Size-extensivity of linear variational wave functions

2 SIZE-EXTENSIVITY OF LINEAR VARIATIONAL WAVE FUNCTIONS 

Size-extensivity holds trivially for exact wave functions. For approximate wave functions, however, size-extensivity is not always observed. We now examine size-extensivity for the linear variational model of Section 4.2.3. We shall find that, for this simple model, size-extensivity may be imposed by a careful construction of the variational space for the compound wave function. For ease of presentation, we assume that aU wave functions are real. 

We begin our discussion of size-extensivity by considering the fragments. The fragment wave functions are written in the form 

Let us now consider the compound system. We assume that the compound wave function is written as a linear combination of determinants in the direct-product space of the fragment spaces 

The compound wave function is thus determined in exactly the same manner as the fragment wave functions but in the direct-product space of the fragment spaces. We shall now demonstrate thaL for noninteracting systems A and B, the variationally optimized energy of the compound system in the direct-product space is equal to the sum of the energies of the fragments 

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