Download Symmetry Groups

Symmetry Groups
The set of symmetry elements of a molecule
behave in exactly the same manner as an
abstract mathematical group.
There is an entire mathematical subject,
Group Theory, from which we will take
certain results.
Definition of a Group
A set of operations (in this case symmetry
operations), [A, B, C, …] for which
multiplication is defined, A B = X, is a
Group if:
1)There is an identity operation, E,
A E = A , for all operations in the set
2)For A and B, operations in the set,
then, A B = X, X is an operation in the
set.
3)For all A there is an operation in the set,
C, such that A C = E. C is known as the
inverse of A, A-1.
4)Multiplication is Associative
A (B C) = (A B) C
It is a fact that if one takes the set of
symmetry operations of a molecule and
writes down the result of every possible
‘multiplication’, then the set is a Group.
Example:
NH3
N
3
1
2
Operations: E 2C3 3σ = E C3 C32 σ1 σ2 σ3
(note: σ labeled by the atom they contain)
consider operating with σ1 and then C3 on
the molecule.
N
C3 σ1
3
1
2
N
N
C3
3
2
1
3
2
1
which is the same result as σ3
N
N
σ3
3
3
1
2
therefore C3 σ1 = σ3
2
1
consider operating with C3 and then σ1 on
the molecule.
N
N
σ1C3
1
3
1
2
3
2
which is the same result as σ2
C3 σ1 = σ3 , σ1C3 = σ2 … this is ‘defining
multiplication’, 36 cases to be defined.
Note that C3 σ1 ≠ σ1C3 , multiplication is
not (necessarily) commutative.
Multiplication tables:
H2O: E C2 σv(yz) σv(xz)
consider the effect on the cartesian axes:
E (x, y, z)
C2 (x, y, z)
σv(yz) (x, y, z)
σv(xz) (x, y, z)
 (x, y, z)
 (-x, -y, z)
 (-x, y, z)
 (x, -y, z)
1st Operation 
2nd
E
C2
Op. E
E
C2
↓
E
C2
C2
σv(yz) σv(yz) σv(xz)
σv(xz) σv(xz) σv(yz)
σv(yz)
σv(yz)
σv(xz)
E
C2
σv(xz)
σv(xz)
σv(yz)
C2
E
NH3 : E C3 C32 σ1 σ2 σ3
Easier to do as shown above.
E (1, 2, 3)
C3 (1, 2, 3)
C32 (1, 2, 3)
σ1 (1, 2, 3)
σ2 (1, 2, 3)
σ3 (1, 2, 3)
2nd E
Op. C3
↓
C32
σ1
σ2
σ3






(1, 2, 3)
(2, 3, 1)
(3, 1, 2)
(1, 3, 2)
(3, 2, 1)
(2, 1, 3)
1st Operation 
E
C3 C32
E
C3
C3 2
C3
C32 E
C32 E
C3
σ1
σ3
σ2
σ2
σ1
σ3
σ3
σ2
σ1
σ1
σ1
σ2
σ3
E
C3
C32
σ2
σ2
σ3
σ1
C32
E
C3
σ3
σ3
σ1
σ2
C3
C32
E
Can be done in Cartesian coordinates but
involves sin(120) and cos(120).
C3 (x, y, z) 
(xcos(120)+ysin(120), -xsin(120)+ycos(120), z)
etc.
Just more cumbersome.