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Unformatted text preview: C566 Master Lecture Notes Lecture 7 Summary of Molecular Symmetry (1) Symmetry operations were defined (identity, C n rotation, plane of reflection, point of inversion). A molecule possesses a symmetry element IF, after a symmetry operation is done on the molecule, it is indistinguishable from its initial state. (2) Molecules can be categorized by point group- Molecules that possess the same symmetry elements fall in the same point group. (3) For each point group, character tables have been constructed to summarize the effects of symmetry operations on other molecular properties (translations, rotations, vibrations, electronics). These will ultimately be used to identify whether various spectroscopic transitions will be allowed or forbidden, based on symmetry. From Lecture 6 notes: 2 3 3 2 2 2 2 1 1 2 1 v M v M v M T ] ) ( ) ( [ 2 1 2 2 3 3 2 2 1 2 2 1 x x k x x k V Euler-Lagrange equation of motion: i i i i v T dt d dx V v L dt d dx L j j ij i x A x V j j ij i v m v T Simplified problem: CO 2 , M 1 = M 3 = m M 2 = M k 1-2 = k 2-3 = k M 1 M 2 M 3 x 1 k 1-2 x 2 k 2-3 x 3 Note that x 2 is in both terms of the potential energy function. Remove translational motion by setting mx 1 + Mx 2 + mx 3 = 0, and all oscillatory motion will be about the center of mass: x 2 = (x 1 + x 3 ) m/M The kinetic term then becomes: ) 2 ( 2 1 3 1 2 3 2 1 2 2 3 2 1 v v v v M m mv mv T This dynamic coupling term, v 1 v 3 , makes solving the classical equations of motion difficult. difficult....
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- Spring '11