Quantitate the description of molecular symmetry by using numbers to represent symmetry operation th

1
Representations.

• Quantitate the description of molecular symmetry
  by using numbers to represent symmetry operation
  these numbers are called Representations.
• The C2v Point Group consists of the following
  elements
• E C2 s(xz)
  s(yz)
• What is the effect of C2 rotation on a px
  orbital?
• What single number might represent this
  transformation?
• C2 px -1px
• How about s(xz)? s(xz) px 1px
• And s(yz)? s(yz) px -1px
• And E? E px 1px
• We say that x belongs to the B1 representation of
  C2v because this set of numbers represents the
  effect of the group operations on a px orbital.

z
z
- ? x C2 ?
-
C2v E C2 s(xz) s(yz) B1 1
-1 1 -1 x
2
Representations

• What set of numbers represent the effect of the
  operations on a py?
• How about a pz?
• The full set of representations is included in
  the Character Table of the group (see Appendix
  C).
• The numbers in this table formally called The
  Characters of the Irreducible Representations.
  NOT irreproducible!

C2v E C2 s(xz) s(yz) B2 1
-1 -1 1 y
C2v E C2 s(xz) s(yz) A1 1
1 1 1 z
s orbital is totally symmetric and always belongs
to the A1 representation.
d orbitals. rotations about an axis.
3
Representation Problem

• What are the representations of the z-axis in
  C4v?
• What happens to an arrow along the y-axis when a
  C4 operation (clockwise) is performed?
• it is converted into the x-axis (note also x ?
  -y)
• So, what simple number represents this
  transformation?
• Can NOT be represented by a simple number, but by
  a matrix.

C4v E 2C4 C2 4sv A1 1
1 1 1 z
4
Matrices

• A matrix is an array of numbers enclosed within
  brackets.
• In symmetry we are most interested in multiplying
  matrices. In order to do this, two matrices must
  be conformable. That is, the number of columns in
  the first is equal to the number of rows in the
  second.
• The product of any two matrices is found by RC
  an element in the rth row and cth column of the
  product is formed by multiplying together the
  elements of the rth row of matrix 1 by the cth
  column of matrix 2.

( )


1 -5 8 1 0 0 1
3 4 2 7 0 -1
6 9 1 6 3 9
10 17
2
A (a r) (b u) (c x) D (d r) (e
u) (f x) B (a s) (b v) (c y) E
(d s) (e v) (f y) C (a t) ( b
w) (c z) F (d t) ( e w) (f z)
( )
r s t u v w x y z
( )
( )
a b c A B C d e
f D E F
5
Matrices

• Evaluate the following matrices
• Note that matrix multiplication is
  non-communative.
• Evaluate the following

( )
( )
1 2 1 1 5 5 3 4
2 2 11 11
( )

( )
( )
( )
1 1 1 2 4 6 2 2
3 4 8 12



0 1 0 x y -1 0 0
y -x 0 0 0 z
z

6
Back to the Representation

• What effect does a clockwise C4 rotation have on
  the x and y axes?
• new x old y
• new y -old x
• What is effect of syz on x and y-axes?
• new x -old x
• new y old y
• We will make use of one other result which comes
  from matrix algebra the character of the matrix,
  which is simply the sum of the diagonal elements.
• Notice that the character expresses the extent to
  which x is converted to itself and y is converted
  to itself in the original equations.
• Remember Character Tables? The numbers in these
  tables are the characters of the matrixes which
  represent the group operations!

( )
( )
expressed as a matrix
( )
0 1 x y -1 0
y -x

( )
( )
-1 0 x -x 0 1
y y
( )

( )
( )
0 1 -1
0 -1 0 0
1
character 0
character 0
7
Reducible Representations.

• Is the set of numbers, 3 3 1 1, an
  irreducible representation of C2v?
• No, but they are a reducible representation of
  C2v they can be obtained by adding 2A1 A2.
• How can the reducible representation, 3 -1 -1
  -1, be obtained?
• Much of the use of Group Theory to solve real
  problems involves generating a reducible
  representation and then reducing it to its
  constituent irreducible representations.

A1 1 1 1 1 A1
1 1 1 1 A2 1 1 -1
-1 2A1 A2 3 3 1 1
the reducible representation can be reduced to
its component irreducible representation, A2 B1
B2
A2 1 1 -1 -1
B1 1 -1 1 -1 B2
1 -1 -1 1 A2 B1 B2 3 -1 -1
-1
8
Reducible Representations.

• The number of times an irreducible representation
  occurs in the reducible representation is given
  by a reduction formula

(1/h) ??R ?I N
number of symmetry operations in the class
order of the group (number of operations)
character of the reducible representation
character of the irreducible representation
h 1 2 3 6
Reduce the following in C3v ?
4 1 -2
C3v 1E 2C3 3sv A1 1 1
1 A2 1 1 -1 E
2 -1 0
A1 (1/6) (4 1 1) (1 1 2) (-2 1
3) 0 A2 (1/6) (4 1 1) (1 1
2) (-2 -1 3) 2 E (1/6) (4 2
1) (1 -1 2) (-2 0 3) 1
So, ? 2A2 E
9
Group Theory Example

• The use of Group Theory can be summarized in 3
  rules
• Use an appropriate basis to generate a reducible
  representation.
• Reduce it to an irreducible representation.
• Interpret the results.
• Example What orbitals can be hybridized to
  produce a set of three trigonal planar s-bonds in
  BF3?
• What is the character of each matrix representing
  the 6 operations in D3h? (i.e. which of the 3
  vectors are transferred to themselves or left
  unshifted by the operation?)

a1 a2 a3
D3h point group 3 vectors as the basis
D3h E 2C3 3C2 sh 2S3 3sv
? 3 0 1 3 0
1
10
Group Theory Example
D3h E 2C3 3C2 sh 2S3 3sv
? 3 0 1 3 0
1

• This is a reducible representation, so use
  character table to reduce it.

A1 (1/12) 3 0 3 3 0
3 1 A2 (1/12) 3 0 - 3 3
0 - 3 0 E (1/12) 6 0
0 6 0 0 1
? A1 E What orbitals belong to these
symmetry species? A1 s-orbital E
2 p-orbitals Therefore, it is an sp2 hybrid
orbital