24
3 Groups
Table 3.2 Representation
matrices for the (p x ,p y )
basis in C 3v
D(E) =
10
01
D(σ 1 ) =
10
0 −1
D(C 3 ) =
⎛
⎝
−
1
2
−
√
3
2
+
√
3
2
−
1
2
⎞
⎠
D(σ 2 ) =
⎛
⎝
−
1
2
−
√
3
2
−
√
3
2
+
1
2
⎞
⎠
D(C 2
3 ) =
⎛
⎝
−
1
2
+
√
3
2
−
√
3
2
−
1
2
⎞
⎠
D(σ 3 ) =
⎛
⎝
−
1
2
+
√
3
2
+
√
3
2
+
1
2
⎞
⎠
3.2 The Group Structure
The set of symmetry operations of ammonia is said to form a group, G. This is a fundamental mathematical structure consisting of a set of elements and a multiplication
rule with the following characteristics:
• Existence of a unit element, ˆ
E, which leaves all elements unchanged:
ˆ
E ˆ
R = ˆ
R ˆ
E = ˆ
R
(3.5)
In the list of elements the unit element is placed in front. As a result, the first row
and first column of the multiplication table will simply repeat the ordered list of
symmetry elements on which the table was based.
• Existence of an inverse element, ˆ
R −1 , for every element ˆ
R:
ˆ
R ˆ
R
−1 = ˆ
R
−1 ˆ
R = ˆ
E
(3.6)
Hence every action can also be undone. If this were not the case, we would have
produced a monster that can only continue to grow. In the multiplication table, a
unit element at position ij indicates that ˆ
R i and ˆ
R j are mutual inverses. Clearly,
for the unit element, as well as for the reflection planes, ˆ
R and ˆ
R −1 coincide,
while, for the ˆ
C 3 axis, the inverse is equal to ˆ
C 2
3 , which completes a full turn.
• Closure:
∀ ˆ
R ∈ G & ˆ
S ∈ G ⇒ ˆ
R ˆ
S ∈ G& ˆ
S ˆ
R ∈ G
(3.7)
In the multiplication table, this is apparent by the fact that no entries are left open.
• Associativity:
ˆ
R( ˆ
S ˆ
T)= ( ˆ
R ˆ
S) ˆ
T
(3.8)
It is difficult to imagine that such a simple set of rules can give rise to such a powerful structure on which entire properties of molecules will depend. Note that, compared with a number system, a group is an even more primitive concept since it
contains only one operation, multiplication, by means of which two elements of the
group may be combined. In contrast, the set of numbers allows for addition and multiplication, which results in two kinds of unit elements, zero for addition and unity
for multiplication. A closer look at the multiplication table of the group shows that it
3 Groups
Table 3.2 Representation
matrices for the (p x ,p y )
basis in C 3v
D(E) =
10
01
D(σ 1 ) =
10
0 −1
D(C 3 ) =
⎛
⎝
−
1
2
−
√
3
2
+
√
3
2
−
1
2
⎞
⎠
D(σ 2 ) =
⎛
⎝
−
1
2
−
√
3
2
−
√
3
2
+
1
2
⎞
⎠
D(C 2
3 ) =
⎛
⎝
−
1
2
+
√
3
2
−
√
3
2
−
1
2
⎞
⎠
D(σ 3 ) =
⎛
⎝
−
1
2
+
√
3
2
+
√
3
2
+
1
2
⎞
⎠
3.2 The Group Structure
The set of symmetry operations of ammonia is said to form a group, G. This is a fundamental mathematical structure consisting of a set of elements and a multiplication
rule with the following characteristics:
• Existence of a unit element, ˆ
E, which leaves all elements unchanged:
ˆ
E ˆ
R = ˆ
R ˆ
E = ˆ
R
(3.5)
In the list of elements the unit element is placed in front. As a result, the first row
and first column of the multiplication table will simply repeat the ordered list of
symmetry elements on which the table was based.
• Existence of an inverse element, ˆ
R −1 , for every element ˆ
R:
ˆ
R ˆ
R
−1 = ˆ
R
−1 ˆ
R = ˆ
E
(3.6)
Hence every action can also be undone. If this were not the case, we would have
produced a monster that can only continue to grow. In the multiplication table, a
unit element at position ij indicates that ˆ
R i and ˆ
R j are mutual inverses. Clearly,
for the unit element, as well as for the reflection planes, ˆ
R and ˆ
R −1 coincide,
while, for the ˆ
C 3 axis, the inverse is equal to ˆ
C 2
3 , which completes a full turn.
• Closure:
∀ ˆ
R ∈ G & ˆ
S ∈ G ⇒ ˆ
R ˆ
S ∈ G& ˆ
S ˆ
R ∈ G
(3.7)
In the multiplication table, this is apparent by the fact that no entries are left open.
• Associativity:
ˆ
R( ˆ
S ˆ
T)= ( ˆ
R ˆ
S) ˆ
T
(3.8)
It is difficult to imagine that such a simple set of rules can give rise to such a powerful structure on which entire properties of molecules will depend. Note that, compared with a number system, a group is an even more primitive concept since it
contains only one operation, multiplication, by means of which two elements of the
group may be combined. In contrast, the set of numbers allows for addition and multiplication, which results in two kinds of unit elements, zero for addition and unity
for multiplication. A closer look at the multiplication table of the group shows that it