The symmetry of a molecule plays a crucial role in determining its vibrational frequencies and infrared IR spectra. Group theory, a mathematical method used to analyze the symmetry properties of molecules, can be employed to predict and explain the observed spectral features of a given molecule.Vibrational frequencies are associated with the normal modes of vibration of a molecule. These normal modes can be classified based on their symmetry properties using group theory. Infrared spectra arise from the absorption of IR radiation by a molecule, which causes a change in its vibrational energy levels. For a vibrational mode to be IR active i.e., observable in the IR spectrum , it must induce a change in the molecular dipole moment.Here is a step-by-step analysis using group theory to predict and explain the observed spectral features of a given molecule:1. Determine the point group of the molecule: The first step is to identify the molecule's point group, which describes its symmetry elements e.g., rotation axes, mirror planes, inversion centers . Common point groups include C2v, D3h, and Td.2. Identify the symmetry operations: List all the symmetry operations associated with the point group, such as identity E , rotations Cn , reflections , and inversions i .3. Construct the character table: The character table is a tabular representation of the symmetry properties of the point group. It lists the irreducible representations irreps and their corresponding characters for each symmetry operation.4. Determine the reducible representation for the vibrational modes: For a molecule with N atoms, there are 3N degrees of freedom. For non-linear molecules, 3N-6 of these correspond to vibrational modes 3N-5 for linear molecules . The reducible representation for these vibrational modes can be determined by considering the transformation properties of the Cartesian coordinates x, y, z of each atom under the symmetry operations.5. Decompose the reducible representation into irreducible representations: Using the character table, decompose the reducible representation obtained in step 4 into a linear combination of irreducible representations. This will give the symmetry species of the vibrational modes.6. Identify the IR-active modes: To be IR active, a vibrational mode must transform as one of the components of the molecular dipole moment x, y, or z . Compare the irreducible representations obtained in step 5 with those of the dipole moment components in the character table. The vibrational modes that have the same symmetry species as the dipole moment components are IR active.7. Predict the spectral features: The number of IR-active vibrational modes determines the number of peaks in the IR spectrum. The symmetry species of these modes can provide information about their relative intensities and frequencies.In summary, the symmetry of a molecule impacts its vibrational frequencies and IR spectra by determining the symmetry species of its vibrational modes. Group theory can be used to predict and explain the observed spectral features of a given molecule by identifying the IR-active vibrational modes and their corresponding symmetry species.