Size consistency and size extensivity

In quantum chemistry, size consistency (or strict separability) is a property that guarantees the consistency of the energy behavior when interaction between the involved molecular system is nullified (for example, by distance).

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 is equal to the sum of the energy of A plus the energy of B taken by themselves ($E(A-B)=E(A)+E(B)$). This property 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.

An other important concept in electronic structure theory is size extensivity. Size consistency and size extensivity are sometimes used interchangeably in the literature, However there are very important distinctions to be made between them. Hartree-Fock, coupled cluster, many-body perturbation theory (to any order), and full configuration interaction (CI) are size extensive but not always size consistent. For example the Restricted Hartree-Fock model is not able to described correctly the dissociation curves of H2 and therefore all post HF methods will fail in that matter (except FCI). Sometimes numerical errors can cause a method that is formally size-consistent to behave in a non-size-consistent manner.

Size-extensivity, is a more mathematically formal characteristic which refers to the correct (linear) scaling of a method with the number of electrons.

Core-extensivity is yet another related property, which extends the requirement to the proper treatment of excited states.