A high-precision study of the hydrogen molecular ion (HMI) H[subscript 2][superscript +] is presented using the variational method in Hylleraas coordinates. Hylleraas coordinates with three non-linear parameters are adopted so that distance scales of the wave function may be described independently for all three radial coordinates r[subscript 1], r[subscript 2] and r[subscript 12]. To achieve high precision, multi-precision arithmetic software QD (quad-double) is included in the code. To obtain high efficiency, the Message-Passing Interface (MPI) is used in computation. The ground state energy eigenvalue is calculated to a precision of 2 parts in 10[superscript 34], which represents a more than 3 orders of magnitude improvement over the best published values. High-precision variational calculations are also carried out for the first excited states of S and P symmetries, with the corresponding achieved precision of about 2 parts in 10[superscript 31] and 3 parts in 10[superscript 29] , respectively. Our results lay a sound foundation for studying high-order relativistic and quantum electrodynamic (QED) effects in this system and set a benchmark for developing other theoretical methods.

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