sizeextensivity

Sizeextensivity, sometimes written as size-extensivity, is a property of a physical quantity or a computational method in which the quantity scales linearly with the size of the system. In particular, for two non-interacting subsystems A and B, a size-extensive quantity Q satisfies Q(A+B) = Q(A) + Q(B). Equivalently, for a homogeneous system, Q(N) ∝ N, meaning the value grows proportionally with the number of constituents.

In quantum chemistry and many-body theory, sizeextensivity is crucial for reliable scaling to larger systems. Energy is the prototypical example: a size-extensive method yields an energy for a system of two well-separated fragments that equals the sum of the fragment energies. This ensures correct dissociation behavior and predictable results as system size increases. Full configuration interaction (FCI) and many coupled-cluster methods are size-extensive, while truncated configuration interaction methods (for example, CISD) are not.

Sizeextensivity is related to, but distinct from, size-consistency. Both concern how a method handles non-interacting subsystems, but size-consistency focuses on exact additivity of energies across subsystems, whereas size-extensivity emphasizes proper linear scaling with system size in general. In practice, non-size-extensive methods can yield energies that do not grow proportionally with system size, leading to errors in large systems and in reaction-energy calculations.

Overall, sizeextensivity is a foundational criterion in evaluating electronic-structure methods, ensuring scalable, additive, and physically reasonable results as systems grow or are divided into fragments.