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Ammonia with non-Cartesian basis vectors N H3H3 H2H2 H1H1 Note that these matrices are not block-diagonalized.

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Similarity Transformation Q takes all symmetry operation matrices for entire point group and block- diagonalizes them Finding Q is hard. Go take a class on group theory.

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Block-diagonalized Symmetry operations [1] is singly degenrate matrix which forms representation z Doubly-degenerate matrix is representation x,y

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Converting to Characters

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Character Table C 3v EC31C31 C32C32 v’v’ v ’’ v ’’’ A1A1 111111 A2A2 111 E2 000 C 3v E2C 3 3v3v A1A1 111 A2A2 11 E2 0

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x0x0 y0y0 z0z0 x1x1 y1y1 z1z1 x2x2 y2y2 z2z2 x3x3 y3y3 z3z3 x0x0 1 Y0Y0 Z0Z0 1 X1X1 1 Y1Y1 Z1Z1 1 X2X2 1 Y2Y2 Z2Z2 1 X3X3 1 Y3Y3 Z3Z3 1 v ’= Whole molocule transformations

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x0x0 y0y0 z0z0 x1x1 y1y1 z1z1 x2x2 y2y2 z2z2 x3x3 y3y3 z3z3 x0x0 1/ 2 Y0Y0 Z0Z0 1 X1X1 1 Y1Y1 Z1Z1 1 X2X2 1 Y2Y2 Z2Z2 1 X3X3 1 Y3Y3 Z3Z3 1

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Laser Molecular Spectroscopy CHE466 Fall 2009 David L. Cedeño, Ph.D. Illinois State University Department of Chemistry Elements of Symmetry.

Laser Molecular Spectroscopy CHE466 Fall 2009 David L. Cedeño, Ph.D. Illinois State University Department of Chemistry Elements of Symmetry.

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