we easily get the eigenvalues $E_\alpha $ and the eigenvectors $|E_{\alpha} \rangle $ of the Hubbard dimer (5). Here, we only mention that the space of 16 eigenvectors $|E_{\alpha} \rangle $ can be devided into some subspaces numbered by $n=\sum\limits_{i,\sigma
}n_{i,\sigma }$. The subspace, belonging to $n=0$ and $n=4$ is 1-dimensional ($E_0=0,$ $|E_0\rangle =|0\rangle ;$ $E_4=2U,$ $|E_4\rangle =|4\rangle )$. There are, however, two 2-dimensional subspaces corresponding to $n=1$ (as e.g. $E_{11}=-t,$ $|E_{11}\rangle =\frac 1{\sqrt{2}}(|11\rangle +|13\rangle
),$ etc.) and two 2-dimensional subspaces corresponding to $n=3$ (as e.g. $%
E_{31}=t+U,$ $|E_{31}\rangle =\frac 1{\sqrt{2}}(|31\rangle +|33\rangle ),$ etc.). The subspace, belonging to $n=2$ consists of one 4-dimensional subspace (as e.g. $E_{21}=0,$ $|E_{21}\rangle =\frac 1{\sqrt{2}}(|23\rangle
+|24\rangle ),$ etc.) and two 1-dimensional subspaces ($E_{25}=E_{26}=0,$ $%
|E_{25}\rangle =|22\rangle ,$ $|E_{26}\rangle =|25\rangle $). The complete set of the eigenvalues $E_{\alpha} $ and eigenvectors $|E_{\alpha} \rangle $ of the Hubbard dimer is given in the Appendix A. It allows to express the dimer Hamiltonian (5) in the equivalent form

$$H^d=\sum\limits_{\alpha} E_{\alpha} |E_{\alpha} \rangle \langle E_{\alpha} |.$$

The next important step in our calculations is the possibility to express the annihilation operators $c_{1(2),\sigma }$ as linear combinations of transition operators between dimer states $P_{\alpha ,\beta }=|E_{\alpha}
\rangle \langle E_{\beta} |.$ 