2012 Spring AIC 03 Group Theory 2 Web

8/2/2019 2012 Spring AIC 03 Group Theory 2 Web

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Inorganic Chemistry

Yunho Lee, Ph.D.02/14/2012

Department of Chemistry

Korea Advanced Institute of Science and Technology

Class 02Advanced Inorganic Chemistry

Symmetry & Group TheoryChemical Applications of Group Theory,

3rd ed.F. Albert Cotton, 1990


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Inorganic Chemistry

A mathematical Group is a collection of elements that areinterrelated according to certain rules.

3. The product (combination) of any two elements and thesquare of each element must be an element in the group.

1. Identity element (E): one element in the group must

commute with all elements and leave them unchanged.

4. The associative law of multiplication must hold.

2. Every element must have a reciprocal.

AB or BA are they the same?Multiplication is commutative.

xy = yx and 3 x 6 = 6 x 3However, in the group theory, the commutative law doesnot in general hold. AB = C while BA = D, where C and Dare two more elements in the group.

EX = XE = X

A(BC) = (AB)C

AA-1 = A-1A = E

(ABC..XY)-1

= X-1

Y-1

..C-1

B-1

A-1

Identity

Inverse

Closure

Associativity


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Point Groups

A The set of symmetry operations describing themolecules overall symmetry is called the point group

of the molecule. Group with low order symmetry: C1, Cs, Ci Group with high order symmetry: Cv, Dh, Td, Oh, Ih D group: Dnh, Dnd, Dn

C or S2n group: Cnh, Cnv, S2n, Cn 1. E,2. 4C3, 4C32,

3. 6C2,4. 3C4, 3C4

3,5. 3C4

2,6. i,7. 3S4, 3S4

3,8. 4S6, 4S6

5,9. 3h,

10. 6d

Oh

48 operations


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Point Groups1. E2. 4C3, 4C3

23. 6C24. 3C4, 3C435. 3C4

2 (=3C2)6. i7. 3S4, 3S4

3

8. 4S6, 4S659. 3h10. 6d

Oh

48 operations

A group may be separated into smaller sets called classes.

To arrange the operations of a symmetry group into sets ofequivalent operations. Theses sets will be the classes.


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Inorganic Chemistry

Classes

- Similarity transformations refer to a matrixtransformation that results in a similarity.

If A and X are in a group,

B = X-1 A X A and B are conjugate.

(i) Every element is conjugate with itself.

(ii) If A is conjugate with B, then B is conjugate with A.

(iii) If A is conjugate with B and C, then B and C are conjugate with

each other.

A group may be separated into smaller sets called classes.

To arrange the operations of a symmetry group into sets ofequivalent operations. Theses sets will be the classes.

A complete set of elements that are conjugate to oneanother is called a class of the group.


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Inorganic Chemistry

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Representations of C2vPoint Groups

E C2 v(xz) v(yz)

1 0 0

0 1 0

0 0 1

1 0 0

0 -1 0

0 0 1

-1 0 0

0 -1 0

0 0 1

-1 0 0

0 1 0

0 0 1

C2v E C2 v(xz) v(yz)B1 1 -1 1 -1 x, Ry xz

B2 1 -1 -1 1 y, Rx yx

A1 1 1 1 1 z x2, y2, z2

A2 1 1 -1 -1 Rz xy

Character table for C2v


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C2v E C2 v(xz) v(yz)

B1 1 -1 1 -1 x, Ry xzB2 1 -1 -1 1 y, Rx yx

A1 1 1 1 1 z x2, y2, z2

A2 1 1 -1 -1 Rz xy


Area IArea II Area III

Schoenflies symbolfor the group

Classes

Character Table of Groups


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Properties of Characters of Irreducible Rep. in the Point Group

1. Order (h)2. Classes3.# of irreducible representations = # of classes4. Dimension: The sum of (Dimension)2= h5.The sum of (Character)2 # of operations in the class = h

Area I Character Table of Groups

6. Irreducible representations are orthogonal each other! The sum of the products of any two representationsmultiplied by the number of operations in each class

equals zero.

7. A totally symmetric representation is included in all groups.


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1. Order (h): The total # of symmetry operations in the group

E, C2, v(xz), v(yz) h = 42. Classes: symmetry operations arranged by class3. # of irreducible representations = # of classes

E C2 v(xz) v(yz)

B1 1 -1 1 -1 x, Ry xz

B2 1 -1 -1 1 y, Rx yx

A1 1 1 1 1 z x2, y2, z2

A2 1 1 -1 -1 Rz xy


Dimension for eachirreducible representation

4. The sum of (Dimension)2

= h

For each irreducible rep,

5. The sum of (Character)2 #

of operations in the class = h



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Character table for D3h

D3h E 2C3 3C2

h 2S3 3v

A1 1 1 1 1 1 1

A2 1 1 -1 1 1 -1

E 2 -1 0 2 -1 0

A1 1 1 1 -1 -1 -1

A2 1 1 -1 -1 -1 1

E 2 -1 0 -2 1 0

D3h group

1.Order, h = 1 + 2 + 3 + 1 + 2 + 3 = 122. 6 Classes3. 6 Classes = 6 irreducible representations

5. The sum of (Character)2 # of operations in the class=1 (1)2+ 2 (1)2+ 3 (-1)2+ 1 (1)2+ 2 (1)2+ 3 (-1)2= 12

4. The sum of (Dimension)2= 1 + 1 + 4 + 1 + 1 + 4 = 12



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Molecular Vibrations of Water- Modes of vibration of molecules can be determined by symmetry!-

Water has C2vsymmetry- C2 axis is parallel to z axis.

x

y

E C2 v(xz) v(yz)

1 0 0

0 1 0

0 0 1

1 0 0

0 -1 0

0 0 1

-1 0 0

0 -1 0

0 0 1

-1 0 0

0 1 0

0 0 1


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Inorganic Chemistry

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1 0 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0 0

0 0 1 0 0 0 0 0 0

0 0 0 1 0 0 0 0 0

0 0 0 0 1 0 0 0 0

0 0 0 0 0 1 0 0 0

0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 0 1

x

y

z

=x

y

z

x

y

z

=x

y

z

x

y

z

=x

y

z

O

Ha

Hb

O

Ha

Hb

E operation

Character () = 9

Molecular Vibrations of Water


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Inorganic Chemistry

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-1 0 0 0 0 0 0 0 0

0 -1 0 0 0 0 0 0 0

0 0 1 0 0 0 0 0 0

0 0 0 0 0 0 -1 0 0

0 0 0 0 0 0 0 -1 0

0 0 0 0 0 0 0 0 1

0 0 0 -1 0 0 0 0 0

0 0 0 0 -1 0 0 0 0

0 0 0 0 0 1 0 0 0

=x

y

z

=x

y

z

=x

y

z

O

Ha

Hb

O

Ha

Hb

C2 operation

Character () = -1

x

y

z

x

y

z

x

y

z



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Inorganic Chemistry

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=x

y

z

=x

y

z

=x

y

z

O

Ha

Hb

O

Ha

Hb

v(xz)operation

Character () = 3

x

y

z

x

y

z

x

y

z

1 0 0 0 0 0 0 0 0

0 -1 0 0 0 0 0 0 0

0 0 1 0 0 0 0 0 0

0 0 0 1 0 0 0 0 0

0 0 0 0 -1 0 0 0 0

0 0 0 0 0 1 0 0 0

0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 0 -1 0

0 0 0 0 0 0 0 0 1



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Inorganic Chemistry

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-1 0 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0 0

0 0 1 0 0 0 0 0 0

0 0 0 0 0 0 -1 0 0

0 0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 0 1

0 0 0 -1 0 0 0 0 0

0 0 0 0 1 0 0 0 0

0 0 0 0 0 1 0 0 0

=x

y

z

=x

y

z

=x

y

z

O

Ha

Hb

O

Ha

Hb

Character () = 1

x

y

z

x

y

z

x

y

z

v(yz)operation



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Inorganic Chemistry

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- Irreducible rep. represents all molecular motions; 3 translations,3 rotations, and 3 vibrations.


A1 1 1 1 1 z x2, y2, z2

A2 1 1 -1 -1 Rz xy

B1 1 -1 1 -1 x, Ry xzB2 1 -1 -1 1 y, Rx yz

9 -1 3 1

# ofirred. rep. ofa given type

=

# ofoperationsin the class

Characterof red.rep.

Characterof irred.

rep. [ ]

Order

Reducing reducible representation to irreducible representations!

How many A1, A2, B1 and B2 are in ?



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# of

irred. rep. ofa given type

=

# ofoperationsin the class

Characterof red.rep.

Characterof irred.

rep.

[ ]Order

Reducing reducible representation to irreducible representations!

How many A1, A2, B1 and B2 are in ?

# of A1 =(191) + (1-11) + (131) + (111)

4

= 3

# of A2 =(191) + (1-11) + (13-1) + (11-1)

4= 1

# of B1 =(191) + (1-1-1) + (131) + (11-1)

4 = 3

# of B2 =(191) + (1-1-1) + (13-1) + (111)

4= 2

= 3A1 + A2 + 3B1 + 2B2


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Inorganic Chemistry

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- 3 translations: A1 + B1 + B2z x y directions!

- 3 rotations: A2 + B1 + B2R

z

Ry

Rx

directions!

- 3 vibrations: 2A1 + B1

Irreducible rep. represents all molecular motions;= 3A1 + A2 + 3B1 + 2B2 Molecular Vibrations of Water


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Inorganic Chemistry

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MOs of Water


A1 1 1 1 1 z x2

, y2

, z2

A2 1 1 -1 -1 Rz xy

B1 1 -1 1 -1 x, Ry xz

B2 1 -1 -1 1 y, Rx yz

2 0 2 0

1.Determine the point group: C2v2.Assign x, y, z coordinate to the atoms3.Construct a reducible representation for two H atoms!4. Reduce red. ref. to irred. ref.;

Symmetry-Adapted Linear Combination.5. Identify the AOs of the central atom with same symmetry

6. Combine the AOs of the central atom and others with matchingsymmetry and energy to form MOs!7. # of AOs = # of MOs

= 2 0 2 0= A1 + B1


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A1 (1/2) (Ha + Hb)

B1 (1/2) (Ha - Hb)

MOs of Water


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MOs of Water


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1:2s + (Ha + Hb) Bonding2:2Px + (Ha Hb) Bonding3:2Pz + (Ha + Hb)

Slightly bonding

1

2

3

4:2PyNonbonding5:2s - (Ha + Hb) Antibonding6:2Px - (Ha Hb)

Antibonding

MOs of Water


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MOs of Water


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MOs of Water

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2012 Spring AIC 03 Group Theory 2 Web

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