Modelling rotations, vibrations, and rovibrational couplings in astructural molecules – a case study based on the H⁺₅ molecular ion

Abstract

One-dimensional (1D) and two-dimensional (2D) models are investigated, which help to understand the unusual rovibrational energy-level structure of the astronomically relevant and chemically interesting astructural molecular ion H + 5 . Due to the very low hindering barrier characterising the 1D torsion-only vibrational model of H + 5 , this model yields strongly divergent energy levels. The results obtained using a realistic model for the torsion potential, including the computed (near) degeneracies, can be rationalised in terms of the model with no barrier. Coupling of the torsional motion with a single rotational degree of freedom is also investigated in detail. It is shown how the embedding-dependent rovibrational models yield energy levels that can be rationalised via the 2D vibrational model containing two independent torsions. Insight into the complex rovibrational energy level structure of the models and of H + 5 is gained via variational nuclear motion and diffusion Monte Carlo computations and by the analysis of the wavefunctions they provide. The modelling results describing the transition from the zero barrier limit to the large barrier limit should prove to be useful for the important class of molecules and molecular ions that contain two weakly coupled internal rotors. IntroductionThe simplest and exactly solvable quantum chemical models that are traditionally employed to understand highresolution spectra of gas-phase molecules are based on the harmonic oscillator (HO) and rigid rotor (RR) approximations of the vibrations and the rotations, respectively. In cases when the results based on the RRHO approximation, perhaps after a slight extension based on second-order perturbation theory [1][2][3][4][5], provide a good qualitative and even a semiquantitative understanding of spectral regularities, the molecule of interest can be considered to be 'semirigid'. These are molecules for which the electronic state that is being investigated contains a single, well-defined, and relatively deep minimum. For semirigid molecules, the timescales for the vibrational and rotational motions are sufficiently different to allow their approximate separation, the vibrational spacing decreases as the vibrational excitation increases, the vibrational states have well-defined symmetries provided by the point group characterising the unique equilibrium structure, and the rotational states can be assigned to a certain vibrational state. The RRHO treatment is familiar to most chemists as excellent textbooks exist which describe slightly anharmonic molecular vibrations [6,7] as

Quasistructural molecules

“…For a quasistructural molecule, all of the following should hold (to a greater or lesser extent) at the same time: (a) consideration of a single minimum on the PES, even though the molecule may possess only one minimum determinable by methods of electronic‐structure theory (this is the case, for example, for the quasistructural H and CH molecular cations treated in detail below), is insufficient to interpret the structure and the dynamical behavior of the molecule and its observable (high‐resolution) spectra, (b) when the structure is averaged, principally over the vibrational ground state(s), it may be significantly (i.e., even qualitatively) different from the equilibrium BO one (“quasistructurality” manifests itself in the effective structure of the molecule, whereby the equilibrium and the ground‐state rotational constants may become drastically different), due to strongly anharmonic zero‐point motions and the significant occupancy of low‐energy (ro)vibrational states, (c) rotational and vibrational spacings are of the same order of magnitude (this is most easily satisfied for light, H‐containing molecules) and the internal nuclear dynamics is thus exceedingly complex, (d) spectroscopic characterization of the molecule must rely on MS groups involving permutation and inversion symmetry operations and feasible and unfeasible motions in the Longuet‐Higgins sense, and (e) the set of assigned rovibrational energies of the molecule exhibits unconventional (in the most striking cases “negative”) rotational‐energy contributions (this is one of the most clear signs of quasistructural behavior). Note right away that (1) case (b) can happen for molecules considered to be semirigid, and (2) in a few publications we called quasistructural molecules “astructural.” Nevertheless, we now believe that the new terminology proposed here should replace the old one. A feasible alternative would be to greatly extend the IUPAC definition of “fluxional molecules.”…”

“…As predicted by all sophisticated electronic‐structure computations, the equilibrium structure of the H ion resembles that of a solvated ion, with the core H ion, formed by a strong 3‐center–2‐electron (3c–2e) bond, solvated by a H 2 molecule, placed perpendicularly to the H plane. The height of the torsional barrier, hindering the internal turnaround of the seemingly loosely attached H 2 subunit, is rather low, ∼80 cm −1 …”

“…In particular, on the vibrationally averaged PES the ∼60 cm −1 electronic barrier, hindering the proton hopping motion between the two sides, disappears, resulting in no difference in the “left” and “right” H 2 units for H [H 2 –H–H 2 ] + . Unlike in the case of semirigid molecules, the lack of a clear structure makes the rovibrational energy‐level set of H challenging to anticipate as well as to interpret …”

“…The peculiar rovibrational energy‐level structure of H can be reproduced surprisingly well in a four‐dimensional (4D) model considering only the torsional vibration and the three rotational dofs . This model is described in detail in Appendices A and B to this paper.…”

“…Figure 4 of Reference reported the correlation between the reduced energies and the reduced barriers of a two‐dimensional vibration–rotation model developed to explain the peculiar rovibrational energy‐level structure of H . The one‐dimensional vibrational (along the torsional dof, φ ) part of the model Hamiltonian is and the potential curve, for H, is approximated as …”