Chemistry 362
Fall 2015
Dr. Jean M. Standard
September 30, 2015
Notes on Raman Spectroscopy
Introduction
Raman spectroscopy is a complementary technique to infrared spectroscopy. It can often be used to observe
vibrational modes that are inactive in infrared spectroscopy. There is also a technique to observe rotational Raman
spectra.
Raman spectroscopy is not based on absorption of photons, but rather on scattering of photons by the molecular
sample. A very small amount of light is scattered by the sample, typically less than one part in one million.
Because only a very small fraction of light is scattered, Raman spectroscopy has only become common since the
advent of very powerful light sources such as lasers, which can be used to produce the incident beam.
The Raman Equation
In Raman spectroscopy, an incident light beam of a given frequency is sent into the sample. Light is scattered by the
molecular sample. The specific frequencies of the scattered light are measured using a detector. The basic equation
for Raman spectroscopy is
hν + Ei = hν # + E f ,
where ν is the frequency of the incident photon, ν " is the frequency of the scattered photon, Ei is the initial energy
state of the molecule, and E f is the€final energy state of the molecule. This equation simply balances the initial and
final energies of the total system (molecule plus photons).
€
€
€
The scattered photon
may
be
either
lower
or
higher
frequency
than
the
incident
photon.
Thus, if ν " < ν , the
€
scattered photon is lower in energy than the incident photon, which implies that the molecule gained energy. Such
transitions are called Stokes transitions.
€
If ν " > ν , then the scattered photon is higher in energy than the incident photon, and the molecule lost energy.
These transitions are referred to as anti-Stokes transitions.
Raman Selection Rules
The selection rules for polyatomics say that the quantum number v i has to change by an integer (but not zero)
during a vibrational Raman transition. However, the other quantum numbers for all the other modes stay the same;
only one mode has to change. Therefore, the selection rules can be given as
€
€
Δv i = ± 1
and Δv j = 0 for j ≠ i ,
€
where v i is the quantum number for the ith vibrational mode of the polyatomic molecule.
€
€
Raman Shifts: Vibrational Raman Spectra
Now, using the Raman equation from above, we can develop an expression for the Raman frequency shift Δν
relative to the incident frequency in a vibrational Raman spectrum. Rearranging the Raman equation, we have
h(ν " − ν ) = Ei − E f .
€
Solving for the Raman shift, Δν = ν $ − ν , leads to the equation
€
Δν = ν $ − ν =
€
Ei − E f
h
.
Now, using the vibrational selection rule Δv i = ± 1 along with the expression for the vibrational energy in the
harmonic oscillator limit, the Raman
€ shift for a transition from v""i → v"i is
€
Δv = ν 0 i ( v$i − v$i ) = ∓ ν 0i .
€
In this equation, ν 0i is the harmonic vibrational frequency of the ith vibrational mode of the polyatomic molecule.
€
Example: Raman Spectrum of Carbon Tetrachloride
An example
vibrational Raman spectrum of CCl4 is shown in Figure 1.
€
Figure 1. Raman spectrum of liquid CCl4. (From R. A. Alberty and R. J. Silbey, Physical
Chemistry, 2nd Ed., John Wiley and Sons, New York, 1997, p. 475.)
The Raman spectrum of CCl4 shows the Raman shifts (in cm–1) measured relative to the incident photon frequency
(shown in the figure at 0 cm–1). The peaks to the left have lower scattered photon frequencies, and correspond to
Stokes transitions. The peaks to the right have higher scattered photon frequencies, and correspond to anti-Stokes
transitions. The anti-Stokes transitions are lower in intensity than the Stokes transitions because they involve
transitions from higher energy levels that have lower populations.
The different Raman shifts of approximately 230, 310, and 470 cm–1 correspond to the harmonic vibrational
frequencies of three of the vibrational modes of the CCl4 molecule that exhibit Raman activity.
2
Raman Shifts: Rotational Raman Spectra
For rotational Raman transitions, the selection rule for diatomic, linear, and spherical top molecules is that ΔJ = ±2 .
The selection rule for symmetric (both oblate and prolate) top molecules is more complicated because of the K
quantum number. If K = 0 , the selection rule is that ΔJ = ±2 , while if K ≠ 0 , the selection rule is that ΔJ = ±1, ± 2
.
€
The Raman€equation for diatomic, linear, and
for a transition from J "€→ J " is
€ spherical top molecules
€
Δν = ν $ − ν = − 2cBe ( 2J $ + 3) for a Stokes transition,
€
Δ
ν
=
ν
$
−
ν
=
2cB
2
J
$
−
1
and
) for an anti-Stokes transition.
e(
€
Raman Activity: Polarizability Changes
Another criterion for€a transition to be observed in a Raman spectrum is that there is a change in the polarizability of
the molecule during the vibrational motion. For example, the "breathing" mode in CH4, in which all the C-H bonds
expand and contract together is Raman active since the polarizability changes as the size of the molecule changes
upon vibration. However, the breathing mode of CH4 would not be infrared active, since the dipole moment of the
molecule does not change.
As a special case, if a molecule has a center of symmetry, such as CO2 or C2H2, then all the modes that are infrared
active are Raman inactive. Modes that are not active in the infrared are active in the Raman spectrum. The rest of
this handout describes a few simple examples of the vibrational modes of polyatomic molecules and their Raman/IR
activity.
EXAMPLE 1: Carbon Dioxide
Carbon dioxide has 4 vibrational modes since it is linear. These are summarized in the table below.
Mode
symmetric stretch
antisymmetric stretch
degenerate bend (2 modes)
Frequency (cm–1)
1390
2350
670
IR activity
Raman active
IR active
IR active
The symmetric stretch of CO2 is Raman active. This is because CO2 has a center of symmetry and the symmetric
stretch is not IR active because the dipole moment does not change during the vibration; therefore, the mode must be
Raman active. Alternately, it is easy to see that the symmetric stretch mode changes polarizability of the molecule,
so it is Raman active.
Example 2: Water
Water has 3 vibrational modes since it is nonlinear. These vibrations are summarized in the table below.
Mode
symmetric stretch
antisymmetric stretch
H-O-H bend
Frequency (cm–1)
3660
3760
1590
IR activity
IR, Raman active
IR, Raman active
IR, Raman active
As we have seen before, all the modes of water are IR active, since the dipole moment changes during each of the
vibrations. Water does not possess a center of symmetry, so modes that are IR active are not necessarily Raman
inactive. We have to consider how the polarizability varies during each vibration. It is not hard to imagine that all
the vibrational modes lead to changes in the polarizability; hence, all are also Raman active.
3
Example 3: Acetylene
Acetylene has 7 vibrational modes, since it is linear. These vibrations are summarized in the table below.
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)
Frequency (cm–1)
3370
3290
1970
610
730
IR activity
Raman active
IR active
Raman active
Raman active
IR active
Acetylene has a center of symmetry, so any modes that are IR inactive will be Raman active. We already
established that the antisymmetric C-H stretch and the degenerate antisymmetric C-C-H bend were the only IR
active modes. The others, including the symmetric C-H stretch, the C-C stretch, and the degenerate C-C-H bend, are
all Raman active.
4