1. 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

2. e.g. diatomic

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

4. |JM>; the set of spherical harmonics Y JM (  ) Diatomics & Linear Molecules M the projection of L onto the lab frame Z-axis

5. For Polyatomic molecules there is a third angle, , 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 3 rd dimension in the wavefunction and thus a 3 rd quantum number. Eigenfunctions: Wigner rotation functions

6. a, b, c refer to internal x,y,z axes ordered such I a  I b  I c. No general set of exact eigenfunctions. L a, L b, and L c do not commute.

7. However, they do commute with L 2 and L z, 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 I a =I b =I c A ball, CH 4 ; Prolate top I b =I c A cigar, CH 3 CN; light atoms off of symmetry axis Oblate top I a =I b ; 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

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

14. Selection Rules  0 ≠0  J=0,±1  M=0,±1  K=0

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

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

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