David Hoffman Notes Spectra - Basics of Vibrational and...

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Basics of Vibrational and Rotational Spectroscopy David Hoffman 4/26/2010 Contents 1 Introduction 1 2 Diatomics 2 2.1 Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.3 Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3 Polyatomics 7 3.1 Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.2 Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4 Raman 9 1 Introduction In the last two lectures we briefly went over the basics of vibrational and rotational spec- troscopy and today we’ll go into a little more detail about where the selection rules came from and extending the techniques we developed for diatomic molecules to polyatomic molecules. Let’s return again to the Born-Oppenheimer Approximation. This approximation basically says that the electrons in a system move infinitely fast compared to the nuclei. This means that no matter how fast the nuclei are moving the electrons are always at equilibrium. As a result of this the Hamiltonian of the system is separable into an electronic component and a nuclear component, H = H e ( R ) + H n (1.1) Where I’ve made it clear that the electronic component depends on the nuclear coordinates. This means that the wave functions are separable as well, i.e., | Ψ i = | ψ e i| ψ n i (1.2) 1
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A B z y x Figure 2.1: Diatomic Within the BO approximation the nuclear Hamiltonian is further separable into center of mass motions and internal motions. If we assume that the timescale of vibrations is much larger than that for rotations we can use the Rigid Rotor approximation to say that the internal motions are separable into vibrations and rotations. So finally we have, H = H e ( R ) + H t + H r + H v (1.3) We’ll be ignoring the translational Hamiltonian as it doesn’t really matter. This means that our wave function can be written as, | Ψ i = | ψ e i| ψ v i| ψ r i (1.4) 2 Diatomics Let’s focus on diatomics now where A and B are not necessarily the same species. 2.1 Rotations Here there are two rotational degrees of freedom, one about z and the other about x . Note that we’re ignoring rotation about y because we’re considering the nuclei to be point masses. Also note that in this case both rotations are degenerate with each other. Remember that even though we consider the nuclei to be moving much slower than the electrons the nuclei are still quantum particles and the rotational Hamiltonian is still, H r = ˆ L 2 2 I (2.1.1) And we’ve already solved this problem, which means that our allowed energy levels are, E J = J ( J + 1) ~ 2 2 I (2.1.2) This would be a good time to introduce the concept of wavenumbers (cm -1 ). Wavenumbers are a means of measuring energy, they’re used all the time in vibrational and rotational spectroscopy. We’ll use the tilde to indicate if a quantity is in wavenumbers. In this case the energy would be, e E J = e B 0 J ( J + 1) (2.1.3) 2
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Where, e B 0 = h 8 π 2 Ic (2.1.4) We also know the wave functions corresponding to these energies and those are the spherical harmonics Y M J ( θ,φ ) 2.2 Vibrations Chandler went over this last lecture but we can approximate the vibrational motion as a Harmonic Oscillator, at least at low energies and we’ve already solved this problem too. Our energy levels, in wavenumbers, are, e E v = ± v + 1 2 ² e ν (2.2.1)
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This note was uploaded on 05/14/2010 for the course CHEM 120A taught by Professor Whaley during the Spring '07 term at University of California, Berkeley.

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David Hoffman Notes Spectra - Basics of Vibrational and...

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