Using a Computer to Help Understand How Symmetry Principles Reduce Calculation
Louis Kijewski
Department of Chemistry, Medical Technology, and Physics, Monmouth University, West Long Branch, NJ 07764
Abstract
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More Information: Keywords, JCE Citation
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.
Downloads
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3oscillatorsmcd.zip |
ZIP archive |
9.5 MB |
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ZIP archive contains
Three animation (.avi ) files: NormalMode1.avi, NormalMode3.avi, NormalMode5.avi;
ThreeMassOscillators.hbk (Mathcad book) and ThreeMassOscillators folder that contains the following fully interactive with Mathcad 2001i files:
contents.mcd, DP1GENERATOR.MCD, DP3DP4GENERATION.MCD, Dp5Dp6Generation.mcd, EquationsForSymmetryVectors.mcd, FifthNormalMode.mcd, FirstNormalMode.mcd, FourthNormalMode.mcd, GOT.mcd, NumberOfIrreps.mcd, PlottingVectors.mcd, SecondNormalMode.MCD, SixthNormalMode.mcd, ThirdNormalMode.mcd, ThreeMassOscillators.mcd, welcome.mcd
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3oscillatorspdf.zip |
ZIP archive |
1.3 MB |
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ZIP archive contains the following non-interactive documents (Acrobat .pdf) derived from Mathcad 2001i files above:
contents.pdf, DP1GENER.pdf, DP3DP4GE.pdf, Dp5Dp6Ge.pdf, EquationsforSymmetryVectors.pdf, FifthNor.pdf, FirstNor.pdf, FourthNo.pdf, GOT.pdf, NumberOfIrreps.pdf, Plotting.pdf, SecondNo.pdf, SixthNor.pdf, ThirdNor.pdf, ThreeMassOscillators.pdf, welcome.pdf |
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JCE Subscribers only: Institutional IP number access or name and password required
Commentary
Editor's Commentary
Other Information
KeywordsComputer-Based Instruction; Physical Chemistry; Quantum Chemistry; Mechanics; Group Theory / Symmetry; Sprectroscopy; Symbolic Mathematics; Upper-Division Undergraduate JCE Citation
Kijewski, Louis. J. Chem. Educ. 2005, 82, 174.
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