The Peierls substitution is given by the following formula: $$t_{12}\rightarrow t'_{12}=\exp\left[\int_{{\bf R}_{1}}^{{\bf R}_{2}}{\bf A}({\bf r})\cdot d{\bf r}\right]t_{12}.$$
What are the meanings of the ${\bf R}_{1}$ and and ${\bf R}_{2}$? For example, let us assume that I have a NaCl crystal with body center cubic structure. The fractional coordinates are $(0, 0, 0)$ for Na and $(\frac{1}{2}, \frac{1}{2}, \frac{1}{2})$ for Cl, and the lattice vectors are $({\bf a},{\bf b},{\bf c})$. Now, the $t_{12}$ is defined as the hopping from the $p$ orbital of Cl [in the cell ${\bf R}(lmn)=l{\bf a}+m{\bf b}+n{\bf c}$] to the $s$ orbital of Na [in the cell of ${\bf R}(uvw)=u{\bf a}+v{\bf b}+w{\bf c}$]. Now my question is: If we want to do Peierls substitution, should I use ${\bf R}_{1}=l{\bf a}+m{\bf b}+n{\bf c}+\frac{1}{2}{\bf a}+\frac{1}{2}{\bf b}+\frac{1}{2}{\bf c}$ and ${\bf R}_{2}=u{\bf a}+v{\bf b}+w{\bf c}$, or should I use ${\bf R}_{1}=l{\bf a}+m{\bf b}+n{\bf c}$ and ${\bf R}_{2}=u{\bf a}+v{\bf b}+w{\bf c}$?
