Infrared (IR) spectroscopy probes vibrational transitions in molecules by
measuring the absorption of IR radiation.
Not all vibrational transitions are observable — whether a transition appears
in an IR spectrum is determined by selection rules.
The transition dipole moment
Because the intensity of a spectroscopic transition is proportional to the square of the Transition Moment
\[
\lvert {\int \psi_{v'}^* \, \hat{\mu} \, \psi_v d \tau \rvert} ^2 \neq 0
\]
an IR transition is allowed only if the integral is nonzero:
\(\hat{\mu}\) is the dipole moment operator and
\(\psi_v\), \(\psi_{v'}\)
are vibrational wavefunctions for the upper and lower state, respectively.
Using symmetry, this condition can be expressed as a direct product:
\[
\Gamma(\psi_v)\times\Gamma(\hat{\mu})\times\Gamma(\psi_{v'})
\supset A_1.
\]
If the totally symmetric representation is present, the transition is
IR-allowed; if not, the transition is forbidden by symmetry.
Selection rule for the harmonic oscillator
For a one-dimensional harmonic oscillator, the dipole moment operator is
proportional to the displacement:
\(\hat{\mu}\propto x\).
Because harmonic oscillator wavefunctions alternate in parity
(even for even \(v\), odd for odd \(v\)),
the integral
\(\langle \psi_{v'} | x | \psi_v \rangle\)
is nonzero only when the parity changes.
This leads directly to the vibrational selection rule:
\[
\Delta v = \pm 1.
\]
In the harmonic approximation, only transitions between adjacent vibrational
levels are allowed, producing a single fundamental band in the IR spectrum
(since all of the overtone bands will occur at the same frequency.)
Big idea: IR absorption requires a changing dipole moment, and for the
harmonic oscillator this restricts observable transitions to
\(\Delta v=\pm1\).
Real molecular vibrational spectra are more complex than predicted by the
harmonic oscillator model.
Experimental IR spectra often contain additional weak bands beyond the
fundamental transition.
Anharmonicity and overtones
When the vibrational potential is anharmonic (as in the Morse potential),
the vibrational energy levels are no longer equally spaced:
\[
G_v
=
\omega_e\left(v+\frac{1}{2}\right)
-
\omega_e x_e\left(v+\frac{1}{2}\right)^2.
\]
Anharmonicity relaxes the strict harmonic selection rule
\(\Delta v=\pm1\), allowing
overtone transitions with
\(\Delta v=\pm2, \pm3, \ldots\)
(although these are expected to be weaker than any bands with
\( \Delta v = \pm 1 \) ).
Determining \(\omega_e\) and \(\omega_e x_e\)
The presence of overtones allows experimental determination of both the
harmonic vibrational constant
\(\omega_e\)
and the anharmonicity constant
\(\omega_e x_e\).
For example, the transition wavenumbers are approximately:
-
Fundamental:
\(\tilde{\nu}_{0\rightarrow1}
= \omega_e - 2\omega_e x_e\)
-
First overtone:
\(\tilde{\nu}_{0\rightarrow2}
= 2\omega_e - 6\omega_e x_e\)
By measuring the positions of these bands, one can solve for
\(\omega_e\) and
\(\omega_e x_e\),
gaining insight into both bond strength and anharmonicity.
Deriving the spacing expression \(\Delta G_{v+1/2}\)
When vibrational energy levels are anharmonic, the spacing between adjacent
levels is no longer constant.
A convenient quantity to examine is the difference between successive
vibrational term values:
\[
\Delta G_{v+1/2} \equiv G_{v+1} - G_v.
\]
Starting from the anharmonic expression for the vibrational term values,
\[
G_v
=
\omega_e\left(v+\frac{1}{2}\right)
-
\omega_e x_e\left(v+\frac{1}{2}\right)^2,
\]
write expressions for two adjacent levels:
\[
G_{v+1}
=
\omega_e\left(v+\frac{3}{2}\right)
-
\omega_e x_e\left(v+\frac{3}{2}\right)^2,
\]
\[
G_v
=
\omega_e\left(v+\frac{1}{2}\right)
-
\omega_e x_e\left(v+\frac{1}{2}\right)^2.
\]
Taking the difference
Subtracting \(G_v\) from \(G_{v+1}\) gives
\[
\Delta G_{v+1/2}
=
\omega_e
-
\omega_e x_e
\left[
\left(v+\frac{3}{2}\right)^2
-
\left(v+\frac{1}{2}\right)^2
\right].
\]
Evaluate the difference of squares:
\[
\left(v+\frac{3}{2}\right)^2
-
\left(v+\frac{1}{2}\right)^2
=
(v^2+3v+\tfrac{9}{4})
-
(v^2+v+\tfrac{1}{4})
=
2(v+1).
\]
Substituting this result yields
\[
\Delta G_{v+1/2}
=
\omega_e
-
2\omega_e x_e (v+1).
\]
Why this expression is useful
This linear relationship between
\(\Delta G_{v+1/2}\)
and
\((v+1)\)
allows experimental determination of both
\(\omega_e\)
and
\(\omega_e x_e\).
A plot of
\(\Delta G_{v+1/2}\)
versus
\((v+1)\)
is a straight line with:
-
intercept \(\omega_e\),
-
slope \(-2\omega_e x_e\).
Big idea: anharmonicity causes vibrational level spacings to decrease
systematically with increasing
\(v\), and this behavior provides a direct route to
extracting molecular force constants from real spectra.
Big idea: real IR spectra contain overtones because molecular vibrations
are anharmonic, and these additional transitions provide deeper information
about molecular bonding than the harmonic model alone.
Worked example: using overtone bands to extract \( \omega_e \) and \( \omega_e x_e \)
In an anharmonic oscillator (Morse-like), the vibrational term values are well approximated by
\[
G_v=\omega_e\left(v+\frac{1}{2}\right)-\omega_e x_e\left(v+\frac{1}{2}\right)^2.
\]
A transition wavenumber is a difference in term values:
\(\tilde{\nu}_{0\rightarrow v}=G_v-G_0\).
Using this, the first few bands are
-
Fundamental:
\(\tilde{\nu}_{0\rightarrow 1}=\omega_e-2\omega_e x_e\)
-
First overtone:
\(\tilde{\nu}_{0\rightarrow 2}=2\omega_e-6\omega_e x_e\)
-
Second overtone:
\(\tilde{\nu}_{0\rightarrow 3}=3\omega_e-12\omega_e x_e\)
Example data (HCl)
The IR spectrum of HCl shows a strong fundamental band near
\(2886\ \text{cm}^{-1}\), a weak first overtone near
\(5668\ \text{cm}^{-1}\), and a very weak second overtone near
\(8347\ \text{cm}^{-1}\).
| v' ← v'' |
frequency (cm-1) |
| 1 ← 0 |
2886 |
| 2 ← 0 |
5668 |
| 3 ← 0 |
8347 |
Use the fundamental and first overtone to solve for
\(\omega_e\) and \(\omega_e x_e\).
Let \(A=\omega_e\) and \(B=\omega_e x_e\).
\[
\tilde{\nu}_{0\rightarrow 1}=A-2B=2886
\]
\[
\tilde{\nu}_{0\rightarrow 2}=2A-6B=5668
\]
Step 1: Solve for \(B=\omega_e x_e\)
Multiply the first equation by 2 and subtract from the second:
\[
(2A-6B) - (2A-4B) = 5668 - 5772
\quad\Rightarrow\quad
-2B = -104
\quad\Rightarrow\quad
B = 52\ \text{cm}^{-1}.
\]
Step 2: Solve for \(A=\omega_e\)
\[
A = 2886 + 2B = 2886 + 104 = 2990\ \text{cm}^{-1}.
\]
So the extracted constants are:
\[
\omega_e \approx 2990\ \text{cm}^{-1},
\qquad
\omega_e x_e \approx 52\ \text{cm}^{-1}.
\]
If you want the dimensionless anharmonicity constant,
\(x_e = (\omega_e x_e)/\omega_e \approx 52/2990 \approx 1.74\times10^{-2}\).
Check using the second overtone
Predict the second overtone position:
\[
\tilde{\nu}_{0\rightarrow 3}
= 3\omega_e - 12\omega_e x_e
= 3(2990) - 12(52)
= 8970 - 624
= 8346\ \text{cm}^{-1},
\]
which matches the observed band near
\(8347\ \text{cm}^{-1}\) very closely.
Big idea: once overtones are observable, the deviation from equal spacing
lets you extract both the harmonic constant \(\omega_e\)
and the anharmonic correction \(\omega_e x_e\) directly from spectral data.
