Application of Group Theory to IR Spectroscopy

Selection Rules:

Selection rules dictate the number of electronic transitions, from one quantum state to another, that are possible for a given molecule. IR spectroscopy probes vibrational transitions from the molecule's ground state, v = 0, to the first excited state, v = 1. The number of degrees of vibrational freedom (normal modes of vibration) for linear and non-linear molecules can be calculated using Equation 1 and Equation 2, respectively.

3N - 5 (1)

3N - 6 (2)

where N = the number of atoms in the molecule

For a normal mode of vibration to be IR active, the molecule's dipole must change. Therefore, any normal modes of vibration where a change in dipole does not occur are IR inactive. The number of active IR modes can be determined using group theory.

Group Theory:

Chemists use group theory to understand the relationship between the symmetry and physical properties of a molecule. While the scope of group theory is too broad to rigorously cover in this video, we will provide the necessary tools needed to apply group theory to simple coordination complexes and show how it can be used to predict the number of IR active vibrational modes. To demonstrate, we will walk through the application of group theory to the molecule cis-Mo(CO)₄[P(OPh)₃]₂.

First, we need to determine the point group of the molecule. Point groups are used to describe the symmetry elements present in a given molecule. To determine the point group of cis-Mo(CO)₄[P(OPh)₃]₂, we can use a flow chart called a symmetry tree, which asks a series of questions about the symmetry elements present in the molecule (Figure 2). Table 1 summarizes all of the symmetry elements included in the symmetry tree. Using the symmetry tree, and assuming that the ligands P(OPh)₃ are point ligands (while ignoring the symmetry of those ligands), we find that cis-Mo(CO)₄[P(OPh)₃]₂ is in the point group C_2v.

Figure 2. Symmetry tree used for point group determination.

Table 1. Symmetry elements used in point group determination.

Symmetry element	Symbol used	Example*
Identity	E
Rotation axis (rotation by 360°/n)	Cn
Horizontal mirror plane (reflection about xy plane)	σh
Vertical mirror plane (reflection about xz or yz plane)	σv
Diagonal mirror plane (reflection between xz and yz planes)	σd
Inversion Center	i
Improper rotation axis (rotation by 360°/n followed by reflection perpendicular to rotation axis)	Sn
*Examples are for an octahedral complex, where ligands 1–6 are equivalent. Upon performing the operation, the resulting molecule should be indistinguishable from its original configuration.

For the next step, we need to introduce character tables, which describe all of the symmetry present within a given point group. The character table for the point group C_2v is shown below.

C2v	E	C2	σv(xz)	σv’(yz)
A1	1	1	1	1	z	x2, y2, z2
A2	1	1	−1	−1	Rz	xy
B1	1	−1	1	−1	x, Ry	xz
B2	1	−1	−1	1	y, Rx	yz

The point group is indicated in the top left-hand corner of the character table. To the right of the point group, all of the symmetry operations inherent to that point group are listed. The subsequent rows list all of the symmetry representations (irreducible representations, represented by Mulliken symbols, i.e.,A₁) contained in that point group, along with the symmetry of functions, which can tell us about the symmetry of atomic orbitals as well as linear movement along the x-, y-, and z-axis.

Using the character table for the point group C_2v, we generate a reducible representation (Γ_red) of the C-O stretching modes in the molecule cis-Mo(CO)₄[P(OPh)₃]₂ (Figure 3). The reducible representation, or the linear combination of irreducible representations, can be generated by applying each of the symmetry operations within the character table to the vibrations within molecule and recording the number of C-O vibrations that remain unchanged (in the same position in space). For example, upon applying the identity symmetry element to the C-O vibrations in cis-Mo(CO)₄[P(OPh)₃]₂, all four of the vibrational arrows remain in the same position. Therefore, the first value in our reducible representation is 4. If we continue this exercise, we generate the reducible representation shown below.

C2v	E	C2	σv(xz)	σv’(yz)
Γred	4	0	2	2

Next, we use the C_2v character table to find the linear combination of irreducible representations that generates Γ_red for the C-O vibrations within cis-Mo(CO)₄[P(OPh)₃]₂. Reduction of the reducible representation can be achieved using the reduction formula shown in Equation 3.

  (3)

where:

n_i = number of times the irreducible representation i occurs in the reducible representation

h = order of the group (total number of symmetry operations)

c = the class of operation

g_c = number of operations in the class

χ_i = character of the irreducible representation for the operations of the class

χ_r = character of the reducible representation for the operations of the class

Using Equation 3 for each of the irreducible representations in the character table C_2v, we find that Γ_red = 2A₁ + B₁ + B₂. All three of the contributing irreducible representations, A₁, B₁, and B₂, are IR active because they transform as either the x-, y-, or z-axis (see the symmetry of functions in the character table). Therefore, we predict that cis-Mo(CO)₄[P(OPh)₃]₂ will exhibit 4 C-O stretching modes in its IR spectrum.

To summarize, the following steps are needed in order to determine the number of IR active vibrational modes in a molecule:

1. Determine the point group of the molecule.

2. Generate a reducible representation of the C-O stretching vibrations within the molecule.

3. Reduce the reducible representation using Equation 3.

4. Identify the number of translational irreducible representations present in the reduced representation from step 3.

If we follow these 4 steps with trans-Mo(CO)₄[P(OPh)₃]₂, we find that the molecule only possesses 1 active C-O vibrational mode.

Figure 3. CO vibrational stretches in cis-Mo(CO)₄[P(OPh)₃]₂.

Figure 5. IR of Mo(CO)₄[P(OPh)₃]_2.

Solution IR in saturated hydrocarbon (cm^-1): 2046 (s), 1958 (s), 1942 (vs).
The fourth resonance can only be seen under high-resolution conditions. Therefore, it is possible, as in this case, that only 3 of the 4 resonances are observed.
Based on the obtained IR, we can conclude that cis-isomer of Mo(CO)₄[P(OPh)₃]₂ was isolated.

In this video, we learned how to use group theory to predict the number of IR active vibrational modes in a molecule. We synthesized the molecule Mo(CO)₄[P(OPh)₃]₂ and used IR to determine which isomer was isolated. We observed that the product had three C-O vibrations in its IR spectrum, which is consistent with the cis-isomer.

Group theory is a powerful tool that is used by chemists to not only predict IR active vibrational modes, but also vibrational, rotational, and other low-frequency modes observed in Raman spectroscopy. Additionally, group theory is implemented in molecular orbital (MO) theory, which is the most widely used model to describe bonding within transition metal complexes. MO diagrams, used by organic and inorganic chemists, can predict and explain a molecule's observed reactivity.

1^st, 2^nd, and 3^rd row metal carbonyl complexes are used widely in inorganic synthesis as metal precursors for more complex organometallic compounds. Some of the most common types of reactions with metal carbonyl complexes include CO ligand substitution, redox at the metal center, and nucleophilic attack at the CO unit. Metal carbonyl complexes themselves are widely used in catalysis. For example, hydroformylation, the industrial production of aldehydes from alkenes, is catalyzed by the metal carbonyl complex HCo(CO)₃ (Figure 6).

Figure 6. Hydroformylation by the metal carbonyl complex HCo(CO)₃.