Physical Chemistry Third Edition

I Matrix Representations of Groups 1301

representation (it is the sum of unity for each diagonal element of the matrix). For

example, from the character table, we see that there is no irreducible representation

with dimension greater than 1 for theC 2 vgroup. There are no degenerate orbitals for

the water molecule.

PROBLEMS

Appendix I

1 Show by matrix multiplication that each of the matrices is

the inverse of the other:

A

⎡

⎣

210

121

012

⎤

⎦ and A−^1 

⎡

⎣

3 / 4 − 1 / 21 / 4

− 1 / 21 − 1 / 2

1 / 4 − 1 / 23 / 4

⎤

⎦

2 a.Verify that each of the matrices in Eqs. (I-4), (I-5), and

(I-6) give the same results as the symmetry operators.

b.Using matrix multiplication, show that

R(̂σyz)R(̂C 2 )R(̂σxz)

3 a.Verify the entire multiplication table for the

one-dimensional representation of Eqs. (I-7) to (I-10).

b.Generate the other two one-dimensional representations

and show that they obey the same multiplication

table.

4 Create another 5×5 matrix with blocks of the same sizes in

the same positions as that of Eq. (I-12) and show by explicit

matrix multiplication that the product of the two matrices

has blocks of the same sizes in the same positions.

5 a.Show that each operator in the groupC 2 vis in a class by

itself.

b.Show that the identity operation is always in a class by

itself, no matter what group is being discussed.

6 Show that the character table for theC 3 vgroup in

Table A.26 of Appendix A conforms to the properties

in the list of theorems.

7 a.Verify the representation of Eqs. (I-20) to (I-22) using

the formulas for the real 3dorbitals from Table 17.3.

b.Obtain five one-dimensional representations from the

matrices of Eqs. (I-20) to (I-22) plus the identity

matrix.

c.Make a character table for these one-dimensional

representations. Determine the number of times each

irreducible representation occurs in this five-dimensional

representation, using Eq. (I-19). Verify this result by

inspection.

8 a.Show that the overlap integral of the oxygen 2sorbital

and symmetry-adapted orbital in Eq. (21.8-1) does not

vanish.

b.Show that the overlap integral of the two

symmetry-adapted orbitals in Eqs. (21.8-1) and (21.8-2)

vanishes.

9 a.Find the maximum degeneracy for orbitals of the

ammonia (NH 3 ) molecule.

b.Find the maximum degeneracy for orbitals of the

methane (CH 4 ) molecule.

10 Find the matrix product

⎡

⎣

123

012

001

⎤

⎦

⎡

⎣

456

450

400

⎤

⎦

11 Construct the multiplication table for the matrices in Eqs.

(I-4)–(I-6) and show that it is identical to the multiplication

table for the point group C 2 v.

12 Carry out the similarity transformation as in Eq. (I-13),

letting the matrix in Eq. (I-4) play the role of B and letting

the matrix in Eq. (I-5) play the role of A. Identify the

resulting matrix as a representative of one of the operations

of the group C 2 v.

13 Benzene belongs to the point groupD 6 h. Identify the

eigenvalues of the following symmetry operations for each

of the orbitals whose regions are depicted in Figure 21.9. If

the orbital is not an eigenfunction, state that fact.

a.̂C 6 (axis perpendicular to the plane of the molecule)

b.̂C 62 (axis perpendicular to the plane of the molecule)

c.̂C 63 (axis perpendicular to the plane of the molecule)

d.̂σh(reflection plane in the plane of the molecule)

14 Determine whether the oxygen 2sorbital and a hydrogen 1s

orbital in the H 2 O molecule in its equilibrium conformation

will have zero overlap.