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Rotational Spectroscopy

Born-Oppenheimer Approximation; Nuclei move on potential defined by solving for electron energy at each set of nuclear coordinates. Hamiltonian for nuclei and electrons is fully separable Transform to center of mass coordinates Consider CofM frame and Lab Frames of reference.

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Rotational Spectroscopy

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  1. Born-Oppenheimer Approximation; • Nuclei move on potential defined by solving for electron energy at each set of nuclear coordinates. • Hamiltonian for nuclei and electrons is fully separable • Transform to center of mass coordinates • Consider CofM frame and Lab Frames of reference Rotational Spectroscopy

  2. e.g. diatomic

  3. Rotation (2 coordinates diatomic 3 non-linear molecule) Translation (3 coordinates) Vibration (3N-5 or 6) I=mR2; What R to choose?

  4. Diatomics & Linear Molecules |JM>; the set of spherical harmonics YJM(q,j) M the projection of L onto the lab frame Z-axis

  5. For Polyatomic molecules there is a third angle, c, which refers to the angle of rotation of the molecule about its own z-axis. (Use Z for lab frame and z for molecular frame) The third angle requires a 3rd dimension in the wavefunction and thus a 3rd quantum number. Eigenfunctions: Wigner rotation functions

  6. a, b, c refer to internal x,y,z axes ordered such IaIbIc. No general set of exact eigenfunctions. La, Lb, and Lc do not commute.

  7. However, they do commute with L2 and Lz, J and M remain “good quantum numbers”. K is not; the eigenfunctions of the Hamiltonian can be expressed as linear combinations:

  8. Simplifications: Spherical Top Ia=Ib=Ic A ball, CH4; Prolate top Ib=Ic A cigar, CH3CN; light atoms off of symmetry axis Oblate top Ia=Ib; Frisbee, Benzene

  9. Simplifications are useful because Hamiltonian is diagonal in Wigner functions (K is a good quantum number). e.g. Prolate top Note two-fold degeneracy in K remains. E depends on |K|. Why?

  10. Note: Definitions of A,B,C vary by constants h and c depending on units

  11. Absorption Intensities • Define dipole moment, m, with a Taylor expansion in internal coordinates, q, and project onto space fixed X,Y,Z axes.

  12. Selection Rules m0≠0 DJ=0,±1 DM=0,±1 DK=0

  13. Multiply by Boltzmann population of level J and sum over all initial M For nonlinear molecule, repeat with sum over K and K’

  14. Discussion • Raman Selection rules • Nuclear spin statistical weights • How can we get multiple bond lengths?

  15. Non-rigid R Expand V(R) (or all q) in a Taylor series about the minimum. Then Rin the Hamiltonian can be expressed as a perturbation about Re with the result for a diatomic or linear molecule

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