On equivalence of two formulas of orbital magnetic susceptibility for tight-binding models

We prove that two formulas of the orbital magnetic susceptibility, namely, Koshino-Ando's formula and G\'{o}mes-Santos and Stauber's formula, are equivalent for the tight-binding models. The difference between these two formulas can be written in a form containing the surface terms arising from the momentum-space integration, and we show that the surface terms are exactly vanishing although the primitive functions are seemingly non-periodic with respect to the translation by the reciprocal lattice vector.