Physical Chemistry Third Edition

20.1 The Born–Oppenheimer Approximation and the Hydrogen Molecule Ion 829

Exercise 20.1

Find the coordinates of the points resulting from the operations:

a.̂i(1,2,3)

b.̂σh(4,−2,−2)

c.̂σvyz(7,−6, 3) wherêσvyzis the reflection operator that reflects through theyzplane.

Rotation operatorsmove a point as though it were part of a rigid body rotating about

a specified axis, which is the symmetry element. By convention rotations are counter-

clockwise when viewed from the positive end of the rotation axis. There are infinitely

many rotation operators among the point symmetry operators since there are infinitely

many lines that pass through the origin and infinitely many possible angles of rota-

tion. We consider only rotation operators that produce a full rotation of 360◦when

applied an integral number of times. A rotation operator that produces one full rotation

when appliedntimes is denoted bŷCn. ThêC 1 operator rotates by 360◦(the same

as doing nothing), âC 2 operator rotates by 180◦, and so on. We can add subscripts

to denote the axis. For example, thêC 4 zoperator rotates by 90◦about thezaxis,

so that

̂C 4 z(x,y,z)(x′,y′,z′)(−y,x,z) (20.1-14)

Figure 20.5 shows the effect of the operatorŝi,̂σh, and̂C 4 zon a point in the first

octant.

EXAMPLE20.2

Find the following locations:

a.̂C 2 z(1, 2, 3) (the axis of rotation is thezaxis).

b.̂C 3 z(1, 0, 1) (the axis of rotation is thezaxis).

0

Intersection

of path with

xy plane

x

y

Z

P
(x,y,z)

C4zP
(y, x, z)

(^) hP
(x, y,z)
iP
(x,y,z)
Figure 20.5 The Effect of the Symmetry Operatorŝi,̂σh, and̂C 4 z.