Molecular Symmetry and Group Theory

One of the key applications of matrices in chemistry is in the characterization of molecular symmetry. In Section 4.3 we saw how it was possible to represent the coordinate transformations associated with rotation and reflection, in terms of matrices. These notions are now explored in the next section, where we develop some of the basic ideas of group theory. 

A group consists of a of elements (c.g. numbers or square matrices), for 

Q Investigate whether the set of integers forms a group under each of the following modes of combination  

A (a) Addition-, the sum of any two integers is an integer (closure satisfied) addition of integers is associative the identity element is zero e.g, 2 + 0 = 0 + 2 = 2) the inverse of any integer n is -n [e.g. 2 + (—2) =0, the identity element], and -n is an integer which is in the set. Since all four criteria are satisfied, the set of integers forms a group of infinite order under addition. 

Demonstrate that the set of numbers Gi = 1, —1,1, — i forms a group of order 4 under multiplication. 

---

Vincent, A. (2000) Molecular Symmetry and Group Theory, 2nd edn., John Wiley, Chichester. 

Ramon M. Sosa Sanchez, Molecular Symmetry and Group Theory in Quantum Mechanics Applications, Univ. Republica, Montevideo, Uruguay, 1976. 

A. Vincent, Molecular Symmetry and Group Theory A Programmed Introduction to Chemical Applications, Second Edition, Wiley-Interscience, New York, 2001. 

R. McWeeny, Coulson s Valence, 3rd Edn, Oxford University Press, Oxford, 1979 A. Szabo and N. S. Ostlund, Modern Quantum Chemistry, Dover, New York, 1996 N. J. B. Green, Quantum Mechanics 1 Foundations, Oxford University Press, Oxford, 1997 D. A. McQuarrie and J. D. Simon, Physical Chemistry A Molecular Approach, University Science Books, Sausalito, CA, 1997 V. M. S. Gil, Orbitals in Chemistry, Cambridge University Press, Cambridge, 2000 A. Vincent, Molecular Symmetry and Group Theory, 2nd Edn, John Wiley Sons, Ltd, Chichester, 2001 A. Rauk, Orbital Interaction Theory of Organic Chemistry, 2nd Edn, John Wiley Sons, Ltd, New York, 2001 D. O. Hayward, Quantum Mechanics for Chemists, Royal Society of Chemistry, Cambridge, 2002 J. E. House, Fundamentals of Quantum Chemistry, 2nd Edn, Elsevier, Amsterdam, 2004 N. T. Anh, Frontier Orbitals A Practical Manual, John Wiley Sons, Ltd, Chichester, 2007 J. Keeler and P. Wothers, Chemical Structure and Reactivity, Oxford University Press, Oxford, 2008. 

Vincent, A. (2001) Molecular Symmetry and Group Theory A Programmed Introduction to Chemical Applications, 2nd edn, John Wiley Sons, Inc., New York, USA. This popular textbook may serve the more advanced reader well, with a slightly deeper mathematical base but a well-staged programmed learning approach. 

The group orbital approach described in this chapter, despite its modest use of group theory, conveniently provides a qualitatively useful description of bonding in simple molecules. Computational chemistry methods are necessary for more complex molecules and to obtain wave equations for the molecular orbitals. These advanced methods also apply molecular symmetry and group theory concepts. 

---