Space size extensivity corrections

Simons, J., Size extensivity correction for complete active space multiconfiguration self-consistent-field configuration interaction energies, J. Phys. Chem. 93, 626-627 (1989). 

These single reference-based methods are limited to cases where the reference function can be written as a single determinant. This is most often not the case and it is then necessary to use a multiconfigurational approach. Multrreference Cl can possibly be used, but this method is only approximately size extensive, which may lead to large errors unless an extended reference space is used. For example, Osanai et al. [8] obtained for the excitation energy in Mn 2.24 eV with the QCISD(T) method while SDCI with cluster corrections gave 2.64 eV. Extended basis sets were used. The experimental value is 2.15 eV. 

The results for N2 are summarized in Table 3. The bond is streched from 2.15 a.u. up to 3.00 a.u. Opposite to the previous molecules, even at the equilibrium geometry CCSD does not perform as well as the other approaches. For the [6e/6o] active space, it differs from FCI by 12.9 mHa, while CAS-SDCI does it by 8.3 mHa, (SCf CAS-SDCI by 3.9 mHa and ec-CCSD by 2.0 mHa or 1.8 mHa, depending on the external correcting source. Again, one may observe that the minimal dressing of CAS-SDCI not only makes it size-extensive, but also improves the absolute value of the yielded energy. 

The majority of systems studied for the importance of core correlation have employed the SCF-based size extensive methods because they are computationally less expensive. The ACPF calculations are considerably more expensive and it is often difficult to demonstrate convergence with expansion of the active space. The development of efficient open-shell CCSD(T) algorithms has largely superseded the MCPF and CPF methods because the triples correction is found to be important. For N2, the triples correction increases the CCSD core contribution by 40% giving good agreement between the CCSD(T) and ACPF results. As the carbon atom results illustrate, even when there are small errors in the valence... 

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