  We extend the calculation of dark matter direct detection rates via
electronic transitions in general dielectric crystal targets, combining
state-of-the-art density functional theory calculations of electronic band
structures and wave functions near the band gap, with semi-analytic
approximations to include additional states farther away from the band gap. We
show, in particular, the importance of all-electron reconstruction for
recovering large momentum components of electronic wave functions, which,
together with the inclusion of additional states, has a significant impact on
direct detection rates, especially for heavy mediator models and at
$\mathcal{O}(10\,\text{eV})$ and higher energy depositions. Applying our
framework to silicon and germanium (that have been established already as
sensitive dark matter detectors), we find that our extended calculations can
appreciably change the detection prospects. Our calculational framework is
implemented in an open-source program $\texttt{EXCEED-DM}$ (EXtended
Calculation of Electronic Excitations for Direct detection of Dark Matter), to
be released in an upcoming publication.
