Developments during the last two decades in nuclear motion theory made it possible to obtain variational solutions to the time-independent, nuclear-motion Schrödinger equation of polyatomic systems as “exact” as the potential energy surface (PES) is. Nuclear motion theory thus reached a level whereby this branch of quantum chemistry started to catch up with the well developed and widely applied other branch, electronic structure theory. It seems to be fair to declare that we are now in the fourth age of quantum chemistry, where the first three ages are principally defined by developments in electronic structure techniques (G. Richards, Nature, 1979, 278, 507). In the fourth age we are able to incorporate into our quantum chemical treatment the motion of nuclei in an exact fashion and, for example, go beyond equilibrium molecular properties and compute accurate, temperature-dependent, effective properties, thus closing the gap between measurements and electronic structure computations. In this Perspective three fundamental algorithms for the variational solution of the time-independent nuclear-motion Schrödinger equation employing exact kinetic energy operators are presented: one based on tailor-made Hamiltonians, one on the Eckart–Watson Hamiltonian, and one on a general internal-coordinate Hamiltonian. It is argued that the most useful and most widely applicable procedure is the third one, based on a Hamiltonian containing a kinetic energy operator written in terms of internal coordinates and an arbitrary embedding of the body-fixed frame of the molecule. This Hamiltonian makes it feasible to treat the nuclear motions of arbitrary quantum systems, irrespective of whether they exhibit a single well-defined minimum or not, and of arbitrary reduced-dimensional models. As a result, molecular spectroscopy, an important field for the application of nuclear motion theory, has almost black-box-type tools at its disposal. Variational nuclear motion computations, based on an exact kinetic energy operator and an arbitrary PES, can now be performed for about 9 active vibrational degrees of freedom relatively straightforwardly. Simulations of high-resolution spectra allow the understanding of complete rotational–vibrational spectra up to and beyond the first dissociation limits. Variational results obtained for H₂O, H⁺ ₃, NH₃ , CH₄, and H₂CCO are used to demonstrate the power of the variational techniques for the description of vibrational and rotational excitations. Some qualitative features of the results are also discussed.

In classical laser fields with frequencies resonant with the electronic excitation in molecules it is by now known that conical intersections are induced by the field and are called light-induced conical intersections (LICI). As optical cavities have become accessible, the question arises whether their quantized modes can also lead to the appearance of LICIs. A theoretical framework is formulated for the investigation of LICIs of diatomic molecules in such quantum light. The eigenvalue spectrum of the dressed states in the cavity is studied with particular emphasis on the investigation of absorption spectra, i.e., transitions between these dressed states, measured by employing a weak probe pulse. The theory is exemplified for the sodium dimer molecule. The dependence of the spectra on the light-matter coupling strength in the cavity and on the frequency of the cavity mode are studied in detail. The computations demonstrate strong non-adiabatic effects caused by the appearing LICI.

First-principles computations and experimental measurements of transition energies are carried out for vibrational overtone lines of the triatomic hydrogen ion H3+ corresponding to floppy vibrations high above the barrier to linearity. Action spectroscopy is improved to detect extremely weak visible-light spectral lines on cold trapped H3+ ions. A highly accurate potential surface is obtained from variational calculations using explicitly correlated Gaussian wave function expansions. After nonadiabatic corrections, the floppy H3+ vibrational spectrum is reproduced at the 0.1\,\,cm-1 level up to 16\,600\,\,cm-1.

Calibration-quality ab initio adiabatic potential energy surfaces (PES) have been determined for all isotopologues of the molecular ion ${\mathrm{H}}_3^{+}$ . The underlying Born–Oppenheimer electronic structure computations used optimized explicitly correlated shifted Gaussian functions. The surfaces include diagonal Born–Oppenheimer corrections computed from the accurate electronic wave functions. A fit to the 41 655 ab initio points is presented which gives a standard deviation better than 0.1 cm⁻¹ when restricted to the points up to 6000 cm⁻¹ above the first dissociation asymptote. Nuclear motion calculations utilizing this PES, called GLH3P, and an exact kinetic energy operator given in orthogonal internal coordinates are presented. The ro-vibrational transition frequencies for ${\mathrm{H}}_3^{+}$ , H₂D⁺, and ${\mathrm{HD}}_2^{+}$ are compared with high resolution measurements. The most sophisticated and complete procedure employed to compute ro-vibrational energy levels, which makes explicit allowance for the inclusion of non-adiabatic effects, reproduces all the known ro-vibrational levels of the ${\mathrm{H}}_3^{+}$ isotopologues considered to better than 0.2 cm⁻¹. This represents a significant (order-of-magnitude) improvement compared to previous studies of transitions in the visible. Careful treatment of linear geometries is important for high frequency transitions and leads to new assignments for some of the previously observed lines. Prospects for further investigations of non-adiabatic effects in the ${\mathrm{H}}_3^{+}$ isotopologues are discussed. In short, the paper presents (a) an extremely accurate global potential energy surface of ${\mathrm{H}}_3^{+}$ resulting from high accuracy ab initio computations and global fit, (b) very accurate nuclear motion calculations of all available experimental line data up to 16 000 cm⁻¹, and (c) results suggest that we can predict accurately the lines of ${\mathrm{H}}_3^{+}$ towards dissociation and thus facilitate their experimental observation.