W2013CHM2311 Part 3a Notes

That the transforma0on matrix for the vyz opera0on

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Unformatted text preview: y’ = 0x  ­1y + 0z x N H H H z’ = 0x + 0y + 1z In matrix nota.on: Ⱥ1 0 0Ⱥ Ⱥ x Ⱥ Ⱥ (1)( x) + (0)( y ) + (0)( z ) Ⱥ Ⱥ x Ⱥ Ⱥ0 − 1 0Ⱥ × Ⱥ y Ⱥ = Ⱥ(0)( x) + (−1)( y ) + (0)( z )Ⱥ = Ⱥ− y Ⱥ Ⱥ Ⱥ Ⱥ Ⱥ Ⱥ Ⱥ Ⱥ Ⱥ Ⱥ0 0 1Ⱥ Ⱥ z Ⱥ Ⱥ (0)( x) + (0)( y ) + (1)( z ) Ⱥ Ⱥ z Ⱥ Ⱥ Ⱥ Ⱥ Ⱥ Ⱥ Ⱥ Ⱥ Ⱥ Matrix Representa0ons of Point Groups A more General Example: An object in the C2v point group contains the following symmetry elements: E, C2, σv(xz), and σv(yz). Determine the transforma0on matrices for each of the corresponding symmetry opera0ons. Step 1: Draw Cartesian coordinate system Follow the right hand rule to determine the axes x, y, z z y x Step 2: Imagine a sphere in this coordinate system, centered around the coordinates (0, 0, 0) z y x Matrix Representa0ons of Point Groups (C2v Example) Step 3: Pick a point on the sphere (e.g. (1, 1, 1)) and determine how each symmetry opera0on will affect the x, y, and z coordinates for that point on the sphere. C2 σv(xz) y x σv(yz) Matrix Representa0ons of Point Groups (C2v Example) Step 4: Determine the appropriate transforma.on matrix. We will label the new coordinates that result from the transforma0on as: x = new x y = new y z = new z Thus, we can define the transforma0on matrix as follows: [New coordinates] = [Transforma0on matrix][Old coordinates] OR Ⱥ x'Ⱥ Ⱥ x Ⱥ Ⱥ y 'Ⱥ = [transforma tion matrix ] × Ⱥ y Ⱥ Ⱥ Ⱥ Ⱥ Ⱥ Ⱥ z ' Ⱥ Ⱥ z Ⱥ Ⱥ Ⱥ Ⱥ Ⱥ Iden0ty Opera0on: E (C2v Example) The x, y, and z coordinates are unchanged by the iden0ty opera0on. Therefore, x’ = x × 1 y’ = y × 1 z’ = z × 1 The matrix equa0on corresponding the E opera0on on the coordinates is: Ⱥ x'Ⱥ Ⱥ1 0 0Ⱥ Ⱥ x Ⱥ Ⱥ y 'Ⱥ = Ⱥ0 1 0Ⱥ Ⱥ y Ⱥ Ⱥ Ⱥ Ⱥ Ⱥ Ⱥ Ⱥ Ⱥ z ' Ⱥ Ⱥ0 0 1Ⱥ Ⱥ z Ⱥ Ⱥ Ⱥ Ⱥ Ⱥ Ⱥ Ⱥ Thus, we have determined that the iden0ty opera0on can represented by the following transforma0on matrix: E: Ⱥ1 0 0Ⱥ Ⱥ0 1 0Ⱥ Ⱥ Ⱥ Ⱥ0 0 1Ⱥ Ⱥ Ⱥ 2 ­Fold Rota0on: C2 (C2v Example) Upon rota0ng around C2, the new x, y, and z coordinates are given by: x’ =  ­x = x ×  ­1 y’ =  ­y = y ×  ­1 z’ = z = z × 1 The matrix equa0on corresponding to the C2 opera0on on the coordinates is: Ⱥ x'Ⱥ Ⱥ− 1 0 0Ⱥ Ⱥ x Ⱥ Ⱥ y 'Ⱥ = Ⱥ 0 − 1 0Ⱥ × Ⱥ y Ⱥ Ⱥ Ⱥ Ⱥ Ⱥ Ⱥ Ⱥ Ⱥ z ' Ⱥ Ⱥ 0 0 1Ⱥ Ⱥ z Ⱥ Ⱥ Ⱥ Ⱥ Ⱥ Ⱥ Ⱥ Thus, we have determined that the transforma0on matrix for the C2 opera0on must be: C2 : Ⱥ− 1 0 0Ⱥ Ⱥ 0 − 1 0Ⱥ Ⱥ Ⱥ Ⱥ 0 0 1Ⱥ Ⱥ Ⱥ Reflec0on: σ (C2v Example) Reflec.on through the xz mirror plane, σv(xz) Upon reflec0ng through the xz plane, the new x, y, and z coordinates are given by: x’ = x = x × 1 y’ =  ­y = y ×  ­1 z’ = z = z × 1 The matrix equa0on corresponding to the σv(xz) opera0on on the coordinates is: Ⱥ x'Ⱥ Ⱥ1 0 0Ⱥ Ⱥ x Ⱥ Ⱥ y 'Ⱥ = Ⱥ0 − 1 0Ⱥ × Ⱥ y Ⱥ Ⱥ Ⱥ Ⱥ Ⱥ Ⱥ Ⱥ Ⱥ z ' Ⱥ Ⱥ0 0 1Ⱥ Ⱥ z Ⱥ Ⱥ Ⱥ Ⱥ Ⱥ Ⱥ Ⱥ Thus, we have determined that the transforma0on matrix for the σv(xz) opera0on must be: σ v ( xz ) : Ⱥ1 0 0Ⱥ Ⱥ0 − 1 0Ⱥ Ⱥ Ⱥ Ⱥ0 0 1Ⱥ Ⱥ Ⱥ Matrix Representa0on of C2v Following the same procedure, we can determine that the transforma0on matrix for the σv(yz) opera0on must be: σ v ( yz ) : Ⱥ− 1 0 0Ⱥ Ⱥ 0 1 0Ⱥ Ⱥ Ⱥ Ⱥ 0 0 1Ⱥ Ⱥ Ⱥ Therefore, the four transforma0on matrices for four symmetry opera0ons in the C2v point group are: Ⱥ1 0 0Ⱥ E : Ⱥ0 1 0Ⱥ Ⱥ Ⱥ Ⱥ0 0 1Ⱥ Ⱥ Ⱥ Ⱥ− 1 0 0Ⱥ C2 : Ⱥ 0 − 1 0Ⱥ Ⱥ Ⱥ Ⱥ 0 0 1Ⱥ Ⱥ Ⱥ Ⱥ1 0 0Ⱥ σ v ( xz ) : Ⱥ0 − 1 0Ⱥ Ⱥ Ⱥ Ⱥ0 0 1Ⱥ Ⱥ Ⱥ Ⱥ− 1 0 0Ⱥ σ v ( yz ) : Ⱥ 0 1 0Ⱥ Ⱥ Ⱥ Ⱥ 0...
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This note was uploaded on 03/28/2014 for the course CHM 2311 taught by Professor Richardson during the Winter '09 term at University of Ottawa.

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