Subject Chemistry Paper No and Title Module No and Title Module Tag 8/ Physical Spectroscopy 23/ Normal modes and irreducible representations for polyatomic molecules CHE_P8_M23
TABLE OF CONTENTS 1. Learning Outcomes 2. Introduction 3. Vibrational Modes of Molecules 4. Summary
1. Learning Outcomes After going through this module, you should be able to: (a) Visualize the vibrations of linear and bent triatomic molecules. (b) Assign the normal modes to irreducible representations. 2. Introduction Diatomic molecules possess one mode of vibration, i.e. stretching. Polyatomic molecules have many more bonds, bond angles and torsions, so that there are many more modes of vibration. It is often difficult to visualize these normal modes. Group Theory not only helps in determining the modes of vibration, but it also helps in finding out which of these is infrared active. 3. Vibrational Modes of Molecules Vibrations in molecules comprise changes in bond lengths and/or bond angles and dihedral angles. An N-atomic molecule has 3N degrees of freedom since each atom needs three coordinates to define its position completely. Three of these are independent degrees of translational motion in the three directions. Three others comprise rotational motion, leaving a total of 3N - 6 vibrational degrees of freedom. However, linear molecules have only two rotational degrees of freedom since rotation about the internuclear axis has zero moment of inertia. This leaves 3N 5 vibrational degrees of freedom for linear molecules. Let us now have a look at the vibrations of two triatomic molecules. EXAMPLES 1. The H 2 O molecule This molecule has three atoms and is nonlinear, hence it has (3 3-6) = 3 modes of vibrational motion. There are two bonds, so there should be two stretching modes. Either the two bonds should be stretching at the same time, or one should compress while the other stretches. The first is called symmetric stretch (ν 1 ) and the other is the antisymmetric stretch (ν 3 ). There is also one bond angle and hence one bending mode (ν 2 ). These are depicted in Figure 1.
Note that the bending mode is at a lower frequency because it is easier to bend a molecule than to stretch a bond. O O O H H H H H H ν 1 ν 2 ν 3 3657 cm -1 1595 cm -1 3756 cm -1 Figure 1: Vibrational modes of water For diatomic molecules, the only mode of vibration has symmetry Σ g + (homonuclear) or Σ + (heteronuclear). Let us see if the normal modes of water also form bases for irreducible representations of the C 2v point group to which the molecule belongs. The symmetry elements of water are as follows: C 2v E C 2 (z) σ v (xz) σ v (yz) A 1 1 1-1 -1 Z x 2, y 2, z 2 A 2 1 1-1 -1 R z xy B 1 1-1 1-1 x, R y xz B 2 1-1 -1 1 y, R x yz With this convention, we find that the first two vibrations transform as A 1 and the third as B 2. For example, the result of a C 2 operation on ν 2 and ν 3 is as under: C O 2 O H H H H ν 2 ν 2
H O H C 2 H O H ν 3 ν 3 ꞌ and the character of C 2 is +1 for ν 2 and -1 for ν 3, since the arrows change direction for ν 3, i.e. ν 3 ꞌ = -ν 3. The vibrations are labelled with the highest symmetry first, or in the order they appear in the character table. Here, the first two vibrations transform as A 1 (most symmetrical), where the one with the higher frequency (symmetric stretch) is labelled as ν 1 and the other one (bending mode) as ν 2. The asymmetric stretch is labelled as ν 3. The vibrational motion that occurs for the three vibrational modes of water is shown in Figure 2. Water, being a nonlinear triatomic molecule, has a nonzero equilibrium dipole moment. During the symmetric stretching mode, the dipole moment increases when the bond stretches, and decreases when the bond contracts. Thus, the dipole moment changes during the vibration and the symmetric stretching mode is infrared active. -Q Q = 0 +Q Symmetric stretching mode ν 1 Bending mode ν 2 Antisymmetric stretching mode ν 3 Figure 2: Vibrational modes of water
In the bending mode, the dipole moment reduces when the bond angle increases (left) and increases when the angle becomes smaller. Hence, this mode is also infrared active. In the antisymmetric stretch, the dipole moment vector changes direction from the two-fold axis to the left or right as the molecule vibrates, and thus this mode is also infrared active. Therefore, all three bands should be infrared active. The infrared spectrum of liquid water is shown in Figure 3. Figure 3: The infrared spectrum of liquid water (from webbook.nist.gov)
The small peak near 1600 cm -1 corresponds to the bending mode, whereas the broad peak corresponds to the two symmetric stretches, which are close in frequency. Figure 4: The gas phase infrared spectrum of water (from webbook.nist.gov) The gas phase spectrum has many more features (Fig. 4). This is because the rotational fine structure is lost in the liquid phase where the molecules are too close to each other for rotation to be possible. The rotational fine structure is clearly visible in the bending mode at 1595 cm -1. 2. The CO 2 molecule This one is also triatomic, but linear and has (3 3-5) = 4 modes of vibrational motion (Fig. 5).
Figure 5: Vibrational modes of CO 2 Carbon dioxide belongs to the D h point group. The character table is shown below: On examination of the symmetric stretch, we find that under the various operations of the point group it transforms as: and hence forms a basis for the Σ g + representation. Similarly, the asymmetric stretch transforms as Σ u + as shown below. The bending modes (Fig. 5) transform as the doubly degenerate Π u representation. Since carbon dioxide is a linear, symmetric molecule, it has no dipole moment. In the symmetric stretching mode, either the two C=O bonds stretch or contract simultaneously. This does not break the symmetry of the charge distribution and this vibration is infrared inactive. CO 2 has two IR active bending modes: the in-plane and the out-of-plane bending of the carbon-oxygen bonds.
When the molecule bends, the molecule starts resembling water and acquires a dipole moment. This dipole moment is in a direction perpendicular to the bond axis and this mode is thus infrared active. In the antisymmetric stretch, one bond stretches while the other is compressed and vice versa. There is thus a periodic alteration in the dipole moment and the vibrational mode is also IR active. ν 1 -Q Q = 0 +Q Symmetric stretching mode Bending mode ν 2 ν 3 Asymmetric stretching mode Figure 6: The vibrational modes of carbon dioxide The modes are numbered in the same way as bent molecules: ν 1 = symmetric stretch, ν 2 = bending mode, ν 3 = antisymmetric stretch. Carbon dioxide thus has three infrared active modes, the two bending modes and an antisymmetric stretch. However, since the two bending modes are degenerate, only two absorptions should be observed. Part of the infrared spectrum of carbon dioxide is shown in Figure 7.
Figure 7: The infrared spectrum of carbon dioxide (from webbook.nist.gov) The peak at 673 cm -1 is due to the degenerate bending modes. The one at 2350 cm -1 corresponds to the antisymmetric stretching mode. The small peaks at high wavenumber are due to combination bands (simultaneous excitation of two or more vibration modes). These peaks have a distinctive shape, called PR contour, which is encountered in diatomic molecules. In fact, this is a characteristic of all linear molecules. The appearance of a PR contour in the IR spectrum of a molecule confirms that the molecule is linear. The contour of the bending mode is also distinctive and called a PQR contour. We have already stated that the dipole moment change for this mode is perpendicular to the bond axis. Such modes are called perpendicular ( ) modes as opposed to parallel ( ) modes where the dipole moment change is parallel to the bond axis, such as the antisymmetric stretching mode. The rotational selection rules also differ for the two modes. For the perpendicular mode, the selection rule is J = 0, ±1, i.e. we can observe a pure vibrational transition, whereas for parallel modes, the selection rule is J = ±1, i.e. a vibrational transition has to be accompanied by a rotational transition. For the fundamental υ = 0 1 transition, the transition wavenumber is given by ~ ν S ( υ') S( υ") = F( J') + G( υ') F( J") G( υ") = ω (1 2x ) = ω (1) spectral = e e 0 where we have substituted Jꞌ = J, υꞌ = 1 and υ = 0 in the expressions for the rotational and vibrational terms. This equation is valid for all J values. Thus, the Q branch consists of coincident
lines at the band centre, and we expect a very intense line at the band centre. However, as shown in Figure 8, the Q-branch appears as a somewhat broad absorption centered around ω 0. Figure 8: PQR band contour This is due to the difference in the rotational constants for the ground and excited vibrational ~ ~ levels. On writing B " and B ', respectively, for the two rotational constants, equation (1) gets modified to ~ ~ ~ ν = ω + ( B' B") J"( J" 1) (2) spectral 0 + ~ ~ The second term is slightly negative since B " > B ', and increases with increasing J. This explains the slight shift to the left with increasing J. In summary, the infrared spectrum of carbon dioxide shows two bands, one bending mode at 667 cm -1 with a PQR contour, and another at 2350 cm -1 showing a PR contour, corresponding to the asymmetric stretch motion. We may now define normal modes as independent, harmonic vibrations which: 1. leave the centre of mass unmoved; 2. involve all atoms moving in phase (coherent motion);
3. transform as an irreducible representation of the molecular point group. 4. Summary In summary, The number of modes of vibration is 3N-6 for bent molecules and 3N-5 for linear molecules. Bending modes occur at lower frequency and antisymmetric stretches at the highest frequency. The modes of vibration are bases for the irreducible representations of the point group of the molecule. For linear molecules, parallel modes display a PR contour in the rotation-vibration spectra, and perpendicular modes display a PQR contour.