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A. III. Molecular Symmetry
and Group Theory
F. Albert Cotton, Chemical Application of Group Theory, Wiley, 1990.
M. Tinkam, Group Theory and Quantum Mechanics, McGraw-Hill, 1964.
Peter F. Bernath, Spectra of Atoms and Molecules, Ch 2, 3, 4.
P. R. Bunker, Molecular Symmetry and Spectroscopy, NRC Research Press, 1998.
1
(a) Groups: Definitions and Theorems

2
A group is a set of elements A, B, C, …such that a form of
multiplication may be defined that associates a third element
with any ordered pair. The multiplication must satisfy the
following requirements:
1. The group is closed.
If A, B ∈ G, then AB ∈ G, ∀ A, B.
2. The associative law holds.
A(BC) = (AB)C
3. The identity element E exists such that
EA = AE = A, ∀ A.
4. For every element A in the group there is an inverse A-1 such
that
AA-1 = A-1 A = E
More Definitions






3
The order of a group is the number of elements in the
group. The order can be infinite, but for the most part,
we will deal with finite groups.
An Abelian group is commutative.
AB = BA, ∀ A, B
A subgroup is a group whose elements are a subset of a
larger group, using the same multiplication rule.
If A is conjugate to B, then B = X-1AX, for some X ∈
G.
A complete set of elements that are conjugate to one
another are called a class.
A cyclic group of order n has elements A, A2, A3, A4,
A5, … An-1, and E (= An).
Mappings between Groups


4
Two groups G and G′ are isomorphic if there is a oneto-one correspondence between the elements A, B, … of
one group and the elements A′, B′, … of the other group
such that AB = C implies A′B′ = C′ and vice versa.
Two groups G and G′ are homomorphic if there is a
one-to-many correspondence between each element A
of one group and two or more elements A1′, A2′, … of
the other group such that AB = C implies Ai′Bj′ ∈ {Ck′}
for all Ai′, Bj′. (Here, “many” means 1, 2, or more. So
an isomorphism is the special case of a homomorphism,
where “many” = 1.)
Theorems and Examples

A group multiplication table gives the product
(column element) × (row element)
G1 E
E

5
G2 E A
E
A
Theorem: If A, B, C ∈ G, and B≠C, then AB≠AC and
BA≠CA.

Theorem: Each element is represented once and only
once in each row and in each column of the group
multiplication table.
G3 E A B
E
A
B




6
What choices do we have to fill in this table?
Is the group cyclic?
Abelian?
Are there subgroups?
(b) Molecular Symmetry



7
Symmetry operations leave the molecule – or more
precisely, the Hamiltonian of the molecule – unchanged.
Several different kinds of molecular symmetry:
1. Geometric symmetry that reflects the shape of the
molecule
2. Permutation of identical nuclei.
3. Inversion of the molecule, the is (x, y, z)  (-x, -y, -z)
for all electrons and nuclei.
4. Permutation of the electrons.
5. Rotation of the molecular about its center of mass (3
Euler angles).
6. Translation of the molecule (x, y, z) (x+a, y+b, z+c).
7. Time invariance t  t+a
8. Time reversal symmetry t  -t.
Each kind of symmetry yields useful concepts.
Symmetries 2. +3. are equivalent to 1. for rigid molecules.
(b) Molecular Symmetry


Point groups
Permutation-inversion
group theory
The Pauli Principle
Conservation of
angular momentum
Conservation of linear momentum
Conservation of energy
Microscopic reversibility

8
Symmetry operations leave the molecule – or more
precisely, the Hamiltonian of the molecule – unchanged.
Several different kinds of molecular symmetry:
1. Geometric symmetry that reflects the shape of the
molecule
2. Permutation of identical nuclei.
3. Inversion of the molecule, the is (x, y, z)  (-x, -y, -z)
for all electrons and nuclei.
4. Permutation of the electrons.
5. Rotation of the molecular about its center of mass (3
Euler angles).
6. Translation of the molecule (x, y, z) (x+a, y+b, z+c).
7. Time invariance t  t+a
8. Time reversal symmetry t  -t.
Each kind of symmetry yields useful concepts.
Symmetries 2. +3. are equivalent to 1. for rigid molecules.
Point Groups
9
Symmetry Element
Operation
Plane (σv, σh)
Reflection in the plane
Center of symmetry (i)
Inversion through the center
Proper axis (Cn … (Cn)n=E)
Rotation by a multiple of 360˚/n
Improper axis (Sn … (Sn)2n=E)
One or more reptitions of a
rotation followed by a reflection
in a plane ⊥ to the rotation axis.
Flow chart for
determining
Point Group
Symmetry
Bernath, p 52
10
Example: Dichloromethane
1
b
z
a
2
x
y
11
(c) Representations of Groups


v  x, y,z

A representation of an abstract group is any group composed
of concrete mathematical entities that is homomorphic to the
original group.
We will be concerned with representations of symmetry
groups by n×n square matrices, where n is the dimensionality
of the representation.
A vector may be used to generate a matrix representation.

12
new vector
Aij

Vector, v
Matrix
The vector generating the representation may have any
number of components.
Functions expressed in terms of basis functions may also be
used.
Types of Representations





– all 1’s.
– isomorphic
– dimensionality is reduced by
similarity transformations.
Irreducible representations
– no further reduction in
dimensionality is possible.
The character of a representation is the sum of the diagonal
n
elements:
The identical representation
A faithful representation
Reducible representations
 A   Aii
i1
Γ1(R)
0

Γ2(R)
0
Γ3(R)
13
C2V Example

Representation matricies based on r  x, y,z 
1 0 0
1 0 0
1 0 0
1 0 0








E  0 1 0 C2  0 1 0  V  0 1 0  V   0 1 0





0 0 1

0 0 1

0 0 1

0 0 1






Dimensionality? 3
Faithful?
YES
Identical?
NO
Reducible?
YES
14
Similarity Transformations
Γ1(R)
0
Γ2(R)
0
Γ3(R)
15
The Great Orthogonality Theorem (GOT)



Let R denote the group operations, A, B, C, …
Let Γi(R) be the matrix for the operation R in the ith irreducible
representation. It has the elements Let Γi(R)mn.
The theorem states
h

R

R

 i  mn  j  mn l mmnnij
i
R
*
where is the order of the group, and li is the dimensionality of
the ith irreducible representation.

16
Theorems derived from the GOT
1. The sums of the squares of the dimensions of the irreducible
representations of a group is equal to the order of the group:
l
2
i
h
i
2. The sum of the squares of the characters of any irreducible
2
representation equals h:
 R h
  
i
R
 whose components are the characters of two
3. The vectors
different irreducible representations are orthogonal:
if i≠j.
   i R j R  0
R
4. In a given representation (reducible or irreducible), the
characters of all matrices belonging to operations of the same
class are identical.

5. The number of irreducible representations is equal to the
number of classes.
17
Example: Character Table for C3V
C3V
A1
A2
E
18
Reducing Reducible Representations
We choose a similarity transformation, such that when it is
applied to each representation matrix of a reducible
representation, each representation become block diagonal as
at left.
In general the irreducible representation Γj appears aj times:

Γ1(R)
0

 R   a j  j R
Γ2(R)
j
0
Γ3(R)
To find aj the proceed as follows:

  R R    a  R R
i

R
j
R
j
R
 ai h
19

i
  a j   j R i R   a j hij
j
Thus
j
ai 
1
  R i R
h R
j
byGOT
Example

C3V E
2C3
3σV
A1
1
1
1
A2
1
1
-1
E
2
-1
0
20
Suppose that for a redicuble representation, χ(R) = 5 2 -1.
Find the irreducible representations that comprise this
reducible rperesentation.
(d) Group Theory and Quantum Mechanics

Transformation operators:
Operate on the coordinate system: x Rx

21
(d) Group Theory and Quantum Mechanics

Transformation operators:
Operate on a function:

22
f x  PR  f x
The Group of the Schroedinger Equation
23
The Group of the Schroedinger Equation
24
Labeling of Quantum States



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25
Wavefunctions transform under group operations according to
the irreducible representations of the group.

Label quantum states accordingly!
The irreducible representations have a systematic
nomenclature (with some exceptions) that is used in
spectroscopy and quantum chemistry.
See standard character tables in the back of Bernath and
elsewhere.
Examples:

Atoms: S, P, D, … (full rotation group)

Homonuclear diatomics: Σg+, Πu, … (D∞h)

CH2Cl2: A1, A2, B1, B2 (C2V)

CH3F: A1, A2, E (C3V)

CH3OH (rigid): A′, A″ (CS)

CH3OH (with internal rotation): A1, A2, E (G6)

CH4: A1, A2, E, T1 [F1], T2 [F2] (Td)
The Direct Product

Suppose that X1, X2, X3, …, Xm, and Y1, Y2, Y3, …, Yn are two sets of
functions that are basis for representations of the group:
m
n
PR X i    ji RX j and PRYk   lkY  RYl
X 
l1
j1
Γ(X) and Γ(Y) are


where
the representations derived from the two sets of basis
functions.
Then we can operate on a product of functions Zs = XiYk:

m
n
PR X iYk     jiX  RlkY  RX jYl
j1 l1
m
n
Z
   ji,lk
RX jYl
j1 l1

26

and Γ(Z) is the representation generated by the product Zs.
We use the direct product notation: Γ(Z) = Γ(X) ⊗ Γ(Y)
Characters of Direct Product Representations


If Γ(X)(R) is an m×m matrix and Γ(Y)(R) is an n×n, then
Γ(Z)(R)is an mn×mn matrix, and might be a reducible
representation.
m n
Z
The character of Γ(Z)(R) is  Z R 
ji,lk
R

j1 l1
m
n
   jiX  RlkY  R
j1 l1
 Z R   X RY R

27
Office Hour & Problems
Friday, Jan 21, 3-4 PM in KNCL 317
Identifying Non-Zero Matrix Elements
28
Identifying Non-Zero Matrix Elements


29
Theorem: If Z = XY, then Γ(Z) contains the totally symmetric
representation only when Γ(X) and Γ(Y) contain at least one
irreducible representation in common.
This theorem allows for the quick identification of non-zero
matrix elements.
Symmetry-Adapted Linear Combinations



30
The procedure described below is sometimes called the
“basis function generating machine.”
It enables us to start with any function defined in the relevant
coordinate space and project out a function that transforms as
any desired irreducible representation of the applicable group.
(If the starting function was chosen poorly, then you might get
zero, but then one just needs to try again with a different
starting function.)
The procedure is good for finding molecular orbitals or
vibrational wavefuntions with the desired symmetry
properties.
Symmetry-Adapted Linear Combinations


Suppose that we have a set of li orthonormal functions
ϕ1i, ϕ2i, … that form a basis for the ith irreducible representation Γi
The action of the group operation PR produces a linear combination of the
other basis functions:
P  i   i i R
R

Multiply both sides by

t
 R
*
j
st 
s st
s
and sum over the group operations:
R PR  ti     sisti Rsjt R
 sjt 
R
R s

*
   si  sti Rsjt R
*
*
s
R
   siijsstt 
s
  siijtt 
31
h
li
h
li
Symmetry-Adapted Linear Combinations

Based on the above result, we define the projection operator

lj
*
j
P   st R PR
h R
which has the action Psjt ti  siijtt 
Now consider a a general function  that is a linear
j
st 
combination of the various basis functions:


Then
P   a P    a   

i,t
i,t
j
st 
i
t
 atj  sj
j
i
st  t
i
t

   ati ti
i
s ij tt 
i,t
If we set s′ = t′, then Psjs  asj  sj
Thus we can project any desired symmetry component out of a
 general function.

32

Projections with Characters Only


The problem is that the full representation matrices are not
available in charater tables. Furthermore, the full
representation matrices are only defined to within a similarity
transformation.
Define
lj
*
j
j
j
P   Pss 
s
h
   R P
ss
s
R
R
lj
P    j RPR
h R
j
This projection operator projects a given function onto the
subspace of the basis, it will generate symmetrized basis
functions of the desired irreducible representation, but where

the dimensionality of the represention is 2 or higher, we don’t
have control over which of the degenerate functions we will
get, and the degenerate functions that we do get my not be
orthogonal.
33
Office Hour & Problems
Friday, Jan 21, 3-4 PM in KNCL 317
Example:
π Orbitals of the Cyclopropenyl Radical.

34
Find the symmetry-adapted linear combinations.