Periodic GFN1-xTB Tight Binding: A Generalized Ewald Partitioning Scheme for the Klopman-Ohno Function

A novel formulation is presented for the treatment of electrostatics in the periodic GFN1-xTB tight-binding model. Periodic GFN1-xTB is hindered by the functional form of the second-order electrostatics, which only recovers Coulombic behavior at large interatomic distances and lacks a closed-form solution for its Fourier transform. We address this by introducing a binomial expansion of the Klopman-Ohno function to partition short- and long-range interactions, enabling the use of a generalized Ewald summation for the solution of the electrostatic energy. This approach is general and is applicable to any damped potential of the form |R + c|^-. Benchmarks on the X23 molecular crystal dataset and a range of prototypical bulk semiconductors demonstrate that this systematic treatment of the electrostatics eliminates unphysical behavior in the equation of state curves. In the bulk systems studied, we observe a mean absolute error in total energy of 35 meV/atom, comparable to the machine-learned universal force field, M3GNet, and sufficiently precise for structure relaxation. These results highlight the promising potential of GFN1-xTB as a universal tight-binding parametrization.