6.4 Product Symmetrization and the Pauli Exchange-Symmetry
123
orbitals into a t 2g and an e g shell. However, these coupled descriptions are not yet
sufficient, since they make a distinction between electron 1, which resides in the
t 2g orbital, and electron 2, which was promoted to the e g level. The fundamental
symmetry requirement that electrons must be indistinguishable is thus not fulfilled.
The operator that permutes the two electrons is represented as ˆ
P 12 :
ˆ
P 12
Γγ(1, 2)
=
Γγ(2, 1)
=
γ a
γ b
Γ a γ a Γ b γ b |Γγ
Γ a γ a (2)
Γ b γ b (1)
(6.33)
The |Γγ(1, 2) and |Γγ(2, 1) states will have exactly the same symmetries, since
the factors in the direct product commute:
Γ a × Γ b = Γ b × Γ a
(6.34)
As a result, ˆ
P 12 commutes with the spatial symmetry operators, and we can symmetrize the coupled states with respect to the electron permutation. The permutation
operator is the generator of the symmetric group, S 2 , which has only two irreps, one
symmetric and one antisymmetric, corresponding, respectively, to the plus and minus combination in Eq. (6.35).
|Γγ;±±=
1
√
2
Γγ(1, 2)
±
Γγ(2, 1)
=
1
√
2
γ a
γ b
Γ a γ a Γ b γ b |Γγ
×
Γ a γ a (1)
Γ b γ b (2)
±
Γ b γ b (1)
Γ a γ a (2)
(6.35)
These states have distinct permutation symmetries, and spatial symmetry operators
cannot mix + and − states. This is a very general property of multi-particle states,
to which no exceptions are known.
On the other hand the permutation symmetry of multi-electron wavefunctions is
restricted by the Pauli principle.
Theorem 11 The total wavefunction should be antisymmetric with respect to exchange of any pair of electrons. Hence, in the symmetric group S 2 , or, for an nelectron system, the symmetric group, S n , the total wavefunction should change sign
under odd permutations, i.e. under permutations that consist of an odd number of
transpositions of two elements, and should remain invariant under even permutations.
Until now we have limited ourselves to the spatial part of the wavefunction. So
far, only the antisymmetrized part obeys the Pauli principle. However, the principle
places a requirement only on the total wavefunction. This also involves a spin part,
which should be multiplied by the orbital part. Anticipating the results of Chap. 7,
we here provide the spin functions for a two-electron system. Spin functions are
characterized by a spin quantum number, S, and a component, M S , in the range
123
orbitals into a t 2g and an e g shell. However, these coupled descriptions are not yet
sufficient, since they make a distinction between electron 1, which resides in the
t 2g orbital, and electron 2, which was promoted to the e g level. The fundamental
symmetry requirement that electrons must be indistinguishable is thus not fulfilled.
The operator that permutes the two electrons is represented as ˆ
P 12 :
ˆ
P 12
Γγ(1, 2)
=
Γγ(2, 1)
=
γ a
γ b
Γ a γ a Γ b γ b |Γγ
Γ a γ a (2)
Γ b γ b (1)
(6.33)
The |Γγ(1, 2) and |Γγ(2, 1) states will have exactly the same symmetries, since
the factors in the direct product commute:
Γ a × Γ b = Γ b × Γ a
(6.34)
As a result, ˆ
P 12 commutes with the spatial symmetry operators, and we can symmetrize the coupled states with respect to the electron permutation. The permutation
operator is the generator of the symmetric group, S 2 , which has only two irreps, one
symmetric and one antisymmetric, corresponding, respectively, to the plus and minus combination in Eq. (6.35).
|Γγ;±±=
1
√
2
Γγ(1, 2)
±
Γγ(2, 1)
=
1
√
2
γ a
γ b
Γ a γ a Γ b γ b |Γγ
×
Γ a γ a (1)
Γ b γ b (2)
±
Γ b γ b (1)
Γ a γ a (2)
(6.35)
These states have distinct permutation symmetries, and spatial symmetry operators
cannot mix + and − states. This is a very general property of multi-particle states,
to which no exceptions are known.
On the other hand the permutation symmetry of multi-electron wavefunctions is
restricted by the Pauli principle.
Theorem 11 The total wavefunction should be antisymmetric with respect to exchange of any pair of electrons. Hence, in the symmetric group S 2 , or, for an nelectron system, the symmetric group, S n , the total wavefunction should change sign
under odd permutations, i.e. under permutations that consist of an odd number of
transpositions of two elements, and should remain invariant under even permutations.
Until now we have limited ourselves to the spatial part of the wavefunction. So
far, only the antisymmetrized part obeys the Pauli principle. However, the principle
places a requirement only on the total wavefunction. This also involves a spin part,
which should be multiplied by the orbital part. Anticipating the results of Chap. 7,
we here provide the spin functions for a two-electron system. Spin functions are
characterized by a spin quantum number, S, and a component, M S , in the range