also $\beta $ and $\delta $ (see (32), (33)). Such an expansion can be performed to any power of $x$, however, the most simple form we obtain when we restrict ourselves to the linear approximation, resulting in the terms proportional to $t$ and $\frac{%
t^2}U.$ The accuracy of this expansion can easily be verified assumming e.g. a typical value of the ratio $\frac{W}{U}=\frac 15$ (or less). Because the bandwidth of the conduction band for the sc lattice is $W=12t,$ it results in a small value of the expansion parameter $x=\frac tU=\frac 1{60}$ in this case. It, however, means that the linear approximation is quite reasonable because all higher terms in the expansion, proportional to $x^n$ ($%
n=2,3,...),$ produce $60$ times smaller contribution.

To present the results of the Taylor expansion including all the terms proportional to $t$ and $\frac{t^2}U$ let us first define several auxiliary quantities

$$P^{(1)}=\prod\limits_{I=1}^MP_I^{(1)},$$

$$P^{(2)}=\frac tU\sum\limits_{I=1}^MP_1^{(1)}\cdot ...\cdot
P_{I-1}^{(1)}P_I^{(2)}P_{I+1}^{(1)}\cdot ...\cdot P_M^{(1)}$$

where

$$P_I^{(1)}=1-\frac 12\left( n_{I,1}^b+n_{I,2}^b\right) +\frac
14n_{I,1}^bn_{I,2}^b,$$

$$P_I^{(2)}=\sum\limits_{\sigma} \sum\limits_{\alpha =1}^2\left( a_{I,\alpha
,\sigma }^{+}b_{I,\overline{\alpha },\sigma }+b_{I,\alpha ,\sigma }^{+}a_{I,%
\overline{\alpha },\sigma }\right)$$

and ($\alpha =1,$ $2$)

$$\widetilde{a}_{I,\alpha ,\sigma }=a_{I,\alpha ,\sigma }\left( 1-\frac 12n_{I,%
\overline{\alpha }}^b\right) ,$$

$$\widetilde{\widetilde{a}}_{I,\alpha ,\sigma }=\underline{b}_{I,\overline{%
\alpha },\sigma }+a_{I,\overline{\alpha },-\sigma }^{+}a_{I,\alpha ,-\sigma
}b_{I,\alpha ,\sigma }+a_{I,\overline{\alpha },\sigma }\frac{n_{I,\alpha 