Size consistency and size extensivity

In quantum chemistry, size consistency and size extensivity are concepts relating to how the behaviour of quantum-chemistry calculations changes with the system size. Size consistency (or strict separability) is a property that guarantees the consistency of the energy behaviour when interaction between the involved molecular subsystems is nullified (for example, by distance). Size extensivity, introduced by Bartlett, is a more mathematically formal characteristic which refers to the correct (linear) scaling of a method with the number of electrons.^[1]

Let A and B be two non-interacting systems. If a given theory for the evaluation of the energy is size-consistent, then the energy of the supersystem A + B, separated by a sufficiently large distance such that there is essentially no shared electron density, is equal to the sum of the energy of A plus the energy of B taken by themselves: ${\displaystyle E(A+B)=E(A)+E(B).}$ This property of size consistency is of particular importance to obtain correctly behaving dissociation curves. Others have more recently argued that the entire potential energy surface should be well-defined.^[2]

Size consistency and size extensivity are sometimes used interchangeably in the literature. However, there are very important distinctions to be made between them.^[3] Hartree–Fock (HF), coupled cluster, many-body perturbation theory (to any order), and full configuration interaction (FCI) are size-extensive but not always size-consistent. For example, the restricted Hartree–Fock model is not able to correctly describe the dissociation curves of H₂, and therefore all post-HF methods that employ HF as a starting point will fail in that matter (so-called single-reference methods). Sometimes numerical errors can cause a method that is formally size-consistent to behave in a non-size-consistent manner.^[4]

Core extensivity is yet another related property, which extends the requirement to the proper treatment of excited states.^[5]

References

1. Bartlett, R. J. (1981). "Many-Body Perturbation Theory and Coupled Cluster Theory for Electron Correlation in Molecules". Annual Review of Physical Chemistry. 32: 359. Bibcode:1981ARPC...32..359B. doi:10.1146/annurev.pc.32.100181.002043.
2. Taylor, P. R. (1994). "Coupled-cluster Methods in Quantum Chemistry". In Roos, Björn O. (ed.). Lecture Notes in Quantum Chemistry: European Summer School. Lecture Notes in Chemistry. Vol. 64. Berlin: Springer-Verlag. pp. 125–202. doi:10.1007/978-3-642-57890-8_3. ISBN 978-3-642-57890-8.
3. "Size-Extensivity and Size-Consistency". Uam.es. 1995-01-20. Archived from the original on 2017-06-06. Retrieved 2014-02-01.
4. Van Dam, Huub; Van Lenthe, Joop; Pulay, Peter (1998). "The size consistency of multi-reference Møller–Plesset perturbation theory". Molecular Physics. 93 (3): 431. Bibcode:1998MolPh..93..431V. doi:10.1080/002689798169122.
5. Mukhopadhyay, S.; Chaudhuri, Rajat; Mukhopadhyay, Debasis; Mukherjee, Debashis (1990). "A comparative study of core-extensive and core—valence-extensive coupled-cluster theories for energy differences: Excitation energies". Chemical Physics Letters. 173 (2–3): 181. Bibcode:1990CPL...173..181M. doi:10.1016/0009-2614(90)80074-N.