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30_580ln_fa08 - MIT OpenCourseWare http/ocw.mit.edu 5.80...

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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 .
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Fall, 2008 Page 1 Lecture #3 0 : What is in a Character Table and How do we use it? Last time matrix representations of symmetry operators representations of group — same multiplication table as symmetry operators characters of matrix representations (all we need for most applications) generate representation from convenient set of objects (basis vectors) GOT character table irreducible representations generalization of odd/even notation symmetry label for multi-dimensional GOAL integral with several non-commuting symmetry operators reduction of reducible representations generate and reduce reducible representations how do we get and use the fancy labels to the right of characters (a, b, c) (x, y, z) [conventions for x, y, z, I a I b I c for a, b, c] selection rules: pure rotation and rotation-vibration and Raman. nature of various types of vibration. Example: D 3h totally symmetric E 2C 3 (z) 3C 2 ( ) σ h (xy) 2S 3 (z) 3 σ v 1 1 1 1 1 1 A 1 1 1 1 1 1 –1 E A 2 2 –1 0 2 –1 0 1 1 1 –1 –1 –1 A 1 ′′ 1 1 –1 –1 –1 1 E A ′′ 2 2 –1 0 –2 1 0 ( rotational level symmetries and perturbations ) Rotations, Translations, IR selection rules, p–orbitals electronic selection rules (magnetic dipole) R z (x,y) z (R x ,R y ) Polarizability, Raman Selection Rules, d–orbitals x 2 + y 2 , z 2 (x 2 – y 2 , xy) (xy, yz) order of group g = 12 = ν n 2 ν (n ν is order of ν -th irreducible representation) equal to number of classes: 1 + 2 + 3 + 1 + 2 + 3 R z “belongs to” A 2 , z (or T z ) belongs to A ′′ 2 5.80 Lecture #3 0
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Fall, 2008 Page 2 Use picture to generate representation z T z E C 3 C 2 σ h S 3 σ v R z 1 1 –1 1 +1 –1 A 2 y R z T z 1 1 –1 –1 –1 1 A 2 show with cartoons why R z A 2 from recall , ″↔σ h these characters x 1,2 ↔σ v (x,y) means symmetry operation transforms x into y (must generate 2D representation using x and y) Selection rules: integrand must contain totally symmetric representation. ψ i O p ψ f d τ 0 Direct Product: Γ ψ i ) ⊗ Γ O ) must include Γ ( ψ f ) because direct product of any irreducible ( ( p representation with itself contains the totally symmetric representation.
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