Chemistry Reference
In-Depth Information
Z
Z
C
2
′(
X
)
y
z
y
C
2
′
(
X
)
x
z
′
C
2
′(
X
)
x
X
Y
X
Y
Figure 5.14
The C
2
(X) operation for the x, y, z basis of a D
4h
square planar complex.
operations need be considered when looking for the transformation of a given basis, and
suitable choices have been made in Table 5.9.
The d-orbital functions
xy
,
xz
,
yz
,
x
2
y
2
are treated by constructing
them from the results for
x
,
y
and
z
. For instance, the change to the
x
,
y
,
z
basis after the
C
2
(
X
) operation is illustrated in Figure 5.14. Algebraically this can be written:
−
y
2
and 2
z
2
−
x
2
−
x
→
x
,
y
→−
y
and
z
→−
z
(5.31)
These formulae use arrows rather than equals signs to indicate the result of a transfor-
mation. They interpret the result of the operation (the vectors marked
x
,
y
and
z
in
Figure 5.14) in terms of the initial basis. Immediately, Equation (5.31) demonstrates that
p
x
has a character of 1 while p
y
and p
z
each have character
1.
For the d-orbitals we simply obtain the transformed functions using the behaviour of the
x
,
y
,
z
basis so that:
−
xy
→
x
(
−
y
)
=−
xy
character
−
1
xz
→−
xz
character
−
1
yz
→
(
−
y
)(
−
z
)
=
yz
character
1
(5.32)
x
2
−
y
2
→
x
2
−
(
−
y
)
2
=
x
2
−
y
2
character
1
and
2
z
2
−
x
2
−
y
2
→
2
z
2
−
x
2
−
y
2
character
1
From this example and Table 5.8 there are three possible outcomes for the d
xy
,d
yz
and
d
xz
-orbital functions:
1.
A function is unaffected by the operation
. This can be the case even though the
x
,
y
and
z
vectors have altered; for example, following the
C
2
operation
x
and
y
become
−
x
and
y
, but for their product the two minus signs give a plus and so
xy
is unaffected. In
these cases a character of 1 for the function is assigned.
2.
A function is transformed to its own negative
. This means that the areas of positive
and negative phase will have been switched for the function. This could also have been
achieved by a multiplication by
−
−
1. Hence, the character in this situation is
−
1, e.g.
yz
σ
v
(
XZ
) operation.
3.
A function is transformed into one of the other d-functions
. The d-functions are effec-
tively our basis set in this analysis; so, if a function is transformed completely into
another basis function, then a character of 0 is taken, e.g.
xz
under the
under the
σ
d
(
LZ
) operation
becomes
yz
and so would have character 0.
