?) 2. Second term ( el ):  * vib Y* J ,M * el | el | vib Y J ,M el  =  * vib | vib  Y* J ,M | Y J ,M  * el | el ( r j )| el 
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2 Note that in the integral over the nuclear motion wavefuctions, d R = R 2 sin dR d d , so it is proper to renormalize the vibrational wavefunction, now being rotated about a sphere, to vib /R. The rotational portion: The whole frame of the molecule is rotating, so electron positions are rotating with the molecule ( el ). The position portion of the electronic contribution to the dipole moment can be represented by the same spherical harmonics, and we have J = 1, and (usually) 0. The vibrational component- distortion of the internuclear coordinate does not
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C566lecture18_000 - C566 Master Lecture Notes Lecture 18...

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