Subject Chemistry
Paper No and Title 8/ Physical Spectroscopy
Module No and Title 25/ Vibrational Raman spectroscopy. Selection rules. Mutual Exclusion Principle. Polarization of Raman lines. Module Tag CHE_P8_M25
CHEMISTRY PAPER No. : 8 (PHYSICAL SPECTROSCOPY) MODULE NO. : 25 (VIBRATIONAL RAMAN SPECTROSCOPY. SELECTION RULES. MUTUAL EXCLUSION PRINCIPLE. POLARIZATION OF RAMAN LINES.)
TABLE OF CONTENTS
1. Learning Outcomes 2. Introduction
3. Vibrational Raman Spectroscopy
4. Polarizability ellipsoids
5. Rule of Mutual Exclusion
6. Polarization of Raman Lines
7. Summary
CHEMISTRY PAPER No. : 8 (PHYSICAL SPECTROSCOPY) MODULE NO. : 25 (VIBRATIONAL RAMAN SPECTROSCOPY. SELECTION RULES. MUTUAL EXCLUSION PRINCIPLE. POLARIZATION OF RAMAN LINES.)
1. Learning Outcomes
After going through this module, you should be able to:
(a) Understand the vibrational Raman spectra (b) Understand the variation of the polarizability ellipsoid of a molecule during a vibration of triatomic molecules. (c) Understand and apply the Rule of Mutual Exclusion. (d) Understand the polarization of Raman lines.
2. Introduction
When a beam of monochromatic radiation passes through a liquid or gaseous substance, it might get scattered. Either the frequency of the radiation is unchanged (Rayleigh scattering) or the scattered radiation has lower frequency (Stokes lines) or higher frequency (anti-Stokes lines). Just as for rotational Raman spectroscopy, the important molecular property is the polarizability which must change during the vibration. The polarizability ellipsoid should either change in shape or size during a vibration for it to be Raman active.
In this module, we study vibrational Raman spectroscopy.
3. Vibrational Raman Spectroscopy
For every vibrational mode one writes:
2 G(υ) = ωe (υ +1/ 2) − ωe χ e (υ +1/ 2) (υ = 0 ,1 , ...) (1)
where ωe is the equilibrium vibrational frequency in reciprocal centimetre and χ e is the anharmonicity constant. The selection rule is:
CHEMISTRY PAPER No. : 8 (PHYSICAL SPECTROSCOPY) MODULE NO. : 25 (VIBRATIONAL RAMAN SPECTROSCOPY. SELECTION RULES. MUTUAL EXCLUSION PRINCIPLE. POLARIZATION OF RAMAN LINES.)
Δυ = 0, ±1, ±2, … (2)
which are the same as for Raman and IR spectroscopy. The probability of Δυ = ±2, ±3, … decreases rapidly and may be ignored. Application of the selection rules to equation (1) yields:
υ = 0 → υ = 1 ~ (3) ν = ωe (1− 2χ e )
We restrict ourselves to fundamental transitions only. We would expect Raman lines to appear at distances from the exciting line corresponding to each active fundamental vibration. Therefore,
~ ~ ~ ν fundamental =ν ex ±ν (4)
The vibrational Raman spectrum of a molecule will then show a series of reasonably intense lines. The minus represents the situation where the molecule has gained energy (Stokes lines), and the plus indicates when the molecule has lost energy (anti-Stokes lines). Stokes lines refer to the transition υ = 0 → 1 and anti-Stokes lines refer to the υ = 1 → 0 transition (Fig. 1). Since there are very few molecules initially in the excited state at room temperature, anti-Stokes lines are much weaker in intensity.
CHEMISTRY PAPER No. : 8 (PHYSICAL SPECTROSCOPY) MODULE NO. : 25 (VIBRATIONAL RAMAN SPECTROSCOPY. SELECTION RULES. MUTUAL EXCLUSION PRINCIPLE. POLARIZATION OF RAMAN LINES.)
Figure 1: Energy level diagram showing the states involved in Raman signal. The line thickness is roughly proportional to the signal strength from the different transitions. Source: http://en.wikipedia.org/wiki/Raman_ spectroscopy
4. Polarizability Ellipsoids
In this section we try to show pictorially the polarizability ellipsoids as changed by various vibrational modes. Note that the ellipsoids shown here are only two-dimensional representations.
Consider first the hydrogen molecule. Its polarizability ellipsoid is shown in Figure 2.
CHEMISTRY PAPER No. : 8 (PHYSICAL SPECTROSCOPY) MODULE NO. : 25 (VIBRATIONAL RAMAN SPECTROSCOPY. SELECTION RULES. MUTUAL EXCLUSION PRINCIPLE. POLARIZATION OF RAMAN LINES.)
Figure 2: Polarizability ellipsoid of the hydrogen molecule
As the bond is stretched, the bond becomes more polarizable and the polarizability ellipsoid shrinks in size. Note that here there is a change in size of the polaizability ellipsoid. During compression, it expands in size, and we thus have a means of determining the strengths of bonds of homonuclear molecules.
The rotational selection rules are the same as those for pure Raman rotations, i.e. ΔJ = 0, ±2, and we observe an OS pattern instead of the PR pattern observed in infrared spectroscopy.
Polyatomic Molecules
Polyatomic molecules have a larger number of vibrational degrees of freedom as compared to diatomic molecules. An N-atomic molecule has a total of 3N degrees of freedom corresponding to three coordinates required to describe the position of each atom. Three of these correspond to translation of the molecule as a whole. Linear molecules have two rotational degrees of freedom as well, leaving the remaining 3N - 5 degrees of freedom for vibration. Non-linear molecules have three rotational degrees of freedom and consequently there are 3N – 6 vibrational degrees of freedom.
CHEMISTRY PAPER No. : 8 (PHYSICAL SPECTROSCOPY) MODULE NO. : 25 (VIBRATIONAL RAMAN SPECTROSCOPY. SELECTION RULES. MUTUAL EXCLUSION PRINCIPLE. POLARIZATION OF RAMAN LINES.)
We first consider a linear triatomic molecule, carbon dioxide, which has four vibrational degrees of freedom. Two of these are degenerate bending modes. The symmetric stretch mode is similar to the stretching mode of a diatomic molecule, and the polarizability ellipsoid varies in size as the molecule vibrates, and this mode of vibration is consequently Raman active (Fig. 3). In fact, the rate of change of polarizability with the normal coordinate is very large and symmetric vibrations are very intense in Raman spectroscopy.
Vibrational modes of carbon dioxide -Q Q = 0 +Q Symmetric stretching mode
ν1
Bending mode
ν2
CHEMISTRY PAPER No. : 8 (PHYSICAL SPECTROSCOPY) MODULE NO. : 25 (VIBRATIONAL RAMAN SPECTROSCOPY. SELECTION RULES. MUTUAL EXCLUSION PRINCIPLE. POLARIZATION OF RAMAN LINES.)
Antisymmetric stretching mode
ν3
Figure 3: Changes in the polarizability ellipsoid of the carbon dioxide molecule during its vibrations
Consider next the bending mode. Only bending in plane is shown in Figure 3. The molecule changes its appearance from a linear molecule to a bent one during the course of the vibration. Obviously, there is a change in polarizability as the molecule vibrates. In the antisymmetric stretch, too, the polarizability ellipsoid decreases in size during the vibration. However, both these modes are Raman inactive, as we can see from application of Group Theory. Both these vibrations have a symmetric polarizability change and are Raman inactive.
We now consider the bent triatomic molecule. The variation of the polarizability ellipsoid (Fig. 4) clearly indicates that all modes are Raman active.
Vibrational modes of water
-Q Q = 0 +Q Symmetric stretching mode
ν1
CHEMISTRY PAPER No. : 8 (PHYSICAL SPECTROSCOPY) MODULE NO. : 25 (VIBRATIONAL RAMAN SPECTROSCOPY. SELECTION RULES. MUTUAL EXCLUSION PRINCIPLE. POLARIZATION OF RAMAN LINES.)
Bending mode
ν2
Antisymmetric stretching mode
ν3
CHEMISTRY PAPER No. : 8 (PHYSICAL SPECTROSCOPY) MODULE NO. : 25 (VIBRATIONAL RAMAN SPECTROSCOPY. SELECTION RULES. MUTUAL EXCLUSION PRINCIPLE. POLARIZATION OF RAMAN LINES.)
Figure 4: Changes in the polarizability ellipsoid during the normal vibrations of the water molecule
5. Rule of Mutual Exclusion
For carbon dioxide, the bending and antisymmetric modes are infrared active, while the symmetric stretch mode is Raman active. This behaviour is typical of all centrosymmetric molecules. Modes that are infrared active are Raman inactive and vice versa. This is the Rule of Mutual Exclusion, which states that no normal mode can be both infrared and Raman active in a molecule that possesses a centre of symmetry.
In fact, even for molecules which do not possess a centre of symmetry, symmetric modes are weak in infrared and strong in Raman, whereas bending and asymmetric modes are weak in Raman.
If there is no centre of symmetry, then some vibrations will be both Raman and IR active. Hence, if some vibrations are observed to give coincident Raman and IR absorptions, it is certain that the species has no centre of symmetry. Also, all symmetric modes of centrosymmetric molecules are strong in Raman, and this helps in identifying symmetric bands. The symmetric bands can also be identified by studying their polarization.
6. Polarization of Raman Lines
We consider a beam of light incident in the x-direction. Assume that this is linearly polarized and all its electric vectors point in the y-direction. Let the direction of observation of the scattered radiation be z (see Fig. 5).
CHEMISTRY PAPER No. : 8 (PHYSICAL SPECTROSCOPY) MODULE NO. : 25 (VIBRATIONAL RAMAN SPECTROSCOPY. SELECTION RULES. MUTUAL EXCLUSION PRINCIPLE. POLARIZATION OF RAMAN LINES.)
Figure 5: Polarization of Raman lines
Let us first consider a molecule, such as CH4, whose polarizability ellipsoid is isotropic, i.e. spherical in shape. The scattered radiation will then have the same polarization as the incident radiation, since the molecule’s polarizability has no preferred direction. What happens if the molecule’s polarizability ellipsoid is anisotropic? As an example, take CH3Cl. The molecule is more polarizable along the C-Cl bond direction and rotates the plane of polarization of the radiation along the C-Cl bond direction. However, in a sample of gaseous chloroform molecules, the molecules are oriented with their C-Cl bonds in random directions, so that the scattered radiation is completely depolarized.
The depolarization ratio is defined as the ratio of intensity measured perpendicular to the incident radiation and that measured parallel to it. In the present example, this would
be ρ = I ⊥ / I // = I x / I y . Calculations show that a scattered beam is polarized if 0 ≤ ρ ≤ ¾.
When we consider vibrations of molecules, the same concept can be applied. All molecules have a symmetric stretching mode and this is always Raman active and very intense. Thus all molecules have at least one strong Raman mode, which cannot be said about infrared activity, since homonuclear diatomic molecules do not have any infrared active mode. In the case of methane, for example, this symmetric stretching mode involves the simultaneous elongation and CHEMISTRY PAPER No. : 8 (PHYSICAL SPECTROSCOPY) MODULE NO. : 25 (VIBRATIONAL RAMAN SPECTROSCOPY. SELECTION RULES. MUTUAL EXCLUSION PRINCIPLE. POLARIZATION OF RAMAN LINES.)
compression of all four C-H bonds. The polarizability ellipsoid therefore alternately decreases and increases in size, but remains spherical in shape. For this reason, this mode is called a breathing mode. Applying the same arguments as above, this mode results in polarized radiation. On the other hand, in the asymmetric stretch mode, where one bond elongates while others compress and vice versa, the polarizability ellipsoid loses its spherical symmetry during the vibration, and the scattered radiation is depolarized.
Example 1
A linear molecule has the following fundamental vibrational transitions: 370 cm-1, 520 cm-1, 880 cm-1 and 1550 cm-1 (all Raman active) and the wavelength of the excitation radiation is 500 nm.
Calculate the wavenumbers of scattered radiation in Stokes range.
Solution
The wavenumber of 500 nm = 1/(500×10-7 cm) = 20,000 cm-1
Therefore, the Stokes lines are at 20000 – 370 = 19,630 cm-1, 20000 - 520 = 19,480 cm-1, 20000 – 880 = 19,120 cm-1 and 20,000 – 1550 = 18,450 cm-1.
7. Summary
• The molecular property that must change during a vibration for it to be Raman active is its polarizability. • Thus, Raman spectroscopy is complementary to microwave and infrared spectroscopy, where the property that must vary is the dipole moment. • Anti-Stokes lines are much weaker than Stokes lines. • The Rule of Mutual Exclusion states that no normal mode can be both infrared and Raman active for a molecule possessing a centre of symmetry. • The polarization of Raman lines gives information about the symmetry of a vibration.
CHEMISTRY PAPER No. : 8 (PHYSICAL SPECTROSCOPY) MODULE NO. : 25 (VIBRATIONAL RAMAN SPECTROSCOPY. SELECTION RULES. MUTUAL EXCLUSION PRINCIPLE. POLARIZATION OF RAMAN LINES.)