}(1-n_{i,-\sigma }), \\
b_{i,\sigma } &=&c_{i,\sigma }n_{i,-\sigma }\end{aligned}$$

and spin operators

$$\begin{aligned}
S_i^z &=&\frac 12(n_{i,\uparrow }-n_{i,\downarrow })=\frac 12(n_{i,\uparrow
}^a-n_{i,\downarrow }^a), \\
S_i^{+} &=&c_{i,\uparrow }^{+}c_{i,\downarrow }=a_{i,\uparrow
}^{+}a_{i,\downarrow }, \\
S_i^{-} &=&c_{i,\downarrow }^{+}c_{i,\uparrow }=a_{i,\downarrow
}^{+}a_{i,\uparrow }\end{aligned}$$

where $n_{i,\sigma }^a=a_{i,\sigma }^{+}a_{i,\sigma }$ $(i=1,2;$ $\sigma
=\uparrow $, $\downarrow ).$

With the use of (9)-(13) the Hamiltonian of the Hubbard dimer (8) for large $%
U$ can be presented in the form (we introduce the omitted earlier dimer index $I$)

$$\begin{aligned}
\overline{H}_I^d &=&-t\sum\limits_{\sigma} [a_{I,1,\sigma }^{+}a_{I,2,\sigma
}+a_{I,2,\sigma }^{+}a_{I,1,\sigma }]  \nonumber \\
&&  \nonumber \\
&&+\frac{4t^2}{U\sqrt{1+(\frac{4t}U)^2}}[\overrightarrow{S}_{I,1}\cdot 
\overrightarrow{S}_{I,2}-\frac{n_{I,1}^an_{I,2}^a}4]  \nonumber \\
&&  \nonumber \\
&&+\frac{t^2(1-\sqrt{1+(\frac{4t}U)^2})}{U(1+\sqrt{1+(\frac{4t}U)^2})\sqrt{%
1+(\frac{4t}U)^2}}[2(b_{I,1,\uparrow }^{+}a_{I,1,\downarrow
}^{+}a_{I,2,\downarrow }b_{I,2,\uparrow }  \nonumber \\
&&  \nonumber \\
&&+b_{I,2,\uparrow }^{+}a_{I,2,\downarrow }^{+}a_{I,1,\downarrow
}b_{I,1,\uparrow }) \\
&&  \nonumber \\
&&+(1-n_{I,1}^a)n_{I,2}^b+(1-n_{I,2}^a)n_{I,1}^b-n_{I,1}^bn_{I,2}^b] 
\nonumber \\
&&  \nonumber \\
&&+\frac{t(1-\sqrt{1+(\frac{4t}U})^2)}{2\sqrt{1+(\frac{4t}U)^2}}%
\sum\limits_{\sigma} \sum\limits_{\alpha =1,2}[a_{I,\alpha ,\sigma }^{+}b_{I,%
\overline{\alpha },\sigma }+b_{I,\alpha ,\sigma }^{+}a_{I,\overline{\alpha }%
,\sigma }]  \nonumber\end{aligned}$$

where $n_{I,\alpha }^{a,b}=\sum\limits_{\sigma} n_{I,\alpha ,\sigma }^{a,b},$ $n_{I,\alpha ,\sigma }^b=b_{I,\alpha ,\sigma }^{+}b_{I,\alpha ,\sigma
}=n_{I,\alpha ,\sigma }n_{I,\alpha ,-\sigma }$ $(\alpha =1,2)$, $\overline{%
\alpha }=1$ when $\alpha =2$ and $\overline{\alpha }=2$ when $\alpha =1$. It is very important to stress that the formula (14) has been obtained in a nonperturbative way, starting from the exact form of the dimer Hamiltonian (see (5) or (7)), and excluding from 