Physical Chemistry , 1st ed.

x,y, and zcoordinates, so there are a total of 3N(Nnumber of atoms) to-

tal coordinates for the molecule. Furthermore, one needs to keep track of the

original positive directions of the x,y, and zaxes, as shown in Figure 13.17.

Instead of the 3 3 matrices defined above, we therefore have to construct a

9 9 matrix to describe the complete spatial effect of each symmetry opera-

tion, shown visually in Figure 13.18:

E(H 2 O) 

xH1

yH1

zH1

xO

yO

zO

xH2

yH2

zH2

xH1

yH1

zH1

xO

yO

zO

xH2

yH2

zH2

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 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

13.5 Character Tables 431

yO

xO

zO

O

yH

xH

zH1

H yH

xH

zH2

H

yO

xO

zO

O

yH

xH

zH1

H yH

xH

zH2

H

E

yO

xO

zO

O

yH

xH

zH1

H yH

xH

zH2

H

yO

zO xO

C 2

yO

xO

zO

O

yH

xH

zH1

H yH

xH

zH2

H

yO

zO xO

O

yH1

xH1

zH1

H yH2

xH2

zH2

H

(yz)

yO

xO

zO

O

yH

xH

zH1

H yH

xH

zH2

H

yO

xO

zO

O

yH2

xH2

zH2

H yH1

xH1

zH1

H

(xz)

yH2

zH2xH2

yH1

zH1xH1

O

HH

Figure 13.18 The effect of the four symmetry
operations ofC2von the degrees of freedom of
the H 2 O molecule.Edoes nothing.C 2 and (xz)
exchange the two hydrogens and reverse the di-
rections of some of the coordinate unit vectors.
(yz) does not move any atoms but reverses some
coordinate vectors. You should be able to recon-
cile the above diagrams with the matrices in equa-
tion 13.2.

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