26_580ln_fa08

26_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 #26 Fall, 2008 Page 1 of 7 pages Lecture #26: Polyatomic Vibrations II: s-Vectors, G-matrix, and Eckart Condition Last time: ξ , q, S, Q Q = L 1 DM 1/2 ξ  q  S G –1 ( D –1 ) D –1 (because q = = D 1 S and D q S ) 2T = S G 1 S 2 V 2V = S F S F ij S i S j 0 L ({ S }, { S }) = T({ S }) – V({ S }) d L L secular equation dT S j S j = 0 F − λ G 1 = 0 assuming that S j (t) = A j cos ( λ 1/2 t + ε ) all j = 1, 2, … 3N–6 i.e. that all internal coordinates oscillate at same frequency and relative phase (but with λ 1/2 different amplitudes) ν = 2 π 3N–6 possible different values of λ obtained from (3N – 6) × (3N – 6) secular equation. TODAY: Finish discussion of * secular equation * forms of various transformations * descriptions of each normal coordinate s -Vectors * definition and properties * imposition of translational and rotational constraints * derivation of G from s -Vectors NEXT TIME - SOME EXAMPLES OF s -VECTOR CALCULATIONS Last time, we derived 0 = | F λ G –1 | left multiply by | G | 0 = | GF λ 1 | must diagonalize GF to get 3N – 6 eigenvalues { λ k } Similarity (not unitary) transformation
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5.80 Lecture #26 Fall, 2008 Page 2 of 7 pages λ 1 0 0 0 0 0 0 λ 3N 6 L 1 GFL = = (Although G and F are both real and symmetric, GF is not symmetric, so the diagonalizing transformation is not unitary: L –1 L ) For example, the product of two real and symmetric matrices a b ⎞ ⎛ c d ⎞ ⎛ ac + bd ad bc b a d c bc ad bd + ac is a matrix that is not symmetric. What do we already know about L from prior requirements that T and V must be put into separable forms? Want 2 T = = Q k 2 where Q = L 1 S Q Q k 2 T = q q = S ( D 1 ) D 1 S = S G 1 S [ G 1 ( D 1 ) D 1 ] We had, previously = Q L G 1 L Q So L G –1 L = 1 is required thus L –1 = L G –1 is needed to keep
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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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26_580ln_fa08 - MIT OpenCourseWare http:/ocw.mit.edu 5.80...

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