Physical Chemistry , 1st ed.

6. Do this for all irreducible representations of the point group. This will
   yield a large number of linear combinations. Inspect them, for some of
   them may be equivalent and so all but one of them can be ignored.
   Others may be linear combinations of two (or more) SALCs, and so are
   not mathematically independent. Only the unique combinations should
   be considered further. (The choice of which are the unique combinations
   is often a matter of preference and not an absolute.)
   The following example is based on the six preceding steps.

Example 13.13

Determine the symmetry-adapted linear combinations for the molecular or-

bitals of H 2 O using the atomic orbitals of H1, H2, and O. (The numbers on

the H’s are labels for identification purposes only.)

Solution

H 2 O has C2vsymmetry. Each step from the list above is labeled.

1. The atomic orbitals used to make the SALCs were identified above and
   will be labeled 1sH1,1sH2,1sO,2sO,2px,O,2py,O, and 2pz,O. These orbitals
   are illustrated in Figure 13.20.


2. The following table can be set up:

1 sH1 1 sH2 1 sO 2 sO 2 px,O 2 py,O 2 pz,O

E

C 2

v



v

3, 4. The Esymmetry operation does not change the orbitals. The C 2 and 

v

operations switch the hydrogen 1sorbitals and have varying effects on

the oxygen’s orbitals. The vdoes not affect the hydrogen orbitals but

does have the effect of reversing the 2py,Oorbital. Each symmetry result

is multiplied by the character for the symmetry operation of that ir-

reducible representation. For the A 1 symmetry species, we get the

following:

1 sH1 1 sH2 1 sO 2 sO 2 px,O 2 py,O 2 pz,O

E 1  1 sH1 1  1 sH2 1  1 sO 1  2 sO 1  2 px,O 1  2 py,O 1  2 pz,O

C 2 1  1 sH2 1  1 sH1 1  1 sO 1  2 sO 1  2 px,O 1  2 py,O 1  2 pz,O

v 1  1 sH1 1  1 sH2 1  1 sO 1  2 sO 1  2 px,O 1  2 py,O 1  2 pz,O



v 1  1 sH2 1  1 sH1 1  1 sO 1  2 sO 1  2 px,O 1  2 py, O 1  2 pz,O

Since the A 1 irreducible representation has all 1’s for characters, the re-

sults of the symmetry operations on the atomic orbitals are all being

multiplied by 1.

5. Summing the terms in each column, we get seven linear combinations,
   which are in order:
   A 1 ^14 (1sH1 1 sH2 1 sH1 1 sH2)
   A 1 ^14 (1sH2 1 sH1 1 sH2 1 sH1)
   A 1 ^14 (1sO 1 sO 1 sO 1 sO)

13.9 Symmetry-Adapted Linear Combinations 445

2 px

2 py

2 pz

2 s

1 s

O

H

1 s
H
Figure 13.20 The valence atomic orbitals used
to construct symmetry-adapted linear combina-
tion molecular orbitals of H 2 O. Although the
atomic orbitals do not have C2vsymmetry, the
proper combinations of atomic orbitals will. See
Example 13.13.