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Unformatted text preview: 26Jan2011 Chemistry 21b Spectroscopy Lecture # 10 Rotation of Asymmetric Tops & Centrifugal Distortion Most molecules are asymmetric tops, which means that I a negationslash = I b negationslash = I c . Thus, the rotational Hamiltonian H rot = 1 2 parenleftbigg J 2 a I a + J 2 b I b + J 2 c I c parenrightbigg (10 . 1) cannot be factored as it was for symmetric and spherical tops. Hence, no simple energy expression can be written for this general case, especially when centrifugal distortion is included (which well look at next time). Approximate solutions to the problem can be obtained by expanding the asymmetric top wavefunction in a symmetric rotor basis, which are functions of the same coordinates, namely the Euler angles discussed previously. We also know that the asymmetric top wavefunction, i is an eigenfunction of P 2 with an eigenvalue of J ( J + 1) h 2 . Thus, in our expansion of the asymmetric top wavefunctions in terms of symmetric top functions, only those with the same value of J need be included. Put another way, when casting the problem into matrix form, the Hamiltonian matrix is block diagonal in J . Similarly, we know that i is an eigenfunction of P z with an eigenvalue of M J h . Thus, matrix elements with different values of M J in the potentially infinite sum of symmetric rotor functions over the quantum numbers J,M J ,K , only the sum over the (2 J +1) values of K survive, and i = J summationdisplay K = J c i,JM J K i,JM J K , (10 . 2) where J and M J are the asymmetric top quantum numbers for the total rotational angular momentum and its component along a spacefixed axis. Recall that, in general, we may set up the eigenfunctions to a quantum mechanical problem as a sum over any complete basis set, and that to determine the eigenenergies we must solve the secular equation det[ < m  H  n > E i mn ] = 0 , where E i is the ith eigenvalue and ( m,n ) run over the full extent of the basis set. Here, the block diagonal nature of the symmetric rotor expansion reduces the secular equation to a finite determinant of order (2 J + 1): det[ H K K E i K K ] = 0 (10 . 3) H K K = integraldisplay JM J K H JM J K d , (10 . 4) where K and K each run over the complete J to J range in unit steps. Calculating these matrix elements is possible given the known properties of the symmetric top wavefunctions, but it is quite tedious and so we simply quote the (lengthy!) result: H K K = K ,K ( 1 2 ) h bracketleftbig (2 C A B )( K ) 2 + ( A + B ) J ( J + 1) bracketrightbig 80 + K ,K +2 ( 1 4 ) h ( B A ) [ J ( J + 1) K ( K + 1)] 1 / 2 [ J ( J + 1) ( K + 1)( K + 2)] 1 / 2 + K ,K 2 ( 1 4 ) h ( B A ) [ J ( J + 1) K ( K 1)] 1 / 2 [ J ( J + 1) ( K 1)( K 2)] 1 / 2 (10 . 5) Barring accidental degeneracies, the roots of Eq.Barring accidental degeneracies, the roots of Eq....
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This note was uploaded on 01/03/2012 for the course CH 21b taught by Professor List during the Fall '10 term at Caltech.
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