<s>
In	O
quantum	O
mechanics	O
,	O
an	O
antisymmetrizer	B-Algorithm
(	O
also	O
known	O
as	O
antisymmetrizing	B-Algorithm
operator	I-Algorithm
)	O
is	O
a	O
linear	O
operator	O
that	O
makes	O
a	O
wave	O
function	O
of	O
N	O
identical	O
fermions	O
antisymmetric	O
under	O
the	O
exchange	O
of	O
the	O
coordinates	O
of	O
any	O
pair	O
of	O
fermions	O
.	O
</s>
<s>
After	O
application	O
of	O
the	O
wave	O
function	O
satisfies	O
the	B-General_Concept
Pauli	I-General_Concept
exclusion	I-General_Concept
principle	I-General_Concept
.	O
</s>
<s>
Since	O
is	O
a	O
projection	B-Algorithm
operator	I-Algorithm
,	O
application	O
of	O
the	O
antisymmetrizer	B-Algorithm
to	O
a	O
wave	O
function	O
that	O
is	O
already	O
totally	O
antisymmetric	O
has	O
no	O
effect	O
,	O
acting	O
as	O
the	O
identity	O
operator	O
.	O
</s>
<s>
This	O
implies	O
that	O
in	O
general	O
and	O
therefore	O
we	O
can	O
define	O
meaningfully	O
a	O
transposition	B-Algorithm
operator	O
that	O
interchanges	O
the	O
coordinates	O
of	O
particle	O
i	O
and	O
j	O
.	O
</s>
<s>
Here	O
we	O
associated	O
the	O
transposition	B-Algorithm
operator	O
with	O
the	O
permutation	B-Algorithm
of	O
coordinates	O
π	O
that	O
acts	O
on	O
the	O
set	O
of	O
N	O
coordinates	O
.	O
</s>
<s>
In	O
this	O
case	O
π	O
=	O
(	O
ij	O
)	O
,	O
where	O
(	O
ij	O
)	O
is	O
the	O
cycle	O
notation	O
for	O
the	O
transposition	B-Algorithm
of	O
the	O
coordinates	O
of	O
particle	O
i	O
and	O
j	O
.	O
</s>
<s>
Transpositions	B-Algorithm
may	O
be	O
composed	O
(	O
applied	O
in	O
sequence	O
)	O
.	O
</s>
<s>
This	O
defines	O
a	O
product	O
between	O
the	O
transpositions	B-Algorithm
that	O
is	O
associative	O
.	O
</s>
<s>
It	O
can	O
be	O
shown	O
that	O
an	O
arbitrary	O
permutation	B-Algorithm
of	O
N	O
objects	O
can	O
be	O
written	O
as	O
a	O
product	O
of	O
transpositions	B-Algorithm
and	O
that	O
the	O
number	O
of	O
transposition	B-Algorithm
in	O
this	O
decomposition	O
is	O
of	O
fixed	O
parity	O
.	O
</s>
<s>
That	O
is	O
,	O
either	O
a	O
permutation	B-Algorithm
is	O
always	O
decomposed	O
in	O
an	O
even	O
number	O
of	O
transpositions	B-Algorithm
(	O
the	O
permutation	B-Algorithm
is	O
called	O
even	O
and	O
has	O
the	O
parity	O
+1	O
)	O
,	O
or	O
a	O
permutation	B-Algorithm
is	O
always	O
decomposed	O
in	O
an	O
odd	O
number	O
of	O
transpositions	B-Algorithm
and	O
then	O
it	O
is	O
an	O
odd	O
permutation	B-Algorithm
with	O
parity1	O
.	O
</s>
<s>
where	O
we	O
associated	O
the	O
linear	O
operator	O
with	O
the	O
permutation	B-Algorithm
π	O
.	O
</s>
<s>
permutations	B-Algorithm
with	O
the	O
associative	O
product	O
:	O
"	O
apply	O
one	O
permutation	B-Algorithm
after	O
the	O
other	O
"	O
,	O
is	O
a	O
group	O
,	O
known	O
as	O
the	O
permutation	B-Algorithm
group	O
or	O
symmetric	B-Algorithm
group	I-Algorithm
,	O
denoted	O
by	O
SN	O
.	O
</s>
<s>
In	O
the	O
representation	O
theory	O
of	O
finite	O
groups	O
the	O
antisymmetrizer	B-Algorithm
is	O
a	O
well-known	O
object	O
,	O
because	O
the	O
set	O
of	O
parities	O
forms	O
a	O
one-dimensional	O
(	O
and	O
hence	O
irreducible	O
)	O
representation	O
of	O
the	O
permutation	B-Algorithm
group	O
known	O
as	O
the	O
antisymmetric	O
representation	O
.	O
</s>
<s>
The	O
antisymmetrizer	B-Algorithm
is	O
in	O
fact	O
a	O
character	O
projection	B-Algorithm
operator	I-Algorithm
and	O
is	O
quasi-idempotent	O
,	O
</s>
<s>
Either	O
Ψ	O
does	O
not	O
have	O
an	O
antisymmetric	O
component	O
,	O
and	O
then	O
the	O
antisymmetrizer	B-Algorithm
projects	O
onto	O
zero	O
,	O
or	O
it	O
has	O
one	O
and	O
then	O
the	O
antisymmetrizer	B-Algorithm
projects	O
out	O
this	O
antisymmetric	O
component	O
Ψ	O
 '	O
.	O
</s>
<s>
The	O
antisymmetrizer	B-Algorithm
carries	O
a	O
left	O
and	O
a	O
right	O
representation	O
of	O
the	O
group	O
:	O
</s>
<s>
with	O
the	O
operator	O
representing	O
the	O
coordinate	O
permutation	B-Algorithm
π	O
.	O
</s>
<s>
If	O
a	O
wave	O
function	O
is	O
symmetric	O
under	O
any	O
odd	O
parity	O
permutation	B-Algorithm
it	O
has	O
no	O
antisymmetric	O
component	O
.	O
</s>
<s>
Then	O
the	O
product	O
is	O
symmetric	O
under	O
the	O
transposition	B-Algorithm
(	O
k	O
,	O
q	O
)	O
and	O
hence	O
vanishes	O
.	O
</s>
<s>
Notice	O
that	O
this	O
result	O
gives	O
the	O
original	O
formulation	O
of	O
the	O
Pauli	B-General_Concept
principle	I-General_Concept
:	O
no	O
two	O
electrons	O
can	O
have	O
the	O
same	O
set	O
of	O
quantum	O
numbers	O
(	O
be	O
in	O
the	O
same	O
spin-orbital	O
)	O
.	O
</s>
<s>
Permutations	B-Algorithm
of	O
identical	O
particles	O
are	O
unitary	B-Algorithm
,	O
(	O
the	O
Hermitian	O
adjoint	O
is	O
equal	O
to	O
the	O
inverse	O
of	O
the	O
operator	O
)	O
,	O
and	O
since	O
π	O
and	O
π−1	O
have	O
the	O
same	O
parity	O
,	O
it	O
follows	O
that	O
the	O
antisymmetrizer	B-Algorithm
is	O
Hermitian	O
,	O
</s>
<s>
If	O
it	O
were	O
otherwise	O
,	O
measurement	O
of	O
could	O
distinguish	O
the	O
particles	O
,	O
in	O
contradiction	O
with	O
the	O
assumption	O
that	O
only	O
the	O
coordinates	O
of	O
indistinguishable	O
particles	O
are	O
affected	O
by	O
the	O
antisymmetrizer	B-Algorithm
.	O
</s>
<s>
the	O
Slater	O
determinant	O
is	O
created	O
by	O
the	O
antisymmetrizer	B-Algorithm
operating	O
on	O
the	O
product	O
of	O
spin-orbitals	O
,	O
as	O
below	O
:	O
</s>
<s>
To	O
see	O
the	O
correspondence	O
we	O
notice	O
that	O
the	O
fermion	O
labels	O
,	O
permuted	O
by	O
the	O
terms	O
in	O
the	O
antisymmetrizer	B-Algorithm
,	O
label	O
different	O
columns	O
(	O
are	O
second	O
indices	O
)	O
.	O
</s>
<s>
The	O
operators	O
appearing	O
in	O
these	O
two	O
antisymmetrizers	B-Algorithm
represent	O
the	O
elements	O
of	O
the	O
subgroups	O
SNA	O
and	O
SNB	O
,	O
respectively	O
,	O
of	O
SNA+NB	O
.	O
</s>
<s>
When	O
A	O
and	O
B	O
interact	O
,	O
the	O
Pauli	B-General_Concept
principle	I-General_Concept
requires	O
the	O
antisymmetry	O
of	O
the	O
total	O
wave	O
function	O
,	O
also	O
under	O
intermolecular	O
permutations	B-Algorithm
.	O
</s>
<s>
The	O
total	O
system	O
can	O
be	O
antisymmetrized	O
by	O
the	O
total	O
antisymmetrizer	B-Algorithm
which	O
consists	O
of	O
the	O
(	O
NA	O
+	O
NB	O
)	O
!	O
</s>
<s>
The	O
operator	O
represents	O
the	O
coset	O
representative	O
τ	O
(	O
an	O
intermolecular	O
coordinate	O
permutation	B-Algorithm
)	O
.	O
</s>
<s>
Obviously	O
the	O
intermolecular	O
antisymmetrizer	B-Algorithm
has	O
a	O
factor	O
NA	O
!	O
</s>
<s>
fewer	O
terms	O
then	O
the	O
total	O
antisymmetrizer	B-Algorithm
.	O
</s>
