Size-extensivity of exact wave functions

Consider a system of two noninteracting molecular fragments A and B. For such a system, the Hamiltonian operator may be written in the form 

Strictly speaking, the Hamiltonian (4.3.1) is an idealized operator that cannot be realized in practice, but - in the limit of an infinite separation between the fragments - it provides an exact representation of the true operator. 

The exact solution for the combined system represented by the Hamiltonian (4.3.1) satisfies the Schrodinger equation 

We shall consider how this solution is related to the exact solutions for the fragments 

The relations (4.3.8) and (4.3.9) constitute the requirements for size-extensivity. The wave function is said to be multiplicatively separable and the energy additively separable. 

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