Chemistry Reference
In-Depth Information
6
Antisymmetry of Many-electron
Wavefunctions
The effects of symmetry on the many-electron wavefunctions arise from
the physical identity and, therefore, from the indistinguishability of the
electrons, and are of fundamental importance in the treatment of the
many-particle systems. Use of suitable density functions allows us to pass
from the abstract 4N-dimensional mathematical space of the N-electron
wavefunctions to the three-dimensional
þ
spin space where physical
experiments are done (McWeeny, 1960).
6.1 ANTISYMMETRY REQUIREMENT
AND THE PAULI PRINCIPLE
Let x
1
and x
2
be two fixed points in the space-spin space and
(x
1
,x
2
)
a normalized two-electron wavefunction. Then, since electrons are iden-
tical particles, we have from first principles that
Y
<
:
2
dx
1
dx
2
¼
probability of finding electron 1
at dx
1
and electron 2 at dx
2
¼jYð
x
2
jYð
x
1
;
x
2
Þj
ð
6
:
1
Þ
2
dx
1
dx
2
¼
probability of finding electron 2
at dx
1
and electron 1 at dx
2
;
x
1
Þj
