Chemistry Reference
In-Depth Information
orbitals into a
t
2
g
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
2
g
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
P
12
:
P
12
Γγ(
1
,
2
)
=
γ
b
Γγ(
2
,
1
)
=
Γ
a
γ
a
(
2
)
Γ
b
γ
b
(
1
)
(6.33)
Γ
a
γ
a
Γ
b
γ
b
|
Γγ
γ
a
The
states will have exactly the same symmetries, since
the factors in the direct product commute:
|
Γγ(
1
,
2
)
and
|
Γγ(
2
,
1
)
Γ
a
×
Γ
b
=
Γ
b
×
Γ
a
(6.34)
As a result,
P
12
commutes with the spatial symmetry operators, and we can sym-
metrize 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 mi-
nus combination in Eq. (
6.35
).
√
2
Γγ(
1
,
2
)
±
Γγ(
2
,
1
)
1
|
Γγ
;±=
γ
b
1
√
2
=
Γ
a
γ
a
Γ
b
γ
b
|
Γγ
γ
a
×
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
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.
+
and
−
Theorem 11
The total wavefunction should be antisymmetric with respect to ex-
change of any pair of electrons
.
Hence
,
in the symmetric group S
2
,
or
,
for an n-
electron 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 permuta-
tions
.
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
