Chemistry Reference
In-Depth Information
Tab l e 7 . 5
The reduction of the reducible representation
for the three H(1s) AOs in BH
3
.
D
3h
E2C
3
3C
2
σ
h
2S
3
3
σ
v
h
=
12
30
1 30
1
C
h
−1
C
g
c
χ
i
(
C
)
χ
(
C
)
A
1
3
3
3
3
12
1
A
2
3
−
33
−
3
0
0
E
6
0
6
0
12
1
A
1
3
3
−
3
−
3
0
0
A
2
3
−
3
−
3
3
0
0
E
6
0
−
6
0
0
0
From the reducible representation we must identify the irreducible representations for
the H(1s) orbitals so that those that match with the B atom orbital symmetries listed above
can be identified. The reduction is laid out in Table 7.5.
In this table there are zeros for the reducible representation under the symmetry classes
2
C
3
and 2
S
3
, because rotations around the principal axis interchange the H(1s) orbitals. In
the reduction, the corresponding columns are left blank because they cannot contribute to
the sums required by the reduction formula:
h
C
1
n
i
=
g
c
χ
i
(
C
)
χ
(
C
)
(7.33)
We first met this formula in Section 5.5;
h
refers to the order of the group,
g
c
is the number
of operations under class
C
,
χ
(
C
) is the character for the reducible representation of the
basis (obtained in Problem 7.5) and
χ
i
(
C
) is the standard character for the
i
th irreducible
representation in class
C
from the character table.
Table 7.5 shows that the SALCs for the three H(1s) orbitals belong to the
A
1
and
E
irreducible representations. To obtain the orbital patterns for these we use the projection
operator method introduced in Section 6.6. To simplify the process, it is useful first to
work with the rotational subgroup
D
3
, which is part of the
D
3h
operator set. This is self-
contained, because the product of any two operations is still within the subgroup. When
the SALC functions with the symmetry of the subgroup have been obtained we will just
need to check which irreducible labels they conform to in the full
D
3h
point group.
The use of the projection operator in
D
3
is shown in Table 7.6. The easiest approach
when using the projection operator is to take one of the basis functions alone as the gener-
ating vector and work out the functional form for each irreducible representation in turn.
In Table 7.6 we use
s
1
as the generating vector. This gives an
a
1
MO which has all the
H(1s) orbitals in phase with one another. In
D
3h
the
a
1
representation also has character 1
under both types of mirror plane and the improper rotation classes. The sum of all H(1s)
orbitals in phase is easily seen to conform to this.
Projection of
s
1
for the
E
irreducible representation of the
D
3
group gives the SALC:
φ
1
(
e
)
=
N
1
(2
s
1
−
s
2
−
s
3
)
(7.34)
1
