Groups - Group Theory Applied to Chemistry - page 21

Fig. 3.1 Group theory of the
ammonia molecule, with
three sets of labels: x,y,z
label the Cartesian axes,
σ 1 , σ 2 , σ 3 label the symmetry
planes, and A, B, C label the
hydrogen atoms

guish the equivalent hydrogen atoms. In the active view, which we keep throughout,
the atoms will be displaced while the symmetry elements remain tied to the immo-
bile Cartesian frame. We shall thus not label the reflection planes by A,B,C, but we
shall instead denote them as

ˆ ˆ ˆ ˆ

σ 1 reflection plane coincides with the
xz coordinate plane. The action of the symmetry elements will be to permute the
atoms. The threefold axis, rotating counterclockwise about z , moves the atom A to
the position of B, which itself is displaced to the position originally occupied by C.
Finally, C travels to the place previously occupied by atom A. The

σ 1 , σ 2 , σ 3 .The

ˆ

σ 1 plane will
leave A unchanged and will interchange B and C. Now consider the combination
ˆ

σ 1 C 3 of these two elements. We place the structure to the right of the right-justified
operators and then simply work out the action from right to left; hence, first the C 3
axis, and then the plane. This is shown in a pictorial way in Fig. 3.2 . First, the axis
will permute the atoms so that C takes the place of A. Consequently, the σ 1 plane

will now conserve C and interchange A and B. The combined action is itself again
one of the symmetry elements, viz.,

ˆ

σ 2 . The reverse product order yields a different ˆ

result. In summary,

σ 1 C 3 =ˆ

ˆ

σ 2

(3.1)

C 3 ˆ

σ 1 =ˆ σ 3

In this way we can easily work out the full set of binary products, keeping in
mind that applying the threefold rotation three times, or the symmetry planes twice,
leaves every atom in place and thus corresponds to the unit element. The results are
gathered in the 6 6 multiplication Table 3.1 . This table should be read from left

to right, i.e., the product R i R j is found in the i th row and j th column. We may
symbolically denote the matrix elements in the table as

×

M ij = R i R j (3.2)