to obtain a relatively compact form of (27) and (28). They are defined as follows

$$\begin{array}{ll}
\underline{\underline{a}}_{I,\alpha ,\sigma }= & \underline{a}_{I,\alpha
,\sigma }+\beta \left( \underline{b}_{I,\alpha ,\sigma }+a_{I,\alpha
,-\sigma }^{+}a_{I,\overline{\alpha },-\sigma }b_{I,\overline{\alpha }%
,\sigma }\right) \\ 
& +\delta \left[ \underline{b}_{I,\overline{\alpha },\sigma }+a_{I,\overline{%
\alpha },-\sigma }^{+}a_{I,\alpha ,-\sigma }b_{I,\alpha ,\sigma }+a_{I,%
\overline{\alpha },\sigma }\frac{n_{I,\alpha }^a}2\right]
\end{array}$$

where

$$\underline{a}_{I,\alpha ,\sigma }=a_{I,\alpha ,\sigma }\left( 1-\frac \beta
2n_{I,\overline{\alpha }}^a-\frac 12n_{I,\overline{\alpha }}^b\right) ,$$

$$\underline{b}_{I,\alpha ,\sigma }=b_{I,\alpha ,\sigma }\left( 1-n_{I,%
\overline{\alpha }}^a-\frac 12n_{I,\overline{\alpha }}^b\right)$$

and

$$\beta =\frac 14\left( 1-\frac 1{\sqrt{1+\left( \frac{4t}U\right) ^2}}\right)
,$$

$$\delta =\frac{t}{U}\frac 1{\sqrt{1+\left( \frac{4t}U\right) ^2}}.$$

To write down the explicit form of the Hubbard Hamiltonian $\overline{%
\overline{H}}$ (see (24)) we have to insert (27) and (28) into $\overline{%
\overline{H}}$. This operation leads, however, to a very complicated form of $\overline{\overline{H}},$ given in the Appendix C (a simplified form of this Hamiltonian is discussed in the next Section). Here again the only assumption made to derive $\overline{\overline{H}}$ was the reduction of the whole dimer space (A.1) to the subspace of the dimer eigenvectors $\left|
E_0\right\rangle $, $\left| E_{11}\right\rangle $, $\left|
E_{12}\right\rangle $, $\left| E_{13}\right\rangle $, $\left|
E_{14}\right\rangle $, $\left| E_{21}\right\rangle $, $\left|
E_{24}\right\rangle $, $\left| E_{25}\right\rangle $ and $\left|
E_{26}\right\rangle $, corresponding to the lowest lying 