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C566lecture9_000

# C566lecture9_000 - C566 Master Lecture Notes Weeks 5 8...

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C566 Master Lecture Notes Weeks 5 - 8 Vibrational wavefunction now can be written out (to a reasonable approximation, at low vibrational energies) in terms of the separated coordinates, vib (Q i ) = 1 (Q 1 ) 2 (Q 2 ) 3N-6 (Q 3N-6 ) E vib = E 1 + E 2 + …+ E 3N-6 = h 1 ( 1 + 1/2) + h 2 ( 2 + 1/2)+ … + h 3N 6 ( 3N 6 + 1/2) Need to now use this, and what we know about molecular symmetry, to devise our spectroscopic selection rules. Can now characterize the species of various molecular vibrational coordinates. C 2h I C 2 i h A g 1 1 1 1 R z xx , yy , zz , xy B g 1 -1 1 -1 R x , R y xz , yz A u 1 1 -1 -1 T z B u 1 -1 -1 1 T x , T y The C 2 axis corresponds to the z axis in our coordinate system, and for glyoxal (below), the C-C axis is the x-axis. For fun, consider rotation about the x-axis (R x ) = B g C C O O H H R x _ _

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( n ) = A u ( m ) = B u Several Observations: (a) Symmetry species (A g , A u , B g , B u ) are vectors in symmetry operation space e.g., A u = (1, 1, 1, 1) B g = (1, 1, 1, 1) (b) All are mutually orthogonal; A g B g = 1 1 + 1  1 + 1 1 + 1  1 = 0 A g A g = 1 1+1 1+1 1+1 1 = 4 = the order of the point group (c) Direct product: A g B g = (1,1,1,1) (1,-1,1,-1)=(1,-1,1,-1) = B g B g B u = (1,-1,1,-1) (1,-1,-1,1) = (1,1, -1,-1) = A u B u B u = A g Direct products will be used to determine the species of combinations of properties, such as combinations of vibrational modes, electronic-vibration combinations, as well as the overall electronic state of a molecule built up from the occupation of MO’s. C C O O H H C C O O H H _ _