This procedure can easily be performed when acting with the annihilation operators on the basis vectors (6) and using the reciprocal relations to (A.1). In this way we can obtain the dimer representation of the annihilation (creation) operators, given in Appedix B. After this operation we can insert so prepared $c_{I,1(2),\sigma }$ $%
(c_{I,1(2),\sigma }^{+})$ into the Hubbard Hamiltonian (4) to obtain the Hubbard model in the dimer representation for the sc lattice. This representation will be used later to derive the Hubbard Hamiltonian in the large $U$ limit (see Sec. 4).

Hubbard dimer for large $U$
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Looking at the eigenvalues (A.1) of the Hubbard dimer it is easy to see that for large $U$ $(U\gg t)$ the energies $E_{\alpha \text{ }%
}=E_{22},E_{23},E_{31},E_{32},E_{33},E_{34}$ and $E_4$ take on large, positive values, producing in the partition function the terms which can practically be neglected. It means that the mentioned energies are not occupied in the reasonable temperature range ($1$ $eV\sim 11604.5$ $K$) and can be excluded from our considerations. Therefore the dimer Hamiltonian, given by (7), reduces to

$$\overline{H}^d=-tP_{11,11}+tP_{12,12}-tP_{13,13}+tP_{14,14}+\left( -C+\frac
U2\right) P_{24,24}.$$

To bring this expression into a compact (second quanization) form we introduce the Hubbard operators

$$\begin{aligned}
a_{i,\sigma } &=&c_{i,\sigma 