 
Using a Computer to Help Understand How Symmetry Principles Reduce Calculations©  
Louis Kijewski Department of Chemistry, Medical Technology, and Physics Monmouth University 400 Cedar Avenue West Long Branch, NJ 07764 United States mail to: kijewski@monmouth.edu  
Abstract  
This Mathcad document uses symmetry to simplify the evaluation of an eigenvalue problem by reducing a matrix to sparse form, a form with many more zeroes in the off diagonal positions. The sparse matrix is then transformed to block diagonal form with each sub matrix diagonalized separately. The technique can be used for any molecular system with symmetry. Group theory is used to set up the calculations without going through the proofs of the methods. Calculations like matrix multiplication, Taylor expansions, and derivatives are performed by symbolic evaluation in Mathcad. Experiencing the great reduction in the calculation helps students see the value of studying this important field of science. The level of understanding is typically graduate and maybe some undergraduate. The ability to use symmetry principles to get the vibrational frequencies and other information about molecules with symmetry is an important asset to any chemist. The document required Mathcad 2001i or higher.  
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Comments to: kijewski@monmouth.edu  
©Copyright 2004 Journal of Chemical Education  