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Transition dipole moments (TDM) between energy bands of solids deservespecial attention nowadays as intense lasers can easily drive non-adiabatictransitions of excited electron wave packets across the Brillouin zones. TheTDM is required to be continuous, satisfying crystal symmetry, and periodic atzone boundaries. While present day ab-initio algorithms are powerful incalculating band structures of solids, they all introduced random phases intothe eigenfunctions at each crystal momentum k. In this paper, we show how tochoose a ``smooth-periodic'' gauge where TDMs can be smooth versus k,preserving crystal symmetry, as well as maintaining periodic at zoneboundaries. Based on band structure and TDMs in the ``smooth-periodic'' gaugecalculated from ab-initio algorithms, we revisit high-order harmonic generationfrom MgO which exhibits inversion symmetry and ZnO which has broken symmetry.The symmetry properties of TDMs with respect to k ensure the absence ofeven-order harmonics in system with inversion symmetry, while the TDM in the`smooth-periodic'' gauge for ZnO is shown to enhance even harmonics that wereunderestimated in previous simulations. These results reveal the importance ofcorrectly treating the complex TDMs in nonlinear laser-solid interactions whichhas been elusive so far.

How to obtain complex transition dipole moments satisfying crystal symmetry and periodicity from ab-initio calculations