If you are a visitor looking for a very basic introduction to the hydrogen molecular ion's physical properties try The simplest molecule by Alan Carrington, New Scientist, January 14, 1989, p. 53. Also, one nice article about the Born-Oppenheimer calculation is The hydrogen molecular ion revisited by Jean-Philippe Grivet, J. Chemical Education, January 1, 2002, pp. 127-132.

The dipole polarizability of the hydrogen molecular ion, J. M. Taylor, A. Dalgarno, and J.F. Babb, Phys. Rev. A 60 (1999), pp. R2630-R2632. Read the preprint abstract (this link also contains links to get the entire preprint.) This paper reports some benchmark polarizability calculations. Text file approximation to the preprint.

Variational calculations on the hydrogen molecular ion, J. M. Taylor, Zong-Chao Yan, A. Dalgarno, and J. F. Babb, Molecular Phys., vol. 97 (1999), p.25. Read the preprint abstract (this link also contains links to get the entire preprint.)
This paper reports some benchmark energy calculations. We think this may be the first reported calculation of lambda doubling in H₂⁺. Text file approximation to the preprint.

The hyperfine structure of the hydrogen molecular ion, invited talk at the Inauguration Conference of the Asia Pacific Center for Theoretical Physics, Seoul, Korea, June 4-10, 1996. Read the preprint abstract (this link also contains links to get the entire preprint)
or look it up in Current topics in physics, Y. M. Cho, J. B. Hong, and C. N. Yang, eds., World Scientific, Singapore (1998), Volume 2, p. 531-40. Text file approximation to the preprint.

Nuclear spin rotation in the hydrogen molecular ion, Physical Review Letters, vol. 75 (1995) 4377-4380. Read the preprint abstract (this link also contains links to get the entire preprint.)
The magnetic field of the rotating molecule interacts with the nuclear spins directly and through the electronic orbital angular momentum (in second order perturbation theory).

Effective oscillator strengths and transition energies for the hydrogen molecular ion, Molec. Phys., vol. 81 (1994), pp. 17-29; Errata
Some handy quantities for second order perturbation theory with the electric dipole operator.

Spin coupling constants and hyperfine transition frequencies for the hydrogen molecular ion, J. F. Babb and A. Dalgarno, Phys. Rev. A (Rapid Communication), vol. 46 (1992), pp. R5317--R5319
Quantitative exploration of the effective spin coupling Hamiltonian for the simplest molecule.