23_580ln_fa08

23_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 #23 Fall, 2008 Page 1 of 7 pages Lecture #23: Asymmetric Top R = I ± 1 H ROT = 2 T I –1 T J = R + L + S + inverse inertia matrix see Watson, 1 H ROT Mol. Phys. 15 , 479 (1968) forget about L, S, for now a-axis is “light” c-axis is “heavy” = 2 (J – L – S – ) I –1 (J – L – S – ) H ROT 2 2 2 = AJ a + BJ b + CJ c A B C A/ cm ² 1 = h 1 I = ³ m i ( b 2 + c 2 ) c 8 ´ 2 I a i a etc. B, C b,c are perpendicular distances from named axis prolate top I b = I c E ROT p )K 2 ( J,K ) = BJ(J + 1) + (A ² B > 0 oblate top I a E ROT )K 2 = I b o ( ) = BJ(J + + (C ² B < 0 TODAY: Asymmetric Top qualitative pattern for Energy levels 1. Correlation Diagram notation J K a K c and J 2. H ROT in | JKM µ basis set ±K = 0 and ±K = ²2 matrix elements 2B ² A ² C 3. perturbation theory and ±² asymmetry parameter A ² C 4. Wang factorization: 4 symmetry species 5. Townes and Schawlow Tables for Asymmetric Top Levels What do we expect energy levels for an asymmetric top to look like? Intermediate between prolate and oblate limits. Correlation diagram is based on non-crossing rule. Imagine a continuous transformation of a molecule from prolate to oblate limit. Levels belonging to different values of a rigorously good quantum number can cross, all others cannot. J is good, K is not good.
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large BJ(J + 1) +(A – B)K 2 1 BJ(J + 1) 2 2 2 0 double degeneracy 1 1 0 1 0 0 0 J K p K 0 energy rank –J J 5.80 Lecture #23 Fall, 2008 Page 2 of 7 pages PROLATE OBLATE near Prolate small –(B – C)K 2 near Oblate smallest asymmetry splitting at max K within each J +2 largest splitting at min | K | > 0 +1 0 0 1 –1 –2 2 +1 0 0 –1 1 double degeneracy 0 0 Levels within a J can’t cross. K > 0 are doubly degenerate and the degeneracy is lifted as soon as top becomes asymmetric. Near the corresponding limit, high K has small “asymmetry splitting” and low K
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23_580ln_fa08 - MIT OpenCourseWare http/ocw.mit.edu 5.80...

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