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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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