In Chem 465, we learn that the Hamiltonian of a rigid rotor is H=BJ2 where B is the rotational constant and J is the rotational angular momentum. The rotational energies are: E=BJ(J+1) 5 where J=0,1,2⦠is the rotational angular momentum quantum number. Consider a perturbation: š»š»ā = 1 2 ššļ潚½š½Ģ+ + š½š½Ģāļæ½. where Ī» is a constant. The operators š½š½Ģ+ and š½š½Ģā have the following effects on the spherical harmonics ššš½š½ ššš½š½ : š½š½Ģāššš½š½ ššš½š½ = š½š½ššš½š½ā1 ššš½š½ā1 , š½š½Ģāšš0 0 = 0 š½š½Ģ+ššš½š½ ššš½š½ = (š½š½ + 1)ššš½š½+1 ššš½š½+1 (1) [5 pts] Prove that the first-order perturbation vanishes based on symmetry arguments. (2) [10 pts] For the ground rotational state (J=0), compute the second-order contribution to its energy due to the perturbation by the J=1 state. (3) [3 pts] For the ground rotational state (J=0), prove that the second-order contribution to its energy due to the perturbation by the J=2 state vanishes
Selection rules only permit transitions between consecutive rotational levels: ĪJ=J±1, and require the molecule to contain a permanent dipole moment. Due to the dipole requirement, molecules such as HF and HCl have pure rotational spectra and molecules such as H2 and N2 are rotationally inactive.
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