We here give a review of our studies of hydrogen molecular ion (H2+) based on the accurate solutions of the Schrödinger equation (SE) and Dirac equations (DE) obtained by the free-complement (FC) methodology developed in our laboratory. We summarize the results of non-relativistic and relativistic studies of H2+ and its isotopomers HD +, and HT +, under the Born-Oppenheimer (BO) and non-BO treatments and with and without external magnetic field. H2+ is a simple one-electron molecule, and so has basic importance in quantum chemistry. Further, it is stable and of rich history of studies, particularly in interstellar science. For the non-relativistic SE, the convergence speed to the exact solution of the FC method is faster than that of the «exact» expansion, exhibiting high efficiency of the FC method. For the relativistic DE, not only accurate energy upper bounds but also lower bounds are calculated. The potential energy curves are also calculated at the non-relativistic and relativistic levels for all isotopomers, and chemically interesting information such as spectroscopic constants and transition frequencies are provided. The non-BO problem is also successfully solved for all isotopomers, and extremely accurate 1 1S and 1 3P energies, expectation values of interparticle distances are calculated for the ground and excited vibrational states. In the magnetic field calculation, our method is accurate in any strengths and any directions of the magnetic field. The gauge-origin problem is also investigated, and has shown that the gauge-origin dependence of the energy becomes smaller and smaller when the FC order is increased. All these results are clear example that the FC method combined with the variational principle gives very accurate analytical wave functions of H2+ in any cases and situations. Our investigations on H2+ and its isotopomers revealed the significant and accurate deterministic powers of the FC method in any situations of the problems: this methodology is reliable and stable to provide exact solutions for various types of non-relativistic and relativistic equations and problems appearing in the atomic and molecular physics and chemistry.
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