Coupled Cluster Methods
Size Consistency and Size Extensivity
The full CI method is the matrix mechanics solution of the Schrödinger equation in a given finite basis set, and therefore, shares the properties of size extensivity and size consistency with the exact complete CI method. In order for a method to be classified as being size extensive, the energy calculated thereby must scale linearly with the number of particles. In other words, the term extensive is used in the same context as in classical thermodynamics. SCF and CC methods are also size extensive. Size consistency is a more difficult term which has suffered from a loose definition in the literature. A method is described as size consistent if the energy of two well separated (i. e. in the limit of infinite separation) subsystems A and B is equal to EA + EB, the sum of the energies of the two systems computed independently. In addition, the term size consistency typically implies that the method must predict a qualitatively correct dissociation curve. CC methods based on size consistent reference and the unrestricted SCF method are both size consistent. Restricted SCF methods cannot produce a correct dissociation curve when the resulting fragments are open-shel, and, thus, are not size consistent in these cases.
As mentioned above full CI is size extensive and size consistent, but
most truncated CI methods are neither. In order to illustrate this point,
consider the energy of two hydrogen molecules at infinite separation as
treated by the CISD method. The energy of the two molecules computed separately
using this method is not equal to the energy of the two molecules taken
together in a ``super molecule'' calculation because the former can include
up to double excitations on each molecule, which is equivalent to including
up to quadruple excitations in the latter. Because truncated CI's are
found to lack the property of size extensivity and will, therefore, decrease
in accuracy with increasing system size, numerous techniques have been
used to adjust truncated CI energies to make them size extensive. The
easiest and most common treatment for correcting the CI energy is the
Davidson correction. The single[14] and multi-reference[15] forms of the Davidson correction are
 |
= | (1-c02)ESD | (49) |
 |
= |  |
(50) |
In the single reference equation C0 is the CI coefficient of the reference and ESD is the correlation energy recovered by the CISD procedure. In the multi-reference equation things become more complicated as the CI coefficient of the reference is replaced by a sum over the CI coefficients of the various references and ESD is replaced by the difference between the full MRCI energy and a CI performed with only the reference, EMR.