126
6 Interactions
Here, σ ∈ S n is an element of the permutation group of the n electron labels, and
sgn(σ ) is its parity. 2 Equation (6.43) indicates that this permutation can equally well
be applied to the component labels, since the determinant is invariant under matrix
transposition. We can now calculate the matrix element in the symmetry operator:
Ψ | ˆ
R|Ψ =
1
n!
σ,π∈S n
sgn(σ )sgn(π)
f σ 1 α(1)| ˆ
R|f π 1 α(1)
···
f σ n α(n)| ˆ
R|f π n α(n)
=
1
n!
σ,π∈S n
sgn(σ )sgn(π)
D
Γ a
σ 1 π 1
(R) ···D
Γ a
σ n π n
(R)
=
λ∈S n
sgn(λ)
D
Γ a
1λ 1
(R) ···D
Γ a
nλ n
(R)
= det
D
Γ a
(6.44)
In this equation we have used the result from Eq. (2.8), which identified matrix
elements over symmetry operators as elements of the representation matrix. The
double summation over permutations covers the permutation group twice and could
be reduced to a single sum. The result indicates that the spatial symmetry of the halffilled shell ground state transforms as the determinant of the irrep of the shell. This
is also called the determinantal representation. For the e g shell the determinantal
irrep is A 2g . The shell ground state is thus a 3 A 2g .
6.5 Matrix Elements and the Wigner–Eckart Theorem
A general interaction element is a bracket around an operator. Each of the three ingredients, bra, ket, and operator, can be put in symmetry-adapted form, so that it
transforms according to a given irrep. Moreover, provided that the symmetry adaptation is done properly, not only the irrep itself but also the subrepresentation is well
defined. Altogether, the matrix element will thus be characterized by six symmetry
labels, as: ψ Ω
ω |O Λ
λ |φ Γ
γ . The labels imply that the symmetry behaviour of each of
these ingredients is fully known:
ˆ
R
ψ
Ω
ω
=
ω ′
¯
D
Ω
ω ′ ω (R)
ψ
Ω
ω ′
ˆ
R
φ
Γ
γ
=
γ ′
D
Γ
γ ′ γ (R)
φ
Γ
γ ′
ˆ
R
O
Λ
λ
R
−1 =
λ ′
D
Λ
λ ′ λ (R)
O
Λ
λ ′
(6.45)
2 Any permutation can be expressed as a sequence of transpositions of two elements. If the total
number of transpositions is even, sgn(σ ) =+1; if it is odd, sgn(σ ) =−1. See also Sect. 3.3.
6 Interactions
Here, σ ∈ S n is an element of the permutation group of the n electron labels, and
sgn(σ ) is its parity. 2 Equation (6.43) indicates that this permutation can equally well
be applied to the component labels, since the determinant is invariant under matrix
transposition. We can now calculate the matrix element in the symmetry operator:
Ψ | ˆ
R|Ψ =
1
n!
σ,π∈S n
sgn(σ )sgn(π)
f σ 1 α(1)| ˆ
R|f π 1 α(1)
···
f σ n α(n)| ˆ
R|f π n α(n)
=
1
n!
σ,π∈S n
sgn(σ )sgn(π)
D
Γ a
σ 1 π 1
(R) ···D
Γ a
σ n π n
(R)
=
λ∈S n
sgn(λ)
D
Γ a
1λ 1
(R) ···D
Γ a
nλ n
(R)
= det
D
Γ a
(6.44)
In this equation we have used the result from Eq. (2.8), which identified matrix
elements over symmetry operators as elements of the representation matrix. The
double summation over permutations covers the permutation group twice and could
be reduced to a single sum. The result indicates that the spatial symmetry of the halffilled shell ground state transforms as the determinant of the irrep of the shell. This
is also called the determinantal representation. For the e g shell the determinantal
irrep is A 2g . The shell ground state is thus a 3 A 2g .
6.5 Matrix Elements and the Wigner–Eckart Theorem
A general interaction element is a bracket around an operator. Each of the three ingredients, bra, ket, and operator, can be put in symmetry-adapted form, so that it
transforms according to a given irrep. Moreover, provided that the symmetry adaptation is done properly, not only the irrep itself but also the subrepresentation is well
defined. Altogether, the matrix element will thus be characterized by six symmetry
labels, as: ψ Ω
ω |O Λ
λ |φ Γ
γ . The labels imply that the symmetry behaviour of each of
these ingredients is fully known:
ˆ
R
ψ
Ω
ω
=
ω ′
¯
D
Ω
ω ′ ω (R)
ψ
Ω
ω ′
ˆ
R
φ
Γ
γ
=
γ ′
D
Γ
γ ′ γ (R)
φ
Γ
γ ′
ˆ
R
O
Λ
λ
R
−1 =
λ ′
D
Λ
λ ′ λ (R)
O
Λ
λ ′
(6.45)
2 Any permutation can be expressed as a sequence of transpositions of two elements. If the total
number of transpositions is even, sgn(σ ) =+1; if it is odd, sgn(σ ) =−1. See also Sect. 3.3.