The best way to represent cluster-expansion wavefunctions is to write them
as operator products. That is, we must define an operator which, when
applied to 
produces cluster functions. Here, again, we follow
reference [29] in defining this operator. (This section assumes
a knowledge of second-quantization methods --
see appendix A for an overview of these techniques.) We will
require that the cluster functions be antisymmetric under index permutation
(related to the Pauli principle for fermions )
and strongly orthogonal to the occupied one-electron states
Here, 
and 
are the set of coordinates (including
spin) of electrons one and two, respectively. With these requirements, we
may define
The summation in this equation runs over all pairs of a and b. One can
see (after a moment of study) that an operator that will replace 
by 
can be written as
with 
and 
.
Products of cluster functions may be produced by successive application of
cluster operators
(Note: Since the creation and annihilation operators are restricted to
unoccupied and occupied spaces, respectively, different 
operators will commute.) Now we will attempt to re-write the rather long
four-electron wavefunction (2.7) from the previous
section in which we included all clusters of one and two electrons.
First, we must define the one-electron counterpart of
This reduces the four-electron interaction equation to
We may introduce further definitions
and
This reduces the equation even more
If we recall the form of the power-series expansion of the exponential
function, we see that this includes all of the above terms. If we remember
that 
and 
commute, then all of the terms from
the above equation match those from the exponential! Thus, the general
expression for (2.16) is
which is a rather nice reduction from the original equation.
In summary, the fundamental idea behind the cluster expansion is the use of
an ``exponential ansatz'' , which, when applied to
the reference function, produces a new wavefunction containing cluster
functions which correlate the motions of a specified number of electrons.
If the exponentiated operator (the cluster operator) includes contributions
from all possible electron groupings (that is, 
), then the exact wavefunction (within the given
one-electron basis) will be obtained from the reference wavefunction.
