Chemistry Reference
In-Depth Information
i
and implies that the irreducible representation is
ungerade
. The three p-orbitals each
point along a reference axis direction, i.e. along equivalent metal-ligand bonds, and
so they remain degenerate in this complex geometry, as confirmed by this irreducible
representation assignment.
For the d-orbitals we will apply the reduction formula. To make the job easier note that
these functions are not changed by the inversion centre, since they all contain only even
products of the
x
,
y
and
z
basis. This means that the d-orbitals have
gerade
symmetry, and
so we only include irreducible representations with the 'g' subscript in the reduction. The
application of the reduction formula is laid out in Table 5.16, which shows that
=
e
g
+
(d)
t
2g
(5.55)
As we found in the
T
d
case, the five d-orbitals are split into a degenerate set of three
(
t
2g
) and a degenerate pair (
e
g
). In this case, however, the reference axis system is aligned
with the metal-ligand bonds, and so the
e
g
orbitals interact more strongly with the ligand
set than those of the
t
2g
, making the former higher in energy, as shown in Figure 5.21. The
energy gap between the
t
2g
and
e
g
levels is referred to as the ligand field splitting parameter
and is given the symbol
o
, and the size of this gap depends on the type of ligand used in
the complex.
5.8.4 Trigonal Bipyramidal,
D
3h
Complexes formed with five identical ligands may have the trigonal bipyramid struc-
ture shown in Figure 5.22a. This geometry belongs to the
D
3h
point group and the set
of example symmetry elements we will employ for the following analysis are defined in
Figure 5.22b. In the complex, two of the ligands are opposite, or
trans
, to one another,
defining the principal
C
3
symmetry axis, which is assigned as the
Z
-direction. The other
three ligands are in the equatorial plane with L M L angles of 120
◦
. The choice of the
reference
X
and
Y
directions is less clear for
D
3h
than in the earlier examples, and here
we make the choice that
X
will be along an M
L bond with
Y
placed to complete the
right-handed axis system.
We begin with the effect of each operation on an
x
,
y
,
z
basis which is initially aligned
with the reference axis system at the central M atom. The
C
3
axis is along the
Z
direction,
and so the
z
basis function is unaffected by the rotation. However, the
x
and
y
basis vectors
are rotated by 120
◦
. This means that the transformed
x
and
y
vectors are made up of a linear
combination of both of the original vectors. The general formula for the transformation of
x
and
y
by a rotation is discussed in Section 4.7. For this rotation we obtain
√
3
2
1
2
x
x
=
x
cos( 120)
−
y
sin( 120)
=−
−
y
(5.56)
√
3
2
1
2
y
y
=
x
sin( 120)
+
y
cos( 120)
=
x
−
(5.57)
The angles are given in degrees and primes have been added to indicate the basis vectors
after the
C
3
1
rotation.
