Infrared (IR) spectroscopy is a powerful tool used to identify functional groups and determine the structure and symmetry of molecules. For metal carbonyl complexes, the stretching vibrations of the carbonyl (CO) ligands are particularly useful because they typically give strong, sharp bands in the IR spectrum in the range of 1800-2150 cm\(^{-1}\). The number and intensity of these CO stretching bands observed in the IR spectrum depend on the molecular symmetry of the complex.

According to group theory, a vibrational mode is considered IR active if it causes a change in the molecule's dipole moment during the vibration. By analyzing the symmetry of the molecule (its point group) and reducing the representation for the CO stretching vibrations, we can determine the irreducible representations corresponding to these modes. Any mode whose irreducible representation transforms as one of the Cartesian coordinates (x, y, or z) in the character table is IR active.

Predicting CO Bands by Symmetry Analysis

We will analyze the symmetry of the given isomers and predict the number of IR active CO stretching bands for each.

Set (i): Trigonal Bipyramidal Isomers, Fe(CO)\(_{\mathbf{4}}\)L

Isomer A: axial‐Fe(CO)\(_{\mathbf{4}}\)L

• In this isomer, the ligand L occupies one of the axial positions in the trigonal bipyramidal structure. The four CO ligands consist of one axial CO and three equatorial COs.
• Assuming the ligand L is different from CO, the point group symmetry for this complex is C\(_{\text{3v}}\). The C\(_{\text{3}}\) axis passes through the L-Fe-CO axis, and there are three mirror planes (\(\sigma_v\)) containing this axis and each of the equatorial CO ligands.
• The representation for the CO stretching vibrations in a C\(_{\text{3v}}\) complex with one axial and three equatorial CO groups reduces to the irreducible representations: \(2A_1 + E\).
• In the C\(_{\text{3v}}\) point group, the \(A_1\) and \(E\) irreducible representations correspond to IR active modes.
• The \(2A_1\) modes represent two distinct symmetric stretching vibrations (one primarily axial, one primarily equatorial, which can mix), and the \(E\) mode represents a degenerate asymmetric stretching vibration of the equatorial COs. These give rise to 3 distinct IR bands.
• Therefore, isomer A is expected to show 3 CO stretching bands in its IR spectrum.

Isomer B: equatorial‐Fe(CO)\(_{\mathbf{4}}\)L

• In this isomer, the ligand L occupies one of the equatorial positions in the trigonal bipyramidal structure. The four CO ligands consist of two axial COs and two remaining equatorial COs.
• Assuming the ligand L is different from CO, the point group symmetry for this complex is C\(_{\text{2v}}\). A C\(_{\text{2}}\) axis passes through the Fe atom and the equatorial ligand trans to L. There are also two mirror planes.
• The representation for the CO stretching vibrations in a C\(_{\text{2v}}\) complex with two axial and two equatorial CO groups reduces to the irreducible representations: \(2A_1 + B_1 + B_2\).
• In the C\(_{\text{2v}}\) point group, the \(A_1\), \(B_1\), and \(B_2\) irreducible representations correspond to IR active modes.
• The \(2A_1\) modes represent two distinct symmetric stretches, while \(B_1\) and \(B_2\) represent asymmetric stretches. These give rise to 4 distinct IR bands.
• Therefore, isomer B is expected to show 4 CO stretching bands in its IR spectrum.

Set (ii): Octahedral Isomers, Mo(CO)\(_{\mathbf{3}}\)L\(_{\mathbf{3}}\)

Isomer C: fac‐Mo(CO)\(_{\mathbf{3}}\)L\(_{\mathbf{3}}\)

• In the facial (fac) isomer, the three CO ligands are located on one face of the octahedron, adjacent to each other around the metal center. The three L ligands occupy the opposite face.
• Assuming the ligand L is different from CO, the point group symmetry for this complex is C\(_{\text{3v}}\). A C\(_{\text{3}}\) axis passes through the center of the two opposite faces (where COs and Ls are located), and there are three mirror planes containing this axis and one ligand from each set (CO and L).
• The representation for the CO stretching vibrations in a C\(_{\text{3v}}\) complex with three facial CO groups reduces to the irreducible representations: \(A_1 + E\).
• In the C\(_{\text{3v}}\) point group, the \(A_1\) and \(E\) irreducible representations correspond to IR active modes.
• The \(A_1\) mode is a symmetric stretch of all three COs, and the \(E\) mode is a doubly degenerate asymmetric stretch. These give rise to 2 distinct IR bands.
• Therefore, isomer C is expected to show 2 CO stretching bands in its IR spectrum.

Isomer D: mer‐Mo(CO)\(_{\mathbf{3}}\)L\(_{\mathbf{3}}\)

• In the meridional (mer) isomer, the three CO ligands are arranged in a plane containing the metal center, with two CO ligands trans to each other and the third CO ligand cis to both. The three L ligands occupy the remaining positions.
• Assuming the ligand L is different from CO, the point group symmetry for this complex is C\(_{\text{2v}}\). A C\(_{\text{2}}\) axis passes through the metal bisecting the angle between two cis ligands (or passing through the metal and one ligand if that's how the axes are defined). There are also two mirror planes.
• The representation for the CO stretching vibrations in a C\(_{\text{2v}}\) complex with three meridional CO groups reduces to the irreducible representations: \(2A_1 + B_2\).
• In the C\(_{\text{2v}}\) point group, the \(A_1\) and \(B_2\) irreducible representations correspond to IR active modes.
• The \(2A_1\) modes represent two distinct symmetric stretches, and the \(B_2\) mode represents an asymmetric stretch. These give rise to 3 distinct IR bands.
• Therefore, isomer D is expected to show 3 CO stretching bands in its IR spectrum.

Summary of CO Band Counts

• For isomer A (axial‐Fe(CO)\(_{\text{4}}\)L, C\(_{\text{3v}}\) symmetry), the number of CO bands is 3.
• For isomer B (equatorial‐Fe(CO)\(_{\text{4}}\)L, C\(_{\text{2v}}\) symmetry), the number of CO bands is 4.
• For isomer C (fac‐Mo(CO)\(_{\text{3}}\)L\(_{\text{3}}\), C\(_{\text{3v}}\) symmetry), the number of CO bands is 2.
• For isomer D (mer‐Mo(CO)\(_{\text{3}}\)L\(_{\text{3}}\), C\(_{\text{2v}}\) symmetry), the number of CO bands is 3.

Thus, the expected number of CO bands are: A, 3 and B, 4; C, 2 and D, 3.