Chemistry Reference
In-Depth Information
{−
. The total number of components, hence the
dimension of the spin-space for a given
S
, is equal to 2
S
S,
−
S
+
1
,
−
S
+
2
,...,S
−
1
,S
}
1. This number is called
the spin multiplicity. For a two-electron system,
S
can be 0 or 1; hence, there is one
singlet state, and there are three components belonging to a triplet state. In a
+
|
SM
S
notation they are given by:
√
2
α(
1
)β(
2
)
−
β(
1
)α(
2
)
1
|
0
,
0
=
|
1
,
1
=
α(
1
)α(
2
)
(6.36)
√
2
α(
1
)β(
2
)
β(
1
)α(
2
)
1
|
1
,
0
=
+
|
−
=
1
,
1
β(
1
)β(
2
)
These functions also exhibit permutation symmetry: the triplet functions are sym-
metric under exchange of the two particles, while the singlet function is antisym-
metric under such an exchange. The total wavefunction can thus always be put in
line with the Pauli principle by combining the coupled orbital states with spin states
of opposite permutation symmetry. Altogether we can thus construct four states:
1
T
1
g
,
1
T
2
g
,
3
T
1
g
,
3
T
2
g
. This set of four states, totalling 24 wavefunctions, forms a
manifold
, representing all the coupled states resulting from the
(t
2
g
)
1
(e
g
)
1
configu-
ration. The dimension of the manifold is equal to the product of the six possible
t
2
g
substates (including spin), and the four possible
e
g
substates. In this case, where the
coupling involves electrons belonging to different shells, the Pauli principle does
not restrict the total dimension of the manifold, since all combinations remain pos-
sible. All states can be written as linear combinations of Slater determinants. As an
example, for the
1
T
1
g
z
|
state, one writes:
1
T
1
g
z
=
√
2
(
1
)
ζ(
2
)
+
(
2
)
ζ(
1
)
√
2
α(
1
)β(
2
)
β(
1
)α(
2
)
1
1
−
√
2
(α)(ζβ)
−
√
2
(β)(ζα)
1
1
=
(6.37)
The situation is different when coupling two equivalent electrons; these are elec-
trons that belong to
the same shell
. In this case, the coupled states are already eigen-
functions of the exchange operator as a result of the special symmetrization prop-
erties of the coupling coefficients for direct squares. Equation (
6.10
) will take the
following form:
γ
a
γ
b
γ
a
γ
a
(
1
)
γ
b
(
2
)
=
γ
a
(
1
)
γ
a
(
2
)
|
Γγ(
1
,
2
)
=
γ
a
γ
b
|
Γγ
γ
a
γ
a
|
Γγ
γ
a
(
1
)
γ
b
(
2
)
+
γ
b
(
1
)
γ
a
(
2
)
+
γ
a
γ
b
|
Γγ
γ
b
γ
a
|
Γγ
γ
a
<γ
b
(6.38)
