24s_580ln_fa08

24s_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 #24S Fall, 2008 Page 1 of 9 pages Lecture #24 Supplement: Energy levels of a Rigid Rotor * 1 1 1 I c ω c 2 = P a 2 + P b 2 + P c 2 W r = 2 I a ω a 2 + 2 I b ω b 2 + 2 2I a 2I b 2I c W r h = AP a 2 + BP b 2 + CP c 2 where a, b, c denote the directions of the three principal axes of inertia (fixed in the molecule), P is the total angular momentum vector, with components P a , P b , P c , and the labeling of the axes is chosen so that I a < I b < I c , or A > B > C, in terms of the rotational constants A = 8 π h 2 I a , B = 8 π h 2 I b , C = 8 π h 2 I c . The magnitude of the total angular momentum vector is quantized, and P 2 = P a 2 + P b 2 + P c 2 = J(J + 1) 2 J = 0,1,2, The orientation of P with respect to a space fixed z-axis is also quantized. The projection of P on this z-axis can have only the values given by P z = M where, for each given value of J, the “azimuthal quantum number” M takes on the 2J + 1 values M = J, J – 1, …, 0, …, –J. It is convenient to distinguish the following types of rotors: Moments of Inertia Symmetry Rotational Constants Examples I a = 0; I b = I c Linear A = ; B = C HCl, CO 2 I a < I b = I c Prolate A > B = C CH 3 Cl, C 2 H 6 , Football symmetric top I a = I b < I c Oblate A = B > C CH 3 CF 3 , C 6 H 6 , Frisbee symmetric top I a = I b = I c Spherical top A = B = C CH 4 , SF 6 I a < I b < I c Asymmetric top A > B > C H 2 O, C 2 H 4 * Professor Dudley Herschbach, Department of Chemistry, Harvard University
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5.80 Lecture #24S Fall, 2008 Page 2 of 9 pages Symmetric Top Spectra For symmetric top molecules, the component of P along the symmetry axis is also quantized, and takes the values P s = K h with K = J, J – 1, …, 0, …, –J; here s denotes the a-axis for prolate tops and the c-axis for oblate tops. Thus the rotational energy of a symmetric top is given by W r = BP 2 + (A B)P a 2 , prolate case = BP 2 + (C – B)P c 2 , oblate case and the energy level patterns 6 5 4 3 2 1 J 6 5 4 3 2 1 0 J 0 K=0 K=1 K=2 K=0 K=1 K=2 Prolate Oblate Since the energy is independent of the sign of K, levels with the same absolute magnitude of K coincide, so that all levels for which K is greater than zero are doubly degenerate, and there are only J + 1 distinct energy values for each possible value of J. For each particular value of K, there is an infinite series of levels with different values of J. These are identical in spacing with the linear molecule levels except that the series must start with J = K rather than J = 0. For a symmetric top, the selection rules for J and M are identical to those for a linear molecule, namely:
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24s_580ln_fa08 - MIT OpenCourseWare http:/ocw.mit.edu 5.80...

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