Chapter 3 - Group Theory

A Group is a collection of elements which is:

i) closed under some single-valued associative binary operation

ii) contains a single element satisfying the identity law

iii) and has a reciprocal element for each element in the group

Collection: a specified # of elements (finite or infinite)

Elements: the consitituents of the group (i.e., symmetry operations)

Binary Operation: the combination of two elements of a group to yield another element in the group. The combination may be mathematical (addition, subtraction, etc.) or qualitative as in the successive application of two symmetry operations on an object.

Single-valued: the combination of two elements yields a unique result

Closed: the combination of any two group elements must always yield another element belonging to the group.

Associative: the associative law of combination must hold for the group.

(AB)C = A(BC)

Group Theory 2

In general, however, elements of a group do not have to commute (but they can):

AB ≠ BA

Identity Law: there must be an element in the group which when combined with any element in the group will leave them unchanged. This element is called the identity or unit element and it commutes with all elements of the group. It is given the symbol E.

EA = A AE = A EE = E

Reciprocal Element: for each element A in a group there must be an element called the reciprocal, A1, such that the following holds:

AA1 = A1A = E

In general, group multiplication is not commutative, i.e., AB ≠ BA. However, it can be and a group in which multiplication is completely commutative is called an Abelian Group.

Group Theory 3

Group Multiplication Table (matrix operations)

G3 E A B

E E A B

A A B E

B B E A

Each row and each column in a group multiplication table lists each of the group elements ONCE and ONLY ONCE. It therefore follows that no two columns or rows may be identical!

Consider a “real” C3 table using symmetry elements:

C3 E C3 C32

E E C3 C32

C3 C3 C32 E

C32 C3

2 E C3

(column) × (row)

Abelian group

Group Theory 4

Consider the two different ways we can set up a 4 × 4 table:

G41 E A B C

E E A B C

A A E C B

B B C E A

C C B A E

Note that each element times itself generates E.

G42 E A B C

E E A B C

A A B C E

B B C E A

C C E A B

Note that this group table above is cyclic, that is, the group is generated by one element:

A = A A3 = C

A2 = B A4 = E

Group Theory 5

Note that the G3 (C3) example above was also cyclic.

There is only one group combination possible for the G5

group, which turns out to be cyclic as well:

G5 E A B C D

E E A B C D

A A B C D E

B B C D E A

C C D E A B

D D E A B C

Note the diagonal lining up of the elements in cyclic groups (symmetry of a matrix sort).

Group Theory 6

SubGroups

A subgroup is a self-contained group of elements residing within a larger group.

h(order of main group )g(order of subgroup )

=k ( integer )

G6 E A B C D F

E E A B C D F

A A E D F B C

B B F E D C A

C C D F E A B

D D C A B F E

F F B C A E D

G3 E D F

E E D F

D D F E

F F E D

Group Theory 7

Group Theory 8

Classes

Assume that A and X are elements of a group and we perform the following operation:

X1AX = B

Where B is another element in the group. B is then called the similarity transform of A by X. If this relationship holds, then A and B are said to be conjugate.

The following is true for elements that are related by similarity transforms:

1) Every element is conjugate with itself

A = X1AX

(X may be equal to the identity element E)

2) If A is conjugate with B, then B is conjugate with A

Thus, if we have:

X1AX = B

Then there must exist another element Y such that:

Y1BY = A

3) Finally, if A is conjugate to both B and C, then B and C must also be conjugate to each other.

A group of elements that are conjugate to one another is called a Class of Elements.

Group Theory 9

To determine which elements group together to form a class you have to work out all the similarity transforms for each element in the group. Those sets of elements that transform into one another are then in the same class.

Consider the C3v symmetry point group “matrix”:

C3v E C3 C32 v

1 v2 v

3

E E C3 C32 v

1 v2 v

3

C3 C3 C32 E v

2 v3 v

1

C32 C3

2 E C3 v3 v

1 v2

v1 v

1 v2 v

3 E C3 C32

v2 v

2 v3 v

1 C32 E C3

v3 v

3 v1 v

2 C3 C32 E

Lets determine the classes of symmetry operations for this point group. Lets start with the similarity transforms for the vertical mirror planes:

v1v

1v11 = v

1

v2v

1v21 = v

3

v3v

1v31 = v

2

Group Theory 10

Let’s see how this works graphically:

v2v

1v21 = v

3

Group Theory 11

v3v

1v31 = v

2

Group Theory 12

C3v1C3

1 = v3

Group Theory 13

If we continue these similarity transforms we find that the various symmetry operations for C3v break down into the

following classes:

E

C3, C32

v1, v

2 , v3

If we examine the character tables in Cotton we find that the symmetry operations are listed and grouped together in these very same classes:

Corollary: the orders of all the classes must be integral factors of the order of a group.

Order of a point group = # of symmetry operations

Group Theory 14

Matrix Operations

Consider the following matrix:

a11 a12 a13 a14 . . . a1n

a21 a22 a23 a24 . . . a2n

a31 a32 a33 a34 . . . a3n

. . .

an1 an2 an2 an2 . . . amn

Character: sum of diagonal elements

In order to multiply two matrices they must be conformable, i.e., to multiply matrix A by matrix B, the number of columns in A must equal the number of rows in B.

(a11 × b11) + (a12 × b21) = c11

3 × 2 2 × 4 = 3 × 4

Row

Column

Row Column

Group Theory 15

The symmetry operations can all be represented mathematically as 3 × 3 square matrices.

To carry out the symmetry operation, you multiply the symmetry operation matrix times the coordinates you want to transform. The x, y, z coordinates are written in vector format as a 3 × 1 matrix:

For example, the inversion operation take the general coordinates x, y, z to x, y, z. In matrix terms we would write:

x(new) = (1)(x) + (0)(y) + (0)(z)

y(new) = (0)(x) + (1)(y) + (0)(z)

z(new) = (0)(x) + (0)(y) + (1)(z)

Group Theory 16

Symmetry Operation Matrices:

E

i

(xy)

(xz)

Group Theory 17

Cn × h = Sn

(yz)

Cn

Sn

Group Theory 18

Cn h

Group Theory 19

Group Representations

The set of four matrices that describe all of the possible symmetry operations in the C2v point group that can act on

a point with coordinates x, y, z is called the total representation of the C2v group.

E C2 xz yz

Note that each of these matrices is block diagonalized, i.e., the total matrix can be broken up into blocks of smaller matrices that have no off-diagonal elements between blocks.

These block diagonalized matrices can be broken down, or reduced into simpler one-dimensional representations of the 3-dimensional matrix.

If we consider symmetry operations on a point that only has an x coordinate (e.g., x, 0, 0), then only the first row of our total representation is required:

C2v E C2 xz yz

1 1 1 1 x

Group Theory 20

We can do a similar breakdown of the y and z coordinates to setup a table:

C2v E C2 xz yz

1 1 1 1 x

1 1 1 1 y

1 1 1 1 z

These three 1-dimensional representations are as simple as we can get and are called irreducible representations.

There is one additional irreducible representation in the C2v point group. Consider a rotation Rz :

The identity operation and the C2 rotation

operations leave the direction of the rotation Rz unchanged. The mirror planes, however,

reverse the direction of the rotation (clockwise to counter-clockwise), so the irreducible representation can be written as:

C2v E C2 xz yz

1 1 1 1 Rz

4 Classes of symmetry operations =

Group Theory 21

4 Irreducible representations!!

Now lets consider a case where we have a 2-dimensional irreducible representation. Consider the matrices for C3v

E C3 vIn this case the matrices block diagonalize to give two reduced matrices. One that is 1-dimensional for the z coordinate, and the other that is 2-dimensional relating the x and y coordinates.

Multidimensional matrices are represented by their characters (trace), which is the sum of the diagonal elements.

Since cos(120º) = 0.50, we can write out the irreducible representations for the 1- (z) and 2-dimensional “degenerate” x and y representations:

C3v E 2C3 v

1 1 1 z

2 1 0 x,y

Group Theory 22

As with the C2v example, we have another irreducible

representation (3 symmetry classes = 3 irreducible representations) based on the Rz rotation axis. This

generates the full group representation table:

C3v E 2C3 v

1 1 1 z

2 1 0 x,y

1 1 1 Rz

Group Theory 23

Character Tables

Schoenflies symmetry symbol

Mulliken Symbol Notation

1) A or B: 1-dimensional representations

E : 2-dimensional representations

T : 3-dimensional representations

2) A = symmetric with respect to rotation by the Cn axis

B = anti-symmetric w/respect to rotation by Cn axis

Symmetric = + (positive) character

Anti-symmetric = (negative) character

Characters of the irreducible

representations

Mullikensymbols

x, y, zRx, Ry, Rz

Squares & binary

products of the coordinates

Group Theory 24

3) Subscripts 1 and 2 associated with A and B symbols indicate whether a C2 axis to the principle axis produces a symmetric (1) or anti-symmetric (2) result.

If C2 axes are absent, then it refers to the effect of vertical mirror planes (e.g., C3v)

4) Primes and double primes indicate representations that are symmetric ( ) or anti-symmetric ( ) with respect to a h mirror plane. They are NOT used

when one has an inversion center present (e.g., D2nh or

C2nh).

5) In groups with an inversion center, the subscript “g” (“gerade” meaning even) represents a Mulliken symbol that is symmetric with respect to inversion.

Group Theory 25

The symbol “u” (“ungerade” meaning uneven) indicates that it is anti-symmetric.

6) The use of numerical subscripts on E and T symbols follow some fairly complicated rules that will not be discussed here. Consider them to be somewhat arbitrary.

Square and Binary Products

These are higher order “combinations” or products of the primary x, y, and z axes.

Group Theory 26

The Great Orthogonality Theorem

Group Theory 27

i (R)mn The element in the mth row and nth column of

the matrix corresponding to the operation R in the ith irreducible representation i.

i (R)mn* complex conjugate used when imaginary or

complex #’s are present (otherwise ignored)

h the order of the group

li the dimension of the ith representation

(A = 1, B = 1, E = 2, T = 3)

delta functions, = 1 when i = j, m = m’, or n = n’; = 0 otherwise

The different irreducible representations may be thought of as a series of orthonormal vectors in h-space, where h is the order of the group.

Group Theory 28

Because of the presence of the delta functions, the equation = 0 unless i = j, m = m’, or n = n’. Therefore, there is only one case that will play a direct role in our chemical applications:

if i ≠ j

if m ≠ m’ n ≠ n’

Group Theory 29

Five “Rules” about Irreducible Representations:

1) The sum of the squares of the dimensions of the irreducible representations of a group is equal to the order, h, of a group.

For example, consider the D3h point group:

l(A1’)2 + l(A2’)2 + l(E’)2 + l(A1”)2 + l(A2”)2 + l(E”)2

(1)2 + (1)2 + (2)2 + (1)2 + (1)2 + (2)2 = 12

2) The sum of the squares of the characters in any irreducible representation is also equal to the order of the group h.

For example, for the E’ representation in D3h:

h = 12 (order of group)

g = # of symmetry operations R in a class

Dimensions:

A or B = 1E = 2T = 3

Group Theory 30

(E)2 + 2(C3)2 + 3(C2)2 + (h)2 + 2(S3)2 + 3(h)2

(2)2 + 2(-1)2 + 3(0)2 + (2)2 + 2(-1)2 + 3(0)2 = 12

3) The vectors whose components are the characters of two different irreducible representations are orthogonal.

For example, multiply out the A2’ and E’ representations

in D3h:

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

2 + (-2) + 0 + 2 + (-2) + 0 = 0

4) In a given representation the characters of all matrices belonging to operations in the same class are identical.

5) The number of irreducible representations in a group is equal to the number of classes in the group.