  The leading relativistic and QED corrections to the ground state energy of
the three-body system (epe) are calculated numerically using a Hylleraas
correlated basis set. The accuracy of the nonrelativistic variational ground
state wave function is discussed with respect to the convergence properties at
the increase of the basis dimension and to the variance of the energy
expectation value. Recent progress in the numerical procedure used to calculate
expectation values for products of various physical operators is presented. It
is shown that the nonrelativistic ground state energy can be calculated with an
accuracy below the level width. The corrections to this energy include the
lowest order Breit interaction, the vacuum polarization potential, one and two
photon exchange contributions, the annihilation interaction, and spin-spin
contact terms. The relativistic effects and the residual interactions
considered here decrease the one electron binding energy from the
nonrelativistic value of 0.012 005 070 232 980 10(3) a.u. to 0.011 981 051
246(2) a.u..
