If we always dealt with Slater determinants rather than with CSF's, then there would be no more to say about the first-order interacting space. Unfortunately, CSF's complicate the matter. The reason for using CSF's is, of course, that we have fewer N-electron basis functions. To make the interacting space as small as possible, we must be careful about how we construct our CSF's.
Before describing how spin- and space-adaptation of the interacting space generally complicates matters, we should first note that for closed-shell systems, everything remains fairly straightforward for CSF's. i There are four types of doubly-excited configurations: (a) both electrons from a given spatial orbital may be excited to the same virtual spatial orbital, (b) two electrons from the same orbital can be excited to different orbitals, (c) two electrons in different orbitals can be excited to the same orbital, and (d) two electrons in different orbitals can be excited into two different orbitals. For case (a), there is only one possible CSF, one which has no unpaired electrons. For cases (b) and (c), there is only one possible way to obtain a singlet CSF. For case (d), there are two possible ways to couple the four unpaired electrons to obtain a singlet state. In all cases, the CSF's consist of determinants which differ from the reference by exactly two spin orbitals, so that they yield nonzero matrix elements with the Hamiltonian. It can be shown for cases (b)-(d) that these nonzero determinantal matrix elements do not cancel each other when combined to form the CSF matrix element.
Thus we cannot use the interacting space concept to reduce the dimensions of the CI problem when the reference is closed-shell. In that case, all CSF's representing doubly-excited configurations will have nonzero Hamiltonian matrix elements (assuming of course that they have the proper spin and space symmetry, but we always make this selection). However, when the reference is open-shell, then restricting the CI space to the interacting-space does remove CSF's.
Both Bunge
and McLean and Liu give complicated methods for obtaining
the the minimum number of first-order interacting space CSF's.
One drawback of both methods is that they also give the non-interacting
portion of the CI space, which is not of interest. In both approaches,
the N-electron functions spanning the interacting space are made
eigenfunctions not only of 
. Bunge's CSF's are eigenfunctions of
as well since they are introduced in the context of the carbon atom,
which is radially symmetric.
In Bunge's method, the
non-interacting space is determined first, and then the interacting space
consists of those CSF's which must be added in order to span the
entire CI space. Bunge uses an orthonormal set 
of
eigenfunctions of , , 
, and 
. The subscript k
designates a configuration, which Bunge defines as ``an ordered set of N
indices (i,l) associated with the spin-orbitals 
,''
which are of the form
Since there are 4l+2 different spin-orbitals associated with each (i,l)
pair, one can form several Slater determinants 
belonging to
each configuration. The Slater determinants are formed from
symmetry-adapted spin orbitals, so they are eigenfunctions of , , and
parity. We can obtain eigenfunctions of and (the set
) by applying the
projection operator 
to
the determinants and orthogonalizing sucessively the resulting
functions.
It may be useful to introduce a few more definitions given in Bunge's paper.
She defines a class of configurations as ``the particular partition
of N (number of electrons) among the spin-orbitals with different l
values. Within each class, a subclass is defined by the particular
set of partitions of the `quantum number' i among the spin-orbitals with
the same l value. Thus, the subclasses 
,
, 
, and 
form the class
(3s,2p,1d).'' She also defines an n-excited determinant as a determinant
which differs by n spin-orbitals from the Hartree-Fock reference (or, for
open shells, it is a determinant which differs by n spin-orbitals from the
reference determinant it differs
least from). An n-excited configuration is a configuration differing
from the HF one by n sets of (i,l) indices.
Now we summarize Bunge's method of constructing CSF's so as to minimize the
first-order interacting space. The operator 
, when
applied to any Slater determinant 
, yields a CSF which can be
expressed as a sum over all 
determinants which span the space
(note that not all determinants with the
correct 
and 
belong to this space):
If the configuration K is degenerate, then we project another determinant
. If the projected CSF is linearly independent, then the two CSF's
are Schmidt orthogonalized. Otherwise, a different determinant is
projected. The process is repeated until the correct number of
's have been found. In order to minimize the size of the
interacting space, the following trick is used: one begins by projecting
the 's which are triply or quadruply excited. This will express
the determinant projected as a sum of CSF's, of which only one has the
correct symmetry.
Now the original determinant has a zero
matrix element with the Hamiltonian, and all the CSF's with the wrong symmetry
also have a zero matrix element. Therefore the CSF with the correct
symmetry must also have a zero matrix element.
Thus the interacting space is minimized by maximizing the noninteracting
space. As many triply and quadruply excited determinants as possible are
projected to form CSF's. When no more of these can be found which give
linearly independent projections, then the doubly excited determinants are
projected, and each of these projections is assumed to be Hartree-Fock
interacting.
McLean and Liu find an even more effective way to reduce the size of the
interacting space. They use a matrix method similar to Gaussian
elimination to transform the equation
where H contains the Hamiltonian matrix elements between the reference and the
other N-electron basis functions (which may be chosen as Slater
determinants or as CSF's), C is a coefficient matrix, and M contains
integrals over molecular orbitals. It seems that this approach is
equivalent to that of Bunge except that it also finds rotations
among the interacting CSF's which cause some of them to become
noninteracting.
Our method of determining the interacting space
is technically very
different from either of the two methods mentioned before.
First, our wavefunctions do not take advantage of degenerate point groups,
as do the other two methods mentioned. Secondly,
we do not need to construct the CSF's--we already have them from
the Distinct Row Table (DRT) program, in the form of Gelfand tableaux. Our
method is simply to take each CSF and ask if it is in the interacting
space or not. It turns out that the simple idea of counting spin flips as
excitations, which works so easily for Slater determinants, also works for
Gelfand states if the open-shell electrons are at the top of the tableaux.
Slater's rules make it abundantly clear why this works for Slater
determinants, but it is not at all clear why it works for Gelfand states.
A clue to this, however, may lie in the fact that, in the typical example
from the literature,
the interacting Gelfand states consist
entirely of
doubly-excited determinants, whereas the noninteracting ones are expanded
as doubly-excited determinants (which give matrix element terms which
``happen'' to cancel)
and higher-excited determinants. It may be possible to show that a
triply or higher excited configuration in a Gelfand representation must,
when expanded in the space of Slater determinants,
contain at least one triply or higher excited determinant in its
expansion, and that this will guarantee that all nonzero matrix element terms
arising from doubly-excited determinants in the expansion will cancel.
On the other hand, it seems that in Bunge's method, the interacting CSF's
may include triply and higher excited determinants in their expansion.