6.4 Product Symmetrization and the Pauli Exchange-Symmetry
125
Now, if the product representation belongs to the symmetrized square, Γ ∈[Γ a ] 2 ,
this result simplifies to:
|Γγ=
γ a
γ a γ a |Γγ
γ a (1)
γ a (2)
+
γ a <γ b
γ a γ b |Γγ
γ a (1)
γ b (2)+
γ b (1)
γ a (2)
(6.39)
Hence, this function is symmetric under the ˆ
P 12 operator. It also obeys the normalization condition of Eq. (6.15). It should always be multiplied by an antisymmetric singlet spin-function in order to obey the Pauli exclusion principle. On the
other hand, if the product representation belongs to the antisymmetrized square,
Γ ∈{Γ a } 2 , the coupled state is given by:
|Γγ=
γ a <γ b
γ a γ b |Γγ
γ a (1)
γ b (2)
−
γ b (1)
γ a (2)
(6.40)
This function is antisymmetric under the ˆ
P 12 operator, and should be multiplied by a
symmetric triplet spin-function in order to obey the Pauli principle. As an example,
for the (e g ) 2 configuration the allowed states are:
e g × e g =[A 1g + E g ]+{A 2g }⇒
1 A 1g +
1 E g +
3 A 2g
(6.41)
For equivalent electrons the Pauli principle thus really does function as an exclusion
principle, since the coupled states are either triplets or singlets, depending on their
symmetrization. The dimension of the manifold is given by the binomial coefficient,
where q is the number of equivalent substates (including spin), and n is the number
of electrons:
q
n
=
q!
n!(q − n)!
(6.42)
For the (e g ) 2 problem, one has n = 2 and q = 4; there are thus six two-electron
states in this configuration (see Eq. (6.41)).
As a special result, we examine the symmetry of the maximal spin-multiplicity
ground state of a system with a half-filled shell. The shell consists of the components
|f 1 ···|f n , transforming according to the irrep Γ a , and each will be occupied by
one electron with α spin. The ground state corresponds to a single determinant:
|Ψ =|f 1 α ···f 2 α|=
1
√
n!
σ ∈S n
sgn(σ )
f 1 α(σ 1 ) ···f n α(σ n )
=
1
√
n!
σ ∈S n
sgn(σ )
f σ 1 α(1) ···f σ n α(n)
(6.43)
125
Now, if the product representation belongs to the symmetrized square, Γ ∈[Γ a ] 2 ,
this result simplifies to:
|Γγ=
γ a
γ a γ a |Γγ
γ a (1)
γ a (2)
+
γ a <γ b
γ a γ b |Γγ
γ a (1)
γ b (2)+
γ b (1)
γ a (2)
(6.39)
Hence, this function is symmetric under the ˆ
P 12 operator. It also obeys the normalization condition of Eq. (6.15). It should always be multiplied by an antisymmetric singlet spin-function in order to obey the Pauli exclusion principle. On the
other hand, if the product representation belongs to the antisymmetrized square,
Γ ∈{Γ a } 2 , the coupled state is given by:
|Γγ=
γ a <γ b
γ a γ b |Γγ
γ a (1)
γ b (2)
−
γ b (1)
γ a (2)
(6.40)
This function is antisymmetric under the ˆ
P 12 operator, and should be multiplied by a
symmetric triplet spin-function in order to obey the Pauli principle. As an example,
for the (e g ) 2 configuration the allowed states are:
e g × e g =[A 1g + E g ]+{A 2g }⇒
1 A 1g +
1 E g +
3 A 2g
(6.41)
For equivalent electrons the Pauli principle thus really does function as an exclusion
principle, since the coupled states are either triplets or singlets, depending on their
symmetrization. The dimension of the manifold is given by the binomial coefficient,
where q is the number of equivalent substates (including spin), and n is the number
of electrons:
q
n
=
q!
n!(q − n)!
(6.42)
For the (e g ) 2 problem, one has n = 2 and q = 4; there are thus six two-electron
states in this configuration (see Eq. (6.41)).
As a special result, we examine the symmetry of the maximal spin-multiplicity
ground state of a system with a half-filled shell. The shell consists of the components
|f 1 ···|f n , transforming according to the irrep Γ a , and each will be occupied by
one electron with α spin. The ground state corresponds to a single determinant:
|Ψ =|f 1 α ···f 2 α|=
1
√
n!
σ ∈S n
sgn(σ )
f 1 α(σ 1 ) ···f n α(σ n )
=
1
√
n!
σ ∈S n
sgn(σ )
f σ 1 α(1) ···f σ n α(n)
(6.43)