Reflections on size-extensivity, size-consistency and generalized extensivity in many-body theory

An overview is presented of the notions of size-extensivity and size-consistency, which play an important role in the discussion of many-body methods in physics and chemistry. We also introduce the concept of generalized extensivity, which has been used before implicitly in the literature, and provide an operational definition, which can be tested numerically in a similar way as size-consistency. A numerical example illustrates the concept of generalized extensivity as applied to the EOM-CCSD, STEOM-CCSD and extended-STEOM-CCSD electronic structure methods for excited states. In another line of thought it is argued that algebraic proofs are often more easily constructed than diagrammatic proofs to demonstrate proper separation into non-interacting fragments. The algebraic line of reasoning is illustrated for single reference methods and applied to discuss separability properties for a variety of Hilbert space multi-reference coupled cluster methods.