The smallest proton-bound dimer H₅⁺: theoretical progress

Abstract

The protonated hydrogen dimer, H₅⁺, is the smallest system including proton transfer, and has been of long-standing interest since its first laboratory observation in 1962. H₅⁺ and its isotopologues are the intermediate complexes in deuterium fractionation reactions, and are of central importance in molecular astrophysics. The recently recorded infrared spectra of both H₅⁺ and D₅⁺ reveal a rich vibrational dynamics of the cations, which presents a challenge for standard theoretical approaches. Although H₅⁺ is a four-electron ion, which makes highly accurate electronic structure calculations tractable, the construction of ab initio-based potential energy and dipole moment surfaces has proved a hard task. In the same vein, the difficulties in treating the nuclear motion could also become cumbersome due to their high dimensionality, floppiness and/or symmetry. These systems are prototypical examples for studying large-amplitude motions, as they are highly delocalized, interconverting between equivalent minima through internal rotation and proton transfer motions requiring state-of-the-art treatments. Recent advances in the computational vibrational spectroscopy of the H₅⁺ cation and its isotopologues are reported from full quantum spectral simulations, providing important information in a rigorous manner, and open perspectives for further future investigations.

This article is part of a discussion meeting issue ‘Advances in hydrogen molecular ions: H₃⁺, H₅⁺ and beyond’.

1. Introduction and overview

The study of protonated hydrogen systems has fasci- nated chemists and physicists over the years [1–17]. The smaller pure hydrogenic species of the series, H⁺, H⁺₂ and H₃⁺, are stable, and are playing a major role in the hydrogen-dominated chemistry of the interstellar medium, by initiating the chains of reactions that lead to the production of many of the complex molecular species observed in the interstellar medium. The H⁺₂ ion is involved in the formation of H₃⁺, which in turn acts as proton donor via the universal proton-hop reactions [18,19]. The H₃⁺ ion was discovered in interstellar space in 1996 [20], while higher-order hydrogenic H₃⁺(H₂)_n cluster ions, although they are stable and have been also mentioned in relation to astrochemistry [21], have not been detected yet in space. However, the smallest proton-bound dimer H₅⁺ and its deuterated analogues have been proposed as intermediates in the symmetric proton transfer from H₃⁺ to H₂, and in the deuterium fractionation in interstellar clouds [12,22,23], respectively. Besides such an astrophysical significance, such cations can also be of fundamental interest as benchmark systems for quantum theories of structure, energy, dynamics and superfluidity. They are also of interest in modelling nucleation dynamics in planetary and interstellar conditions, for understanding solvation mechanics in liquids, and for initiating nuclear fusion reactions, energetic Coulomb explosions, as well as possible energy storage media [22,24–26].

Owing to the long-range ion–molecule interaction, such H₃⁺(H₂)_n clusters grow easily into higher molecular species, and odd-numbered clusters up to H⁺₄₇ have been earlier observed [2], while more recently such clusters up to H⁺₁₁₉ have been reported in ultracold helium nanodroplets [27]. Both experimental and theoretical studies have proposed that the structure of the odd-numbered cluster ions consists of a tightly bound H₃⁺ core surrounded by weakly bound H₂ molecules arranged in solvation shells. The shell model structure for the H⁺_2n+1 clusters has been supported by the observed infrared (IR) predissociation spectroscopy [4], mass spectra [28] and measurements of dissociation enthalpies and entropies [29–32].

Apart from the first vibrational predissociation spectra of these ions reported by Okumura et al. [4,33], more recently new IR spectra have been recorded by Duncan and co-workers [14,16,32]. These latter measurements have been also extended for both H₅⁺ and D₅⁺ into the mid- and far-IR regions, below the one-photon dissociation threshold, using the expanded frequency coverage of the FELIX free electron laser and multiple-photon dissociation spectroscopy. The new experimental data have refocused attention on the unusual vibrational dynamics of these odd-numbered hydrogen ions, and have stimulated recent theoretical studies on their energetics, structures, vibrational frequencies and IR spectra [15,34–49].

Numerous theoretical studies have been carried out, with a majority of the earlier ones aiming to determine equilibrium structures and dissociation energies of the small ions from electronic structure calculations at different levels of theory, such as MP2 and CCSD(T), as well as various density functional theory (DFT) approaches [3,6–10,50,51]. Most of our knowledge on the minima and distinct stationary points of the title ions became initially available from the ab initio calculations by Yamaguchi et al. [50], while a few attempts to provide local or global representations of the potential energy surface (PES) can be found in the literature for only the H₅⁺ [8,52–57] and more recently the H⁺₇ ions [44,58,59]. Apart from the earlier site–site interactions fitted to MP4 [52], a local representation at CASSCF/MR-CISD [53], a first attempt at a global description based on the DIM-PT1 approach [8], as well as an interpolation CCSD(T)-MP2 scheme [54], later on three PESs have been developed, and are available in the literature for the H₅⁺ system [55–57]. Two of them employed different fitting procedures to CCSD(T) ab initio data using triatomic-in-molecules, generalized many-body expansion and permutationally invariant functional forms for the parametrization of the surface [55,57]. The third one is an on-the-fly DFT-based representation of the PES [56] using the improved parametrized B3(H) functional, designed for hydrogen-only systems [6,59,60]. From a theoretical perspective, the existence of several, nearly isoenergetic, low-lying stationary points, the presence of low energy barriers on their potential surface, the highly anharmonic character of their vibrations and the permutation–inversion symmetry make the study of such ions challenging [61–63]. Clearly, a detailed knowledge of the structure and energetics of such species requires a faithful interplay between advanced experimental techniques and state-of-the-art computational approaches.

In this study, we survey the current status on the smallest H₃⁺(H₂) cluster and its deuterated analogues (containing both isotopologues and isotopomers). We review our recent work on H₅⁺ in §2, together with new data from multi-reference correlation interaction (MRCI) electronic structure calculations, and quantum multiconfigurational time-dependent Hartree (MCTDH) computations on high-lying vibrational levels of selected H₅⁺ isotopologues, such as H₄D⁺, H₃D⁺₂, D₃H⁺₂, D₄H⁺ and D₅⁺. A systematic analysis of their vibrational dynamics up to dissociation is provided from fully coupled quantum computations, employing ab initio-based potential energy (PES) and dipole moment (DMS) surfaces. Some concluding remarks and direction of future studies are summarized at the closing section.

2. The H₅⁺ ion in detail

(a). Electronic structure calculations

Figure 1 shows the relative energies of the 10 low-lying stationary points localized on the H₅⁺ PES at various levels of theory, such as MRCI(+Q), CCSD(T), MP2 and DFT ones, as well as their comparison with the values from analytical ab initio-based potentials [54,55,57], and from on-the-fly DFT approaches [56]. The total energy of the optimized global minimum 1C_2v structure corresponding to an isosceles H₃⁺ subunit bound to an H₂ molecule (see insets in figure 1) was calculated to be −2.532127 a.u. at the MRCI(+Q)/aug-cc-pV6Z level. This new reported value is just 32 cm⁻¹ lower than the best-known value available in the literature of −2.53198 a.u. at CCSD(T)/CBS[DTQ] level of theory. In turn, the next four low-lying stationary points, namely 2D_2d, 3C_2v, 4D_2h and 5C_2v (see inset plots in figure 1), correspond to the transition states for H₂–H₂ proton exchange, for non-planar (2D_2d) and planar (4D_2h) configurations, and H₂, H₃⁺ internal rotations (3C_2v and 5C_2v), with energies just 57, 99, 177 and 1547 cm⁻¹, respectively, above the value of the 1C_2v from the present MRCI(+Q)/aug-cc-pV6Z calculations. These energy values are found to be very close to the those previously available from CCSD(T)/CBS, which estimated values of 64, 95, 182 and 1543 cm⁻¹, respectively. In the inset panel of figure 1, we display a magnified plot for the four lower PESs' stationary points, and as we can see, small differences are obtained between the DFT/B3(H), MP2, CCSD(T) and MRCI(+Q) energies, while larger deviations are those from the DFT/B3LYP calculations, and somehow the higher barrier values for the 2D_2d and 4D_4h predicted by the analytical CCSD(T) fit by Aguado et al. [57]. One should note the remarkable performance of the DFT/B3(H); among the several DFT functionals used, only the B3(H) reproduces the structure and energetics of all stationary points of H₅⁺ in quantitative agreement with the high-level CCSD(T)/CBS and MRCI(+Q) data.

Figure 1.

Relative interaction energies of the 10 lower stationary points with respect to the total energy of the 1C_2v global potential minimum of the H₅⁺ PES (see text). The 2D_2d, 3C_2v and 5C_2v are first-order saddle-points, while the 4D_2h is a second-order one. (Online version in colour.)

(b). Potential energy surfaces

As mentioned above, only three potential energy surfaces have been recently reported for the H₅⁺ ion, and have been found to provide a globally realistic representation of the underlying interactions [55–57]. Figure 2 presents one-dimensional plots (see panels a,b) from two of them: the analytical CCSD(T) parametrized surface by Xie et al. [55], and the on-the-fly DFT/B3(H) PES generated by carrying out electronic structure calculations at each point of the whole grid. The R- and z-coordinates correspond to the distance between the centres of mass of the H₂ units, and the distance of the proton from the origin of the body-fixed system (see panels, and inset plot in figure 2), respectively. The presented curves are the minimum energy paths in each R- or z-coordinate with the H₅⁺ fixed at C_2v configurations. Although the two potentials show similar behaviour along the R- and z-coordinates, one can see that the DFT/B3(H) one shows a deeper global minimum than that obtained from the analytical surface by Xie et al. [55]. In particular, a well depth of 9.47 kcal mol⁻¹ is estimated by the DFT/B3(H), and a value of 8.30 kcal mol⁻¹ from the analytical PES by Xie et al., while the CCSD(T)/CBS calculations predict 8.65 kcal mol⁻¹. Further, the B3(H) curve seems to be less anharmonic than the analytical PES, with differences in both the repulsive and attractive regions. In the R-coordinate, the repulsive part of the curves is the same up to energies of ≈ 1200 cm⁻¹, while significant differences are obtained for larger R distances. In the z-coordinate, differences are present even in the low-lying aspects of the PES, such as the internal proton transfer barrier that is underestimated by 20 cm⁻¹ from the B3(H) PES, and they are getting larger as the energy increases, while the external proton exchange barrier, associated with a rotation of the H₃⁺ (see 5C_2v structure above) is overestimated by 200 cm⁻¹ from the B3(H) PES. Such PESs' differences are clearly expected to influence the zero-point energies (ZPEs), the binding D₀ energies and vibrational dynamics, as well as spectral features of the system. In figure 2c, a contour plot of the DFT/B3H surface in the (R,z)-plane is shown, together with the probability density distribution of the ground (n = 0) vibrational state of the H₅⁺ ion, as calculated from quantum nuclear computations, discussed below.

Figure 2.

Minimum energy paths (a,b) and contour plot (c) of the indicated potential surfaces along the R- and z-coordinates (see inset panel). In (c), the probability distribution of the ground vibrational n = 0 energy level for the H₅⁺ is also shown. (Online version in colour.)

(c). Post-harmonic quantum treatments

Given that realistic and accurate representations of the potential are available, for a spectroscopic characterization of the H₅⁺ ion, further connections between the potential and vibrational structures, as well as the sensitivity of the computed properties to the underlying potential, should be explored. The presence of five light hydrogen atoms and the possibility for proton permutation associated with large-amplitude motions make vibrational calculations challenging. So far, various quantum treatments going beyond harmonic approximation have been reported on ground-state energies and properties, as well as on a few vibrational excited states of H₅⁺ and isotopologues. These investigations include diffusion Monte Carlo (DMC) [15,34], path-integral Monte Carlo (PIMC) [35,36], vibrational configuration-interaction (VCI) by using reaction path Hamiltonian (RPH) version of MULTIMODE [14,32], reduced- and full-dimensional variational calculations [46,49,64], as well as MCTDH ones [38,39,41,42] employing all three potential energy surfaces available [55–57].

Some of these studies have also taken into account both quantum and temperature effects on the ground-state properties of these systems. Figure 3 summarizes results from PIMC calculations [35,36] on the radial distributions of the thermal equilibrium states of H₅⁺ and D₅⁺ ions at T = 10 K. For comparison reasons, we also display in the figure the H–H bond lengths corresponding to the four lower 1C_2v, 2D_2d, 3C_2v and 4D_2h structures (see inset panels) on the potential surface, as well as a snapshot from the last MC step of the PIMC calculations of the H₅⁺. For both H₅⁺ and D₅⁺, the radial distributions show large delocalization, indicating the importance of including nuclear quantum anharmonic effects on the structural zero-point states of these systems. Such distributions show that H₅⁺ ions expand in the configuration space, and their ground vibrational states could be described as a transition between the four low-lying energy structures including both internal proton transfer and H₂ rotations around the C₂-axis of H₅⁺, which is completely different from the global minimum 1C_2v structure of its Born–Oppenheimer PES. Furthermore, this result is common for all three PESs available, and, as a consequence of such unusual structural behaviour of its ground vibrational state, the H₅⁺ ion has been classified as an astructural or structureless molecule [46]. Radial distributions of the thermal equilibrium states from the PIMC calculations at T = 10 K have been also found in very good agreement with those reported from DMC and MCTDH calculations (see contour plot distribution shown in figure 2c) using the same PESs [34,39,42,55,56]. This finding together with PIMC results at higher temperatures indicate that quantum effects are much more important than thermal effects for such isotopes [36].

Figure 3.

Quantum radial distribution functions of the thermal equilibrium states of H₅⁺ and D₅⁺ ions at T = 10 K as a function of the H–H and D–D bond distances, respectively. (Online version in colour.)

In addition, various studies in the literature have also reported ZPEs and binding (D₀) energies for H₅⁺ and its deuterated isotopologues [15,34–36,39,47]. Figure 4 depicts the ZPE and D₀ values from the deuterated isotopologues of interest in this work, as they were calculated from full-dimensional 9D MCTDH calculations [39]. As can be seen, ZPE values are decreasing as the number of D atoms is increasing, and the energies of the structures with an H atom in the central position are lower than those with a D atom as the central one.

Figure 4.

Potential energy curve of the H₅⁺ fixed at its 1C_2v orientation along the intermolecular distance of the H₃⁺ and H₂ fragments. Binding energies and ZPE for the isotopologues of interest in this study are also shown. (Online version in colour.)

The importance of the ZPE values of all 12 H₅⁺ isotopologues has been extensively discussed in connection with their implications in reactive scattering, such as deuterium fractionation in the interstellar medium [12,13,22,34,65–67]. Owing to their lower ZPEs, deuterated species are thermodynamically more stable; thus their relative abundance is expected to be higher. In this study below, we explore the evolution of energies and wavefunctions focused on the dissociation pathways in which the proton is transferred between the two diatomic H₂ or D₂ molecules; we are interested in the effect of deuteration, in particular of the central proton, and its spectroscopic implications. We should also point out that asymmetrically deuterated H₅⁺ species have non-zero dipole moments, and hence can be detected by rotational spectroscopy [13]. Thus, in figure 5, we display the general structures of different symmetric and asymmetric deuterated isotopologues of H₅⁺, as well as their dissociated fragments, which are of interest in the present study in order to explore the effect of deuteration on the proton exchange process. The upper panel of the figure corresponds to H₅⁺ or D₅⁺ that dissociates to the H₃⁺/D₃⁺ + H₂/D₂ with D₀ energies [39] of 2235 and 2406 cm⁻¹, respectively, while the middle and lower panels correspond to the partial deuterated H₄D⁺ or D₄H⁺, and H₃D⁺₂ or D₃H⁺₂ ions, with their corresponding dissociation pathways, and thresholds at 2190 or 2475 and 2430 or 2186 cm⁻¹, respectively.

Figure 5.

General structures and dissociation pathways of the different H₅⁺ isotopologues studied here. (Online version in colour.)

(d). IR spectra simulations and features' assignment

Given the importance of and interest in H₅⁺ due to its possible detection in the interstellar medium, laboratory-based spectra for both H₅⁺ and D₅⁺ ions have been recently recorded in the region of 300–4500 cm⁻¹ from single-photon and resonance-enhanced multi-photon dissociation experiments [14,32] using a free electron laser. It is clear that astrochemical observation of such molecular ions requires knowledge of vibrational features that should be manifested in their rotational and rovibrational spectra. Thus, the recent mid- and far-IR spectra [14,32] have stimulated low-lying vibrational state calculations [38,39,46,49,64], and further theoretical spectral simulations [14,37,41,42,45,68,69] for these ions. In particular, reduced- and full-dimensional calculations have been carried out in order to provide assignments of the experimentally observed features based on variational, RPH-MULTIMODE and MCTDH methods [14,32,37,41,42,45,68,69].

Figure 6 presents the calculated 4D and 9D MCTDH spectra for the H₅⁺ ion [41,45] using the time-dependent approach within the MCTDH package [70] with ab initio-based potential energy and dipole moment surfaces [55,56], together with the experimental recorded ones [14,32]. Both spectra show rich vibrational patterns, with 10 intense bands below its dissociation threshold, which represent a challenge for state-of-the-art theoretical methods to provide a consistent assignment with the experimental ones. So, in order to assign these features and to obtain the main vibrational character of the state(s) contributing to each of them, 120 vibrationally excited states and their energies were computed from time-independent block improved relaxation (BIR) calculations [45] within the MCTDH methodology [71]. By analysing the probability distributions and their energies, the predominant spectral features were found [45] to correspond to the progressions in the H₃⁺–H₂ stretch (v_R = 1 − 4) of the shared proton (v_z = 1) vibration (see the first five contour plots of the first row of the inset panels, and their energy values in rectangles in figure 6), while the calculated features at 1182, 1876 and 2139 cm⁻¹ correspond to the internal rotation of the H₃⁺ unit by exciting the H₃⁺–H₂ stretching mode (see contour plots of the corresponding first three wavefunctions of the second row in the inset panels of figure 6). All these states are expected to play a central role in the dissociation mechanism of H₅⁺ at low temperatures. Other spectral peaks above dissociation have been also assigned to asymmetric/symmetric H₂ stretch modes and combinations with the shared proton v_z one (see last column wavefunctions' plots in figure 6). We should note that, as expected, the reduced 4D MCTDH spectrum could only reproduce specific spectral peaks [41,42] and has served to guide the full-dimensional computationally expensive spectral simulations [45]. One can see in figure 6 that the predictions of the 4D model are in excellent agreement with the full 9D results for the bands' assignment of the shared proton stretching states.

Figure 6.

Simulated spectrum of H₅⁺ from 4D (cyan line) and 9D (black line) MCTDH computations [41,45], and its comparison with the single- and multiple-photon dissociation spectrum from [14,32]. The assignment of spectral features with the probability density distribution of the corresponding vibrational state is also shown (see text). (Online version in colour.)

In this vein, we adopted this 4D model as described in [41] to further investigate the effect of deuteration on the spectral features of the proton transfer vibration. Figure 7 shows the spectra of H₅⁺ and D₅⁺ computed from 4D MCTDH simulations [41], as well as their stick spectra and those of the partially deuterated H₄D⁺, H₃D⁺₂, D₃H⁺₂ and D₄H⁺ isotopologues (figure 5), as they were computed from the present time-independent BIR MCTDH calculations, together with their assignment to specific proton stretch vibrations.

Figure 7.

Calculated spectra of the H₅⁺ (solid black line) and D₅⁺ (solid green line) ions in the 250–3000 cm⁻¹ spectral region from the 4D MCTDH theoretical simulations [41], together with the assigned spectral lines of the proton stretch vibration progressions for the H₅⁺ (dashed black lines), H₄D⁺ (solid magenta lines), H₃D⁺₂ (solid light green lines), D₃H⁺₂ (solid orange lines), D₄H⁺ (solid indigo lines) and D₅⁺ (dashed green lines) isotopologues, computed in the present study. Arrow lines indicate the dissociation D₀ energy of each isotopologue. Comparison with the experimentally observed spectra [14,32] of H₅⁺ (solid blue line) and D₅⁺ (solid violet line) (shifted by 139.6 and 78.1 cm⁻¹ according to 9D MCTDH calculations [39,41], respectively, to facilitate comparison among the plots) is also displayed. (Online version in colour.)

Figure 8 shows contour plots of the probability density distribution for each assigned vibrational energy level and each isotopologue. In a first glance, one can see that the energies of spectral lines obtained from time-dependent [41] and time-independent calculations for the H₅⁺ and D₅⁺ species are in excellent accord for the whole spectral range. By analysing the nodal structure of the corresponding wavefunction (figure 8), we assigned all energy levels shown in figure 7 as progressions in the proton transfer mode. In particular, the fundamental v_z = 1 is found at energies of 505, 340, 496, 323, 498 and 316 cm⁻¹ for each of the H₅⁺, H₄D⁺, H₃D⁺₂, D₃H⁺₂, D₄H⁺ and D₅⁺ isotopologues. These values have been found to be somewhat higher by 139.6 and 78.1 cm⁻¹ for the H₅⁺ and D₅⁺, respectively, by comparison with values from full 9D MCTDH calculations [39,41], due to the coupling with bending and torsional motions of the ions. In turn, at energies of 830 and 567 cm⁻¹, we found the dark IR v_R = 1 state for the H₅⁺ and D₅⁺, respectively, while the next bright band of lines is found at energies of 1189, 959, 1110, 883, 1048 and 822 cm⁻¹ for each of the H₅⁺ isotopologues under study, and assigned to the (v_z = 1, v_R = 1). This state couples the shared-proton mode with the intermolecular R stretch one, and as one can see in figures 7 and 8 progressions of such states with v_z = 1 and v_R = 2, 3 and 4 are manifested in the spectra of all isotopologues studied, indicating a transition from the normal-mode to local-mode states. Again, at higher energies we found another transition from such local mode to normal mode again with the v_z = 3 state at energies of 2768, 2203, 2740, 2064 and 1862 cm⁻¹ for all species studied, except the D₄H⁺ one. This state is above the dissociation threshold for the lighter H₅⁺, H₄D⁺ and H₃D⁺₂ ions, and as can be seen manifests its signature in the experimentally recorded spectra of both H₅⁺ and D₅⁺ [14]. Such changes in the underlying dynamics are common for all isotopologues studied here, generating a complex pattern in the spacing of the lines, and could generate a difference in the interpretation of the quantum number assignments for the bands, if individual normal-mode treatments are employed [14,39–41,45].

Figure 8.

Contour plots of the probability density distributions of the indicated vibrational states assigned to the proton stretch vibration progressions for the H₅⁺, H₄D⁺, H₃D⁺₂, D₃H⁺₂, D₄H⁺ and D₅⁺ isotopologues (from left to right panels, respectively). (Online version in colour.)

On the basis of the plots in figure 8, we can see the evolution of the indicated vibrational states assigned to proton-stretch excitations up to energies of 2800 cm⁻¹. Although the two potential minima in the z-coordinate are equivalent (figure 2b), nuclear vibrations break the symmetry in the case of partially deuterated species, and this can be seen in the plots of probability density distributions of the H₃D⁺₂ and D₃H⁺₂ species. For these two isotopologues, the central node of the wavefunction is shifted to non-zero values for the z-coordinate (see two middle columns of figure 8), and as the v_R excitation increases the symmetry disappears, with the central H or D to be localized closer to the H₂ or D₂ moieties. For the remaining isotopologues, one can see a gradual expansion of the wavefunction in R-coordinate as expected when moving from D₅⁺, D₄H⁺, H₄D⁺ to H₅⁺, while as vibrational excitation increases the probability amplitude of the wavefunction is extended along the dissociating R-coordinate.

3. Conclusion

We briefly assessed the current status of the intriguing H₅⁺ system, the smallest proton-bound dimer, on the basis of our recently reported studies. Very recently, new IR spectra have been experimentally recorded for both H₅⁺ and D₅⁺, refocusing attention on the unusual spectroscopic properties of these ions, and recalling the interest of further theoretical investigations. Advances in computer technology matched with algorithm developments have provided the first theoretical models based on sophisticated anharmonic treatments.

Regarding electronic structure calculations on this four-electron ion, we reviewed our previously published results, with emphasis on recently developed PESs, which are characterized by several nearly isoenergetic low-lying stationary points, and the presence of low energy barriers for internal proton interconversion. We examined anew the energetics of such PES configurations by means of current state-of-the-art ab initio calculations at MRCI level of theory, and discussed the accuracy of the available PESs. A consideration of this issue relies on the improving and fitting in order to describe with higher accuracy the underlying intermolecular interactions. Recently, data-driven approaches, also known as machine learning techniques [72], have been introduced with the primary objective to achieve both accuracy and computational efficiency by training the machines with existing data using suitable algorithms, such as artificial neural networks. Attempts in this direction are a promising route to follow, as they could provide new avenues to construct complicated PESs, and are an essential step in developing PESs for higher-order hydrogenic systems.

Full-dimensional rovibrational state calculations and spectral simulations for such fluxional and highly symmetric species have proved very challenging. The existence of large-amplitude motions makes the assignment of spectral features based on harmonic analysis doubtful, and requires more sophisticated treatments. In this regard, we reviewed our studies from reduced- and full-dimensional time-dependent and time-independent calculations within the MCTDH framework. On the basis of our results, we have been able to assign the main bands of the observed H₅⁺ and D₅⁺ spectra to progressions in the H₃⁺–H₂ stretch mode with the proton-shared and internal H₃⁺ rotation. Here, we extend these calculations to explore the effect of deuteration on the proton stretch vibrations from 4D MCTDH computations. We were able to observe specific transitions from normal to local modes and vice versa, which manifest their fingerprints in the spectra. However, full nine-dimensional spectral simulations have shown that progressions of the H₃⁺–H₂ stretch and the internal rotation of H₃⁺ are also main features in the observed spectra of H₅⁺ and D₅⁺ cations, so including such motions should also be considered for specific deuterated species. Clearly, further explorations of the nature of such spectral features involving vibrations along the H₃⁺ and H₂ reaction coordinate should also contribute to deepen our understanding of the mechanisms and their implications in the deuterium fractionation in the interstellar medium.

Data accessibility

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Author's contributions

R.P. carried out the electronic structure calculations. A.V. performed the quantum vibrational computations. Both authors participated in data analysis. R.P. conceived of and designed the study, and drafted the manuscript. A.V. critically revised the manuscript. Both authors read and approved the manuscript.

Competing interests

The authors declare that they have no competing interests.

Funding

This work has been supported by MINECO grants no. FIS2014-51933-P and FIS2017-83157-P, ‘CSIC for Development’ (i-COOP) ref: ICOOPB20214, COST Action CM1405(MOLIM), and by the Research Headquarters Address Bogota - DIEB, National University of Colombia, HERMES code: 37338.