3. Size-consistency, size-extensivity, and strict separability¶

Those three concepts are used to qualify and differentiate quantum chemistry models from one another. A lot of confusion exists in the literature and it is thus important to explain what is meant when using those words.

3.1. Size-consistency¶

The first concept, that we will refer to as size-consistency, has to do with the ability of a model to properly describe the entire potential energy surface of a system. For example at the equilibrium geometry but also when all the elements are far apart as well as the intermediate regions (bond-breaking processes). The size-consistency concept (as defined here) is thus more of a system dependent problem than a property of an approximate quantum chemistry model. Of course the FCI solutions are always size-consistent. However, HF is not always size-consistent (e.g. hydrogen dissociation) and thus all approximate post-HF methods will inherit from this issue. When HF is failing, one should instead consider multi-configuration and multi-reference methods to provide a size-consistent description of the system under consideration.

Note

Unfortunately, the term size-consistent is also often used to describe the concept explained below (size-extensivity, and strict separability) which are related but fondamentally different. This leads to a lot of confusion…

3.2. Size-extensivity¶

This term was introduced in analogy to size-extensive properties in thermodynamics. A quantum chemistry model (like CC, and FCI) should provide size-extensive energies in the sense that the energy should grow linearly with the number of electrons in the system.

3.3. Strict separability¶

This is the easiest concept to define clearly and the one we are really interested in here.

The idea is that if a system is composed of non-interacting fragments A and B, such that

\[H^{AB} | \Psi^{AB} \rangle = E^{AB} | \Psi^{AB} \rangle\]

\[H^{A} | \Psi^{A} \rangle = E^{A} | \Psi^{A} \rangle, \qquad \text{and} \qquad H^{B} | \Psi^{B} \rangle = E^{B} | \Psi^{B} \rangle,\]

and such that we can split the Hamiltonian of the total system as

\[H^{AB} = H^A + H^B,\]

then, a wave-function model that is strictly separable will fulfill the following conditions on the resulting wave-function and energy.

\[| \Psi^{AB} \rangle = | \Psi^{A} \rangle | \Psi^{B} \rangle \qquad \text{and} \qquad E^{AB} = E^{A} + E^{B}\]

The key here is to understand that when talking about non-interacting system we mean that the total Hamiltonian can be decomposed into the sum of the Hamiltonian of the fragments.

Warning

If one is to compute the CC energy of, e.g., a water molecule for a very large H–O bond length (e.g. 20 Angstroms), the relation above does not hold unless we specify excplicitly to the computer program that the Hamiltonian should be written as two independent contributions. Otherwise, the CC energy of the total system will suffers from the lack of size-consitency of the HF solution. CC energies are therefore not always size-consistent (as defined above).

The size-extensivity and strict separability properties described here are mathematical properties and one should be careful when testing those properties numerically.

3.4. Why do we want size-extensivity?¶

So why is it important for a good quantum chemistry model to satisfy the strict separability and the size-extensivity conditions above?

The first and most important reason is that those properties are properties of the exact solution to the electronic Schrödinger equation. For the other reasons, I cannot put it better than that:

“An important advantage of a size-extensive method is that it allows straightforward comparisons between calculations involving variable numbers of electrons, e.g. ionization processes or calculations using different numbers of active electrons. Lack of size-extensivity implies that errors from the exact energy increase as more electrons enter the calculation.”

Taken from: “Size-Extensivity and Size-Consistency” (1996) by T. Daniel Crawford.

One of the main troubles of truncated CI models is that they do not satisfy the strict separability and size-extensivity properties. On the other hand, truncated CC methods do satisfy those properties. In the appendix CI and CC for the hydrogen dimer we propose an illustration of the strict separability problem for a dimer of hydrogen in a minimal basis.

Note

HF and MP theories also satisfy the strict separability and size-extensivity requirements.