25_580ln_fa08

25_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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± ± ² ³ ´ ´ ´ µ · ± ± ± ± ± ± ³ µ ´ ´ ´ ´ ´ ´ = ± ² ³ ³ ³ ³ ³ ³ ³ ³ ³ ´ µ ± ² ³ ³ ³ ´ µ 5.80 Lecture #25 Fall, 2008 Page 1 of 8 pages Lecture #25: Polyatomic Vibrations: Normal Mode Calculations Pure rotation spectrum ¸ A, B, C ¸ some information about molecular shape Vibrational spectrum: Qualitative Quantitative GROUP THEORY NORMAL MODES 1. What point group? 1. Derive force constants from spectrum and 2. How many vibrations of each symmetry geometry. species? 2. Predict spectrum from geometry and force 3. bend vs. stretch character for modes of constants. each symmetry species 3. Isotope effects. 4. which modes are IR and Raman active? 4. Pictures of normal modes in terms of 5. band types (a–, b–, c–): rotational contours internal coordinate displacements. predicted. 5. beyond normal modes. A. perturbations B. IVR Wilson's FG matrix Method. Approximately 3 lectures. Not treated in Bernath, i.e., Bernath does what ab initio calculations do - eigenvalues of mass weighted Cartesian f matrix. Lecture #1 (#25): Formal Derivation of GF matrix secular equation as condition for the existence of a transformation from Cartesian displacements to normal coordinates that permits 1 1 2 k 2 k oscillators) Lecture #2 (#26): How do we actually obtain the G matrix? Eckart condition (a compromise): Vibration-rotation separation. Lecture #3 (#27): Examples. Beyond the Harmonic Approximation. Lots of matrices and transformations - introduce all of the actors now! 2 Q k 2 Q ± k H to be written as H = (sum of separate harmonic ¹ k + COORDINATES 3N CARTESIAN x 1 ² x 1 e z N ² z N e DISPLACEMENTS ± ± ± · 3N = ² y 1 e 1 y 1 ² z 1 e z 1 BODY 3N MASS WEIGHTED CARTESIAN DISPLACEMENTS 1/2 m 1 groups of 3 m 1 m 1 0 m 2 1 q = M 1/2 0 m N 3N
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± ² ³ ´ ± ² ³ 5.80 Lecture #25 Fall, 2008 Page 2 of 8 pages remove C. M. translation and rotation 3N–6 INTERNAL DISPLACEMENTS S = Dq = B *** (to be defined later)*** see WILSON-DECIUS-CROSS bond stretch, bend, torsion 3N–6 NORMAL DISPLACEMENTS Q ± L –1 S today we will show formally the condition for the existence of L .
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25_580ln_fa08 - MIT OpenCourseWare http:/ocw.mit.edu 5.80...

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