Practical Decomposition of Irreducible Representations: Applications to Molecular Vibrations and Molecular Orbitals

The use of symmetry is widespread in chemistry, as it is used for predictions of the number of allowed and forbidden absorptions in electronic, vibrational, and rotational spectroscopies; for predictions of the combination of atomic orbitals to produce molecular orbitals; and in many other chemical applications. One critical step in these applications is the determination of the irreducible representations contained in a reducible one. To expand the tools available in this area, in addition to providing a new method for addressing irreducible representations in point group theory, an open source language program is presented here. Moreover, this program can even be applied to point groups that have complex characters, avoiding the "doubling problem" which occurs in the treatment of these groups. This software has immediate spectroscopic applications, and as an example, special attention is given here to vibrational spectroscopy. The use of this program, in both undergraduate and graduate classes, has the potential to reduce significantly the difficulty in understanding the applications of group theory in chemistry, especially for molecules which belong to symmetry groups with a large number of elements.