The whole ``interacting space'' idea comes out
of Rayleigh-Schrodinger perturbation theory.
The ``first-order'' interacting space is so named because it contains those
N-electron basis functions which contribute to the perturbation theory
first-order wavefunction correction.
Consider the Moller-Plesset partition of the Hamiltonian,
where 
is the Hartree-Fock Hamiltonian,
and
For simplicity, let's first assume that we choose Slater determinants,
rather than CSF's, as our N-electron basis.
In that case our reference will
be a single Slater determinant 
.
If we expand the first-order correction to the wavefunction in terms of
the other eigenfunctions of (which represent Slater
determinants formed from different occupations of the Hartree-Fock
orbitals), we have
where the superscript on 
represents the perturbation
order and the subscript indexes the eigenvalue of .
The coefficients are given by
or
Now we can see that 
contributes to 
only if
is nonzero.
If a Slater determinant contributes to the first-order interacting space if
it has a nonzero matrix element through the perturbation 
, then why does the literature on the interacting space claim that it
must have a nonzero matrix element through the full Hamiltonian 
?
We can see that this is nearly the same condition as follows.
Since 
, we
could rewrite 
as
The reference is an eigenfunction of , and Slater
determinants formed from the same one-particle basis are orthogonal,
so the second term
becomes zero. Thus we can say that for a Slater determinant to
contribute to , it must have a non-zero matrix element with
the reference through the Hamiltonian .
In the case of CSFs, these same arguments hold, so that the requirement
is equivalent to
except in the case that
and 
contain common Slater determinants. In this
case, is generally sufficient
to ensure .
Now clearly any determinant which differs from the reference by more than 2 spin orbitals will have a zero Hamiltonian matrix element with it, since the Hamiltonian contains no higher than two-body interactions. Thus if we insist on looking at spin orbitals, we have a fairly simple criterion for judging what is not in the interacting space, at least for Slater determinants. For closed-shell systems, the interacting space consists of all doubles. For open-shell cases, some ``doubles'' are not included, namely those in which two electrons have been excited and one of the open-shell electrons has changed its spin. However, if we count spin-flips as excitations (or if we focus at spin orbitals), then once again all doubles are in the first order interacting space.