W3 - Supplementary Notes for Week 3 and 4 Changyi Li...

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Unformatted text preview: Supplementary Notes for Week 3 and 4 Changyi Li January 31, 2010 Selection Rules As shown in the Einstein Coe cient section of week 2's supplementary notes, the transition probability is given by the Fermi's Golden Rule, P = 2 π ~ |h f | V | i i| 2 ρ (1) where f is the nal state and i is the initial state. ρ is the density of states for the nal state and V is the perturbation to the original Hamiltonians of the system. In the case shown in week 2, V =--→ μ ·-→ E , therefore P ∝ |h f |-→ μ | i i| 2 (2) or the transition dipole moment squared. In other words, if our transition dipole integral turns out to be 0, we will have 0 probability of such a transition, or forbidden by symmetry. The rules by which we determine which transitions are allowed are called the selection rules. The selection rules for rudimentary systems such as particle-in-a-box, harmonic potential, anharmonic potential with cubic and quartic anharmonicity, hydrogen-like atom are listed in the lecture slides. Due to the importance of the carbonyl bond, formaldehyde is used as an example in the lecture slide and the π → π * transition is used as an example. In case one example is not su cient, here I will show (with help from Sunney's omitted slides) how the selection rules apply to the n → π * transition. The problem start out the same way: determine the planes of symmetry of the C=O bond, for which there are 2: the xy plane and the xz plane. σ signi es a symmetry operation called re ection , which is to re ect the molecule with respect to the plane. For example, for the operationre ect the molecule with respect to the plane....
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This note was uploaded on 10/27/2010 for the course BI 110 taught by Professor Richards,j during the Winter '08 term at Caltech.

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W3 - Supplementary Notes for Week 3 and 4 Changyi Li...

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