$P^{(1)}\overline{\overline{H}}^{(1)}$, contains the terms proportional to $t$ whereas the term in the parentheses is proportional to $%
\frac{t^2}U$ (see (35) and (42)). In the expression for $\overline{\overline{%
H}}^{(2)}$(see (42)) the first term describes the antiferromagnetic, Heisenberg intradimer interaction (see also the second term in (15)). The apearence of this term may suggest that such an interaction should also arise between different dimers. This is of course the case. However, because of applied procedure such terms do not appear explicitly. In the last analysis our method treats all interactions within and between dimers, on the same quality level, i.e. all interactions are taken into account. The magnetic interdimer interactions, represented by the fourth and fifth term in (42), have formally the same structure as the Heisenberg interactions but instead of the scalar products of spin operators there are the products of spin operators and ”hopping spin” operators (defined in C.1). All the terms presented in (42) are correct because they originate from the exact decomposition of the Hubbard Hamiltonian (1) into a set of interacting dimers (4) where each dimer problem has been exactly solved (exact dimer representation of the construction operators after applying the projection procedure, given by (27) and 