Method to deal with random phases generated in ab-initio algorithms

Transition dipole moments between energy bands of solids deserve special attention nowadays as non-adiabatic transition can be easily achieved by strong laser. Driven by strong laser with long wavelength, the excited electron wavepacket can even go across the first Brillouin zone. It requires continuous and periodic transition dipole moments if people want to deal with the interaction of strong laser with crystal in a finite space. While, it is hard to get such kind of transition dipole moments from the ab-initio algorithms because random phases will be introduced into the eigenfunction for each k point. In this paper, we show how to get the continuous and periodic k-dependent dipole moments in "smooth-periodic" gauge. Based on the accurate transition dipole moments, high-order harmonic generations from MgO with inversion symmetry and ZnO with broken symmetry are revisited. The symmetry properties of transition dipole moments with respect to k ensure the absence of even-order harmonics in system with inversion symmetry, even though multiband excitation exists in the laser-driven transition. The harmonic spectrum from ZnO is also improved with accurate transition dipole moments compared to the previous simulation.