Quantum Mechanics for Engineers 

© Leon van Dommelen 

N.16 A single Slater determinant is not exact
The simplest example that illustrates the problem with representing a
general wave function by a single Slater determinant is to try to
write a general twovariable function as a Slater determinant
of two functions and . You would write
A general function cannot be written as a combination of the
same two functions and at every value
of . However well chosen the two functions are.
In fact, for a general antisymmetric function , a single
Slater determinant can get right at only two nontrivial values
and . (Nontrivial here means that
functions and should not just be multiples of
each other.) Just take and
. You might object that in general, you have
where , , , and are
some constants. (They are or values at or
, to be precise). But if you plug these two expressions
into the Slater determinant formed with and and
multiply out, you get the Slater determinant formed with and
within a constant, so it makes no difference.
If you add a second Slater determinant, you can get right at two
more values and . Just take the second Slater
determinant's functions to be and
, where is the
deviation between the true function and what the first Slater
determinant gives. Keep adding Slater determinants to get more and
more values right. Since there are infinitely many
values to get right, you will in general need infinitely
many determinants.
You might object that maybe the deviation from the single
Slater determinant must be zero for some reason. But you can use the
same ideas to explicitly construct functions that show that this
is untrue. Just select two arbitrary but different functions
and and form a Slater determinant. Now choose two locations
and so that and
are not in the same ratio to each other. Then add additional Slater
determinants whose functions
you choose so that
they are zero at and . The so constructed function
is different from just the first Slater determinant. However, if
you try to describe this by a single determinant, then it could
only be the first determinant since that is the only single
determinant that gets and right. So a single determinant
cannot get right.