Drude weight in presence of nonlocal potentials

The nonlocal potential contributes an extra term to the velocity operator; I show here that such term affects the formal expression of the Drude weight in a nontrivial way. Notably, the present main result fixes a disturbing discrepancy in the Dreyer-Coh-Stengel sum rule [Phys. Rev. Lett. {\bf 128}, 095901 (2022)].