Periodic GFN2-xTB in CP2K: Multipolar Ewald Electrostatics, k-Point Sampling, and Transferability Benchmarks for...

Extended tight-binding methods are attractive for condensed-phase simulations because they retain an explicit electronic Hamiltonian at a cost far below conventional density functional theory. Their transferability to periodic materials, however, cannot be assessed reliably without a genuinely periodic Hamiltonian, Brillouin-zone sampling, and symmetry-aware crystal calculations. We present a periodic implementation of the second-generation geometry, frequency, and noncovalent interaction parametrization of extended tight binding (GFN2-xTB) in CP2K. The central theoretical ingredient is a multipolar Ewald formulation for the self-consistent shell charges, atomic dipoles, and quadrupoles of GFN2-xTB, together with analytical forces and stress contributions for geometry optimization, cell optimization, and molecular dynamics. For the ranking of ice polymorphs in the DMC-ICE13 data set, periodic GFN2-xTB on a converged 33k-point mesh improves the relative polymorph energies over GFN1-xTB, reducing the mean absolute error from 8.01 to 3.46 kJ mol-1. Because GFN1-xTB and GFN2-xTB differ in several coupled, jointly parametrized terms, this whole-model comparison does not isolate the contribution of multipolar electrostatics. For the LC10 set of cubic covalent and ionic solids, GFN2-xTB yields valid equation-of-state minima for all ten systems, with lattice-constant and cohesive-energy mean absolute errors of 0.062 Å and 1.30 eV per atom, respectively. Periodic GFN2-xTB is evaluated for molecular crystal lattice energies and cell parameters using the X23b data set. Using native Bloch 23-mesh cell optimization with full symmetry reduction, followed by a 33-mesh energy evaluation on the optimized cells, periodic GFN2-xTB gives a mean absolute lattice-energy error of 14.09 kJ mol-1 and a mean absolute relaxed-cell volume error of 5.84% for the full X23b relaxed-cell set. The benchmarks position periodic GFN2-xTB as an efficient electronic structure level for screening, pre-optimization, and large exploratory simulations. This work creates the foundation for periodic anisotropic electrostatics in the extended tight-binding framework and their efficient application in CP2K.