Group Theory with Mathcad

Frank Rioux
Department of Chemistry
Saint John's University
College of Saint Benedict
Saint Joseph, MN 56374


Mathcad has a powerful array of matrix manipulation commands that make it an ideal programming environment for applying mathematical group theory to chemistry problems. To illustrate this capability Mathcad will be used to do symmetry analyses on a number of examples involving vibrational and electronic spectroscopy, and chemical bonding.

The examples treated in this series will deal only with molecules that belong to the finite point groups. Most examples will deal with molecules having moderate to high symmetry - C2v to Ih. Obviously all finite groups will not be considered, but the most frequently encountered symmetries will receive sufficient attention. Fortunately Mathcad worksheets are easy to prepare and edit, so the worksheets included in this series can serve as templates for a large number of additional exercises. Once a worksheet has been created for a particular point group, it can be used for any molecule with that symmetry after minor modifications are made.

Among the exercises included are:

  • Analysis of the Vibrational and Electronic States of C60.

  • Cubane, Tetrahedrane, and XeF4: Molecular Structure from Vibrational Spectroscopy

  • Analysis of the Electronic Spectrum of a Transition Metal complex.

  • Hybridization of Atomic Orbitals with Group Theory

  • Miscellaneous Exercises
  • For each of the major areas listed above, a number of additonal exercises have been recommended. These exercises only require editing of the template Mathcad worksheet.


  • Harris, D. C.; Bertolucci, M.D. Symmetry and Spectroscopy: An Introduction to Vibrational and Electronic Spectroscopy, Dover Publications, Inc.: New York, 1989.

  • Cotton, F. A. Chemical Applications of Group Theory, 3rd Ed., Wiley Interscience: New York, 1990.
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