Energies and polarizabilities of the hydrogen molecular ions

The nonrelativistic energies of the hydrogen molecular ions ${\mathrm{H}}_2^{+},$ ${\mathrm{D}}_2^{+},$ ${\mathrm{T}}_2^{+},$ ${\mathrm{HD}}^{+},$ ${\mathrm{HT}}^{+},$ and ${\mathrm{DT}}^{+}$ are evaluated for the ground and the first-excited P states, using variational wave functions in Hylleraas coordinates. The dipole polarizabilities of the ground states of the ions are determined precisely. Our results for the dipole polarizabilities represent the most accurate calculations reported so far. For ${\mathrm{H}}_2^{+},$ for example, the dipole polarizability is $3.168725802{67\left(1\right)a}_0^3,$ with the uncertainty of three parts in ${10}^{12}.$