群论在红外光谱学中的应用

The five symmetry elements are identity (E), mirror planes (σ), centers of inversion (i), proper rotation axes (Cn), and improper rotation axes (Sn). Identity means no change occurs. Mirror planes reflect atoms to identical configurations. Centers of inversion reflect every atom through a point. Proper rotation axes rotate molecules to identical configurations, while improper rotation axes combine rotation with reflection through a perpendicular mirror plane.

Group theory uses character tables specific to each point group to predict IR-active vibrational modes. Reducible representations are generated by applying symmetry operations to molecular properties like CO stretches. The reduction formula calculates coefficients of irreducible representations. Modes transforming as x, y, or z axes are IR-active because they change the dipole moment. This method allows chemists to predict the exact number of CO stretches expected in an infrared spectrum.

The cis isomer belongs to the C2v point group with a principal C2 rotation axis and two mirror planes containing that axis. The trans isomer belongs to the D4h point group with a principal C4 rotation axis, four C2 axes perpendicular to it, and a perpendicular mirror plane. These different symmetries result in different numbers of IR-active CO stretches: the cis isomer shows four IR-active modes while the trans isomer shows only one.

A symmetry tree identifies the symmetry operations present in a molecule to classify it into a point group. The procedure checks whether the molecule is linear, counts rotation axes with order greater than 2, identifies the principal axis with the highest n value, and determines the presence of perpendicular C2 axes and mirror planes. For example, BF3 has a C3 principal axis, three perpendicular C2 axes, and a perpendicular mirror plane, placing it in the D3h point group.

Group theory generates symmetry-adapted linear combinations of atomic orbitals representing ligands in transition metal complexes. Scientists create reducible representations of ligand atomic orbitals and reduce them to irreducible representations. The symmetry representations of the metal center and ligands are compared in molecular orbital diagrams. Orbitals with matching symmetry overlap to form bonding and antibonding molecular orbitals, enabling prediction of complex geometry and bonding properties.

IR spectroscopy detects vibrations that change the dipole moment, while Raman spectroscopy detects vibrations involving changes in polarizability of the electron cloud. For example, a symmetrical CO2 stretch is IR-inactive because it does not change the dipole moment, but it is Raman-active because electrons moving away from nuclei alter polarizability. Group theory identifies both IR-active and Raman-active modes using the same general method of applying symmetry operations.

First, synthesize the complex using a Schlenk line to handle air-sensitive reagents, then acquire an IR spectrum of the product dissolved in hexanes. Count the number of CO stretches observed in the spectrum. Compare this to group theory predictions: the cis isomer shows four IR-active CO stretches while the trans isomer shows one. The isolated Mo(CO)4[P(OPh)3]2 complex is identified as cis if peaks appear at 2046, 1958, and 1942 cm-1, with a possible fourth peak at higher resolution.