74
4 Representations
Table 4.6 Ground or
positional representation of
the four equatorial ligand
sites in a square pyramidal
complex; the sites are ordered
as in Fig. 4.3(a)
P(E) =
⎛
⎜
⎜
⎜
⎜
⎝
1000
0100
0010
0001
⎞
⎟
⎟
⎟
⎟
⎠
P(σ 1 ) =
⎛
⎜
⎜
⎜
⎜
⎝
1000
0001
0010
0100
⎞
⎟
⎟
⎟
⎟
⎠
P(C 4 ) =
⎛
⎜
⎜
⎜
⎜
⎝
0001
1000
0100
0010
⎞
⎟
⎟
⎟
⎟
⎠
P(σ 2 ) =
⎛
⎜
⎜
⎜
⎜
⎝
0010
0100
1000
0001
⎞
⎟
⎟
⎟
⎟
⎠
P(C 2
4 ) =
⎛
⎜
⎜
⎜
⎜
⎝
0010
0001
1000
0100
⎞
⎟
⎟
⎟
⎟
⎠
P(σ 3 ) =
⎛
⎜
⎜
⎜
⎜
⎝
0001
0010
0100
1000
⎞
⎟
⎟
⎟
⎟
⎠
P(C 3
4 ) =
⎛
⎜
⎜
⎜
⎜
⎝
0100
0010
0001
1000
⎞
⎟
⎟
⎟
⎟
⎠
P(σ 4 ) =
⎛
⎜
⎜
⎜
⎜
⎝
0100
1000
0001
0010
⎞
⎟
⎟
⎟
⎟
⎠
We are now ready to start the proof. The total induction space is invariant under
the operations of the group G. As an example, we can act with an operator of the
group on one of the functions on one of the sites:
ˆ
g|γm γ ; κ ˆ
g ˆ
g κ |γm γ ; a=
λ
P λκ (g) ˆ
g λ
ˆ
g
−1
λ ˆ
g ˆ
g κ
|γm γ ; a
(4.76)
where we have placed the subelement of ˆ
g in square brackets. This subelement
belongs to H A , under the protection of the P λκ (g) prefactor, which will be nonzero
only for values of ˆ
g for which this is indeed the case. We can thus introduce the
representation matrix for the local on-site transformations:
ˆ
g|γm γ ; κ=
λ
m ′
γ
P λκ (g) ˆ
g λ |γm
′
γ ; aD
γ
m ′
γ m γ
g
−1
λ gg κ
=
λ
m ′
γ
P λκ (g)|γm
′
γ ; λD
γ
m ′
γ m γ
g
−1
λ gg κ
(4.77)
This result provides the matrix transformation that shows how the basis functions
of the total induction space are transformed under the operations of G. We shall
denote this matrix as D H ↑G . The structure of this matrix is based on the permutational structure of the ground representation, but the zeros are replaced by
small zero blocks of dimension dim(γ ) × dim(γ ), and, instead of the ones, the
D γ (g
−1
λ gg κ ) matrices are inserted. A diagonal element of this matrix will be given
by P κκ (g)D
γ
m γ m γ ( ˆ
g −1
κ ˆ
g ˆ
g κ ).