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Lecture33Notes - Chem 120A Spring 2006 READING Molecular...

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Chem 120A Molecular states 04/17/06 Spring 2006 Lecture 33 READING: Engel (QCS): Ch. 12-13, section B.2 McQuarrie and Simon: Chapter 7.2, 9.3-9.6 Basic energies and spectra In Lecture 30 we saw that the large difference in the masses of nuclei and electrons allows us to apply the Born-Oppenheimer approximation in order to solve the Schr¨odinger equation for a molecule. This allows us to separate the nuclear and electronic degrees of freedom. This separation is equivalent to a separation of time scales, meaning that since the nuclei are so massive the move on a much slower time scale than the electrons. This leads to a separation of the energies that characterize the spectrum associated with absorption and emission of light by a molecule. The magnitudes of the energies for electronic and nuclear (vibrational) excitations are also quite different. Recall that 1 eV = 8065 cm 1 , 1 Hartree = 219 , 474 cm 1 27 eV. Electronic motion is confined to a typical molecular size of 1 ˚ A. Using the particle in a box model, we can approximate the energy for electronic motion, where m is the electron mass: E e ¯ h 2 ma 2 few eV . (1) The nuclei move in the field of the fast moving electrons. In other words, the electronic energies, which depend on the positions of the nuclei, represent an effective external potential in which the nuclei reside. This potential is shown in Fig. 1. There are two types of nuclear motion, vibration and rotation. We have already studied the case of nuclear vibration, which can be modeled by the harmonic oscillator. The potential energy for the harmonic oscillator is proportional to M ω 2 vib x 2 , where ω vib is the angular frequency of vibration and M is the nuclear mass. At equilibrium, we can take x = a (Fig. 1), so that the potential a Figure 1: A Born-Oppenheimer potential for a diatomic molecule with vibrational states shown. A harmonic potential is also shown for comparison. a is the approximate size of the molecule. Chem 120A, Spring 2006, Lecture 33 1
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v=1 v=2 v=3 K=0 1 2 3 ~10 -2 eV ~10 -4 eV } rotational levels vibrational levels Figure 2: A rovibrational sequence for a molecule within the Born-Oppenheimer approximation. Here we have also assumed that the kinetic energy for the nuclei can be separated into vibrational and rotational parts.
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