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Section 14.4: Molecular Models and Molecular Spectra

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When we describe molecules, we do so by describing three energy scales: electronic, rotational, and vibrational.  The electronic energy scale is the largest, on the order of the atomic energy scale of a few eV. The rotational energy scale is far less than that of the electronic, but on a larger than the vibrational energy scale.  Both these scales are between 100 and 1000 times smaller than the electronic energy scale.

The rotational energy scale for a diatomic molecule is modeled quite well by the three-dimensional rigid rotator. This situation can be modeled quantum mechanically by considering the three-dimensional time-independent Schrödinger equation with a fixed radius, R,

ħ2/2μ [(1/R2sin(θ)) ∂/∂θ(sin(θ) ∂/∂θ) + 1/(R2sin2(θ)) ∂2/∂φ2 ] ψ(θ,φ) = Eψ(θ,φ) , (14.8)

which, after dividing by −ħ2/2μ, looks exactly like Eq. (13.22), with the substitution

E = l(l + 1)ħ2/2μR2 = l(l + 1)ħ2/2Imolecule,

where Imolecule = μR2, which gives us the energy spectrum for a three-dimensional rigid rotator. The vibrational energy scale is modeled by a simple harmonic oscillator and hence E = (n + 1/2)ħω.

In the animation, we show three important molecular potentials that closely model the rotational modes:

Kratzer Potential: V(r) = -2V0(α/r − α/2r2)
Morse Potential: V(r) = V0(e-2αr − 2er)
Lennard-Jones Potential: V(r) = V0(α/r12 − α/r6)

where α is a tunable parameter. These potential energy functions are shown in Animation 1Animation 2 , and  Animation 3, respectively. To see the other bound states, simply click-drag in the energy level diagram on the left to select a level. The selected level will turn red.

Transitions between energy levels are also of importance and can be calculated once the energy levels are calculated. As an example of molecular spectra, shown in Animation 4 is an approximation to the P and R branches of the CO2 vibrational spectrum.

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