the considerations unoccupied dimer energy levels in the large $U$ limit. Let us note that using the Taylor expansion in (14) with respect to $x=\frac{t}{U}$ and retaining the terms proportional to $\frac{t^2}U$ we obtain

$$\overline{H}_I^d=-t\sum\limits_\sigma [a_{I,1,\sigma }^{+}a_{I,2,\sigma
}+a_{I,2,\sigma }^{+}a_{I,1,\sigma }]+\frac{4t^2}U[\overrightarrow{S}%
_{I,1}\cdot \overrightarrow{S}_{I,2}-\frac{n_{I,1}^an_{I,2}^a}4].$$

The formula (15) is the well-known $t-J$ model for the Hubbard dimer where the first part in (15), similarly to (14), represents the exact form of the dimer Hamiltonian (5) in the limit $U\rightarrow \infty $.

The same result (14) can also be obtained applying a more general approach which will be used later to derive the Hubbard Hamiltonian for large $U$ in the case of a crystal. Let us note that after elimination of the unoccupied levels the subspace of the eigenvectors of the Hubbard dimer (A.1) for large $U$ consists of the following eigenvectors: $|E_0\rangle $, $|E_{11}\rangle $, $|E_{12}\rangle $, $|E_{13}\rangle $, $|E_{14}\rangle $, $|E_{21}\rangle $, $|E_{24}\rangle $, $|E_{25}\rangle $, and $|E_{26}\rangle $. It means that we can define a projection operator onto this subspace

$$\begin{array}{ll}
P_I= & 
P_{0,0}^{(I)}+P_{11,11}^{(I)}+P_{12,12}^{(I)}+P_{13,13}^{(I)}+P_{14,14}^{(I)}
\\ 
& +P_{21,21}^{(I)}+P_{24,24}^{(I)}+P_{25,25}^{(I)}+P_{26,26}^{(I)}
\end{array}$$

which, in the second quantization form, reads

$$\begin{array}{ll}
P_I= & 1-\frac 12(n_{I,1}^b+n_{I,2}^b)+\frac 14n_{I,1}^bn_{I,2}^b \\ 
&  \\ 
& 
\begin{array}{ll}
& +\frac 18(1-\frac 1{\sqrt{1+(\frac{4t}U)^2}})[4(\overrightarrow{S}%
_{I,1}\cdot 