χ C
xyz
3
À
Á ¼ 2:
As another example σ
yz
d , we have
H 1 H 2 H 3 H 4 C2s C2p x C2p y C2p z
À
Á σ
yz
d
¼ H 1 H 2 H 4 H 3 C2s C2p x C2p z C2p y
À
Á
,
where σ
yz
d represents a mirror symmetry with respect to the plane that includes the xaxis and bisects the angle formed by the y- and z-axes. Thus, we have
σ
yz
d ¼
1 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0
0 0 0 1 0 0 0 0
0 0 1 0 0 0 0 0
0 0 0 0 1 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 1
0 0 0 0 0 0 1 0
0
B
B
B
B
B
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
C
C
C
C
C
A
:
ð19:179Þ
Then, we have
χ σ
yz
d
À Á ¼ 4:
Taking some more examples, for C
z
2 we get
H 1 H 2 H 3 H 4 C2s C2p x C2p y C2p z
À
Á
C
z
2
¼ H 4 H 3 H 2 H 1 C2s À C2p x À C2p y C2p z
À
Á
,
where C
z
2 means a rotation by π around the z-axis. Also, we have
19.5 MO Calculations of Methane
779