Chemistry Reference
In-Depth Information
hydrogen atomic labels
. Interchange of A and B means that, in this row,
the element A is replaced by B and vice versa. Another way to express this is that
“A becomes B, and B becomes A,” and hence
(
A
ABC
A
)
. This interchange is a
transposition
or 2-cycle, which will be denoted as (AB). The operation for the entire
set is then written as a sequence of two disjunct cycles
(
C
)(
AB
)
, where the 1-cycle
indicates that the element C remains unchanged. The 3-cycle (ABC) corresponds
to a cyclic permutation of all three elements:
(
A
→
B
→
A
)
. The successive
application of both operations, acting on the letter string, can be worked out as
follows:
→
B
→
C
→
ABC
(C) (AB)
⇓
BAC
(3.17)
(ABC)
⇓
CBA
The result is to permute A and C, and leave B invariant. This result defines the
product of the two operations as
·
=
(
ABC
)
(
C
)(
AB
)
(
B
)(
AC
)
(3.18)
The multiplication table for the whole group is given in Table
3.4
. The group multi-
plication tables of
S
3
and
C
3
v
clearly have the same structure, but the isomorphism
can be realized in six different ways, as there are six ways to associate the three
letters with the three trigonal sites. It is important to keep in mind that the two
kinds of groups have a very different meaning. The
C
3
v
operations refer to spatial
symmetry operations of the ammonia molecule, while the permutational group is a
set-theoretic concept and acts on elements in an ordered set. As an example, one
might easily identify the
σ
1
reflection plane with the (A)(BC) permutation opera-
tion since it indeed leaves A invariant and swaps B and C. However, as shown in
Fig.
3.2
, when this reflection is preceded by a trigonal symmetry axis, the atom C
has taken the place of A, and the
ˆ
σ
1
plane now should be described as (C)(AB).
For a proper definition of the relationship between nuclear permutations and spatial
symmetry operations, we refer to Sect.
5.4
, where the molecular symmetry group is
introduced.
In
S
3
the number of transpositions, i.e., pairwise interchanges of atoms, is zero
for the unit element, one for the reflection planes, and two for the threefold axes.
Odd permutations
are defined by an odd number of transpositions. The product of
two even permutations is an even permutation, and for this reason, the even per-
mutations alone will also form a group, known as the
alternating group
,
A
n
.Inthe
present example, the alternating group
A
3
is isomorphic to the cyclic group
C
3
.By
contrast, the product of two odd permutations is not odd, but even. So odd permuta-
tions cannot form a separate group.
ˆ
