physics

Size consistency and size extensivity

Size consistency and size extensivity is a physics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Size consistency and size extensivity rather than just read about it. In short: 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).

Key takeaways

Size consistency and size extensivity belongs to physics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Size consistency and size extensivity to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Size consistency and size extensivity from memory before moving on to harder problems.

Reference excerpt

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. 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: E ( A + B ) = E ( A ) + E ( B ) . {\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. 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 (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. Core extensivity is yet another related property, which extends the requirement to the proper treatment of excited states.

References

Worked examples

Example 1 — a first encounter with Size consistency and size extensivity

Start with the simplest possible case. Write down what Size consistency and size extensivity claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In physics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Size consistency and size extensivity before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Size consistency and size extensivity ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Size consistency and size extensivity

In research: Size consistency and size extensivity appears in physics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Size consistency and size extensivity in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Size consistency and size extensivity is common in secondary-school and first-year university syllabi. It links to neighbouring topics Quantum chemistry, Quantum chemistry stubs, so understanding it makes those chapters shorter.
In everyday life: Look for Size consistency and size extensivity outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Size consistency and size extensivity in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Size consistency and size extensivity means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Size consistency and size extensivity out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Size consistency and size extensivity in simple terms?

Why does Size consistency and size extensivity matter?

Because it connects several physics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Size consistency and size extensivity?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Size consistency and size extensivity.