lying dimer states with the use of the projection operator

$$P=P_1P_2...P_M$$

where $P_{I}$ is given by (16) or (17). In analogy to (14) we denote the Hubbard Hamiltonian in the large $U$ limit by $\overline{H}$. Similar to (18) we write

$$\overline{H}=PHP$$

and instead of the form (21) for the Hamiltonian $H$ we can use (4). Taking into account that $P^2=P$ $(P_I^2=P_I,$ $\left[ P_I,P_J\right] =0)$ we obtain

$$\begin{array}{ll}
\overline{H}= & P[\sum\limits_I\overline{H}_I^d-t\sum\limits_{I,\sigma
}\left( \overline{c}_{I,2,\sigma }^{+}\overline{c}_{I+1,1,\sigma }+\overline{%
c}_{I+1,1,\sigma }^{+}\overline{c}_{I,2,\sigma }\right) \\ 
& -t\sum\limits_{I\neq J,\sigma }\left( \overline{c}_{I,1,\sigma }^{+}%
\overline{c}_{J,1\sigma }+\overline{c}_{I,2,\sigma }^{+}\overline{c}%
_{J,2,\sigma }\right) ]\equiv P\overline{\overline{H}}
\end{array}$$

where $\overline{H}_I^d$ is given by (14) and $(\alpha =1,2)$

$$\overline{c}_{I,\alpha ,\sigma }=c_{I,\alpha ,\sigma }P_I,$$

$$\overline{c}_{I,\alpha ,\sigma }^{+}=P_Ic_{I,\alpha ,\sigma }^{+}.$$

Applying the projection operator (16) or (17) to (B.1) - (B.4) and introducing Hubbard- and spin operators (9) - (13) we obtain

$$\overline{c}_{I,1,\uparrow }=\underline{\underline{a}}_{I,1,\uparrow }+\beta
\left[ S_{I,2}^za_{I,1,\uparrow }+S_{I,2}^{-}a_{I,1,\downarrow }\right]
-\delta \left[ S_{I,1}^za_{I,2,\uparrow }+S_{I,1}^{-}a_{I,2,\downarrow
}\right] ,$$

$$\overline{c}_{I,1,\downarrow }=\underline{\underline{a}}_{I,1,\downarrow
}-\beta \left[ S_{I,2}^za_{I,1,\downarrow }-S_{I,2}^{+}a_{I,1,\uparrow
}\right] +\delta \left[ S_{I,1}^za_{I,2,\downarrow
}-S_{I,1}^{+}a_{I,2,\uparrow }\right]$$

where the corresponding expressions for $\overline{c}_{I,2,\sigma }$ ($%
\sigma =\uparrow ,\downarrow $) can easily be obtained by changing the internal dimer index $1\Leftrightarrow 2$ in (27) and (28).

The new operators $\underline{\underline{a}}_{I,\alpha ,\sigma }\left(
\alpha =1,2;\sigma =\uparrow ,\downarrow \right) $ are introduced 