Physical Chemistry , 1st ed.

bitals onto each other (for an overall contribution of 0). The total contribu-

tion to the character is 2, so that d2. The complete set of characters is

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

sp^3 4 1 0 0 2

This is not one of the irreducible representations ofTd, so the great orthogo-

nality theorem must be applied. Doing so shows that above is a combination

ofA 1 and T 2 , or rather,

A 1 T 2

13.11 Hybrid Orbitals 455

E 4

a

cb

d

a

cb

d

C 3 1

a

cb

d

a

bd

c

C 2 0

a

cb

d

b

d a

c

S 4 0

d

a c

b

d 2

a

cb

d

a

db

c

(e)

(c)

(b)

(a)

a

cb

d

(d)

Figure 13.26 Operation of the symmetry
classes ofTdon the sp^3 orbitals. The a,b,c, and d
labels are used only to keep track of the individ-
ual hybrid orbitals. The number of hybrid or-
bitals that do not move when a symmetry opera-
tion occurs is listed in the final column. This set
of numbers is the reducible representation of
the sp^3 orbitals. The great orthogonality theorem
is used to reduce into its irreducible represen-
tation labels.