The theoretical interest arises because an accurate mathematical description, taking into account the quantum motion of all constituents and also the interaction of the electron with the radiation field, is feasible.

The description's accuracy has steadily improved over more than half a century, eventually resulting in a theoretical framework allowing ultra-high-accuracy predictions for the energies of the rotational and vibrational levels in the electronic ground state, which are mostly metastable.

Employing advanced techniques, such as ion trapping and laser cooling, the rotational and vibrational transitions can be investigated in extremely fine detail.

This is a good approximation because the nuclei (proton, deuteron or triton) are more than a factor 1000 heavier than the electron.

The analytical solution of the equation, the wave function, is therefore proportional to a product of two infinite power series.

The analytical solutions for the electronic energy eigenvalues are also a generalization of the Lambert W function[8] which can be obtained using a computer algebra system within an experimental mathematics approach.

was published by the Danish physicist Øyvind Burrau in 1927,[1] just one year after the publication of wave mechanics by Erwin Schrödinger.

[12] The complete mathematical solution of the electronic energy problem for H+2 in the clamped-nuclei approximation was provided by Wilson (1928) and Jaffé (1934).

An additive term ⁠1/R⁠, which is constant for fixed internuclear distance R, has been omitted from the potential V, since it merely shifts the eigenvalue.

In atomic units (ħ = m = e = 4πε0 = 1) the wave equation is We choose the midpoint between the nuclei as the origin of coordinates.

There are wave functions ψg(r), which are symmetric with respect to i, and there are wave functions ψu(r), which are antisymmetric under this symmetry operation: The suffixes g and u are from the German gerade and ungerade) occurring here denote the symmetry behavior under the point group inversion operation i.

Similarly, asymptotic expansions in powers of ⁠1/R⁠ have been obtained to high order by Cizek et al. for the lowest ten discrete states of the hydrogen molecular ion (clamped nuclei case).

For general diatomic and polyatomic molecular systems, the exchange energy is thus very elusive to calculate at large internuclear distances but is nonetheless needed for long-range interactions including studies related to magnetism and charge exchange effects.

Note that although the generalized Lambert W function eigenvalue solutions supersede these asymptotic expansions, in practice, they are most useful near the bond length.

[18][19] Once the energy function Etot(R) has been obtained, one can compute the quantum states of rotational and vibrational motion of the nuclei, and thus of the molecule as a whole.

The equation describes the motion of a fictitious particle of mass equal to the reduced mass of the two nuclei, in the potential Etot(R)+VL(R), where the second term is the centrifugal potential due to rotation with angular momentum described by the quantum number L. The eigenenergies of this Schrödinger equation are the total energies of the whole molecule, electronic plus nuclear.

The Born-Oppenheimer approximation is unsuited for describing the dihydrogen cation accurately enough to explain the results of precision spectroscopy.

The full Schrödinger equation for this cation, without the approximation of clamped nuclei, is much more complex, but nevertheless can be solved numerically essentially exactly using a variational approach.

When the solutions are restricted to the lowest-energy orbital, one obtains the rotational and ro-vibrational states' energies and wavefunctions.

The most accurate solutions of the ro-vibrational states are found by applying non-relativistic quantum electrodynamics (NRQED) theory.

For transitions between ro-vibrational levels having small rotational and moderate vibrational quantum numbers the frequencies have been calculated with theoretical fractional uncertainty of approximately 8×10−12.

Specifically, spectroscopically determined pure rotational and ro-vibrational transition frequencies of the particular isotopologue

[22] The level of agreement is actually limited neither by theory not by experiment but rather by the uncertainty of the current values of the masses of the particles, that are used as input parameters to the calculation.

In order to measure the transition frequencies with high accuracy, the spectroscopy of the dihydrogen cation had to be performed under special conditions.

Finally, in 2021, ab initio theory computations were able to provide the quantitative details of the structure with uncertainty smaller than that of the experimental data, 1 kHz.

Some contributions to the measured hyperfine structure have been theoretically confirmed at the level of approximately 50 Hz.

The dihydrogen ion is formed in nature by the interaction of cosmic rays and the hydrogen molecule.

High speed electrons also cause ionization of hydrogen molecules with a peak cross section around 50 eV.