Quantum vibrational study of the proton-bound [Ar–H–He]⁺ mixed noble-gas dimer

Abstract

Proton-bound heteronuclear noble-gas complexes serve as challenging test systems for exploring proton-mediated interactions, weak intermolecular forces, and nuclear quantum effects in chemically simple yet highly anharmonic environments. In this work, we present a new three-dimensional machine-learning potential energy surface (ML-PES) for the noble-gas hydride cation [Ar–H–He]⁺, constructed within a reproducing kernel Hilbert space (RKHS) formalism and trained to high-level CCSD(T)/CBS reference energies. Particular attention is devoted to the generation, training, and validation of the PES to ensure a dense, smooth, and globally consistent representation of the interaction potential over the full configuration space relevant for quantum dynamics. The resulting ML-PES is employed in multi-configuration time-dependent Hartree (MCTDH) quantum calculations to obtain accurate vibrational energy levels for the most abundant isotopologues of [Ar–H–He]⁺. Our results reveal noticeable differences in both energies and level assignments compared to earlier theoretical studies, highlighting the importance of PES quality for reliable spectroscopic modeling. The vibrational energy levels and transition positions reported here provide spectroscopic benchmarks and may support future laboratory measurements and astrophysical observational investigations of these proton-bound noble-gas molecular systems.

A CCSD(T)/CBS machine-learning potential energy surface and MCTDH quantum calculations reveal the strongly anharmonic vibrational structure of the proton-bound mixed noble-gas complex [Ar–H–He]⁺, providing new spectroscopic benchmarks.

The chemistry of noble gases (Ng), long regarded as paradigmatic examples of chemical inertness, has attracted renewed attention following the detection of ionic noble-gas species in the interstellar medium (ISM).^1–6 In particular, the identification of HeH⁺ in the planetary nebula NGC 7027 and of ArH⁺ in both galactic and extragalactic environments^7–9 has unequivocally demonstrated that, under extreme astrophysical conditions, noble gases can actively participate in chemically relevant processes. These discoveries have highlighted noble-gas hydride ions as chemically non-trivial species whose structure and spectroscopy are governed by subtle proton-mediated interactions.

Within this framework, proton-bound homonuclear and heteronuclear triatomic complexes constitute fundamental molecular systems for elucidating how a proton can mediate and stabilize interactions between noble-gas atoms. Homonuclear species, such as HeH⁺He and ArH⁺Ar, exhibit linear and highly symmetric structures. In contrast, heteronuclear complexes display pronounced structural and electronic asymmetry, directly linked to differences in the proton affinities of the constituent atoms. Among noble-gas hydrides, HeH⁺ represents a limiting case, as helium possesses the lowest proton affinity of all elements, whereas heavier noble gases such as argon exhibit substantially higher proton affinities, as established by thermochemical studies.¹⁰ Consequently, in heteronuclear complexes such as ArH⁺He, the proton is preferentially localized toward the atom with higher proton affinity, leading to a strongly asymmetric charge distribution. These characteristics make mixed noble-gas complexes challenging test systems for assessing the interplay between weak intermolecular forces, proton localization, and nuclear quantum effects, as well as for benchmarking the accuracy of PESs and their impact on rovibrational spectroscopy.

From this perspective, heteronuclear proton-bound complexes constitute particularly valuable model systems for probing the chemical behavior of noble-gas hydrides. In contrast to NgH⁺ diatomic molecules, which are dominated by a largely covalent bond, these triatomic species are primarily stabilized by weak van der Waals interactions modulated by the presence of a shared proton. Importantly, such mixed cationic complexes possess permanent electric dipole moments and are therefore rotationally active, making them, in principle, accessible to rotational and/or vibrational spectroscopy. Although their abundance and formation pathways in the ISM remain uncertain, accurate spectroscopic predictions are essential for any future observational searches and their potential detection in extreme astrophysical environments.

Despite their fundamental interest, the available literature on proton-bound heteronuclear noble-gas complexes remains scarce. This limited coverage can be partly attributed to the greater stability of homonuclear systems, which typically exhibit stronger binding and have thus been more accessible both experimentally and theoretically. Early spectroscopic evidence for mixed NgH⁺Ng′ species was reported by Fridgen and Parnis,¹¹ followed by a limited number of theoretical investigations primarily focused on the construction of force fields and PESs.^2,12–17

For the specific case of the [Ar–H–He]⁺ complex, only two PESs are currently available in the literature. The earliest was a two-dimensional PES based on CCSD(T)-F12/AVTZ calculations,¹⁷ and the latest a three-dimensional CCSD(T)/CBS[Q5] PES,¹⁶ both employed in collisional and dynamics calculations. To our knowledge, results on rovibrational coefficients and transitions are only available from CcCR (coupled-cluster with core correlation and relativistic corrections) quadratic force-field (QFF) calculations.¹³

The work presented here builds upon our previous efforts to assess proton-bound noble gas complexes through high-level ab initio potential surfaces, with the specific goal of overcoming the limitations of existing PESs and enabling a more reliable description of rovibrational spectroscopy. The use of a machine-learning PES is essential to achieve a dense, smooth, and globally consistent representation of the interaction potential, while preserving spectroscopic accuracy over the full configuration space required for quantum dynamical calculations. To this end, we employ the MCTDH methodology, which provides an efficient and accurate framework for treating the nuclear quantum dynamics of weakly bound molecular systems, especially those characterized by large-amplitude motions and strong mode coupling.

1. Computational details, results and discussion

1.1. Electronic structure calculations

All ab initio electronic structure calculations were carried out using the MOLPRO 2022 program package.¹⁸ The electronic-structure calculations were performed on the HPC facilities of SGAI-CSIC and CESGA using distributed CPU-based resources. The DENEB software¹⁹ was employed to generate the input geometries and to manage and organize the corresponding output data.

As a first step, the equilibrium geometry of the [Ar–H–He]⁺ was determined at the CCSD(T)/AV6Z level, followed by a harmonic vibrational frequency analysis. In order to identify the most stable isomer, several structural arrangements were explored. In all cases, the geometry optimizations converged to linear configurations with the proton located between the two noble-gas atoms (see upper panel in Fig. 1), corresponding to a total energy of −530.14393297 a.u. The addition of a He atom to the ArH⁺ fragment induces only a minor perturbation of the Ar–H⁺ bond, which is elongated by 0.0118 Å with respect to isolated ArH⁺. In contrast, the He–H⁺ distance is more significantly affected when compared to that in the HeH⁺ cation or in its neutral analogue He–H (see ref. 6 and 20). Replacing the He atom with a heavier noble gas leads to a markedly different behavior: for instance, in the ArH⁺Ar complex,²¹ the coordination of a second Ar atom results in an Ar–H⁺ bond elongation approximately 17% larger than that observed in ArH⁺. The upper panel of Fig. 1 also displays two additional linear local minima of the [ArHHe]⁺, which correspond to the He–Ar–H⁺ arrangement at total energy of −530.14188108 a.u., lying only 0.00205189 a.u. (450.34 cm⁻¹) above the global minimum. In this configuration, the He atom is attached to the Ar atom, with the ArH⁺ bondlength remaining essentially unchanged with respect to the isolated ArH⁺, while the second local minimum corresponds to the Ar–He–H⁺ configuration lying at significantly higher energy of −530.07207673 a.u., 0.07185624 a.u. (15 770.62 cm⁻¹) above the global minimum.

Fig. 1. (Upper panel) optimized equilibrium structures and total energies of the [Ar–H–He]⁺ complex from the CCSD(T)/AV6Z computations. (Lower panel) relative formation/dissociation energies (in cm⁻¹) with respect to the global ArH⁺He minimum (in cm⁻¹) computed at the CCSD(T)/AV6Z level.

The harmonic vibrational frequencies of the optimized ⁴⁰Ar–H⁺–⁴He structure are also obtained at the CCSD(T)/AV6Z level. As in other linear triatomic proton-bound systems, this heteronuclear cation exhibits four vibrational modes, corresponding to the Ar–H⁺ stretching mode (ω₁) at 2542.57 cm⁻¹, the doubly degenerate proton bending mode (ω₂) at 325.49 cm⁻¹, and the He–H⁺ stretch (ω₃) at 233.52 cm⁻¹. These modes yield a harmonic zero-point energy (ZPE) of 1713.535 cm⁻¹. Both the equilibrium geometry and the harmonic frequencies are in very good agreement with previously reported theoretical studies.^2,13,15 We should note that the proton shuttle motion is described more accurately as an Ar–H⁺ stretch vibration, while the He–H⁺ stretching is an order of magnitude smaller, reflecting the relatively weak interaction between the He atom and the ArH⁺.

Several dissociation pathways were examined at the CCSD(T)/AV6Z level in order to identify the relevant fragmentation channels and assess the relative stability of the [ArHHe]⁺ complex. The corresponding dissociation energies are summarized in the lower panel of Fig. 1. Our calculations indicate that the lowest-energy total dissociation channel corresponds to He + Ar + H⁺, located 33 399.37 cm⁻¹ above the global minimum. In contrast, the lowest partial dissociation pathway, leading to ArH⁺ + He, lies only 703.70 cm⁻¹ above the global minimum. This value is in good agreement with previous theoretical estimates reported.^16,17 The next accessible dissociation channel, corresponding to Ar + HeH⁺, is located at 16 923.43 cm⁻¹, more than an order of magnitude higher in energy than the ArH⁺ + He, confirming that ArH⁺ + He is the lower dominant dissociation pathway.

1.2. Potential model: training and validation

For the generation of the PES, coupled-cluster calculations including single and double excitations with perturbative triples, CCSD(T), were performed using the aug-cc-pV5Z (AV5Z) and aug-cc-pV6Z (AV6Z) basis sets.^22,23 Correlation energies were extrapolated to the complete basis set (CBS) limit using the two-point inverse power formula proposed by Schwartz,²⁴ with n = 5 and 6, yielding CCSD(T)/CBS[56] energies. The total CBS energy was obtained as, E_CBS = E_HF/AV6Z + E^corr_CBS.

The Jacobi coordinates (r, R, θ) were used to sample the configurational space, where r denotes the Ar–H⁺ bond distance, R the distance between the He atom and the center of mass of the ArH⁺ fragment, and θ the angle between them. For each configuration, the interaction energy was defined as, ΔE = E_ArH⁺He − E_ArH⁺ − E_He, with the total dissociation limit Ar + H⁺ + He taken as the zero of energy (see Fig. 1). All energies entering this expression correspond to CCSD(T)/CBS[56] values evaluated at the same geometry, with basis set superposition error (BSSE) corrections²⁵ being negligible at the CBS level for this class of systems.

The reproducing kernel Hilbert space (RKHS) ML scheme was employed to represent the PES of the [ArHHe]⁺ system, due to its ability to exactly reproduce the training data, and at the same time ensuring a physically sound description of long-range interactions through an appropriate choice of kernel functions. The potential form is given by

1

where the C_ijk coefficients are obtained by solving the corresponding system of linear equations, and N_r, N_R and N_θ are the number of ab initio data points along each coordinate. The k₁^n,m, k₂^n,m and k₃ are the 1D reproducing kernel functions for the diatom distance-like (r), the He-diatom distance-like (R) and angle-like (θ) variables, respectively, given by , with χ = r or R, χ_> and χ_< are the larger and smaller of the χ, respectively. The angle-like kernel function has the form, . The n and m superscripts refer to the order of smoothness of the kernel function and its asymptotic behavior at large distances, with n = 2, and m = 3 accounting the r⁻⁴ and R⁻⁴ leading dispersion interaction between Ar and proton and He with ArH⁺ molecular ion, respectively. B is the beta function, 2F1 is the Gauss hyper-geometric function, and P_ll the Legendre polynomials with ll = 0–18.

Owing to the triatomic nature of the system, the relevant configurational space can be systematically explored using Jacobi coordinates and structured grids, which provide a controlled and reproducible sampling of the physically relevant regions of the PES. The full CCSD(T)/CBS[56] dataset consists of 11 962 configurations, spanning the ranges r = 1.00–1.90 Å, R = 2.10–7.00 Å, and θ = 0–180°. The whole dataset was divided into two subsets of 3990 points employed for training the model, while the remaining 7972 points were reserved for validation. The spatial distribution of the training set was organized following a structured three-dimensional grid in internal coordinates, including 7 equidistant points along the r-coordinate, 30-points along the R and 19-points along θ.

As illustrated in the upper panel of Fig. 2, the training set is predominantly concentrated in the low-energy region of the potential energy surface. Nearly all configurations (98.6%) lie below the ArH⁺He → Ar + HeH⁺ dissociation channel, indicating that the sampling strategy prioritizes energetically bound and quasi-bound configurations. A fraction of 5.7% of these data sample the energy range of 703.60 cm⁻¹ between the global minimum and the ArH⁺ + He dissociation threshold. The majority of the data (92.90%) are distributed within the energy interval from −32695.68 to −16466.94 cm⁻¹, corresponding to configurations located below the ArH⁺He → Ar + HeH⁺ dissociation limit, and therefore within the dynamically relevant bound region of the PES. In contrast, only a very small fraction of the data (1.41%) lies in the higher-energy region between the Ar + HeH⁺ and below the Ar + He + H⁺ dissociation limits. This distribution clearly confirms that the training set is strongly focused on the energetically most relevant regions of the PES, while still retaining limited coverage of near-dissociative configurations.

Fig. 2. (Upper panel) energy distribution of all configurations included in the training sets of the RKHS ML PES. (Middle panel) the RMSE values (in cm⁻¹) of the RKHS ML-PES as a function of training set size. (Lower panel) correlation plot of the best-performed RKHS ML-PES model against the reference CCSD(T)/CBS[56] energies for the testing set, together with the corresponding average RMSE and MAE values (in cm⁻¹) as a function of energy.

The hold-out cross-validation strategy was applied to obtain the optimal RKHS representation of the [ArHHe]⁺ PES. Four training sets of increasing size, containing 1200, 2280, 2100, and 3990 data, were constructed to ensure full coverage of the corresponding coordinate intervals. The middle panel of Fig. 2 illustrates the evolution of RKHS ML-PES model accuracy as a function of the training set size. The root-mean-square error (RMSE) values were computed using two different testing configurations: the full validation set of 7972 points, and a randomly selected subset of 2625 points, with all configurations lying below the dissociation threshold. In both cases, similar error-reduction trends were observed, highlighting the direct correlation between model performance and the size of the training dataset, particularly when comparing the 1200/2100 and 2280/3990 point datasets.

A systematic decrease in RMSE was observed as the number of sampling points increased, particularly when the angular coordinate θ increases from 10 to 19 points, and the radial coordinate r from 4 to 7. This improvement results in notably low and consistent errors for the 3990 RKHS ML model, which yields RMSE values of 30.62 and 13.22 cm⁻¹ for the 7972- and 2625-point test sets, respectively, in regions below the dissociation threshold. In contrast, the 1200, 2100, and 2290 models exhibit significantly larger errors, ranging from 860 to 2000 cm⁻¹.

Likewise, the correlation plot shown in the lower panel of Fig. 2 displays the performance of the selected RKHS ML-PES compared against the CCSD(T)/CBS[56] reference energy datasets over the full energy range. The RMSE and mean absolute error (MAE), resolved as functions of the energy interval, are also reported in the figure. Overall, the RMSE remains below 36 cm⁻¹ throughout the entire energy range up to the dissociation limit, with the notable exception of the −15000 to −10000 cm⁻¹ interval, where the error increases. This region is sparsely sampled in the training set, accounting for less than 1.5% of the total data, which explains the reduced predictive accuracy. A similar trend is observed for the MAE values, with the error remaining below 7 cm⁻¹ in most energy intervals, while reaching 126 cm⁻¹ in the sparsely sampled region. These results indicate that the RKHS ML-PES provides a highly accurate representation of the low- and intermediate-energy regions most relevant for spectral computations.

1.3. Bound state calculations: quantum effects on vibrations

The nuclear quantum calculations were performed using the MCTDH package of codes.²⁶ The time-dependent Schrödinger equation is solved using the vibrational Hamiltonian expressed in Jacobi coordinates as,

2

where the V(r, R, θ) term is the RKHS ML PES of the [ArHHe]⁺ cation, as given in eqn (1). The nuclear dynamics calculations are performed on a Born–Oppenheimer PES, and non-adiabatic corrections, including the Born–Oppenheimer diagonal correction, are not included in the present study. The Jacobi coordinate system is chosen with the R coordinate along the body-fixed (BF) Z-axis, while the r coordinate is lying in the XZ-plane. The reduced masses are defined as and , with M = m_Ar + m_He + m_H being the total mass of [ArHHe]⁺ system, and m_Ar, m_He and m_H the atomic masses of the ^36,38,40Ar isotopes, ⁴He and H or D, respectively.

The POTFIT algorithm²⁷ was employed to transform the RKHS ML-PES into a sum of products of single-particle operators, required in the MCTDH.^28–30 This procedure yields a set of natural potentials (NPs) that approximate the original PES with controlled accuracy. In the present calculations, the relevant region below the Ar + H⁺ + He dissociation threshold, is considered, with contraction applied on the θ coordinate, and 30 NPs for each of the r and R coordinates, resulting in a potential representation with a RMSE smaller than 0.01 cm⁻¹.

Both the wavefunction and Hamiltonian operator are represented on a discrete grid in the (r, R, θ) coordinates. Convergence is achieved using 51 and 71 harmonic oscillator (HO) discrete variable representation (DVR) functions in each of the r and R coordinates, in the range of [1.0–1.6/1.9] and [2.1–8.5] Å, respectively, while for the angular θ coordinate, 55 Leg DVR functions in the [0,2π] interval are used.

The ground and excited vibrational states of the [^36,38,40ArH⁴He]⁺ and [^36,38,40ArD⁴He]⁺ isotopologues are obtained using the improved relaxation (IR) and the block improved relaxation (BIR) methods, as implemented in the MCTDH package.^26,31,32 A total of 23 vibrational states are computed for each isotopologue using a basis set of (20,20,20) single-particle functions (SPFs) in each coordinate. Convergence in energies below 10⁻⁶ cm⁻¹ was achieved with total propagation time of 312 fs.

All J = 0 states together with their corresponding assignments are listed in Table 1 for the [⁴⁰ArH⁴He]⁺ and [⁴⁰ArD⁴He]⁺, isotopologues. The calculations are restricted to J = 0 in order to isolate the pure vibrational dynamics and to assess the intrinsic effect of the ML PES on the vibrational structure. Rotational-vibrational and Coriolis couplings are therefore not included at this stage, although J > 0 calculations are feasible within the same general framework with an additional computational effort. The computed vibrational energies for all isotopologues studied are provided in Table S1 of the SI. Table 1 also includes a comparison with values reported in previous theoretical studies.¹³

Table 1. Bound vibrational energies for the [⁴⁰ArHHe]⁺ and [⁴⁰ArDHe]⁺ isotopologues calculated with the corresponding CCSD(T)/CBS[56] RKHS ML-PES from the MCTDH computations, and comparison with previous studies,¹³ considering the energy of ZPE as zero. The D_e are given with respect to the total Ar + H⁺ + He dissociation limit energy.

	[40ArHHe]+			[40Ar(H/D)He]+	[40ArDHe]+
	This work			From ref. 13	This work
n/De (ZPE)	−33402.73	θ, R-plot	r, R-plot	—	−33402.73
ZPE/0	−31759.20/0 (0,0,0)			—/0	−32187.34/0 (0,0,0)
1	158.64 (0,0,1)			240.3/219.2	165.31 (0,0,1)
2	271.26 (0,0,2)			429.8/387.9	266.02 (0,2,0)
3	285.37 (0,2,0)			506.4/333.6	287.52 (0,0,2)
4	309.16 (0,0,0)I			—	339.10 (0,0,0)I
5	339.62 (0,0,3)			—	349.95 (0,2,0)
6	352.11 (0,3,2)			—	366.69 (0,0,3)
7	366.11 (0,3,1)I			—	379.85 (0,1,1)
8	374.90 (0,2,3)			—	390.52 (0,2,1)
9	384.63 (0,3,0)			—	405.24 (0,4,1)
10	386.64 (0,0,5)			—	407.38 (0,0,4)
11	391.30 (0,1,3)			—	414.38 (0,4,2)
12	394.65 (0,0,6)			—	422.73 (0,0,4)I
13	404.91 (0,1,5)			—	427.63 (0,1,3)
14	405.85 (0,0,7)			—	433.79 (0,0,6)
15	413.10 (0,1,4)			—	437.04 (0,1,4)
16	415.19 (0,2,5)			—	440.04 (0,2,2)
17	421.78 (−)			—	442.71 (0,3,3)
18	424.31 (−)			—	444.31 (−)
19	433.51 (−)			—	447.02 (−)
—	—	—	—	—
—	2437.38 (1,0,0)			2466.2/1786.8	1770.12 (1,0,0)
—	4671.90 (2,0,0)			4721.7/3466.3	3445.50 (2,0,0)
—	6737.28 (3,0,0)			—	4991.72 (3,0,0)

The anharmonic ZPEs obtained from the RKHS ML-PES are 1646.10, 1644.75 and 1643.53 cm⁻¹ for the [^36,38,40ArH⁴He]⁺ isotopologues, and 1218.76, 1216.99 and 1215.39 cm⁻¹ for the corresponding [^36,38,40ArD⁴He]⁺ ones. As expected, the deuterium substitution decreases the ZPE values. Since no experimental data are currently available in the literature for these systems, comparisons are made with anharmonic ZPEs reported from CcCR quadratic force-field (QFF) calculations.¹³ The present RKHS ML-PES predicts lower ZPEs values than the CcCR QFF results, with differences of approximately 160 and 90 cm⁻¹ in the case of the [^36,38,40ArH⁴He]⁺ and [^36,38,40ArD⁴He]⁺, respectively.

For the fundamental ν₁, ν₂, and ν₃ vibrational modes, the RKHS ML-PES/MCTDH calculations predict the (1,0,0), and (0,0,1) excitation energies at 2440.45/2439.57/2437.38 and 159.11/158.87/158.64 cm⁻¹, respectively, while the ν₂ bending vibration appears at 285.23/285.30/285.37 cm⁻¹ for the [^36,38,40ArH⁴He]⁺ isotopologues. The corresponding CcCR QFF values are 2469.3/2467.7/2466.2, 241.2/240.7/240.3 and 506.7/506.5/506.4 cm⁻¹, respectively. The ν₁ stretching mode frequency is significantly higher than the ν₂ and ν₃ modes, in contrast to the homonuclear NgH⁺Ng proton bound complexes. This behavior reflects the comparatively weak interaction between the He atom and the ArH⁺ core, with the ν₁ frequency being closer to that of the isolated ^36,38ArH⁺, reported at 2612.50/2610.70 cm⁻¹ from high resolution IR absorption spectroscopy.³³

Table 1 also reports 2D probability density distributions of all computed vibrational states into the (r, R) and (θ, R) planes, together with the corresponding 2D contour plot of the underlying RKHS ML-PES. The 1D probability distributions for selected vibrational (ν₁, ν₂, ν₃) states are shown in Fig. 3 and 4. The assignment of each energy level was determined by analyzing the nodal structure of the probability density functions. The ground vibrational state (0,0,0) exhibits no nodes, while the (1,0,0) and (0,0,1) states show a single node along the r and R coordinates, respectively. In turn, the (0,2,0) shows two nodes along θ coordinate, whereas the (0,0,0)^I state is mainly localized in the lower local HeArH⁺ (I) minimum.

Fig. 3. Minimum energy paths of the [ArHHe]⁺ RKHS ML-PES as a function of the angle θ for selected values of the r coordinate, together with selected vibrational energy levels obtained from the MCTDH calculations (see Table 1). The left panel shows the full energy range up to the dissociation channels, whereas the right panel enlarges the low-energy region containing the global ArH⁺He minimum and the local HeArH⁺ (I) minimum. Black dashed lines indicate the partial dissociation thresholds, while the red dashed line corresponds to the energy of the total Ar + H⁺ + He dissociation.

Fig. 4. (Upper panel) 1D cuts of the RKHS ML-PES as a function of the R coordinate for fixed values of θ and r, together with the (0, ν₂, ν₃) or n = 0–19 vibrational levels for the [⁴⁰ArHHe]⁺ and [⁴⁰ArDHe]⁺ isotopologues. Selected 1D probability density distributions of [⁴⁰ArHHe]⁺ vibrational states are also shown along the R coordinate. (Lower panel) same as the upper panel along the r coordinate for fixed values of θ and ArHe (d) distance, together with the (ν₁, 0, 0), with ν₁ = 1–3 vibrational levels. Horizontal solid lines indicate the energies of global and local HeArH⁺ (I) minima, while dashed lines indicate the ArH⁺ + He dissociation threshold.

Higher excited states exhibit increasingly complex nodal patterns, making their assignment more challenging. For example, the (0,3,2) state shows approximately three and two nodes along the θ and R coordinates, respectively. Other states, such as those with n = 13–19, reflect pronounced mode coupling and anharmonicity, displaying an extended nodal structure along the dissociation channel. In contrast, the (ν₁, 0, 0) states lie at much higher energies and remain well localized in the global minimum, with clear assignments to ν₁ = 1, 2, and 3 at energies of 2437.38, 4671.90, and 6737.28 cm⁻¹, respectively.

To illustrate the anisotropy and anharmonicity of the underlying RKHS ML-PES, Fig. 3 displays the computed vibrational levels for the [⁴⁰ArHHe]⁺ and [⁴⁰ArDHe]⁺ together with the potential minimum energy paths as a function of θ for selected fixed values of r. As shown, the PES exhibits a global minimum at the linear ArH⁺He configuration and a secondary local HeArH⁺ (I) minimum, located approximately 495 cm⁻¹ higher in energy and only about 200 cm⁻¹ below the ArH⁺ + He dissociation threshold. This double-minimum topology strongly influences the vibrational structure, leading to partial localization of certain vibrational states in the local well. This feature is particularly relevant for understanding the differences with previous theoretical models. The present global 3D RKHS ML-PES explicitly describes both the strongly bound ArH⁺ core and the weak He–ArH⁺ intermolecular coordinate, including the angular anisotropy and the connection between the global and local minima. In contrast, local harmonic or reduced-dimensional descriptions are less suited to capture the large-amplitude motion and the near-dissociative character of the low-energy vibrational states.

As a direct consequence of this PES topology, the (0, ν₂, ν₃) vibrational manifolds display a high density of states already at low excitation energies. With increasing energy, these states become progressively more delocalized along the R and θ coordinates and approach the ArH⁺ + He dissociation region. Analysis of the corresponding one-dimensional probability distributions up to n = 19 shows that some states remain localized in the global well, others are mainly localized in the local HeArH⁺ (I) well, and several states are distributed over both regions. This behavior is a direct manifestation of the double-minimum topology and strong anharmonicity of the PES, and differs markedly from the vibrational structure of homonuclear proton-bound noble-gas dimers.

In turn, in the upper and lower panels in Fig. 4 we display the (0, ν₂, ν₃) or n = 0–19 and (ν₁, 0, 0) with ν₁ = 0–3 vibrational levels for both [⁴⁰ArH/DHe]⁺ isotopologues, together with selected 1D probability density distributions along the R and r coordinates, respectively. The corresponding RKHS potential curves are also shown as a function of R for fixed θ and r values (upper panel), and as a function of r for θ = 0 and 180° with d_ArHe = 2.8 Å (lower panel). In both cases, the energies of the global and the local (I) minima, as well as the lowest dissociation threshold are indicated. Again, one can clearly observe the high density of (0, ν₂, ν₃) states, with the low lying n = 1, 2 and 5 showing a clear node structure in the R coordinate, while others, such as the n = 16 at higher energy spread all over the R coordinate values, and the n = 13, 14, 15, 17, 18 and 19 in both R and θ range, leading to the He + ArH⁺ dissociation. In contrast, the three (ν₁, 0, 0) states at energies of 2437.38, 4671.90 and 6737.28 cm⁻¹ are found to be well localized in the global potential well with a very clear assignment to ν₁ = 1, 2 and 3, respectively (see also Table 1 2D plots).

Higher-order overtones and combination bands were also reported in previous CcCR QFF calculations¹³ for the (2,0,0), (0,2,0) and (0,0,2), as well as (1,1,0), (1,0,1) and (0,1,1) excitations. Table 1 compares the present anharmonic results with the QFF values in all cases where assignments are possible. Significant discrepancies are observed, particularly for modes involving bending and HeH⁺ stretching motion, reflecting the strong anisotropy and anharmonicity of the underlying PES (see right panel of Fig. 3 and upper panel of Fig. 4). Differences are smaller for the (1,0,0) excitation but increase for higher overtones, consistent with the shape of the RKHS potential along the r coordinate (see lower panel of Fig. 4). These discrepancies can be traced back to the different nature of the underlying potential models. While QFF-based approaches provide an accurate local description around the equilibrium geometry, the present RKHS ML-PES offers a global representation of the interaction landscape, including the weak intermolecular coordinate, the angular anisotropy, the local HeArH⁺ (I) minimum, and the approach to dissociation. The largest deviations therefore appear for bending and intermolecular stretching excitations, which are precisely the modes most sensitive to anharmonicity and large-amplitude motion.

2. Conclusions

In this work, we have presented a new three-dimensional ML-PES for the proton-bound noble-gas hydride cation [ArHHe]⁺, constructed within the RKHS framework and trained to high-level CCSD(T)/CBS reference energies. The heteronuclear [ArHHe]⁺ system represents a particularly challenging test case, as it combines a strongly bound ionic core (ArH⁺) with a weakly interacting noble-gas partner (He), leading to highly anisotropic and markedly anharmonic potential energy landscapes. Particular attention was devoted to the generation, training, and validation of the PES to ensure a dense, smooth, and globally consistent representation of the interaction potential over the full configuration space relevant for nuclear quantum dynamics.

The resulting ML-PES was employed in MCTDH vibrational calculations to obtain accurate energy levels and spectroscopic parameters for the most abundant isotopologues of [ArHHe]⁺. The interplay between proton affinity asymmetry, long-range ion-induced dipole forces, and large-amplitude motion generates a subtle vibrational structure that is extremely sensitive to the quality of the underlying PES. By explicitly analyzing the vibrational states, we were able to assess the impact of PES quality on spectroscopic observables and to quantify differences with respect to previously available studies. The observed differences, particularly for modes involving bending and the weak intermolecular motion, highlight the crucial role of an accurate and well-sampled PES for reliable spectroscopic predictions in weakly bound, proton-mediated systems. The new data indicate that both fundamental and progression bands are dominantly driven by the strength and shape of the underlying interactions. Unfortunately, to date no experimental measurements are available for the [ArHHe]⁺ species, and the present predictions therefore await experimental verification. It should be emphasized that the present results correspond to vibrational energy levels and transition positions obtained for J = 0. Transition intensities are not reported because their calculation requires a dipole moment surface of comparable accuracy to the PES. Similarly, non-adiabatic corrections and full J > 0 rovibrational effects, including rotational-vibrational and Coriolis couplings, have not been included at this stage. These aspects represent natural extensions of the present work toward a complete high-resolution spectroscopic characterization of the [Ar–H–He]⁺ complex.

Overall, the present study provides spectroscopic benchmarks for vibrational transitions for the [ArHHe]⁺ complex, offering valuable insights not only for astrochemically relevant species, but also as a paradigmatic model to assess the interplay between PES quality, anharmonicity, and nuclear quantum effects in mixed noble-gas systems. More generally, the fully quantum-mechanical methodology employed here is readily transferable to other proton-bound noble-gas complexes and related systems, offering a systematic route toward improved potential energy surfaces and predictive quantum dynamical simulations.

Author contributions

The authors confirm contribution to the paper as follows: M. J. MOE (data curation, formal analysis, investigation, methodology, validation, writing – review and editing), A. V. (data curation, formal analysis, investigation, writing – review and editing), R. P. (funding acquisition, project administration, conceptualization, formal analysis, investigation, methodology, validation, writing – original draft preparation, writing – review and editing).

Conflicts of interest

There are no conflicts to declare.

Data availability

The data that support the findings of this study are available within the article and its supplementary information (SI). The ML-PES data are available from the corresponding author upon reasonable request. Supplementary information is available. See DOI: https://doi.org/10.1039/d6ra01445c.