Character Tables

1
Lecture 6
2
Character Tables
Group theory makes use of the properties of
matrices Idea When an operation, O, (proper
rotation, improper rotation, reflection,
inversion) is done on the function, f(x,y,z) O
f(x,y,z) the result is a value taken on by the
original function at some other point
(x,y,z). Or O f(x,y,z) f (x,y,z) We
know the function f(x,y,z) the problem is to
determine the values (x,y,z). The nature of
the operation tells us where to look.
3
Character Tables
Group theory makes use of the properties of
matrices Each operation may be expressed as a
transformation matrix New coordinates
transformation matrixold coordinates

• Example in Cartesian coordinate system,
  reflection in x 0 plane
• Changes the value of x to x (multiplies it by
  -1)
• Leaves y unchanged (multiplies it by 1)
• Leaves z unchanged (multiplies it by 1)

To see the result of the operation at (x, y, z)
look at the original object at (x, y, z).

Results of transformation.
Original coordinates
Transformation matrix
4
Recall Technique of Matrix multiplication
V
M
V
To get an element of the product vector a row in
the operation square matrix is multiplied by the
original vector matrix.
For example
V2 y
M2,1 V1
M2,2 V2
M2,3 V3
y 0 x 1 y 0 z y
5
Character Tables - 2
The matrix representation of the symmetry
operations of a point group is the set of
matrices corresponding to all the symmetry
operations in that group. The matrices record
how the x,y,z coordinates are modified as a
result of an operation.
For example, the C2v point group consists of the
following operations E do nothing.
Unchanged. C2 rotate 180 degrees about the z
axis x becomes x y becomes y and z
unchanged. sv (xz) y becomes y sv (yz) x
becomes -x
sv (yz)
E
C2
sv (xz)
6
Operations Applied to Functions - 1
Transform the coordinates.
Consider f(x) x2 sv (f(x)) sv(x2)
(-x)2 x2 f(x)
or
sv (f(x)) 1 f(x)
f(x) is an eigenfunction of this reflection
operator with an eigenvalue of 1. This is
called a symmetric eigenfunction.
Similarly f(x) x3
sv (f(x)) -1 f(x)
f(x) is an eigenfunction of this reflection
operator with an eigenvalue of -1. This is
called a antisymmetric eigenfunction.
7
Plots of Functions, x2
Reflection yields.
Here f(x) is x2. It can be seen to be a
symmetric function for reflection at x 0
because of mirror plane.
The reflection carries out the mapping shown with
the red arrows.
x2 is an eigenfunction of s with eigenvalue 1
8
Plots of Functions, x3
Reflection yields.
Here f(x) is x3. It can be seen to be a
antisymmetric function for reflection at x 0.
The reflection carries out the mapping shown with
the red arrows.
x3 is an eigenfunction of s with eigenvalue -1
9
Plots of Functions - 2
Reflection yields.
Here f(x) is x3. It can be seen to be a
antisymmetric function for reflection at x 0.
The reflection carries out the mapping shown with
the red arrows.
10
Operations Applied to Functions - 2
Now consider f(x) (x-2)2 x2 4x 4 sv
(f(x)) sv(x-2)2 (-x-2)2 x2 4x 4
f(x) (x-2)2 is not an eigenfunction of this
reflection operator because it does not return a
constant times f(x).
Reflection yields this function, not an
eigenfunction.
Neither symmetric nor antisymmetric for
reflection thru x 0.
11
Lets look at Atomic Orbitals
Reflection
Get the same orbital back, multiplied by 1, an
eigenfunction of the reflection, symmetric with
respect to the reflection. The s orbital forms
the basis of an irreducible representation of the
operation
s orbital
z
12
Atomic Orbitals
s
Reflection
Get the same orbital back, multiplied by -1, an
eigenfunction of the reflection, antisymmetric
with respect to the reflection. The p orbital
behaves differently from the s orbital and forms
the basis of a different irreducible
representation of the operation
p orbital
z
13
Simplest ways that objects can behave for a group
consisting of E and sh , the reflection plane.
Irreducible Representations.
Basis of the Irreducible Reps.
Cs E sh
A A 1 1 -1 x, y,Rz z, Rx,Ry x2,y2,z2,xy yz, xz
s orbital is spherical behaves as x2 y2 z2.
s orbital is A. The s orbital is an
eigenfunction of both E and sh.
pz orbital has a multiplicative factor of z times
a spherical factor. Behaves as A. pz is an
eigenfunction of both E and sh.
14
sp Hybrids
Reflection
Do not get the same hybrid back multiplied by 1
or -1 or some other constant. Not an
eigenfunction.
hybrid
z
The two hybrids form the basis of a reducible
representation of the operation
Recall the hybrid can be expressed as the sum of
an s orbital and a p orbital.


Reduction expressing a reducible representation
as a combination of irreducible representations.
15
Reducible Representations
Use the two sp hybrids as the basis of a
representation
h1
h2
sh operation.
E operation.
h1 becomes h1 h2 becomes h2.
h1 becomes h2 h2 becomes h1.


The reflection operation interchanges the two
hybrids.
The hybrids are unaffected by the E operation.
Proceed using the trace of the matrix
representation.
0 0 0
1 1 2
16
Our Irreducible Representations
Cs E sh
A A 1 1 -1 x, y,Rz z, Rx,Ry x2,y2,z2,xy yz, xz
The reducible representation derived from the two
hybrids can be attached to the table.
G 2 0 (h1, h2)
Note that G A A
17
Return to polynomials f(x) (x-2)2 x2 4x
4 sv (f(x)) sv(x-2)2 (-x-2)2 x2
4x 4 g(x)
Neither f nor g is an eigenfunction of s but,
taken together, they do form an reducible
representation since they show what the s
operator does.
Approaching the problem in the same way as we did
for hybrids we can carry out the reduction this
way u(x) ½ (f(x) g(x)) ½ (f(x) s f(x))
x2 4 symmetric, unchanged by the s operator.
Behaves as A v(x) ½ (f(x) - g(x)) ½ (f(x)
- s f(x)) -4x, antisymmetric, multiplied by -1
by the s operator. Behaves as A
18
Character Table
Symmetry operations, Classes
Point group
x, y, z Symmetry of translations (p orbitals)
Rx, Ry, Rz rotations
Characters 1 symmetric behavior -1 antisymmetric
Mülliken symbols
Each row is an irreducible representation
19
Character Tables - 3
Irreducible representations are not linear
combinations of other representation (Reducible
representations are) of irreducible
representations of classes of symmetry
operations Instead of the matrices, the
characters are used (traces of matrices) A
character Table is the complete set of
irreducible representations of a point group
20
Effect of the 4 operations in the point group
C2v on a translation in the x direction. The
translation is simply multiplied by 1 or -1. It
forms a basis to show what the operators do to
an object.
Operation E C2 sv sv
Transformation 1 -1 1 -1
21
Character Table
Verify this character. It represents how a
function that behaves as x, Ry, or xz behaves for
C2.
22
Another point group, C3v.
x, y, z Symmetry of translations (p orbitals)
Classes of operations
Rx, Ry, Rz rotations
dxy, dxz, dyz, as xy, xz, yz dx2- y2 behaves as
x2 y2 dz2 behaves as 2z2 - (x2 y2) px, py, pz
behave as x, y, z s behaves as x2 y2 z2
23
Symmetry of Atomic Orbitals
24
Naming of Irreducible representations

• One dimensional (non degenerate) representations
  are designated A or B.
• Two-dimensional (doubly degenerate) are
  designated E.
• Three-dimensional (triply degenerate) are
  designated T.
• Any 1-D representation symmetric with respect to
  Cn is designated A antisymmétric ones are
  designated B
• Subscripts 1 or 2 (applied to A or B refer) to
  symmetric and antisymmetric representations with
  respect to C2 ? Cn or (if no C2) to ? sv
  respectively
• Superscripts and indicate symmetric and
  antisymmetric operations with respect to sh,
  respectively
• In groups having a center of inversion,
  subscripts g (gerade) and u (ungerade) indicate
  symmetric and antisymmetric representations with
  respect to i

25
Character Tables

• Irreducible representations are the generalized
  analogues of s or p symmetry in diatomic
  molecules.
• Characters in rows designated A, B,..., and in
  columns other than E indicate the behavior of an
  orbital or group of orbitals under the
  corresponding operations (1 orbital does not
  change -1 orbital changes sign anything else
  more complex change)
• Characters in the column of operation E indicate
  the degeneracy of orbitals
• Symmetry classes are represented by CAPITAL
  LETTERS (A, B, E, T,...) whereas orbitals are
  represented in lowercase (a, b, e, t,...)
• The identity of orbitals which a row represents
  is found at the extreme right of the row
• Pairs in brackets refer to groups of degenerate
  orbitals and, in those cases, the characters
  refer to the properties of the set

26
Definition of a Group

• A group is a set, G, together with a binary
  operation such that the product of any two
  members of the group is a member of the group,
  usually denoted by ab, such that the following
  properties are satisfied
• (Associativity) (ab)c a(bc) for all a, b, c
  belonging to G.
• (Identity) There exists e belonging to G, such
  that eg g ge for all g belonging to G.
• (Inverse) For each g belonging to G, there exists
  the inverse of g, g-1, such that g-1g gg-1
  e.
• If commutativity is satisfied, i.e. ab ba for
  all a, b belonging to G, then G is called an
  abelian group.

27
Examples

• The set of integers Z, is an abelian group under
  addition.
• What is the element e, identity, such that
• ae a?
• What is the inverse of the a element?

0
-a
28
As applied to our symmetry operators.

• For the C3v point group

What is the inverse of each operator? A A-1
E
E C3(120) C3(240)
sv (1) sv (2) sv (3)
E C3(240) C3(120)
sv (1) sv (2) sv (3)
29
Examine the matrix represetation of the C2v point
group
E
C2
sv(yz)
sv(xz)
30
Multiplying two matrices (a reminder)
31
C2
sv(xz)
sv(yz)
E
What is the inverse of C2?
C2

What is the inverse of sv?
sv

32
What of the products of operations?
C2
sv(xz)
sv(yz)
E
C2
E C2 ?

sv C2 ?
sv

33
Classes
Two members, c1 and c2, of a group belong to the
same class if there is a member, g, of the group
such that gc1g-1 c2
34
Properties of Characters of Irreducible
Representations in Point Groups

• Total number of symmetry operations in the group
  is called the order of the group (h). For C3v,
  for example, it is 6.

1 2 3 6

• Symmetry operations are arranged in classes.
  Operations in a class are grouped together as
  they have identical characters. Elements in a
  class are related.

This column represents three symmetry operations
having identical characters.
35
Properties of Characters of Irreducible
Representations in Point Groups - 2

• The number of irreducible reps equals the number
  of classes. The character table is square.

1 2 3 6
3 by 3
1 1 22 6
The sum of the squares of the dimensions of the
each irreducible rep equals the order of the
group, h.
36
Properties of Characters of Irreducible
Representations in Point Groups - 3
For any irreducible rep the squares of the
characters summed over the symmetry operations
equals the order of the group, h.
A1 12 (12 12 ) 6
A2 12 (12 12 ) ((-1)2 (-1)2 (-1)2
) 6
E 22 (-1)2 (-1)2 6
37
Properties of Characters of Irreducible
Representations in Point Groups - 4
Irreducible reps are orthogonal. The sum of the
products of the characters for each symmetry
operation is zero.
For A1 and E 1 2 (1 (-1) 1 (-1))
(1 0 1 0 1 0) 0
38
Properties of Characters of Irreducible
Representations in Point Groups - 5
Each group has a totally symmetric irreducible
rep having all characters equal to 1
39
Reduction of a Reducible Representation
Irreducible reps may be regarded as orthogonal
vectors. The magnitude of the vector is h-1/2 Any
representation may be regarded as a vector which
is a linear combination of the irreducible
representations.
Reducible Rep S (ai IrreducibleRepi) The
Irreducible reps are orthogonal.
Hence S(character of Reducible Rep)(character of
Irreducible Repi) ai h Or ai S(character
of Reducible Rep)(character of Irreducible Repi)
/ h
Sym ops
Sym ops
40
These are block-diagonalized matrices (x, y, z
coordinates are independent of each other)
Reducible Rep
41
C2v Character Table to be used for water
Symmetry operations
Point group
Characters 1 symmetric behavior -1 antisymmetric
Mülliken symbols
Each row is an irreducible representation
42
Lets use character tables! Symmetry and
molecular vibrations
of atoms degrees of freedom Translational modes Rotational modes Vibrational modes
N (linear) 3 x 2 3 2 3N-5 1
Example 3 (HCN) 9 3 2 4
N (non- linear) 3N 3 3 3N-6
Example 3 (H2O) 9 3 3 3
43
Symmetry and molecular vibrations
A molecular vibration is IR active only if it
results in a change in the dipole moment of the
molecule A molecular vibration is Raman
active only if it results in a change in the
polarizability of the molecule
In group theory terms A vibrational mode is IR
active if it corresponds to an irreducible
representation with the same symmetry of a x, y,
z coordinate (or function) and it is Raman
active if the symmetry is the same as A quadratic
function x2, y2, z2, xy, xz, yz, x2-y2 If the
molecule has a center of inversion, no vibration
can be both IR Raman active
44
How many vibrational modes belong to each
irreducible representation?
You need the molecular geometry (point group) and
the character table
Use the translation vectors of the atoms as the
basis of a reducible representation. Since you
only need the trace recognize that only the
vectors that are either unchanged or have become
the negatives of themselves by a symmetry
operation contribute to the character.
45
A shorter method can be devised. Recognize that
a vector is unchanged or becomes the negative of
itself if the atom does not move. A reflection
will leave two vectors unchanged and multiply the
other by -1 contributing 1. For a rotation
leaving the position of an atom unchanged will
invert the direction of two vectors, leaving the
third unchanged. Etc.
Apply each symmetry operation in that point
group to the molecule and determine how many
atoms are not moved by the symmetry
operation. Multiply that number by the character
contribution of that operation E 3 s 1 C2
-1 i -3 C3 0 That will give you the
reducible representation
46
Finding the reducible representation
E 3 s 1 C2 -1 i -3 C3 0
3x3 9
1x-1 -1
3x1 3
1x1 1
( atoms not moving x char. contrib.)
G
47
Now separate the reducible representation into
irreducible ones to see how many there are of
each type
S
A1 1/4 (1x9x1 1x(-1)x1 1x3x1 1x1x1) 3
A2
1/4 (1x9x1 1x(-1)x1 1x3x(-1) 1x1x(-1)) 1
48
Symmetry of molecular movements of water
Vibrational modes
49
Which of these vibrations having A1 and B1
symmetry are IR or Raman active?
50
Often you analyze selected vibrational modes
Example C-O stretch in C2v complex.
n(CO)
2 x 1 2
0 x 1 0
2 x 1 2
0 x 1 0
G
Find vectors remaining unchanged after
operation.
51
Both A1 and B1 are IR and Raman active
A1 B1
A1 1/4 (1x2x1 1x0x1 1x2x1 1x0x1) 1
A2 1/4 (1x2x1 1x0x1 1x2x-1 1x0x-1) 0
B1 1/4 (1x2x1 1x0x1 1x2x1 1x0x1) 1
B2 1/4 (1x2x1 1x0x1 1x2x-1 1x0x1) 0
52
What about the trans isomer?
Only one IR active band and no Raman active bands
Remember cis isomer had two IR active bands and
one Raman active
53
Symmetry and NMR spectroscopy
The of signals in the spectrum corresponds to
the of types of nuclei not related by
symmetry The symmetry of a molecule may be
determined From the of signals, or vice-versa
54
Molecular Orbitals
55
Atomic orbitals interact to form molecular
orbitals Electrons are placed in molecular
orbitals following the same rules as for atomic
orbitals
In terms of approximate solutions to the
Scrödinger equation Molecular Orbitals are linear
combinations of atomic orbitals (LCAO) Y caya
cbyb (for diatomic molecules)
Interactions depend on the symmetry
properties and the relative energies of the
atomic orbitals
56
As the distance between atoms decreases
Atomic orbitals overlap
Bonding takes place if the orbital symmetry
must be such that regions of the same sign
overlap the energy of the orbitals must be
similar the interatomic distance must be short
enough but not too short
If the total energy of the electrons in the
molecular orbitals is less than in the atomic
orbitals, the molecule is stable compared with
the atoms
57
Combinations of two s orbitals (e.g. H2)
58
(No Transcript)
59
Both s (and s) notation means symmetric/antisymme
tric with respect to rotation
s
s
s
60
Combinations of two p orbitals (e.g. H2)
s (and s) notation means no change of sign upon
rotation
p (and p) notation means change of sign upon C2
rotation
61
Combinations of two p orbitals
62
Combinations of two sets of p orbitals
63
Combinations of s and p orbitals
64
Combinations of d orbitals
No interaction different symmetry
d means change of sign upon C4
65
Is there a net interaction?
NO
NO
YES
66
Relative energies of interacting orbitals must be
similar
Weak interaction
Strong interaction
67
Molecular orbitals for diatomic molecules From H2
to Ne2
Electrons are placed in molecular
orbitals following the same rules as for atomic
orbitals Fill from lowest to highest Maximum
spin multiplicity Electrons have different
quantum numbers including spin ( ½, - ½)
68
(No Transcript)
69
O2 (2 x 8e)
1/2 (10 - 6) 2 A double bond
Or counting only valence electrons 1/2 (8 - 4)
2
Note subscripts g and u symmetric/antisymmetric up
on i
70
Place labels g or u in this diagram
su
pg
pu
sg
71
su
sg
g or u?
pg
pu
du
dg
72
Orbital mixing
73
s orbital mixing
When two MOs of the same symmetry mix the one
with higher energy moves higher and the one with
lower energy moves lower
74
Molecular orbitals for diatomic molecules From H2
to Ne2
75
(No Transcript)
76
Bond lengths in diatomic molecules
77
Photoelectron Spectroscopy
78
O2
N2
sg (2p)
pu (2p)
pu (2p)
sg (2p)
pu (2p)
Very involved in bonding (vibrational fine
structure)
su (2s)
su (2s)
(Energy required to remove electron, lower energy
for higher orbitals)