  We study the different ways of introducing light-matter interaction in
first-principle tight-binding (TB) models. The standard way of describing
optical properties is the velocity gauge, defined by linear coupling to the
vector potential.
  In finite systems a transformation to represent the electromagnetic radiation
by the electric field instead is possible, albeit subtleties arise in periodic
systems. The resulting dipole gauge is a multi-orbital generalization of
Peierl's substitution. In this work, we investigate accuracy of both pathways,
with particular emphasis on gauge invariance, for TB models constructed from
maximally localized Wannier functions. Focusing on paradigmatic two-dimensional
materials, we construct first-principle models and calculate the response to
electromagnetic fields in linear response and for strong excitations.
Benchmarks against fully converged first-principle calculations allow for
ascertaining the accuracy of the TB models. We find that the dipole gauge
provides a more accurate description than the velocity gauge in all cases. The
main deficiency of the velocity gauge is an imperfect cancellation of
paramagnetic and diamagnetic current. Formulating a corresponding sum rule
however provides a way to explicitly enforce this cancellation. This procedure
corrects the TB models in the velocity gauge, yielding excellent agreement with
dipole gauge and thus gauge invariance.
