Part 2.5: Character Tables 1. Character table structure – Mulliken symbols – Order – Basis...

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Part 2.5: Character Tables

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• Character table structure– Mulliken symbols– Order– Basis functions

• Properties of Char. Tables• Driving the table

– From the rules– From matrix math

Review

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Character TableTwo-dimensional table compose of elements and irreducible representations of a point group.

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Character Table

Group Symbol Symmetry Elements

Irreducible Representations

Characters Basis Functions

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Mulliken Symbols

Don’t mistake the operation E for the Mulliken symbol E!

A or B: singly degenerateE: doubly degenerateT: triply degenerate

A: symmetric (+) with respect to Cn

B: anti-symmetric (-) with respect to Cn

subscript g: symmetric (+) with respect to isubscript g: anti-symmetric (-) with respect to i

subscript 1: symmetric (+) with respect to C⊥ 2 or sv

subscript 2: anti-symmetric (-) with respect to C⊥ 2 or sv

superscript ‘ : symmetric (+) under sh (if no i)superscript “: anti-symmetric (-) under sh (if no i)

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Mulliken Symbols

Don’t mistake the operation E for the Mulliken symbol E!

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Mulliken Symbols

C2h

D4h

⊥C2

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Mulliken Symbols

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Infinity Character Tables

C∞v D∞h

Infinity tables us Greek rather than Latin letters.

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Infinity Character Tables

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Order (h)

Symmetry Elements

Infinite groups (C∞v , D∞h ) have a infinite order.

h = 1 + 1 + 1 + 1 = 4

order (h)

Order of a group (h) = the number of elements in the group

D3

h = 1 + 2 + 3 = 6h = 1 + 2 + 2 + 2 + 1 + 4 + 4 = 16

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Basis Functions

linear functions, rotations

quadratic functions

Basis Functions

In the C2v point group

px has B1 symmetry

px transforms as B1

px has the same symmetry as B1

px forms a basis for the B1 irrep A1B2B1

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Basis Functions

linear functions, rotations

quadratic functions

Basis Functions

dxz:

B: Anti symmetric with respect to Cn

sub 1: symmetric with respect to sv

B1A2B2

A1 A1

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)]3(),3([),,(),(),(),,(00202

11111

)(11111

11111

,11111

222

2222222

22221

2

22221

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yxyyxxyzxzyzxzRRyxE

xyzxyB

yxzyxB

RA

zzyxzA

CCEC

yx

z

dvv

Basis Functions

linear functions

quadratic functions

cubicfunctions

Lanthanide and Actinide coordination chemistry.

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Character TableGroup Symbol

For the 32 crystallographic point groups. Hermann-Mauguin Symbol

Schönflies symbols

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Character Table

For the 32 crystallographic groups Hermann-Mauguin Symbol

Rhombic-dipyramidal class

one 2-fold axis and 2 mirror planes

2 m m

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Character TableHermann-Mauguin Symbol

4 fold axis, 3-fold rotoinversion axes, and two sets of mirror planes

s that are to the 4 fold axes⊥ s that are to the 2 fold axes⊥

3-fold rotoinversion axis

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Character Table

Hermann-Mauguin Symbol

32 crystallographic classes

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Character Table

Group Symbol Symmetry Elements

Irreducible Representations

Characters Basis Functions

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4. The sum of the squares of the dimensionality of all the irreducible representations is equal to the order of the group.

5. The sum of the squares of the characters multiplied by the number of operations in the class equals the order of the group.

6. The sum of the products of the corresponding characters of any two different irreducible representations of the same group is zero.

1. The characters of all matrices belonging to the operations in the same class are identical in a given irreducible representation.

2. The number of irreducible representations in a group is equal to the number of classes of that group.

Properties of the Character Table

3. There is always a totally symmetric representation for any group.

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Properties of the Character Table1. The characters of all matrices belonging to the operations in the same

class are identical in a given irreducible representation.

Rotational Class

No similar operations.Each operation in its own class.

Rotational ClassReflection Class Reflection Class

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Properties of the Character Table2. The number of irreducible representations in a group is equal to the

number of classes of that group.

4 x 4 table

3 x 3 table10 x 10 table

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Properties of the Character Table3. There is always a totally symmetric representation for any group.

A, A1, A1g, Ag, A’, A’1, (Σ+, Σg+ for infinity groups)

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dimensionality = character under E


h = 1 + 1 + 1 + 1 = 4

12 + 12 + 12 + 12 = 4

h = 1 + 2 + 3 = 6

12 + 12 + 22 = 6

c(E) = characters under E

order (h)

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Properties of the Character Table5. The sum of the squares of the characters multiplied by the number of

operations in the class equals the order of the group.

Order = 1 + 1 + 1 + 1 = 4

(1)2(1) + (-1)2(1) + (-1)2(1) + (1)2(1) = 4

Order = 1 + 2 + 3 = 6

(1)2(1) + (1)2(2) + (1)2(3)= 6

(2)2(1) + (-1)2(2) + (0)2(3)= 6

c(R)= characters under an operationgc = the number of operations in a class

Properties of the Character Table6. The sum of the products of the corresponding characters of any two

different irreducible representations of the same group is zero.

Irreducible representations are orthoganal to each other.

(1)(1)(1) + (-1)(-1)(1) + (-1)(1)(1) + (1)(-1)(1) = 0

(1)(1)(1) + (1)(1)(2) + (-1)(1)(3) = 0

(2)(1)(1) + (-1)(1)(2) + (0)(-1)(3) = 0

ci(R)= characters for irreducible representation igc = the number of operations in a class

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5. The sum of the squares of the characters multiplied by the number of operations in the class equals the order of the group.

6. The sum of the products of the corresponding characters of any two different irreducible representations of the same group is zero.

1. The characters of all matrices belonging to the operations in the same class are identical in a given irreducible representation.

2. The number of irreducible representations in a group is equal to the number of classes of that group.


3. There is always a totally symmetric representation for any group.

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Example Table

4. The sum of the squares under E = order of the group.

5. The sum of the squares x # of operations = order of the group.

6. Irreducible reps are orthoganal S(G1 x G2 x opperation) = 0

1. Classes are grouped.

2. The table is square.

3. There is always a G = 1 representation.

D4h

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• Open an inorganic text book or google– Easy Mode

• From the rules/inspection– Heroic Mode

• From matrix math– Legendary Mode

Derive the character table

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From the Rules: C2v

1. Classes are grouped.-no groups for C2v

Operations: E, C2, σ, σ'

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From the Rules: C2v


- Algebra


2. The table is square.-4 x 4 table

3. There is always a G = 1 representation.-Easiest step


G 1

G 2

G 3

G 4

h = 1 + 1 + 1 + 1 = 4

d2

d3

d4

(1)2 + d22 + d3

2 + d42 = h = 4

d2 = d3 = d4 = 1 or -1Under E always positive.

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From the Rules: C2v


- Algebra

5. The sum of the squares times # of operations = order of the group.

- Algebra





G 1

G 2

G 3

G 4

h = 1 + 1 + 1 + 1 = 4

1(1)2 + 1(e2)2 + 1(e3)2 + 1(e4)2 = h = 4

e2 e3 e4

e2 = e3 = e4 = 1 or -1

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From the Rules: C2v


- Algebra


- Algebra






G 1

G 2

G 3

G 4

1(1)(1) + 1(1)(e2) + 1(1)(e3) + 1(1)(e4) = 0

e2 e3 e4

e2 = e3 = e4 = 1 or -1

(1) + (e2) + (e3) + (e4) = 0

e2 = e3 = e4 = two -1 and one 1

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From the Rules: C2v


- Algebra


- Algebra






G 1

G 2

G 3

G 4

1(1)(1) + 1(1)(e2) + 1(1)(e3) + 1(1)(e4) = 0

e2 = e3 = e4 = 1 or -1

(1) + (e2) + (e3) + (e4) = 0

e2 = e3 = e4 = two -1 and one 1

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From the Rules: C2v


- Algebra


- Algebra






G 1

G 2

G 3

G 4

1(1)(1) + 1(1)(e2) + 1(-1)(e3) + 1(-1)(e4) = 0

e2 = e3 = e4 = 1 or -1

(1) + (e2) + -(e3) + -(e4) = 0

e2 = -1 e3 = 1, e4 = -1or

e3 = -1, e4 = 1

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From the Rules: C2v


- Algebra


- Algebra






G 1

G 2

G 3

G 4

1(1)(1) + 1(1)(e2) + 1(-1)(e3) + 1(-1)(e4) = 0

e2 = e3 = e4 = 1 or -1

(1) + (e2) + -(e3) + -(e4) = 0

e2 = -1 e3 = 1, e4 = -1or

e3 = -1, e4 = 1

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From the Rules: C2v


- Algebra


- Algebra






G 1

G 2

G 3

G 4

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From the Rules: C2v

?

G2

G3

G4

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From the Rules: C2v

A1

G2

G3

G4

G1 = A1

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From the Rules: C2v

A1

?

G3

G4

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From the Rules: C2v

A1

A2

?

?

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From the Rules: C2v

A1

A2

B1

B2

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From the Rules: C3v




1. Classes are grouped.

2. The table is square.

3. There is always a G = 1 representation.

A1

A2

B1

B2

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From Matrix Math1. Assign/pick a point group2. Choose basis function3. Apply operations4. Generate a representation matrix5. Apply similarity transformations6. Generate an irreducible block diagonal matrix7. Character of the irreducible blocks8. Fill in the character table9. Complete the table10.Assign symmetry labels11.Assign basis functions

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Example 1: H2O (C2v)

1. Assign a point group

C2v

Character Table

Steps 2-11

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2. Choose a basis function


Cartesian Coordinates of O

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3. Apply operationsE, C2, sxz, syz

4. Generate a representation matrix

E = C2 = sxz = syz =

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5. Apply similarity transformations6. Generate an irreducible block diagonal matrix


Block diagonal and single number.These representations cannot be reduced any further.


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7. Character of the irreducible blocks

8. Fill in the character table



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9. Complete the table


Rule 2) The number of irreducible representations is equal to the number of classes in the group.

4 classes = 4 irreducible representations. Table must be 4 x 4!

Rule 4) The sum of the squares of the dimensions under E is equal to the order of the group.

Order = 4, Therefore 12 + 12 + 12 + x2 = 4

G1

G2

G3

G4 x

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Rule 5) The sum of the squares times # of operations = order of the group.

G1

G2

G3

G4 1 e2 e3 e4

1(1)2 + 1(e2)2 + 1(e3)2 + 1(e4)2 = h = 4

e2 = e3 = e4 = 1 or -1



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Rule 6) Irreducible reps are orthoganal S(G1 x G2 x opperation) = 0.

G1

G2

G3

G4 1 e2 e3 e4

1(1)(1) + 1(-1)(e2) + 1(1)(e3) + 1(-1)(e4) = 0

(1) - (e2) + (e3) - (e4) = 0

e2 = 1, e3 = 1, e4 = 1 or e2 = 1, e3 = -1, e4 = -1 e2 = -1, e3 = -1, e4 = 1or

Bonus rule: no two G can be the same.



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G1

G2

G3

G4 1 1 -1 -1

1(1)(1) + 1(-1)(e2) + 1(1)(e3) + 1(-1)(e4) = 0

(1) - (e2) + (e3) - (e4) = 0

e2 = 1, e3 = 1, e4 = 1 or e2 = 1, e3 = -1, e4 = -1 e2 = -1, e3 = -1, e4 = 1or

Bonus rule: no two G can be the same.

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10.Assign symmetry labels


Symmetry Labels

1 1 -1 -1

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1 1 -1 -1

Rearrange

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11.Assign Basis Function


x, y, z, Rx, Ry, Rz

xy, xz, yz, x2, y2, z2

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Start with z or pz:z

If the orbital/vector stays the same = 1If the sign/arrow direction flips = -1

E

1 1 1 1

C2

sxz

syz

x, y, z, Rx, Ry, Rz

z

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x or px:x

E, sxz

1 -1 1 -1

C2, syz

z

x, y, Rx, Ry, Rz

x

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y or py:y

E, syz

1 -1 -1 1

C2, sxz

z

y, Rx, Ry, Rz

xy

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z

y

xz

y

x



Rz:Rz

E, C2

1 1 -1 -1

syz, sxz

z

Rx, Ry, Rz

xy

z

z

y

xRotation direction unchanged = 1Rotation direction flips = -1

Rz

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z

y

xz

y

x



Rx:

E, syz

1 -1 -1 1

C2, sxz

z

Rx, Ry

xy

Rotation direction unchanged = 1Rotation direction flips = -1

Rz

x z

y

x

, Rx

Rx

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z

y

xz

y

x



Ry:

E, sxz

1 -1 1 -1

C2, syz

z

Ry

xy

Rotation direction unchanged = 1Rotation direction flips = -1

Rz

, Rx

y

, Ry

z

y

x

Ry

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z

y

x

z

y

x



yz or dyz:yz

E, syz

1 -1 -1 1

C2, sxz

z

xy

Rz

, Rx

, Ry

xy, xz, yz, x2, y2, z2

z

y

x

z

y

x

If the orbital is unchanged = 1If the orbital sign flips = -1

yz

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xz or dxz:

z

xy

Rz

, Rx

, Ry

yzxz

xy

xy or dxy:

B1 A2

s orbital (x2, y2, z2):

A1

x2, y2, z2

z

y

x

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C2v Char. Table from Matrix Math1. Assign/pick a point group2. Choose basis function3. Apply operations4. Generate a representation matrix5. Apply similarity transformations6. Generate an irreducible block diagonal matrix7. Character of the irreducible blocks8. Fill in the character table9. Complete the table10. Assign symmetry labels11. Assign basis functions

z

xy

Rz

, Rx

, Ry

yzxz

xy

x2, y2, z2

C2v

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Example 2: NH3 (C3v)

1. Assign a point group

C3v Character Table

Steps 2-11

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2. Choose a basis function


Hydrogen Atoms (A, B, C)

C A

B

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3. Apply operationsE, C3, C3

2, sv, sv’, sv”


A B CA’B’C’

Starting Position

Ending Position

C3 Representation Matrix

C3

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3. Apply operationsE, C3, C3

2, sv, sv’, sv”


C3

E

C32

sv’

sv

sv’’

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Matrix must be reduced down to either blocks of 1x1 matrices or a matrix that cannot be reduced further.

E C3

C32

sv’

sv

sv’’

Irreducible Matrix Reducible Matrices

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5. Apply similarity transformations


A is a matrix representation for some type of symmetry operation

n is a similarity transform operator

n-1 is the transpose of the similarity transform operator

n-1 • A • n = A’

A’ is the product matrix

A A’

• n =n-1 •

non-block diagonal

block diagonal



n-1 • A • n = A’

AC3 =

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n-1 • A • n = A’A’

n2-1 • A2 • n2 = A2

A2

A1

Irreducible Matrix!

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n-1 • A • n = A’

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C3

E

C32

sv’

sv

sv’’

Irreducible Matrices

Block Diagonal Matrices

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7. Character of the irreducible blocksExample 2: NH3 (C3v)

8. Fill in the character table

C3

E C32

sv’

sv

sv’’

1

-1

1

0

1

-1

10

1

0

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Group Similar Classes

(C3, C32)

(sv, sv’, sv”)gamma = general label for a rep.

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1 1 1G1

2 -1 0G2

G3 x

Rule 2) The number of irreducible representations is equal to the number of classes in the group.

3 classes = 3 irreducible representations. Table must be 3 x 3!

Rule 4) The sum of the squares of the dimensions under E is equal to the order of the group.

Order = 6, Therefore 12 + 22 + x2 = 6

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1 1 1G1

2 -1 0G2

G3 1

Rule 5) The sum of the squares times # of operations = order of the group.

e2 e3

1(1)2 + 2(e2)2 + 3(e3)2 = h = 6

e2 = e3 = 1 or -1

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1 1 1G1

2 -1 0G2

G3 1 e2 e3


1(1)(1) + 2(1)(e2) + 3(1)(e3) = 0

1 + 2e2 + 3e3 = 0

e2 = 1, e3 = -1

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1 1 1G1

2 -1 0G2

G3 1 e2 e3


1(1)(1) + 2(1)(e2) + 3(1)(e3) = 0

1 + 2e2 + 3e3 = 0

e2 = 1, e3 = -1

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1 1 1G1

2 -1 0G2

G3 1 1 -1

Symmetry Labels

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1 1 1A1

2 -1 0EA2 1 1 -1

Rearrange

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x, y, z, Rx, Ry, Rz

xy, xz, yz, z2, x2-y2

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z, Rz

z

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x, y, Rx, Ry

py px

px and py are neither symmetric nor antisymmetric with respect to the C3 operations, but rather go into linear combinations of one another and must therefore be considered together as components of a 2 dimensional representation.

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xy, xz, yz, z2, x2-y2

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From Matrix Math1. Assign/pick a point group2. Choose basis function3. Apply operations4. Generate a representation matrix5. Apply similarity transformations6. Generate an irreducible block diagonal matrix7. Character of the irreducible blocks8. Fill in the character table9. Complete the table10.Assign symmetry labels11.Assign basis functions

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• Character table structure– Mulliken symbols– Order– Basis functions

• Properties of Char. Tables• Driving the table

– From the rules– From matrix math

Outline

Part 2.5: Character Tables 1. Character table structure – Mulliken symbols – Order – Basis...

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Transcript of Part 2.5: Character Tables 1. Character table structure – Mulliken symbols – Order – Basis...