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

Optics;Mesoscale and Nanoscale Physics