Adiabatic corrections for velocity-gauge simulations of electron dynamics in periodic potentials

We show how to significantly reduce the number of energy bands required to model the interaction of light with crystalline solids in the velocity gauge. We achieve this by deriving analytical corrections to the electric current density. These corrections depend only on band energies, the matrix elements of the momentum operator, and the macroscopic vector potential. Thus, the corrections can be evaluated independently from modeling the interaction with light. In addition to improving the convergence of velocity-gauge calculations, our analytical approach overcomes the long-standing problem of divergences in expressions for linear and nonlinear susceptibilities.