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Chapter+4.2 - Chapter 4 Symmetry and Group Theory...

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Click to edit Master subtitle style Chapter 4 Symmetry and Group Theory Applications of Symmetry Lecture 4.2 Lei Li [email protected] 317-278-2202
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D. Properties of Point Groups 1. Symmetry operation of NH3 a. Ammonia has E, 2C3 (C3 and C23) and 3σv b. Point group = C3v 1. Properties of C3v (any group) a. Must contain E a. Each operation must have an inverse; doing both gives E (right to left) c. Any product equals another group member
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I. Matrices A. Why Matrices? The matrix representations of the point group’s operations will generate a character table . We can use this table to predict properties. A. Definitions and Rules 1) Matrix = ordered array of numbers 1) Multiplying Matrices a) The number of columns of matrix #1 must = number of rows of matrix #2 b) Fill in answer matrix from left to right and top to bottom c) The first answer number comes from the sum of [(row 1 elements of matrix #1) X (column 1 elements of matrix #2)] a) The answer matrix has same number of rows as matrix #1 The answer matrix has same number of columns as matrix #2 [ ] 5 2 4 3 or 1 7 2 3 = + + + + = × 54 38 43 27 48 6 24 14 40 3 20 7 8 4 3 7 6 2 5 1 C ij = Σ A ik × B kj
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e) Relevant example: e) Exercise 4-4 II. Representations of Point Groups A. Matrix Representations of C2v 1) Choose set of x,y,z axes a) z is usually the Cn axis b) xz plane is usually the plane of the molecule 2) Examine what happens after the molecule undergoes each symmetry operation in the point group (E, C2, 2 σ ) [ ] [ ] 3 2 1 1 0 0 0 1 0 0 0 1 3 2 1 - = - ×
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3) Transformation Matrix = matrix expressing the effect of a symmetry operation on the x,y,z axes 3) E Transformation Matrix a. x,y,z x,y,z b. What matrix times x,y,z doesn’t change anything? × = z y x ? ? ? ? ? ? ? ? ? z' y' x' transformation matrix = × z y x z y x 1 0 0 0 1 0 0 0 1 E Transformation Matrix
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5) C2 Transformation Matrix a.
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