As we have previously pointed out, full CI --being the matrix formulation of the Schrödinger equation--is an exact theory for nonrelativistic electronic structure problems. If we truncate the CI (either in the one-electron or N-electron space), we no longer have an exact theory. Of course either of these truncations will introduce an error in the wavefunction, which will cause errors in the energy and all other properties. One particularly unwelcome result of truncating the N-electron basis is that the CI energies obtained are no longer size extensive or size consistent.
These two terms, size extensive and size consistent, are used somewhat
loosely in the literature. Of the two, size extensivity is the most
well-defined. A method is said to be size extensive if the energy
calculated thereby scales linearly with the number of particles N. The
word ``extensive'' is used in the same sense as in thermodynamics, when we
refer to an extensive, rather than an intensive property. A
method is called size consistent if it gives an energy 
for two
well separated subsystems A and B. While the definition of size
extensivity applies at any geometry, the concept of size consistency
applies only in the limiting case of infinite separation. In addition,
size consistency usually also implies correct dissociation into
fragments; this is the source of much of the confusion arising from this
term. Thus restricted Hartree-Fock (RHF) is size extensive, but it is not
necessarily size consistent, since it cannot properly describe dissociation
into open-shell fragments. It can be shown that many-body perturbation
theory (MBPT)
and coupled-cluster
(CC) methods are size extensive, but
they will be size consistent only if they are based on reference
wavefunction which dissociates properly.
As previously stated, truncated CI's are neither size extensive nor size consistent. A simple (and often used!) example is sufficient to make the point. Consider two noninteracting hydrogen molecules. If the CISD method is used, then the energy of the two molecules at large separation will not be the same as the sum of their energies when calculated separately. In order for this to be the case, we would have to include quadruple excitations in the supermolecule calculation, since local double excitations could happen simultaneously on A and B.
We would tend to think that size extensivity and size consistency are important, physical properties that all quantum mechanical models should have (indeed, full CI , an exact theory, has these properties), but perhaps they are not as essential as all that. Duch and Diercksen have claimed that ``making size extensivity the most important requirement of quantum chemical methods, although it does not guarantee correct physical description, seems to be based not that much on physical as on esthetical criteria'' [24]. Indeed, they show that quantum mechanics is a ``holistic'' theory, not well-suited toward the description of separated subsystems:
Hilbert space of antisymmetric, many particle functions, describing the total system, can not be decomposed into separate subspaces. Consider two systems,and
, with
and
electrons, respectively. Each system is described by its own function,
antisymmetric in
in
is always ``far'' from the antisymmetric function
, as measured by the overlap
or the norm of the difference
.
Such arguments notwithstanding, it is clear that the fraction of the
correlation energy
recovered by a truncated CI will diminish as the size of
the system increases, making it a progressively less accurate method.
There have been many attempts to correct the CI energy to make it size
extensive. The most widely-used (and simplest) of these methods is
referred to as the Davidson correction
[25], which is
where 
is the basis set correlation energy
recovered by a
CISD procedure.
This correction approximately accounts for the effects of ``unlinked
quadruple'' excitations (i.e. simultaneous pairs of double excitations).
The multireference version [24] of this correction is
where 
is the multireference CI energy and
is the energy
obtained from the set of references (MCSCF energy if the references are
obtained as all references in an MCSCF procedure). We have simply replaced
the CISD
correlation energy in equation (4.26) with the
analogous multireference correlation energy, and we have replaced 
with
the analogous sum of squares of all the reference coefficients.
There are a number of other size extensivity corrections, and most of them do not take any significant amount of computation. Reference [24] provides a nice comparison of several of the more common CI size extensivity correction methods. We should also mention that Malrieu and co-workers have presented a self-consistent dressing of the Hamiltonian which gives size extensive results for selected CI procedures [26].