Character Tables, Lecture notes of Playwriting and Drama

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x

y

z

original coord

transformation

matrix

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z'

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new coord

z x y (1,1,1) i z x y (-1,-1-,1)

Transformation Matrices

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y'

z'

x

y

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Matrix Math

x'

y'

z'

-1(x) + 0(y) + 0(z)

0(x) + -1(y) + 0(z)

0(x) + 0(y) + -1(z)

x'

y'

z'

-x

-y

-z

Number of columns

in first matrix must

equal number of

rows in the second

Row1 x Column

Row1 x Column

Row1 x Column

(yz)

z

x

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H

O

H

reflect

on

yz plane

x

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original coord

transformation

matrix

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y'

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H

O H 1 0 0 1

z

x

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(xz)

z

x

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H

O

H

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x

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H

O

H

reflect

on

xz plane

x

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original coord

transformation

matrix

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x

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new coord

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The Set of Operations

E C 2 σv(xz) σv(yz)

traces

Create Irreducible Representations

E C 2 σv(xz) σv(yz)

what happens on x, y, and z... separately x (^1) -1 1 - y 1 -1 -1 1 z 1 1 1 1 3 -1 1 1

Traits of a Character Table

1. order , h is the total number of symmetry operations in the point group C4v E 2 C 4 C 2 2 σv 2 σd A 1 1 1 1 1 1 z x^2 + y^2 , z^2 A 2 1 1 1 -1 -1 Rz B 1 1 -1 1 1 -1 x^2 – y^2 B 2 1 -1 1 -1 1 xy E 2 0 -2 0 0 (x, y), (Rx, Ry) (xz, yz)

Traits of a Character Table

2. classes are the operations that have different characters for their transformation matrix. If the operations have the same characters they are in the same class(column) C4v E 2 C 4 C 2 2 σv 2 σd A 1 1 1 1 1 1 z x^2 + y^2 , z^2 A 2 1 1 1 -1 -1 Rz B 1 1 -1 1 1 -1 x^2 – y^2 B 2 1 -1 1 -1 1 xy E 2 0 -2 0 0 (x, y), (Rx, Ry) (xz, yz)

Traits of a Character Table

4. the sum of the squares of the dimensions (the characters under E) equals the order of the group C4v E 2 C 4 C 2 2 σv 2 σd A 1 1 1 1 1 1 z x^2 + y^2 , z^2 A 2 1 1 1 -1 -1 Rz B 1 1 -1 1 1 -1 x^2 – y^2 B 2 1 -1 1 -1 1 xy E 2 0 -2 0 0 (x, y), (Rx, Ry) (xz, yz) E.g. 12 + 1^2 + 1^2 + 1^2 + 2^2 = 1 + 2 + 1 + 2 + 2

Traits of a Character Table

5. the sum of the squares of the of the characters of the irreducible representations times the number of operations in a class equals the order of the group (sum after multiplying the number of operations in the class by the square of the character of the class). C4v E 2 C 4 C 2 2 σv 2 σd A 1 1 1 1 1 1 z x^2 + y^2 , z^2 A 2 1 1 1 -1 -1 Rz B 1 1 -1 1 1 -1 x^2 – y^2 B 2 1 -1 1 -1 1 xy E 2 0 -2 0 0 (x, y), (Rx, Ry) (xz, yz) For A 2 1•1^2 + 2•1^2 + 1•1^2 + 2•(-1)^2 + 2•(-1)^2 = 1 + 2 + 1 + 2 + 2

Traits of a Character Table

7. all groups contain an irreducible representation that has all 1’s for each class. C4v E 2 C 4 C 2 2 σv 2 σd A 1 1 1 1 1 1 z x^2 + y^2 , z^2 A 2 1 1 1 -1 -1 Rz B 1 1 -1 1 1 -1 x^2 – y^2 B 2 1 -1 1 -1 1 xy E 2 0 -2 0 0 (x, y), (Rx, Ry) (xz, yz)

We almost have a character table based on group theory E C 2 σv(xz) σv(yz) x 1 -1 1 - y 1 -1 -1 1 z 1 1 1 1 Since character tables must be square… The missing irreducible representation must satisfy the properties of a character table…. We are missing an irreducible representation, and…

We almost have a character table based on group theory E C 2 σv(xz) σv(yz) the missing irreducible representation must be 1 1 -1 - 1 (1) + -1 (1) + 1 (-1) + -1 (-1) = 0 1 (1) + 1 (1) + 1 (-1) + 1 (-1) = 0 1 (1) + -1 (1) + -1 (-1) + 1 (-1) = 0 (1)

(1)

(1)

(1)

• • • = 4 (^1) (1) 1 1 1

(1)

(-1)

(-1)

• • • = 4

The Character Table for the C2v Point Group E C 2 σv(xz) σv(yz) 1 -1 1 -1 x, Ry 1 -1 -1 1 y, Rx 1 1 1 1 z 1 1 -1 - B 1 B 2 A 1 A 2 Rz xz yz x

, y

, z

xy