Relation between the Anderson and s-d Models (Schrieffer-Wolff Transform)

Write the total wave function as the sum of and , where the subscripts refer to the occupation number. In this formalism the Schrodinger equation can be expressed as where and is the projection operator on the different sub-space of occupation , The matrix elements are If we consider only the local moment limit, that and are eliminated, and from the equation we have Then we have The second term on the left side is Thus we obtain which becomes the first three terms in the s-d exchange model, considering that which hold in the subspace. Similarly we can calculate Take the lowest order in , the equivalent effective exchange coupling is together with a potential scattering term where which is valid for and .