How to understand Peierls substitution?

The Peierls substitution is given by the following formula:
$$t_{12}rightarrow t’_{12}=expleft[int_{{bf R}_{1}}^{{bf R}_{2}}{bf A}({bf r})cdot d{bf r}right]t_{12}.$$

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 ${bf R}_{2}=u{bf a}+v{bf b}+w{bf c}$?