116
6 Interactions
Table 6.1 Direct product of E g × T 2g in O h symmetry
O h
ˆ
E
8 ˆ
C 3
6 ˆ
C 2
6 ˆ
C 4
3 ˆ
C 2
ˆ
i
6 ˆ
S 4
8 ˆ
S 6
3 ˆ
σ h
6 ˆ
σ d
E g
2
−1002
2
0
−120
T 2g
30
1−1
−13
−10
−11
E g × T 2g
6
000
−260
0
−20
T 1g
30−11
−131
0
−1
−1
T 2g
30
1−1
−13
−10
−11
is identified as the direct product of the orbital irreps, and is denoted as Γ a × Γ b .
If both irreps are degenerate, the direct product will be reducible. Let c Γ be the
number of times that the irrep Γ occurs in the direct product:
Γ a × Γ b =
Γ
c Γ Γ
(6.7)
By straightforward application of the character theorem one obtains:
c Γ =
1
|G|
R
¯
χ
Γ (R)χ
Γ a ×Γ b (R)
=
1
|G|
R
¯
χ
Γ (R)χ
Γ a (R)χ
Γ b (R)
(6.8)
Here we have, for the first time, a formula with a triad of irreps. This will form the
basis for the symmetry evaluation of general matrix elements. The c Γ coefficients
are obtained by performing product manipulations on the character tables. As an
example, Table 6.1 illustrates the reduction of the E g × T 2g product in O h ,asgiven
in Eq. (6.9). Product tables are given in Appendix E.
E g × T 2g = T 1g + T 2g
(6.9)
Let us now proceed with the two-electron problem and address the next question, which is that, after having determined which symmetry species are present, we
should like to know what the corresponding two-electron wavefunctions look like,
i.e. we should like to construct the SALCs. This construction does not pose any
new problems; the projection operators that were introduced in Sect. 4.5 will do the
job perfectly well. Some notation is important here. The product function will be
written as:
Γγ(1, 2)
=
γ a
γ b
c
Γγ
γ a γ b
Γ a γ a (1)
Γ b γ b (2)
(6.10)
The combination coefficient is itself identified as a matrix element, by multiplying
left and right with the one-electron bra functions and using orthonormality of the
6 Interactions
Table 6.1 Direct product of E g × T 2g in O h symmetry
O h
ˆ
E
8 ˆ
C 3
6 ˆ
C 2
6 ˆ
C 4
3 ˆ
C 2
ˆ
i
6 ˆ
S 4
8 ˆ
S 6
3 ˆ
σ h
6 ˆ
σ d
E g
2
−1002
2
0
−120
T 2g
30
1−1
−13
−10
−11
E g × T 2g
6
000
−260
0
−20
T 1g
30−11
−131
0
−1
−1
T 2g
30
1−1
−13
−10
−11
is identified as the direct product of the orbital irreps, and is denoted as Γ a × Γ b .
If both irreps are degenerate, the direct product will be reducible. Let c Γ be the
number of times that the irrep Γ occurs in the direct product:
Γ a × Γ b =
Γ
c Γ Γ
(6.7)
By straightforward application of the character theorem one obtains:
c Γ =
1
|G|
R
¯
χ
Γ (R)χ
Γ a ×Γ b (R)
=
1
|G|
R
¯
χ
Γ (R)χ
Γ a (R)χ
Γ b (R)
(6.8)
Here we have, for the first time, a formula with a triad of irreps. This will form the
basis for the symmetry evaluation of general matrix elements. The c Γ coefficients
are obtained by performing product manipulations on the character tables. As an
example, Table 6.1 illustrates the reduction of the E g × T 2g product in O h ,asgiven
in Eq. (6.9). Product tables are given in Appendix E.
E g × T 2g = T 1g + T 2g
(6.9)
Let us now proceed with the two-electron problem and address the next question, which is that, after having determined which symmetry species are present, we
should like to know what the corresponding two-electron wavefunctions look like,
i.e. we should like to construct the SALCs. This construction does not pose any
new problems; the projection operators that were introduced in Sect. 4.5 will do the
job perfectly well. Some notation is important here. The product function will be
written as:
Γγ(1, 2)
=
γ a
γ b
c
Γγ
γ a γ b
Γ a γ a (1)
Γ b γ b (2)
(6.10)
The combination coefficient is itself identified as a matrix element, by multiplying
left and right with the one-electron bra functions and using orthonormality of the