Chemistry 362
Fall 2015
Dr. Jean M. Standard
September 16, 2015
Introduction to Molecular Vibrations and Infrared Spectroscopy
Vibrational Modes
For a molecule with N atoms, the number of vibrational modes is 3N-5 for a linear molecule and 3N-6 for a
nonlinear molecule. All diatomic molecules (N=2) have only a single vibrational mode corresponding to the
vibration of the bond. A linear triatomic molecule such as CO2 possesses four vibrational modes, while a nonlinear
triatomic molecule such as H2O posseses only three vibrational modes.
Infrared Activity: Dipole Moment
The dipole moment of the molecule must change during a vibration in order for a spectroscopic transition to occur.
If the dipole moment changes during a vibration, then that mode is said to be infrared active.
Recall that the dipole! moment is a measure of the charge separation in a system. For a system of N point charges,
the dipole moment d is defined as
!
d =
€
N
∑ q r!
i i
,
i=1
!
€
where qi is the charge of the ith particle and ri is the position vector of the ith particle. In quantum mechanics,
because the charge distribution is based on probability, and is therefore smeared out, the dipole moment is not a sum
€
of point charges times the position vector, but rather an integral over the charge distribution times the position
vector.
€
Selection Rules for Infrared Spectroscopy
Harmonic oscillators: Δv i = ±1 (only the fundamental transitions are allowed)
Anharmonic oscillators: Δv i = ±1, ± 2, ± 3, … (the fundamental transitions, as well as overtones and
combinations, are allowed)
€
In these selection rules, the quantity v i is the quantum number for the ith vibrational mode of the polyatomic
€
molecule. Each of the vibrational modes of the molecule is treated as a harmonic (or an anharmonic) oscillator.
Assuming that the Hamiltonian operator is approximately separable, the vibrational energy of the molecule is then a
sum of energies of each of the modes,
€
3N −6(5)
Ev1v 2 v 3 ... =
∑
(
)
hν 0i v i + 12 ,
i=1
where ν 0i is the harmonic frequency of the ith mode.
€
The selection rules for polyatomics state that the quantum number v i has to change by one unit during a
spectroscopic transition (for the fundamental transition). However, the other quantum numbers for all the other
€
modes stay the same–only one mode has to change. Hence, depending on which quantum number v i changes, there
will be 3N-6 (3N-5) distinct fundamental transitions for each polyatomic molecule.
€
€
The rest of this handout describes a few simple examples of the vibrational modes of polyatomic
molecules and their
infrared activity.
EXAMPLE 1: Water
Water has three vibrational modes since it is a nonlinear triatomic molecule. These vibrations are summarized in the
table below.
Frequency (cm–1)
3660
3760
1590
Mode
symmetric stretch
antisymmetric stretch
H-O-H bend
IR activity
IR active
IR active
IR active
A sketch of the vibrational motion that occurs for the symmetric stretching mode of water is shown in Figure 1.
Figure 1. Symmetric stretching mode of water
O
H
Equilibrium
H
O
Bond stretching
H
H
O
H
H
H O H
O
H
H
Equilibrium
Bond contraction
Equilibrium
Water is a nonlinear molecule, so its equilibrium dipole moment is nonzero. For the symmetric stretching mode, the
dipole moment increases when bond stretching occurs, and the dipole moment decreases when bond contraction
occurs. Thus, the dipole moment changes during the vibration and so the symmetric stretching mode is infrared
active.
A sketch of the vibrational motion that occurs for the antisymmetric stretching mode of water is shown in Figure 2.
Figure 2. Antisymmetric stretching mode of water
O
H
Equilibrium
H
H
O
Antisymmetric bond stretching
H
O
H
H
O H
Equilibrium
Antisymmetric bond stretching
H
O
H
H
Equilibrium
When one O-H bond is stretched and the other is contracted, the dipole moment changes direction. It shifts from
being positioned in line with the oxygen atom to the right and the to the left as the molecule vibrates. Therefore, the
antisymmetric stretching mode is infrared active.
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A sketch of the vibrational motion that occurs for the bending mode of water is shown in Figure 3.
Figure 3. Bending mode of water
O
H
H
O
H
H
Angle bending
H
Equilibrium
O
H
O
H
Angle bending
H
O
H
Equilibrium
H
Equilibrium
The molecule begins its bending motion at equilibrium with a nonzero dipole moment. When the angle bends and
becomes larger, the dipole moment is reduced. When the angle bends and becomes smaller, the dipole moment is
increased. This mode is infrared active because the dipole moment changes as the molecule bends.
All three vibrational modes of water are infrared active; therefore, all three should be observed in the infrared
spectrum of water. An example infrared spectrum for liquid water is shown in Figure 4. One large broad peak in the
spectrum at about 3500 cm–1 corresponds to the two stretching modes (their frequencies are very similar); the other
smaller peak at around 1600 cm–1 corresponds to the bending mode.
Figure 4. Infrared spectrum of liquid water (from webbook.nist.gov).
A gas phase infrared spectrum of water is shown in Figure 5. Much more spectroscopic detail is generally observed
in gas phase spectra. For example, what appears to be two peaks centered at about 1600 cm–1 in the gas phase water
spectrum is actually a single band, divided into branches, corresponding to the bending mode. The extra detail
observed in the gas phase spectrum is a result of rotational structure that is washed out in the liquid phase spectrum.
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Figure 5. Infrared spectrum of gas phase water (from webbook.nist.gov).
EXAMPLE 2: Carbon Dioxide
Carbon dioxide has four vibrational modes since it is a linear triatomic molecule. These modes are summarized in
the table below.
Frequency (cm–1)
1390
2350
670
Mode
symmetric stretch
antisymmetric stretch
degenerate bend (2 modes)
IR activity
not IR active
IR active
IR active
In order to explain the IR activity of each of the modes, we have to consider the dipole moment. Because CO2 is
linear and symmetric in structure, its equilibrium dipole moment is zero.
A sketch of the vibrational motion corresponding to the symmetric stretching mode of CO2 is shown in Figure 6.
Figure 6. The symmetric stretching mode of carbon dioxide
O
O
C
C
O
C
O
O
C
Symmetric bond stretching
Equilibrium
Symmetric bond contracting
O C O
O
Equilibrium
O
O
Equilibrium
For the symmetric stretch, the dipole moment does not change, since stretching or contracting the C-O bonds in a
symmetric fashion does not break the symmetry of the molecule. Thus, the dipole moment remains zero throughout
the vibration and the symmetric stretching mode is not IR active.
A sketch of the vibrational motion corresponding to the antisymmetric stretching mode of CO2 is shown in Figure
7.
Figure 7. The antisymmetric stretching mode of carbon dioxide
O
C
O C
O
O
C
O
O
C
Antisymmetric bond stretching
Equilibrium
Antisymmetric bond stretching
C O
O
Equilibrium
O
O
Equilibrium
For the antisymmetric stretching mode, when the C-O bonds are in their equilibrium positions, the dipole moment is
zero. However, when one C-O bond is stretched and the other is contracted, the symmetry of the charge distribution
is broken, and the molecule gains a dipole moment. Therefore, the antisymmetric stretching mode is IR active.
A sketch of the vibrational motion corresponding to one of the degenerate bending modes is shown in Figure 8. The
other degenerate bend is identical to the one shown, but it is at right angles to the one shown and out of the plane.
Figure 8. One of the bending modes of carbon dioxide
O
C
O
C
O
O
O
C
O
O
O
C
O
C
O
Equilibrium
Angle bending
Equilibrium
Angle bending
Equilibrium
The molecule begins its bending motion at equilibrium and the dipole moment is zero. When the angle bends, the
center of negative charge is shifted down so that it is no longer in line with the center of positive charge; thus, the
molecule acquires a dipole moment. This mode is IR active because the dipole moment changes as the molecule
bends.
For carbon dioxide, three of the four vibrational modes are infrared active: the antisymmetric stretch and the two
degenerate bends. Therefore, those three should be observed in the infrared spectrum of CO2; however, since the
degenerate bends occur with the same harmonic vibrational frequency, only two primary peaks will be observed.
An example infrared spectrum for carbon dioxide is shown in Figure 9. The peak at about 670 cm–1 corresponds to
the degenerate bending modes; the other large peak around 2300 cm–1 corresponds to the antisymmetric stretching
mode. The small peaks observed in the range 3600-3800 cm–1 correspond to combination bands.
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Figure 9. Infrared spectrum of gas phase carbon dioxide (from webbook.nist.gov).
EXAMPLE 3: Acetylene
Acetylene has seven vibrational modes since it is a linear tetratomic molecule. These vibrations are summarized in
the table below.
Frequency (cm–1)
3370
3290
1970
610
730
Mode
symmetric C-H stretch
antisymmetric C-H stretch
C-C stretch
symmetric C-C-H bend (2 modes)
antisym C-C-H bend (2 modes)
IR activity
not IR active
IR active
not IR active
not IR active
IR active
A sketch of the vibrational motion corresponding to the symmetric C-H stretching mode of acetylene is shown in
Figure 10.
Figure 10. Symmetric C-H stretching mode of acetylene
H
H
H
C
C
C
C
C
C
H C
H
C
H
H
H
C H
C
Equilibrium
Symmetric C-H bond stretching
Equilibrium
Symmetric C-H bond contracting
H
Equilibrium
Since acetylene is a linear molecule, its equilibrium dipole moment is zero. The dipole moment does not change
during the symmetric C-H stretching mode, since stretching or contracting the C-H bonds in a symmetric fashion
does not break the symmetry of the molecule. Thus, the dipole moment remains zero throughout the vibration and
the symmetric C-H stretching mode is not IR active.
A sketch of the vibrational motion corresponding to the antisymmetric C-H stretching mode of acetylene is shown
in Figure 11.
Figure 11. Antisymmetric C-H stretching mode of acetylene
H
C
C
H C
C
C
C
C
C H
C
C
H
H
H
H
Equilibrium
H
H
Antisymmetric C-H bond stretching
Equilibrium
Antisymmetric C-H bond contracting
H
Equilibrium
In this case, when one C-H bond is stretched and the other is contracted, the symmetry of the charge distribution is
broken, and the molecule gains a dipole moment. Therefore, the antisymmetric C-H stretching mode is IR active.
A sketch of the vibrational motion corresponding to the C-C stretching mode of acetylene is shown in Figure 12.
Figure 12. C-C stretching mode of acetylene
H
H
C
C
C
H
H
H
H
C
C
C
C C
C
H
H
Symmetric C-C bond stretching
Equilibrium
Symmetric C-C bond contracting
H
C
Equilibrium
H
Equilibrium
The molecule begins its stretching motion at equilibrium with a dipole moment of zero. When the C-C bond
stretches or contracts, the dipole moment remains zero. Since there is no change in dipole moment during the C-C
stretch, this mode is not IR active.
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A sketch of the vibrational motion corresponding to one of the degenerate symmetric C-C-H bending modes of
acetylene is shown in Figure 13.
Figure 13. One of the symmetric C-C-H bending modes of acetylene
H
C
C
C
C
H
Equilibrium
H
Symmetric C-C-H bending
H
H
C
C
H
Equilibrium
H
C
C
C
C
Symmetric C-C-H bending
H
H
H
Equilibrium
The symmetric bending of the C-C-H bond produces no change in the dipole moment. It starts at zero and it
remains zero; therefore, the mode is not IR active.
A sketch of the vibrational motion corresponding to one of the degenerate antisymmetric C-C-H bending modes of
acetylene is shown in Figure 14.
Figure 14. One of the antisymmetric C-C-H bending modes of acetylene
H
C
C
H
H
Equilibrium
H
C
C
C
C
C
C
H
H
H
Antisymmetric C-C-H bending
H
Equilibrium
Antisymmetric C-C-H bending
H
C
C
H
Equilibrium
Since both hydrogens move to one side of the molecule during the course of the vibration, the antisymmetric C-C-H
bending modes are IR active.
For acetylene, only three of the seven vibrational modes are infrared active: the antisymmetric C-H stretch and the
doubly degenerate antisymmetric C-C-H bends. Therefore, those three should be observed in the infrared spectrum
of CO2; however, since the degenerate bends occur with the same harmonic vibrational frequency, only two primary
peaks will be observed. An example infrared spectrum for acetylene is shown in Figure 15. The large peak at about
700 cm–1 corresponds to the doubly degenerate antisymmetric C-C-H bending modes. The peak at about 3300 cm–1
corresponds to the antisymmetric C-H stretching mode (the "doublet"-like appearance is due to unresolved rotational
structure). The small peak observed in the range 1200-1400 cm–1 is an overtone or combination band.
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Figure 15. Infrared spectrum of gas phase acetylene (from webbook.nist.gov).