Chemistry Reference
In-Depth Information
in the
XY
plane. This will give an integral of 1/2, and so the three terms sum to 3 and the
normalization factor then reduces this total volume to 1, as required.
In this exercise we have found a normalization factor that brings the
z
2
orbital onto the
same scale as
x
2
y
2
. In Appendix 9 (Table A9.1) it is shown that the normalized functions
for all five d-orbitals are,
−
15
π
1
2
xz
r
2
1
2
15
π
1
2
yz
r
2
1
2
15
π
1
2
xy
r
2
1
2
(5.30)
15
π
1
2
(
x
2
1
4
−
y
2
)
r
2
5
π
1
2
2
z
2
1
4
−
x
2
−
y
2
r
2
Where r is the distance from the nuclear centre. These functions also show that to bring
x
2
−
y
2
to the same scale as xy etc., requires a future factor of 2 to be included. To carry
the full normalization constants in the calculations of the next few sections would be cum-
bersome since symmetry is really only concerned with how the functional forms change
after symmetry operations and the proportions of the original basis set required to obtain
the same result. Hence we will only use relative scaling factors when required. We will
meet normalization factors again in Chapters 6 and 7.
To work out how the d-orbitals are affected by the symmetry of their environment,
we will first analyse a basis of the
x
,
y
and
z
vectors at the central metal atom of each
complex geometry. This will automatically show how the p-orbitals respond to each
operation.
The results can then be used to deduce the functional form of each d-orbital after the
transformation and so find the required character set for the d-orbitals. In general, the
p- and d-orbitals will give reducible representations to which we can apply the reduction
formula to find the irreducible representations for the point group.
5.8.1 Square Planar,
D
4h
The
x
,
y
,
z
basis vectors on the central atom in a
D
4h
complex have already been considered
in Section 4.9, where the orientation of the basis is defined in Figure 4.10. It was shown
that the three vectors, and so the corresponding p-orbitals at a central metal atom, reduce
to the
A
2u
and
E
u
irreducible representations.
Table 5.8 gives the transformations of the
x
,
y
and
z
vectors for a representative operation
of each class of
D
4h
. We now know that the operations in a class give identical characters
for any irreducible representation. This means that only one example from each class of
