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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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 – 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


Anharmonicity

use De = 40, α = 1; then scale by c


Anharmonicity

Energy levels


Anharmonicity

Morse model

dissociated above this

are the generalized Laguerre polynomials


Anharmonicity

Harmonic oscillator model

are the Hermite polynomials


Anharmonicity

Wavefunctions: harmonic oscillator


Anharmonicity

Wavefunctions: Morse oscillator


Anharmonicity

Wavefunctions: harmonic vs. Morse


Anharmonicity

Wavefunctions


Anharmonicity

Wavefunctions


Anharmonicity

Expectation value of position


Anharmonicity

Expectation value of position


Anharmonicity

Expectation value of position


Anharmonicity

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


Anharmonicity

Selection rules


Anharmonicity

Correspondence principle

Where xturn is the maximum value of x


Anharmonicity

Correspondence principle


Anharmonicity

Correspondence principle


Anharmonicity

Correspondence principle


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