Physical Chemistry Third Edition

830 20 The Electronic States of Diatomic Molecules

Solution

a.(−1,−2, 3)

b.(−

1

2

,

1

2

√

3, 1)

Exercise 20.2

Find the following locations:

a.̂C 4 x(1, 2, 3) (the axis of rotation is thexaxis).

b.̂C 3 y(1, 1, 0) (the axis of rotation is theyaxis).

Theidentity operator does nothing. It is denoted bŷEfor the German word

“Einheit,” meaning “unity.”

̂E(x,y,z)(x,y,z) (20.1-15)

The Operation of Symmetry Operators on Functions

Multiplication operators and derivative operators produce new functions when they

operate on mathematical functions. We now define the way in which the symmetry

operator̂Oproduces a new functiong(x,y,z) when it operates on the functionf(x,y,z).

Let̂Obe a symmetry operator such that

̂OPP′ (20.1-16a)

̂O(x,y,z)(x′,y′,z′) (20.1-16b)

We define the new functiongthat is produced when̂Ooperates onf(x,y,z):

̂Of(x,y,z)g(x,y,z) (20.1-17)

such thatghas the same value at pointP′that the functionfhas at pointP:

g(x′,y′,z′)f(x,y,z) (20.1-18)

A function can be an eigenfunction of a symmetry operator. The only eigenvalues that

occur are+1 and−1. Iffis an eigenfunction ofÔ, there are two possibilities. Either

g(x,y,z)f(x,y,z) (20.1-19)

or

g(x,y,z)−f(x,y,z) (20.1-20)

EXAMPLE20.3

Show that the hydrogen-like 1sorbital is an eigenfunction of the inversion operator̂iand

find the eigenvalue.