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PG510

Symmetry and Molecular Spectroscopy

Lecture no. 4

Group Theory:

Point Group Representation

Giuseppe Pileio

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Learning Outcomes

By the end of this lecture you will be able to:

!! Understand the concepts of Representative and Representation

!! Write down the representatives of the symmetry operations

!! Understand the difference between reducible and irreducible representations

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Representatives and Representations

What is a Representative?

A representative of a symmetry operation is an appropriate matrix that represents the operation in a mathematical way

What is a Representation?

A representation of a symmetry point group is a full set of appropriate matrices that represent each operation of the group

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Vectors and matrices

A vector (v) is a collection of numbers written as a row or a column

A matrix (M) is a rectangular array of number on m rows and n columns. When m=n the matrix is called square matrix

where the element in the j-th row and k-th column is indicated by Mjk

M11 M12 . M1!nM21 M22 . M2!n . . . .

Mm1 Mm2 . Mmn

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•! The Scalar product of two vectors is a number:

•! When A.B=0 the two vectors are said to be orthogonal

A.B !!i!1

m

Ai"Bi

•! The product of a matrix (M) and a vector (A) is a vector B with components:

Bj !!i!1

m

Mji"Ai

•! The product of two matrices (M, N) is a matrix P such as:

Pjk !!i!1

m

Mji"Nik

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•! The product of two block diagonal matrices (M and N) is a block diagonal matrix (P) where each block is the product of the corresponding two in M and N matrices:

1 2 3 0 0 0

3 2 1 0 0 0

0 1 2 0 0 0

0 0 0 2 3 0

0 0 0 1 2 0

0 0 0 0 0 2

.

2 0 0 0 0 0

1 0 2 0 0 0

0 1 1 0 0 0

0 0 0 0 2 0

0 0 0 1 2 0

0 0 0 0 0 3

!

4 3 7 0 0 0

8 1 5 0 0 0

1 2 4 0 0 0

0 0 0 3 10 0

0 0 0 2 6 0

0 0 0 0 0 6

i.e.: 1 2 3

3 2 1

0 1 2

.

2 0 0

1 0 2

0 1 1

!4 3 7

8 1 5

1 2 4

! 2 3

1 2".! 0 2

1 2" ! ! 3 10

2 6"

#2$.#3$ ! 6

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Representatives of symmetry operations

Let’s indicate a point in the Cartesian space with the vector (x,y,z) and write the possible transformations of the point in the space

Identity, E

1 0 0

0 1 0

0 0 1

!x

y

z

"x

y

z

Since the identity operation must not change the point at all it can be easily described by the matrix:

E !1 0 0

0 1 0

0 0 1

Inversion, i !

Inversion with respect to the Cartesian origin changes all the signs. This can be done writing a negative identity matrix:

!1 0 0

0 !1 0

0 0 !1"x

y

z

#!x!y!z

i !"1 0 0

0 "1 0

0 0 "1

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Reflections, "!

If the plane is chosen to coincide with one of the principal Cartesian planes (xy, xz or yz) then reflection has to leave the two coordinates whose axis define the plan unchanged while changing the sign of the third one

!"!xy" # 1 0 0

0 1 0

0 0 $1

1 0 0

0 1 0

0 0 !1"x

y

z

#x

y

!z

!"!xz" # 1 0 0

0 $1 0

0 0 1

1 0 0

0 !1 0

0 0 1

"x

y

z

# !x

y

z

!"!yz" # $1 0 0

0 1 0

0 0 1

!1 0 0

0 1 0

0 0 1

"x

y

z

#!xy

z

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Proper Rotations, Cn !

If the rotation axis is along z, then the z coordinate is unchanged. The rotation of x and y in the xy plane can be expressed using trigonometry:

Cn!2"#$"!z" ! cos $ sin $ 0

%sin $ cos $ 0

0 0 1

cos ! sin ! 0

"sin ! cos ! 0

0 0 1

#x

y

z

$x '

y '

z '

(x,y) (x’,y’)

#!

x

y

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Representations of point groups

To write the representation of a group we need to:

i.! Find out the group and its symmetry operations ii.! Choose a convenient Cartesian axis system iii.! Write the representative of each symmetry operation

Example: C2v !

Let z be the C2 axis and xz the plane for " (this is the convention). Since the presence of a C2 and a " implies two " ’s, there must be another " (called " ’) at an angle of $/2 with ", i.e. "’(yz)

" E, C2, "(xz), "(yz)

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According to what said before it is easy to write down the four representatives required:

E !1 0 0

0 1 0

0 0 1

!"!xz" # 1 0 0

0 $1 0

0 0 1

!"!yz" # $1 0 0

0 1 0

0 0 1

C2!!z" " #1 0 0

0 #1 0

0 0 1

The collection of these four matrices is said to be a representation of the C2v group

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

Note also that, for example, "vC

2 =

"’

v. In fact

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Let’s resume:

o! Each symmetry operation has its own representative and the product of these representatives is itself a member of the group o! The product of the diagonal identity matrix with any other representative leaves this latter unchanged o! Matrix product is associative o! Any square matrix whose determinant is not zero has its own inverse (representatives of symmetry operations always have non-zero determinant and are square matrix)

Representations and symmetry groups

! A collection of Representatives also forms a group

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Since the representatives form a group, then we may conjugate two matrices through a third one:

A’ = X-1 A X

and suppose we find the transformation X that operating on A will produce a block diagonal matrix A’

Reducible and Irreducible representations

If we take now a representation of the group i.e. the whole collection of representatives we have:

A’ = X-1 A X B’ = X-1 B X C’ = X-1 C X …

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where A’, B’, C’, … also form a group:

A’ . B’ = X-1 A X X-1 B X = X-1 A.B X = X-1 C X A’ . B’ = C’

The crucial point is that A’, B’, … are all block diagonal matrices and so:

A1' 0 0

0 A2' 0

0 0 A3'

.

B1' 0 0

0 B2' 0

0 0 B3'

!

C1' 0 0

0 C2' 0

0 0 C3'

i.e. A1’ . B1’ = C1’ etc…

"! The collection of each same block in all the representatives also forms a group

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To further clarify:

there is a transformation that reduces a representation to a collection of blocks that do not admit any further reduction

! This representation is said to be reducible

! The smallest representations in which the reducible representation is reduced are said to be irreducible

Although reducible representations (REP) are infinite only a finite number of irreducible (IRREP) exists

Group Theory: The number of IRREP’s is equal to the number of classes

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What did we learn in this lecture?

•! The concept of representation of a point group

•! Representatives of symmetry operations

•! The group of representatives

•! The concepts of reducible and irreducible representations