Physical Chemistry , 1st ed.

symmetry labels of the representation in the new point group. An example is

electron subshells. The quantum number defines which irreducible repre-

sentation a set of wavefunctions belongs to in the Rh(3)point group, the th

representation. One must also determine whether the wavefunction is positive

or negative with respect to inversion, which determines whether the label “g”

or “u” is applicable to the irreducible representation. One then determines the

characters in Rh(3), transfers those characters to the symmetry operations of a

new point group, and using the GOT determines the symmetry labels of the

2 1 orbitals of that subshell. The following example illustrates how this

might be done.

Example 13.10

Determine the symmetry labels of the hydrogen-like porbitals in Tdsymmetry.

Solution

For porbitals,1, and the sign on the porbitals changes upon inversion,

so the character for inversion should be negative. In Rh(3), the irreducible

representation for porbitals is therefore D(1)u. Using the formulas for the char-

acters for the symmetry operations, we can calculate what the characters are

for Td[see the character table for Rh(3)]:

E 3

C 3  1 2 cos  1 2 cos (120°)  0

C 2  1 2 cos  1 2 cos (180°)  1

S 4  1 2 cos  1 2 cos (90°)  1

d 1

Since Tddoes not have a center of inversion, the formula for iis not needed.

The characters for the porbitals are thus

E 8 C 3 3 C 2 6 S 4 6 d

 3 0  11  1

Using the great orthogonality theorem, we can determine what linear

combination of irreducible representations of the Tdpoint group this is. One

finds:

aA 1 ^214 [(1  1 3) (8  1 0) (3  1 1) (6  1 1) (6  1 1)]

^214 (0)  0

aA 2 ^214 [(1  1 3) (8  1 0) (3  1 1) (6  1 1) (6  1 1)]

^214 (0)  0

aE^214 [(1  2 3) (8  1 0) (3  2 1) (6  0 1) (6  0 1)]

^214 (0)  0

aT 1 ^214 [(1  3 3) (8  0 0) (3  1 1) (6  1 1) (6  1 1)]

^214 (24)  1

aT 2 ^214 [(1  3 3) (8  0 0) (3  1 1) (6  1 1) (6  1 1)]

^214 (0)  0

440 CHAPTER 13 Introduction to Symmetry in Quantum Mechanics