The short range order problem

E the energy eigenvalues of the Hamiltonian then writes the Hamiltonian as the sum of 

This expression will be used in Chapter 7 to calculate magnetic interactions and to discuss quantum corrals. 

Now it is not too hard to find the cluster interactions. They are found by the expression [113]  

To get the GPM effective interaction we expand the logarithm in Eq.(5.23), assuming that St r is small, and keep the interactions only the lowest order of scattering on each cluster  

In the limit of c — 0(1), this takes into account the higher order scattering, but overestimates these interaction around c = 0.5. A better method is to just calculate the energy difference AABB — ABAB in Eq.(5.22). 

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