An experiment to search for the electron electric dipole moment (eEDM) on the metastable H 3 1 state of ThO molecule was proposed and now prepared 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 T,P-odd pseudoscalar-scalar electron-nucleus neutral current interaction, an accurate theoretical study of an effective electric field on electron, E eff , and a parameter of the T,P-odd pseudoscalar-scalar interaction, W T ,P , in ThO is required. We report our results for E eff (84 GV/cm) and W T ,P (116 kHz) together with the hyperfine structure constant, molecule frame dipole moment, and H 3 1 → X 1 + transition energy, which can serve as a measure of reliability of the obtained E eff and W T ,P values. Besides, our results include a parity assignment and evaluation of the electric-field dependence for the magnetic g factors in the -doublets of H 3 1 .