High-accuracy calculations of the rotation-vibration spectrum of $H_3^{+}$

Calculation of the rotation-vibration spectrum of $H_3^{+}$, as well as of its deuterated isotopologues, with near-spectroscopic accuracy requires the development of sophisticated theoretical models, methods, and codes. The present paper reviews the state-of-the-art in these fields. Computation of rovibrational states on a given potential energy surface (PES) has now become standard for triatomic molecules, at least up to intermediate energies, due to developments achieved by the present authors and others. However, highly accurate Born–Oppenheimer energies leading to highly accurate PESs are not accessible even for this two-electron system using conventional electronic structure procedures (e.g. configuration-interaction or coupled-cluster techniques with extrapolation to the complete (atom-centered Gaussian) basis set limit). For this purpose, highly specialized techniques must be used, e.g. those employing explicitly correlated Gaussians and nonlinear parameter optimizations. It has also become evident that a very dense grid of ab initio points is required to obtain reliable representations of the computed points extending from the minimum to the asymptotic limits. Furthermore, adiabatic, relativistic, and quantum electrodynamic correction terms need to be considered to achieve near-spectroscopic accuracy during calculation of the rotation-vibration spectrum of $H_3^{+}$. The remaining and most intractable problem is then the treatment of the effects of non-adiabatic coupling on the rovibrational energies, which, in the worst cases, may lead to corrections on the order of several cm⁻¹. A promising way of handling this difficulty is the further development of effective, motion- or even coordinate-dependent, masses and mass surfaces. Finally, the unresolved challenge of how to describe and elucidate the experimental pre-dissociation spectra of $H_3^{+}$ and its isotopologues is discussed. (topical review)