  Detailed derivation of the analytical, reciprocal-space approach of Hessian
calculation within the self-consistent-charge density functional based
tight-binding framework (SCC-DFTB) is presented. This approach provides an
accurate and efficient way for obtaining the SCC-DFTB Hessian of periodic
systems. Its superiority with respect to the traditional numerical force
differentiation method is demonstrated for doped graphene, graphene
nanoribbons, boron-nitride nanotubes, bulk zinc-oxide and other systems.
