${\cal B}_{{\bf k},\sigma }^{\dagger }$ based on one site small polaron operators ${\cal B}_{{\bf r}\sigma }^{\dagger }$ $$\label{polar}{\cal B}_{{\bf k},\sigma }^{\dagger }=\frac 1{\sqrt{NK_{{\bf k}
} }}\sum_{{\bf r}}{\rm e}^{i{\bf k\cdot r}}{\cal B}_{{\bf r},\sigma
}^{\dagger },$$ $$\label{singl}{\cal B}_{{\bf r},\sigma }^{\dagger }=\frac 12\sum_{{\bf a}}(c_{
{\bf r}+{\bf a},\sigma }^{\dagger }Z_{{\bf r}}^{\bar \sigma \bar \sigma }-c_{
{\bf r}+{\bf a},\bar \sigma }^{\dagger }Z_{{\bf r}}^{\sigma \bar \sigma }),$$ $$K_{{\bf k}}=\langle \frac 1N\sum_{{\bf r},{\bf r}^{\prime }}{\rm e}^{-\imath
{\bf k}({\bf r}-{\bf r}^{\prime })}\{{\cal B}_{{\bf r},\sigma },{\cal B}_{
{\bf r}^{\prime },\sigma }^{\dagger }\}\rangle =1+(C_{{\bf g}}+\frac 14
)\gamma _{{\bf k}}.$$ Here $\{ ,\} ,[,]$ stand for an anticommutator and commutator, respectively; $ \langle ...\rangle \equiv Q^{-1}Sp[{\rm 
e}^{-\beta H}...],\hspace{\parindent} Q=Sp{\rm e}^{-\beta H}$. $Sp$ implies taking the trace of an operator, $ \beta =(kT)^{-1}$ is an inverse temperature; $\bar \sigma \equiv -\sigma $; $ 
Z_{{\bf r}}^{\sigma _1\sigma _2}\equiv \left| \sigma _1\right\rangle
\left\langle \sigma _2\right| $ are the Hubbard projection operators for Cu sites states; $\gamma _{{\bf k}}=\frac 14\sum_{{\bf g}}\exp (\imath {\bf
k\cdot g})$.

To calculate the average for commutators and anticommutators such as $K_{
{\bf k}}$, we take into account that these expressions are reduced to the two-site or three-site averages of the Hubbard operators located at different copper sites. In the spherically 