Truncated CI is not Size Extensive

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 *E*_{A} + *E*_{B} 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,S_{A}andS_{B}, withN_{A}andN_{B}electrons, respectively. Each system is described by its own function, antisymmetric inN_{A}particles and inN_{B}. Assuming that both functions are normalized to unity it is easy to show that the product function 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

(tex2html_deferredtex2html_deferred4.tex2html_deferred27 E_DC = ( 1 - _i Ref |c_i|^2 ) (E_MRCI - E_MR) ) |

where

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].