lecture06_2011

lecture06_2011 - 14Jan2011 Chemistry 21b Spectroscopy...

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14Jan2011 Chemistry 21b – Spectroscopy Lecture # 6 – Nuclear Motion in Diatomic Rotors Once the electronic potential energy surfaces have been computed the motion of the nuclei can be determined by solving eq. (4.10). Here for simplicity we will consider the case of a diatomic molecule, over the next couple of weeks we’ll build up the quantum mechanical and group theoretical machinery needed to handle polyatomic systems. If we transform to the center-of-mass coordinates and deFne the internuclear coordinate −→ R = −→ R 1 −→ R 2 , the nuclear kinetic energy term reduces to a term containing the center-of-mass motion, which is not of spectroscopic interest (being a constant, you’ll look more carefully at these degrees of freedom in Ch 21c), and a term describing the relative motion. This last term can be separated further into a part governing the radial motion along R = | −→ R | , and a part containing the angular coordinates, described by the nuclear angular momentum operator J : T N = 1 2 μR 2 ∂R p R 2 ∂R P + J 2 2 μR 2 , (6 . 1) where μ = M 1 M 2 / ( M 1 + M 2 ) is the reduced nuclear mass. If we write Ψ nuc ( −→ R ) = Y JM ( ˆ R ) F ( R ) /R (6 . 2) where ˆ R denotes the angular part of −→ R and use J 2 Y JM ( ˆ R ) = J ( J + 1) Y JM ( ˆ R ) (6 . 3) we obtain b 1 2 μ d 2 dR 2 + E el ( R ) + J ( J + 1) 2 μR 2 E B F ( R ) = 0 . (6 . 4) a) Vibration If the electronic state is bound, that is, stable with respect to dissociation, it has a minimum D e at a certain distance R e . The potential can then be expanded in powers of ( R R e ) about R e (writing V = E el ): V ( R ) = V ( R e ) + p dV dR P R e ( R R e ) + 1 2 p d 2 V DR 2 P R e ( R R e ) 2 + . (6 . 5) At R = R e , the Frst derivative vanishes, and up to quadratic terms: V ( R ) ≃ − D e + 1 2 k ( R R e ) 2 (6 . 6) 47
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Figure 6.1 a) Harmonic approximation to a molecular potential curve E el ( R ): b) Energy levels of a harmonic oscillator (dashed lines), compared with those for an anharmonic potential (solid line). with k = ( d 2 V/dR 2 ) R e . Thus, if rotation is neglected ( J = 0), the purely radial equation (6.4) becomes the equation for the harmonic oscillator. Thus, the solution to eq. (6.4) can in that case be written as E = D e + E vib (6 . 7) with E vib = ω e ( v + 1 / 2) (6 . 8) where ω e = ( k/μ ) 1 / 2 . In the harmonic oscillator approximation, the vibrational levels are equidistant. As Figure 6.1 shows, this is a good approximation for the lowest few vibrational levels, but the higher levels will lie much closer together due to “anharmonicities” in the potential curve. The harmonic oscillator wave functions can be written in terms of Hermite polynomials. Some characteristic wave functions are illustrated in Figure 6.2. Note that the quantum number v of the wave function is the same as the number of nodes. Eq. (6.4) can also be solved analytically for the case that the potential
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lecture06_2011 - 14Jan2011 Chemistry 21b Spectroscopy...

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