C566lecture15_000

C566lecture15_000 - C566 Master Lecture Notes Lecture 15...

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Lecture 15 Spherical Top Energies Classically , with the angular momentum vector J = (J a , J b , J c ), assuming principle inertia axes: 2 2 2 2 2 1 2 1 J I I J I J I J T c c b b a a rot Heuristic conversion to QM Angular momentum is quantized in units of , J 2 rot = 2 J(J+1) rot J = 0, 1, 2, … F(J) = BJ(J+1) where B = 2 /2I; I is the one moment of inertia in the molecule. Summary of our treatment of rotations so far: For a molecule with n atoms, kinetic E, T = n i i n i i i r m v m 1 2 1 2 ) ( 2 1 2 1 which includes square and cross terms that reflect the tensor form of the moment of inertia. However, by rotating axes from arbitrary x,y,z in a way that results in zero cross terms, a,b,c, energy can be expressed as c c b b a a rot I J I J I J T 2 2 2 2 1 This is classical rotational energy, in which the J a,b,c values are the projection of J onto the non- arbitary a, b, c axes. Axes are labeled by convention so I c I b I a [I c = m i (a i 2 + b i 2 ), etc.] Given molecular symmetries, there are cases in which I a = I b = I c (spherical top; T rot surface is a sphere when plotted on a,b,c) I c = I b > I a (prolate symmetric top, rugby ball) or I c < I b = I a (asymmetric top)
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C566lecture15_000 - C566 Master Lecture Notes Lecture 15...

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