64
4 Representations
Table 4.2 Complete set of matrix element strings for the group C 3v
C 3v
ˆ
E
ˆ
C 3
ˆ
C 2
3
ˆ
σ 1
ˆ
σ 2
ˆ
σ 3
D Ω
ij |D Ω
ij
A 1
111
111
6
A 2
111
−1
−1
−16
E 11
1
−
1
2
−
1
2
1
−
1
2
−
1
2
3
E 21
0
+
√
3
2
−
√
3
2
0
−
√
3
2
+
√
3
2
3
E 12
0
−
√
3
2
+
√
3
2
0
−
√
3
2
+
√
3
2
3
E 22
1
−
1
2
−
1
2
−1
+
1
2
+
1
2
3
are listed in Table 4.2, based on the matrices in Table 3.2. In general, this number is always equal to the order of the group since, for every irrep, the number of
{i, j } combinations is equal to the squared dimension of that irrep, and, according
to Eq. (4.43), the sum of these squares is equal to |G|. The strings in Table 4.2 thus
form a set of six linearly independent vectors. This is in accord with our earlier finding that the set of arbitrary functions that form the most general function space for a
group has dimension |G|. The GOT thus offers the complete list of coefficients from
which SALCs may be constructed. How this can be done is shown in the next section. Note that the trace theorem in the previous section is a direct consequence of
the GOT that is obtained by taking diagonal matrix entries ii and kk and summing
over i and k:
χ
Ω |χ
Ω ′
=
i,k
R∈G
¯
D
Ω
ii (R)D
Ω ′
kk (R)
=
|G|
dim(Ω)
δ Ω,Ω ′
ik
δ ik
=
|G|
dim(Ω)
δ Ω,Ω ′ dim(Ω)
= δ Ω,Ω ′ |G|
(4.46)
4.5 Projection Operators
We recapitulate what we have so far: a group G has been identified, and a function space |f was constructed, which is invariant under the action of the group.
Next, the characters were determined for each conjugacy class and arranged in a
character string, |χ, which was mapped onto the irreducible characters in the table. Nonzero brackets determined which irreps are present in the function space.
Now, the final step is to carry out the actual symmetry adaptation and to obtain
the resulting SALCs, say |Φ Ω
i . The SALCs are characterized by two indices: the
4 Representations
Table 4.2 Complete set of matrix element strings for the group C 3v
C 3v
ˆ
E
ˆ
C 3
ˆ
C 2
3
ˆ
σ 1
ˆ
σ 2
ˆ
σ 3
D Ω
ij |D Ω
ij
A 1
111
111
6
A 2
111
−1
−1
−16
E 11
1
−
1
2
−
1
2
1
−
1
2
−
1
2
3
E 21
0
+
√
3
2
−
√
3
2
0
−
√
3
2
+
√
3
2
3
E 12
0
−
√
3
2
+
√
3
2
0
−
√
3
2
+
√
3
2
3
E 22
1
−
1
2
−
1
2
−1
+
1
2
+
1
2
3
are listed in Table 4.2, based on the matrices in Table 3.2. In general, this number is always equal to the order of the group since, for every irrep, the number of
{i, j } combinations is equal to the squared dimension of that irrep, and, according
to Eq. (4.43), the sum of these squares is equal to |G|. The strings in Table 4.2 thus
form a set of six linearly independent vectors. This is in accord with our earlier finding that the set of arbitrary functions that form the most general function space for a
group has dimension |G|. The GOT thus offers the complete list of coefficients from
which SALCs may be constructed. How this can be done is shown in the next section. Note that the trace theorem in the previous section is a direct consequence of
the GOT that is obtained by taking diagonal matrix entries ii and kk and summing
over i and k:
χ
Ω |χ
Ω ′
=
i,k
R∈G
¯
D
Ω
ii (R)D
Ω ′
kk (R)
=
|G|
dim(Ω)
δ Ω,Ω ′
ik
δ ik
=
|G|
dim(Ω)
δ Ω,Ω ′ dim(Ω)
= δ Ω,Ω ′ |G|
(4.46)
4.5 Projection Operators
We recapitulate what we have so far: a group G has been identified, and a function space |f was constructed, which is invariant under the action of the group.
Next, the characters were determined for each conjugacy class and arranged in a
character string, |χ, which was mapped onto the irreducible characters in the table. Nonzero brackets determined which irreps are present in the function space.
Now, the final step is to carry out the actual symmetry adaptation and to obtain
the resulting SALCs, say |Φ Ω
i . The SALCs are characterized by two indices: the