Chemistry Reference
In-Depth Information
Note that if
Γ
k
fulfills the character theorems, then
Γ
k
also does, and hence it, too,
must be irreducible. The two irreps are said to be complex conjugates. They are or-
thogonal to each other, and hence there is no point group operation that can turn a
function belonging to
Γ
k
into a function belonging to
Γ
k
. For this reason, we also
should denote them by two separate labels. However, in the absence of external
magnetic fields, symmetry is not restricted to spatial symmetry, but also includes
time-reversal symmetry. As we have seen in Sect.
2.4
, this symmetry will precisely
turn functions into their complex conjugates and thus also interchange the corre-
sponding conjugate irreps. Complex-conjugate irreps thus remain degenerate under
time reversal, and for this reason, they are usually indicated by means of a brace.
4.4 Matrix Theorem
Determining the symmetry contents of a function space is only a first step. We would
also like to know what are the SALCs that correspond to the different irreps. To carry
out this task, we have to work with the representation matrices themselves. In the
group
C
3
v
the matrices for the one-dimensional irreps
A
1
and
A
2
are trivial since
these are simply equal to the corresponding characters. For the
E
irrep, we need to
determine the generator matrices and perform the proper multiplications in order to
generate all
E
(R)
for the whole group. These matrices are already available from
Table
3.2
for the standard basis of the
p
x
and
p
y
orbitals. An important theorem,
known as the
Great Orthogonality Theorem
(GOT), is due to Schur.
D
Theorem 6
Let Ω and Ω
be two irreducible representations of a group G
,
and
consider vectors formed by taking elements
from the respective repre-
sentation matrices for every element of the group
.
Then these vectors are orthogonal
to each other
,
and their squared norm is equal to the order of the group
,
divided by
the dimension of the irrep
:
{
ij
}
and
{
kl
}
|
G
|
dim
(Ω)
δ
Ω,Ω
δ
ik
δ
jl
G
D
ij
(R)D
Ω
kl
(R)
=
(4.45)
R
∈
The theorem thus proceeds as follows: take a given entry
ij
in the representation
matrix of the irrep
Ω
for every
R
and order these elements to form a vector of
length
. Do the same with another entry,
kl
, for a different representation,
Ω
,
and also arrange these to form a vector. Then take the scalar product of these two
vectors, bearing in mind that, in this process, the complex conjugate of one of them
should be taken (it does not matter which one since the scalar product is always
real). The theorem states that this scalar product is zero unless the same irrep is
taken, and in this irrep the same row and column index are selected. In that case, the
scalar product yields the norm of the vector equal to
|
G
|
/
dim
(Ω)
.
Let us apply this to
C
3
v
. For this group, the total number of
|
G
|
combi-
nations that can be formed according to the GOT procedure is equal to 6. These
{
Ω,i,j
}
