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The static and dynamic electric multipolar polarizabilities and secondhyperpolarizabilities of the H$_2^+$, D$_2^+$, and HD$^+$ molecular ions in theground and first excited states are calculated nonrelativistically usingexplicitly correlated Hylleraas basis sets. The calculations are fullynonadiabatic; the Born-Oppenheimer approximation is not used. Comparisons aremade with published theoretical and experimental results, where available. Inour approach, no derivatives of energy functions nor derivatives of responsefunctions are needed. In particular, we make contact with earlier calculationsin the Born-Oppenheimer calculation where polarizabilities were decomposed intoelectronic, vibrational, and rotational contributions and wherehyperpolarizabilities were determined from derivatives of energy functions. Wefind that the static hyperpolarizability for the ground state of HD$^+$ isseven orders of magnitude larger than the corresponding dipole polarizability.For the dipole polarizability of HD$^+$ in the first excited-state the highprecision of the present method facilitates treatment of a near cancellationbetween two terms. For applications to laser spectroscopy of trapped ions wefind tune-out and magic wavelengths for the HD$^+$ ion in a laser field. Inaddition, we also calculate the first few leading terms for long-rangeinteractions of a hydrogen molecular ion interacting with a ground-state H, He,and Li atoms.

physical review. a, atomic, molecular, and optical physics/physical review, a, atomic, molecular, and optical physics

High-precision nonadiabatic calculations of dynamic polarizabilities and  hyperpolarizabilities for the lowlying vibrational-rotational states of  hydrogen molecular ions