Chemistry Reference
In-Depth Information
This time the coefficient
a
is the required character, so we can use the
z
2
coefficient to
obtain it directly:
1
√
3
=
2
a
√
3
1
2
−
so that
a
=−
(5.52)
The characters for all functions under each example operation are listed in Table 5.15,
which also gives the total characters for the reducible representations of the sets of p- and
d-orbitals.
Table 5.15
The characters generated by the transformations of the p- and d-orbital functional
forms under a symmetry operation from each class in O
h
and the summation giving the total
representations for p- and d-orbitals.
E
3
1
C
4
1
C
2
(=
C
4
2
)
S
4
1
S
6
1
O
h
C
2
i
σ
h
σ
d
x
1
0
0
0
−
1
−
1
0
0
1
0
y
1
0
−
1
0
−
1
−
1
0
0
1
0
z
1
0
0
1
1
−
1
−
1
0
−
1
1
(p)
3
0
−
1
1
−
1
−
3
−
1
0
1
1
xy
1
0
0
−
1
1
1
−
1
0
1
1
xz
1
0
1
0
−
1
1
0
0
−
1
0
yz
1
0
0
0
−
1
1
0
0
−
1
0
x
2
−
y
2
1
−
1
/
21
/
2
−
1
1
1
−
1
−
1
/
2
1
−
1
2
z
2
−
x
2
−
y
2
1
−
1
/
2
−
1
/
2
1
1
1
1
−
1
/
2
1
1
(d)
5
−
1
1
−
1
1
5
−
1
−
1
1
1
Problem 5.8:
In Table 5.14 under the
S
6
1
operation we find the transformations
1
√
3
1
√
3
x
2
−
y
2
→
y
2
−
z
2
and
(2
z
2
−
x
2
−
y
2
)
→
(2
x
2
−
y
2
−
z
2
)
(5.53)
The corresponding characters in Table 5.15 are both
2. By finding the appropriate
linear combination coefficients, confirm that this assignment is correct.
−
1
/
Because we have assembled the reducible representations for complete sets of orbitals,
the character totals obtained are independent of the choice of symmetry elements or
operations from each class in the point group. We can now proceed to using the reduc-
tion formula to find the irreducible labels for p- and d-orbitals in
O
h
. For the p-orbitals,
inspection of the standard character table from Appendix 12 shows that
(p)
=
t
u
(5.54)
The p-orbitals are each reversed by the inversion centre, since the positive- and negative-
phase regions are swapped by this operation. This leads to a
−
3 total character under
