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, …

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

€

V (r) = De 1− e−α (r−r0 )[ ]

2

So we do

Put x = r – r0 and Taylor expand:

€

V (r) = De 1− e−α (r−r0 )[ ]

2

€

V (x) = De 1− 2e−αx + e−2αx( )

€

=Deα2x 2

Comparing to the harmonic oscillator

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V (x) = 12 kx

2

we see that

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k = 2Deα2

€

≈De 1− 2 1−αx + 12α

2x 2[ ] + 1− 2αx + 1

2 4α 2x 2[ ]( )

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De → cDe

€

α →αc

to keep the force constant the same but change the anharmonicity€

α =k

2De

⎛

⎝ ⎜

⎞

⎠ ⎟

1/ 2

=μω2

2De

⎛

⎝ ⎜

⎞

⎠ ⎟

1/ 2

€

V (x) = De 1− e−αx[ ]

2 use De = 40, α = 1; then scale by c

Energy levels

Morse model

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En = 2αDe1/ 2 n +1/2( ) − n +1/2( )

2 α

2De1/ 2

⎡

⎣ ⎢

⎤

⎦ ⎥

€

ψn (x) = Nnyβ −n−1/ 2e−y / 2Ln

2β −2n−1(y)

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β =De

1/ 2

α

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y = 2βe−αx

€

Nn =α (2β − 2n −1)n!

Γ(2β − n)

⎡

⎣ ⎢

⎤

⎦ ⎥

1/ 2

€

Lna (x) are the generalized Laguerre polynomials

€

0 ≤ n <De

1/ 2

α−

1

2dissociated above this

Harmonic oscillator model

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ψn (x) = NnHn (α 1/ 2x)e−αx 2 / 2

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α =kμ

h2

⎛

⎝ ⎜

⎞

⎠ ⎟

1/ 2

€

Nn =1

2n n!( )1/ 2

α

π

⎛

⎝ ⎜

⎞

⎠ ⎟

1/ 4

are the Hermite polynomials

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Hn (x)

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EnMORSE = 2αDe

1/ 2 n +1/2( ) − n +1/2( )2 α

2De1/ 2

⎡

⎣ ⎢

⎤

⎦ ⎥

€

EnHO = 2αDe

1/ 2 n +1/2( )

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

€

μz( )nn'= ψ n'

*μ zψ n−∞

∞

∫

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μz ≈ μ0 +dμ

dx 0

x +d2μ

dx 2

0

x 2

2!+L

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

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Pclassical (x) = π 2E /k − x 2( )

−1

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2E /k = x turn2

Where xturn is the maximum value of x

Correspondence principle

Correspondence principle

Correspondence principle

Correspondence principle