# Which methods are size-consistent / size-extensive, and why?

This is a follow up question from another question I posted here. I am looking for an explanation (or a mathematical proof, if possible) of the size extensivity and size consistency of the following commonly used quantum chemistry methods:

1. Hartree-Fock (i.e. RHF, ROHF, UHF)
2. Density Functional Theory (do all functionals have the same property?)
3. Configuration interaction
4. Complete Active Space SCF (CASSCF)
5. Quadratic CISD, with perturbative triples i.e. QCISD(T)
6. MBPT and coupled cluster methods
7. Local correlation methods (i.e. DLPNO-CCSD, DLPNO-MP2 etc.)

What is the size consistency and size extensivity of each of these methods, and why?

## Configuration Interaction (CI)

A good explanation of size consistency/extensivity for CI is given in Chapter 4, section 6 [1], though note that they describe both properties as forms of size consistency. This answer mostly focuses on truncated CI, but its helpful to contrast its behavior with Full CI.

Full CI is size consistent and size extensive; this makes sense as it should exactly solve the Schrodinger equation (for the finite basis used) and real systems are size consistent/extensive. Truncated CI, however, is neither size consistent nor size extensive.

This is very easy to show for size consistency: consider the example of infinitely separated $$\ce{H2}$$ molecules in a minimal basis. For a single $$\ce{H2}$$, CISD is equivalent to Full CI (each molecule only has two electrons and a single unoccupied spatial orbital), so the energy we get from calculations on the separated molecules individually is just two times the Full CI energy of $$\ce{H2}$$. For the combined system, CISD is no longer equivalent to full CI: without quadruple excitations, we aren't including the case where both $$\ce{H2}$$ molecules are doubly excited. So the energy is not consistent when calculated as a supersystem vs separate monomer calculations. This same argument can be extended to any truncated CI in any basis.

The argument for truncated CI failing to be size extensive can be made very similarly. Instead of two $$\ce{H2}$$ monomers, we can consider a chain of $$N$$ of them. We still consider them all infinitely separated, though this is just so we can compare with the exact energy per monomer. [1] shows that CID energy of $$N$$ minimal basis $$\ce{H2}$$ scales as $$O(N^{1/2})$$, while size extensivity implies that the energy should scale as $$O(N)$$. In the limit as $$N\to\infty$$, this means that the energy per monomer in DCI will actually vanish, which is clearly unphysical.

References:

1. Szabo, Attila, and Neil S. Ostlund. Modern quantum chemistry : introduction to advanced electronic structure theory. Mineola, N.Y: Dover Publications, 1996. Print.

## Coupled Cluster and MBPT methods

The basis for this answer is the Brueckner-Goldstone linked-diagram theorem which applies to many-body perturbation theory (MBPT) and coupled-cluster methods. The original papers by Bruckner (1955) and Goldstrone (1957) are given in my answer explaining what about size-extensivity means.

The most important part to remember is that when using only linked diagrams, a method based on this diagramatic approach will be size-consistent. Size-inconsistency is also sometimes associated with the presence of "unlinked diagrams".

For closed-shell systems, RHF is size-consistent, and therefore size-extensivity of a method to treat correlation beyond RHF is sufficient for guaranteeing size-consistency of the post-RHF method. For systems that dissociate into open-shell fragments, a UHF reference might have spin contamination, so one must be careful when using UHF, but it's still possible to generate a linked-diagram expansion from a UHF reference.

MBPT and coupled cluster methods happen to both be linked-diagram expansions which can be made on top of either an RHF or UHF reference wavefunction, and therefore MBPT and coupled cluster methods derived in this way (e.g. CCSD and CCSD(T)) are size-extensive and therefore in this case, also size-consistent.

• So MBPT and Coupled Cluster are size-extensive as well? I looked at your linked answer, and it seems to suggest that, but I want to be sure. Jan 5 at 11:48
• Yes, size-extensivity is one of the major advantages of truncated CC vs truncated CI! Jan 5 at 13:06