An Harmonic
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Transcript of An Harmonic
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
€
V (x) = 12 kx
2
we see that
€
k = 2Deα2
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≈De 1− 2 1−αx + 12α
2x 2[ ] + 1− 2αx + 1
2 4α 2x 2[ ]( )
€
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
€
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)
€
β =De
1/ 2
α
€
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
€
ψn (x) = NnHn (α 1/ 2x)e−αx 2 / 2
€
α =kμ
h2
⎛
⎝ ⎜
⎞
⎠ ⎟
1/ 2
€
Nn =1
2n n!( )1/ 2
α
π
⎛
⎝ ⎜
⎞
⎠ ⎟
1/ 4
are the Hermite polynomials
€
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−∞
∞
∫
€
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
€
Pclassical (x) = π 2E /k − x 2( )
−1
€
2E /k = x turn2
Where xturn is the maximum value of x
Correspondence principle
Correspondence principle
Correspondence principle
Correspondence principle