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.


Anharmonicity

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


Anharmonicity

use can detect De = 40, α = 1; then scale by c


Anharmonicity

Energy levels can detect


Anharmonicity

Morse model can detect

dissociated above this

are the generalized Laguerre polynomials


Anharmonicity

Harmonic oscillator model can detect

are the Hermite polynomials





Anharmonicity

Wavefunctions can detect


Anharmonicity

Wavefunctions can detect





Anharmonicity

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


Anharmonicity

Selection rules can detect


Anharmonicity

Correspondence principle can detect

Where xturn is the maximum value of x





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