GROUP THEORY
(a) (b)H
H
H
H
H
N
M
Figure 28.1 (a) The hydrogen molecule, and (b) the ammonia molecule.(iii) inversion through the pointM;
(iv) reflection in the plane that passes throughMand has its normal parallel
to the long axis.These operations collectively form the set of symmetry operations for the hydro-
gen molecule.
The somewhat more complex ammonia molecule consists of a tetrahedron withan equilateral triangular base at the three corners of which lie hydrogen atoms
H, whilst a nitrogen atom N is sited at the fourth vertex of the tetrahedron. The
set of symmetry operations on this molecule is limited to rotations ofπ/3and
2 π/3 about the axis joining the centroid of the equilateral triangle to the nitrogen
atom, and reflections in the three planes containing that axis and each of the
hydrogen atoms in turn. However, if the nitrogen atom could be replaced by a
fourth hydrogen atom, and all interatomic distances equalised in the process, the
number of symmetry operations would be greatly increased.
Onceallthe possible operations in any particular set have been identified, itmust follow that the result of applying two such operations in succession will be
identical to that obtained by the sole application of some third (usually different)
operation in the set – for if it were not, a new member of the set would have
been found, contradicting the assumption that all members have been identified.
Such observations introduce two of the main considerations relevant to decid-ing whether a set of objects, here the rotation, reflection and inversion operations,
qualifies as agroupin the mathematically tightly defined sense. These two consid-
erations are (i) whether there is some law for combining two members of the set,
and (ii) whether the result of the combination is also a member of the set. The
obvious rule of combination has to be that the second operation is carried out
on the system that results from application of the first operation, and we have
already seen that the second requirement is satisfied by the inclusion of all such
operations in the set. However, for a set to qualify as a group, more than these
two conditions have to be satisfied, as will now be made clear.