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

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 – r can detect 0 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 can detect De = 40, α = 1; then scale by c


Energy levels can detect


Morse model can detect

dissociated above this

are the generalized Laguerre polynomials


Harmonic oscillator model can detect

are the Hermite polynomials





Wavefunctions can detect


Wavefunctions can detect





Selection rules can detect

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 can detect


Correspondence principle can detect

Where xturn is the maximum value of x





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