  An experiment to search for the electron electric dipole moment (\eEDM) on
the metastable $H^3\Delta_1$ state of ThO molecule was proposed and now in the
final stage of preparation by the ACME collaboration
[http://www.electronedm.org]. To interpret the experiment in terms of \eEDM\
and dimensionless constant $k_{T,P}$ characterizing the strength of the scalar
T,P-odd electron-nucleus neutral current interaction, an accurate theoretical
study of effective electric field on electron, Eeff, and $W_{T,P}$ constants is
required. We report calculation of \Eeff\ (84 GV/cm) and a parameter of T,P-odd
scalar neutral currents interaction, $W_{T,P}$ (116 kHz), together with the
hyperfine structure constant, molecule frame dipole moment and $H^3\Delta_1\to
X^1\Sigma^+$ transition energy, which can serve as a measure of reliability of
the obtained \Eeff\ and $W_{T,P}$ values. Besides, our results include a parity
assignment and evaluation of the electric-field dependence for the magnetic $g$
factors for the $\Omega$-doublets of $H^3\Delta_1$.
