Leading order relativistic corrections to the dipole polarizability of the hydrogen molecular ions

The static dipole polarizability for the hydrogen molecular ions H$_2^+$, HD$^+$, and D$_2^+$ are calculated. These new data for polarizability takes into account the leading order relativistic corrections to the wave function of the three-body system resulted from the Breit-Pauli Hamiltonian of $m\alpha^4$ order. Our study covers a wide range of rotational ($L=0-5$) and vibrational ($v=0-10$) states, which are of practical interest for precision spectroscopy of the hydrogen molecular ions.