We have seen that the wavefunctions of electrons are completely anti
symmetric. By this statement we mean that under the exchange of any two
particles’ coordinates and spins the wavefunction changes sign. Inclusion of
the spin degree of freedom complicates the matters a little bit, but the dis
cussion is fairly simple for two electrons. When the total spin is conserved
eigenfunctions of the Hamiltonian can also be chosen as the eigenfunctions
of the total spin.
More precisely, we can choose the energy eigenfunctions
as the common eigenstates of
H
,
S
2
and
S
z
. In that case the eigenfunctions
will be
Ψ(1
,
2) =
φ
(
r
1
,
r
2
)
χ
spin
where
φ
is the part of the wavefunction describing positions of the two elec
trons and
χ
spin
is the part that describe the “spin motion” of the electrons.
Since that last part is chosen as an eigenstate of
S
2
and
S
z
, it can be written
as
χ
spin
=

S
= 1
, m
s
=
↑↑
m
s
= 1
,
1
√
2
(
↑↓
+
↓↑
)
m
s
= 0
,
↓↓
m
s
=

1
,
and
χ
spin
=

S
= 0
, m
s
= 0
=
1
√
2
(
↑↓  ↓↑
)
.
Since for
S
= 1 case (triplet)
χ
spin
is symmetric under the exchange of par
ticles’ spin, the space part of the wavefunction should be antisymmetric:
φ
triplet
(
r
1
,
r
2
) =

φ
triplet
(
r
2
,
r
1
)
.
For the
S
= 0 case (singlet), spin part is antisymmetric and as a result the
space part should be symmetric:
φ
singlet
(
r
1
,
r
2
) =
φ
singlet
(
r
2
,
r
1
)
.
We should add that this simplification occurs
only if
the total spin,
S
=
S
1
+
S
2
, is a conserved quantity (i.e., it commutes with the Hamiltonian). In
real atoms and molecules, this is not the case due to the spinorbit coupling.
Only the total angular momentum is conserved and not the total spin. As a
result, true eigenstates of realistic Hamiltonians will be a mixture of singlet
and triplet states, i.e., a very complicated wavefunction where spin and space
are “correlated”.
1
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For the case of three or more particles, the breakdown of the wavefunc
tion into a space and spin part will be quite complicated, even when the
total spin is conserved. You may try to see this yourself for the case of three
electrons with total spin quantum number
S
= 1
/
2.
If the complete an
tisymmetry of the whole wavefunction is imposed, no wavefunction can be
written as a product of a space part and a spin part. This is a result of the
noncommutativeness of the different exchange operators for three or more
particles.
As a result in some cases (e.g., when
S
= 1
/
2 restriction is im
posed) you cannot find common eigenfunctions of all the exchange operators.
1
Slater Determinants
Therefore a general method is needed to construct electron wavefunctions
which are completely antisymmetric in all cases. For this purpose, we can
use the complete antisymmetry property of determinants. Since interchang
ing any two rows (or columns) of a determinant changes the sign of the
determinant, completely antisymmetric wavefunctions can be expressed in
the form of a determinant.
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 Spring '11
 starg
 Slater, Slater determinants, oneparticle states

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