29_580ln_fa08

# 29_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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5.80 Lecture # 29 Fall, 2008 Page 1 of 8 pages Lecture # 29 : A Sprint Through Group Theory Bernath 2.3-4, 3.3-8, 4.3-6. I'll touch on highlights Symmetry odd vs. even integrands 0 integrals selection rules for matrix representation of any operator * transition moment * H block diagonalization generation of symmetry coordinates how to deal with totality of exact O , H = 0 approx. O , H ° = 0 convenient C 2 , H ROT a,b,c = 0 symmetries Chapter 2: Molecular Symmetry rotation C n (axis) rotation by 2 π n radians about (specified) axis ( C n C n = C n 2 etc.) reflection σ (plane) reflect thru plane σ v vertical (includes highest order C n axis) σ h horizontal ( to highest order C n axis) σ d dihedral (also vertical, bisects angle between 2 C 2 axes to C n ) contrast to I - inversion in lab (parity) inversion in body ˆ i = C 2 σ h inversion (C 2 axis to plane of σ h ) improper rotation S n = σ h C n = C n σ h (C n axis to plane of σ h ) [i = S 2 ] identity E do nothing Groups: Closure Associative Multiplication Identity Element Inverse of every element R. Rigid isolated molecules — point groups — all symmetry elements intersect at one point [distinct from translational symmetries — periodic lattices CNPI - nonrigid molecules (Complete Nuclear Permutation-Inversion) MS - (Molecular Symmetry Group) subgroup of CNPI, isomorphic with point group, but more insightful (especially when dealing with transitions between different point- group structures)] Point Group notation C s , C i , C n , D n , C nv , C nh , D nh , D nd 1 plane inversion nC 2 C n n σ v C n + σ h C n + nC 2 + σ h C n + nC 2 + σ d
Fall, 2008 Page 2 of 8 pages S n T d O h I h tetrahedral octahedral icosohedral [Flow Chart: Figure 2.11, page 52 of Bernath] K h spherical Bernath Chapter 3. Matrix Representations x which means r = x ˆ i + y ˆ j + zk ˆ = x i e ˆ i r = y z i e ˆ 1 e ˆ 2 e ˆ 3 convenient notation x 1 x 2 x 3 Apply symmetry operator, R , to coordinates of an atom (“Active”) =

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## This note was uploaded on 11/28/2011 for the course CHEM 5.74 taught by Professor Robertfield during the Spring '04 term at MIT.

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29_580ln_fa08 - MIT OpenCourseWare http/ocw.mit.edu 5.80...

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