  Recently an improved limits on the electron electric dipole moment, \eEDM,
and dimensionless constant, $k_{T,P}$, characterizing the strength of the
T,P-odd pseudoscalar$-$scalar electron$-$nucleus neutral current interaction in
the $H^3\Delta_1$ state of ThO molecule were obtained by ACME collaboration
[Science 343, 269 (2014)]. The interpretation of the experiment in terms of
fundamental quantities \eEDM\ and $k_{T,P}$ is based on the results of
theoretical study of appropriate ThO characteristics, the effective electric
field acting on electron, \Eeff, and a parameter of the T,P-odd
pseudoscalar$-$scalar interaction, $W_{T,P}$, given in [J.Chem.Phys.\ 139,
221103 (2013)] by St.Petersburg group. To reduce the uncertainties of the given
limits we report improved calculations of the molecular state$-$specific
quantities \Eeff, 81.5~GV/cm, and $W_{T,P}$, 112~kHz, with the uncertainty
within 7\% of the magnitudes. Thus, the values recommended to use for the upper
limits of the quantities are 75.8~GV/cm and 104~kHz, correspondingly. The
hyperfine structure constant, molecule-frame dipole moment of the $H^3\Delta_1$
state and $H^3\Delta_1\to X^1\Sigma^+$ transition energy which, in general, can
serve as a measure of reliability of the obtained \Eeff\ and $W_{T,P}$ values
are also calculated. Besides we report the first calculation of g-factor for
the $H^3\Delta_1$ state of ThO. The results are compared to the earlier
experimental and theoretical studies, and a detailed analysis of uncertainties
of the calculations is given.
