C 2 (H 2 O)
(13.2)
(yz)(H 2 O) (xz)(H 2 O) Not only do the hydrogen atoms exchange places in (xz) and C 2 , but some
of their unit vectors have also reversed direction. This accounts for the appear-
ance of some negative signs in the product matrices. You should satisfy yourself
that these matrices and the diagrams in Figure 13.18 do coincide with each other.
Each of the 9 9 matrices in equation 13.2 is called a representationof the
corresponding symmetry operation. These representations are complete, but
cumbersome. And this just for a molecule that has three atoms. For Natoms,
the 3N 3 Nmatrix contains 9N^2 terms. Therefore the complete representa-
tion for dimethyl ether, (CH 3 ) 2 O, which also has C2vsymmetry, can be defined
by four 27 27 matrices with each having 27^2 729 numbers in it! To be
sure, most of them are zero (as they are above), but determining which are ex-
actly zero is a chore. Dimethyl ether is still a rather small molecule. We need a
simpler representation.
The representations above can be dramatically simplified, so they are called
reducible representations.Ultimately we are after the simplest possible repre-
sentations of the symmetry operations of a point group, which are called the
irreducible representations.The trick to defining such irreducible representa-
tions is to recognize the patterns in the 9 9 representations above. For ex-
ample, in this case the symmetry operation Emultiplies each coordinate by 1:
E(coordinate) 1 (coordinate) (13.3)xH2
yH2
zH2
xO
yO
zO
xH1
yH1
zH1xH1
yH1
zH1
xO
yO
zO
xH2
yH2
zH20 0 1 0 0 0 0 0 0
0
1
0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0
0 0 0 0 0 1 0 0 0
0
0
0
0
1
0
0
0
0
0 0 0 1 0 0 0 0 0
0 0 0 0 0 0 0 0 1
0 0 0 0 0 0 0
1
0
0 0 0 0 0 0 1 0 0
xH1
yH1
zH1
xO
yO
zO
xH2
yH2
zH2xH1
yH1
zH1
xO
yO
zO
xH2
yH2
zH20 0 0 0 0 0 0 0 1
0 0 0 0 0 0 0 1 0
0 0 0 0 0 0
1
0
0
0 0 0 0 0 1 0 0 0
0 0 0 0 1 0 0 0 0
0
0
0
1
0
0
0
0
0
0 0 1 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0
1
0 0 0 0 0 0 0 0
xH2
yH2
zH2
xO
yO
zO
xH1
yH1
zH1xH1
yH1
zH1
xO
yO
zO
xH2
yH2
zH20 0 1 0 0 0 0 0 0
0
1
0 0 0 0 0 0 0
1
0 0 0 0 0 0 0 0
0 0 0 0 0 1 0 0 0
0
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0 0 0 0 0 0 0 0 1
0 0 0 0 0 0 0
1
0
0 0 0 0 0 0
1
0
0
432 CHAPTER 13 Introduction to Symmetry in Quantum Mechanics