technique onto the lowest lying dimer states we derive in Sec. 4 the Hubbard Hamiltonian for the crystal in this limit (central formula of this paper). A simplified version of the derived Hamiltonian with the use of the Taylor expansion with respect to $x=\frac{t}{U}$ $(x\ll 1)$ and the case $U\rightarrow \infty $ is discussed in Sec. 5.

Exact solution of the Hubbard dimer
===================================

The eigenvalues and eigenvectors of the dimer Hamiltonian (5) can be found by using a standard procedure (cf Refs \[33-35\]). We start with the vectors $%
|n_{1,\uparrow },n_{1,\downarrow };n_{2,\uparrow },n_{2,\downarrow }\rangle $ $(n_{i,\sigma }=0,1;$ $i=1,2$; $\sigma =\uparrow ,\downarrow )$ which constitute the Fock basis of the single-dimer space of states:

$$\begin{array}{lll}
\begin{array}{l}
\begin{array}{l}
|0\rangle =|0,0;0,0\rangle ,
\end{array}
\\ 
\\ 
\begin{array}{l}
|11\rangle =|1,0;0,0\rangle , \\ 
|12\rangle =|0,1;0,0\rangle , \\ 
|13\rangle =|0,0;1,0\rangle , \\ 
|14\rangle =|0,0;0,1\rangle ,
\end{array}
\end{array}
& 
\begin{array}{l}
|21\rangle =|1,1;0,0\rangle , \\ 
|22\rangle =|1,0;1,0\rangle , \\ 
|23\rangle =|1,0;0,1\rangle , \\ 
|24\rangle =|0,1;1,0\rangle , \\ 
|25\rangle =|0,1;0,1\rangle , \\ 
|26\rangle =|0,0;1,1\rangle ,
\end{array}
& 
\begin{array}{l}
\begin{array}{l}
|31\rangle =|0,1;1,1\rangle , \\ 
|32\rangle =|1,0;1,1\rangle , \\ 
|33\rangle =|1,1;0,1\rangle , \\ 
|34\rangle =|1,1;1,0\rangle ,
\end{array}
\\
\\ 
\begin{array}{l}
|4\rangle =|1,1;1,1\rangle .
\end{array}
\end{array}
\end{array}$$

Starting with the vectors (6) 