Two important concepts in electronic structure theory are *
size-consistency* and * size-extensivity*. Though these terms are
sometimes used interchangeably in the literature, there are very important
distinctions to be made between them.

There are two primary definitions of * size-consistency* in use. The
first was employed by Pople
as one criterion for a well-constructed quantum chemical method. If we
imagine two molecules, separated by a large distance (large enough
that we may consider them to be non-interacting) then the energy calculated
for both molecules simultaneously should be exactly twice that calculated
for only one, isolated molecule of , just like the exact energy. This
``non-interacting limit'' description is the original concept of
size-consistency. From this perspective, size-consistency describes what
has been referred to as the ``additive separability'' of the wavefunction. However, a more recently imposed
definition[27] requires that the method not only correctly
describe the fragmentation limit, but the entire process (in a qualitative
sense). That is, the entire potential energy curve mapped out when we
bring our two non-interacting molecules close together must be
correctly described as well. For example, both spin-unrestricted
Hartree-Fock (UHF) and
spin-restricted Hartree-Fock (RHF)
wavefunctions are size-consistent for the separated dimer system
described above. However, for a closed-shell molecule dissociating into
open-shell fragments, a RHF wavefunction does not conform to the second
definition of size-consistency, as we will discuss further below.

* Size-extensivity*, on the other hand, is a more mathematically formal
characteristic which refers to the correct (linear) scaling of a method
with the number of electrons. The term was introduced to electronic
structure theory by Bartlett[25]
, and
is based on analogous ``extensive'' thermodynamic properties. All
Hartree-Fock methods qualify as size-extensive,
as well as many-body perturbation theory and coupled-cluster theories . Truncated configuration interaction methods, however, are not size-extensive.
An important advantage of a size-extensive method is that it allows
straightforward comparisons between calculations involving variable numbers
of electrons, e.g. ionization processes or calculations using different
numbers of active electrons. Lack of size-extensivity implies that errors
from the exact energy increase as more electrons enter the calculation.

Size-extensivity and size-consistency are not mutually exclusive
properties, by any means. At the non-interacting limit, size-extensivity
of a method is a necessary and sufficient condition to ensure
size-consistency, implying that the former is more general than the latter.
However, size-extensivity does * not* ensure correct fragmentation. For
example, we may consider two different fragmentation processes for :

and

The first process is correctly described by both RHF and UHF wavefunctions,
and hence, both methods are size-consistent. However, the second process
is not correctly described by a RHF wavefunction (and, therefore,
perturbation theory and coupled-cluster theory methods which use this as a
reference will not be size-consistent.) Both RHF and UHF are * always*
size-extensive, though. This implies, then, that size-consistency is more
general than size-extensivity, but this is also incorrect. At
non-interacting limits, size-extensivity is a more general property, and
its existence implies that of size-consistency. However, size-consistency
has the additional requirement of correct fragmentation that is not
necessarily dependent on the mathematical scaling of the energy. Chapter 1
of the review by Taylor[27]
deals explicitly with these
concepts, and the interested reader is urged to study this reference.

Last modified: Fri Jan 19 17:13:01 EST 1996