27_580ln_fa08

27_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 #27 Fall, 2008 Page 1 of 8 pages Lecture #27: Polyatomic Vibrations III: s-Vectors and H 2 O Last time: F -matrix: too many F ij ’s even at quadratic-only level Internal coordinates: types 3N–6 independent ones constraints * translation * rotation s -vectors α S t s t α * direction of fastest increase * magnitude resulting from unit displacement in optimum direction ρ ρ s t α α α N ( ) = S t { } α = 1 rigid translation ρ α = ε for all α constraint s t α = 0 no center of mass translation α R α × d rigid rotation by d Ω ρ α ) = ( d Ω Ω s t α × R α constraint s t α × R α e = 0 ECKART α (minimizes vibrational angular momentum) If the normal displacements are built from s t α vectors that satisfy these constraints, then, for infinitesimal displacements from equilibrium, there is no rotation. For large displacements, or for small displacements away from a non-equilibrium configuration, there is a small vibrational angular momentum. This definition of vibrations embeds a specific partitioning between rotation and vibration. TODAY: G from s t α ’s Examples of s t α ’s 1. valence bond stretch r 2. valence angle bend φ G matrix using diagrams and tables from WDC pages 304 and 305 H 2 O FG handout G DD recall | S = B | ξ〉 = D | q = D Μ 1/2 | ξ〉 B = DM 1/2 BM 1/2 = D G = DD = BM 1/2 ( M 1/2 ) B = BM 1 B S t ( d Ω ) = d Ω α
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5.80 Lecture #27 Fall, 2008 Page 2 of 8 pages G tt = 1 B ti B * t i i = 1 3N m i definition of | S = B | ξ〉 → ⎛∂ ∂ξ
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27_580ln_fa08 - MIT OpenCourseWare http/ocw.mit.edu 5.80...

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