Anharmonicity
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In real molecules, highly sensitive vibrational spectroscopy can detect overtones , which are transitions originating from the n = 0 state for which Δn = +2, +3, …. Anharmonicity.

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Anharmonicity

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In real molecules, highly sensitive vibrational spectroscopy can detect overtones, which are transitions originating from the n = 0 state for which Δn = +2, +3, …

Anharmonicity

Overtones are due to anharmonicity. A good approximation of realistic anharmonicity is given by the Morse potential.


Put x = r – r0 and Taylor expand:

Comparing to the harmonic oscillator

we see that

So we do

to keep the force constant the same but change the anharmonicity


use De = 40, α = 1; then scale by c


Energy levels


Morse model

dissociated above this

are the generalized Laguerre polynomials


Harmonic oscillator model

are the Hermite polynomials


Wavefunctions: harmonic oscillator


Wavefunctions: Morse oscillator


Wavefunctions: harmonic vs. Morse


Wavefunctions


Wavefunctions


Expectation value of position


Expectation value of position


Expectation value of position


Selection rules

For anharmonicity, can replace the H.O. wavefunctions with Morse wavefunctions…

…or can keep more terms in the Taylor expansion of the dipole moment


Selection rules


Correspondence principle

Where xturn is the maximum value of x


Correspondence principle


Correspondence principle


Correspondence principle


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