  We report the first results of ab initio relativistic correlation calculation
of the effective electric field on the electron, E_eff, in the ground state of
the HI$^+$ cation. This value is required for interpretation of the suggested
experiment on search for the electron electric dipole moment. The generalized
relativistic effective core potential, Fock-space relativistic coupled cluster
with single and double cluster amplitudes and spin-orbit direct configuration
interaction methods are used, followed by nonvariational one-center restoration
of the four-component wavefunction in the iodine core. The calculated value of
E_eff by the coupled cluster method is E_eff=0.345\times 10^{24}Hz/e*cm.
Configuration interaction study gives E_eff=0.336\times 10^{24}Hz/e*cm (our
final value). The structure of chemical bonding and contributions to E_eff in
HI$^+$ is clarified and significant deviation of our value from that obtained
in Ravaine etal Phys.Rev.Lett., 94, 013001 (2005) is explained.
