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Applications of Symmetry to Integrals

One of the most useful applications of symmetry is to the evaluation of various integrals. It is possible with many integrals that are commonly encountered in chemistry to use symmetry arguments to tell whether an integral must necessarily be zero.

Normal Modes of Vibration

Vibrations in molecules can be excited by the application of radiation at the same frequency as the vibration itself. This gives rise to spectroscopic methods of studying molecular structure, which depend upon being able to predict or at least rationalise the vibrations that are observed. Symmetry considerations play an important role in this.

Symmetry of Normal Modes

The normal modes of a molecule must belong to one of the symmetry species of the molecular point group, and this proves to be a highly convenient method of classifying them, allowing rapid prediction of the form of the vibrational spectra of the molecule.

Interpretation of a Character Table

The characters of the identity operation, E, reveal the degeneracy of the orbitals of each symmetry species.

Character Tables

The symmetry properties of orbitals in polyatomic molecules are denoted by symmetry labels, which look like a, a1, b, eetc. These labels indicate the behaviour of the orbitals in a molecule under the symmetry operations of the point groupto which the molecule belongs. This can be illustrated by a hypothetical example.

Point Groups

The name of the group to which a molecule belongs is determined by the symmetry elements it possesses. Grouping together molecules with the same symmetry elements will automatically group together molecules of the same shape eg CX4 where X = H, F, Cl, Br or I; as long as the four X groups are identical, these molecules will all fall into the same point group.

Key for Assignment of Point Groups

Polar and Chiral Molecules

There are some conclusions that may be drawn about a molecule as soon as its point group has been identified: A polar molecule is one which possesses a permanent electric dipole moment.

Introduction to Symmetry

The symmetry properties of molecules and orbitals are extremely useful in a consideration of their properties. In order to explore the symmetry of an object in a rigorous mathematical fashion, we need to introduce various terms to describe the concepts we shall encounter