Spectroscopy of H ⁺₃ with energies above the barrier to linearity: rovibrational transitions in the range of 10,000–14,000 cm⁻¹

Abstract

Based on a recently extended potential energy surface for H ⁺₃ with a highly reliable form of the topology of the surface far beyond the barrier to linearity, rovibrational frequencies in the range of 10,000–14,000 cm⁻¹ have been derived and are compared with new experiments. The computed transition frequencies reproduce experimental transitions mostly within a few tenths of a wavenumber, if non-adiabatic effects are crudely simulated using different reduced masses for vibrational and rotational motions. Deviations can only be compensated if non-adiabatic effects are treated more rigorously.

1 Introduction

The theoretical investigations of spectra of polyatomic molecules based on first principles are nowadays a formidable task at least for triatomic systems with two or three hydrogen atoms, if one aims at highest, i.e. spectroscopic, accuracy. The deviations between calculated and experimental spectra depend on three different subjects: (a) the solution of the electronic Schrödinger equation, (b) the solution of the nuclear motion problem, and (c) the question whether the Born–Oppenheimer Ansatz is appropriate. Converged results can be reached for few electron systems with few degrees of freedom. The quality of the potential energy surface decides about the absolute accuracy of the rovibrational frequencies: (a) for scattering problems an extended region of the potential energy surface is needed, and consequently the ab initio calculations are mostly of inferior accuracy and (b) in case of spectroscopy a smaller local region of the potential energy surface is probed.

With two electrons and three protons, H ⁺₃ is the simplest polyatomic molecule; its equilibrium geometry is an equilateral triangle. H ⁺₃ is a rich source of information about ion chemistry, planetary atmospheres, unusual spectroscopic effects, etc., [1] and a benchmark molecule for theorists [2]. Most of the investigations have been performed for the electronic ground state which is a singlet state, although the triplet state is not of less interest at least for theoreticians [3]. Since H ⁺₃ is a fairly floppy molecule, it undergoes large-amplitude vibrational motions to such extent that the rovibrational spectrum does not confirm to many of the standard rules of spectroscopy.

In a very recent paper [4], we presented a new potential energy surface fit (termed BCJK) that is based on additional new 5,900 geometries (compared to the CRJK potential defined in Ref. [5]) with emphasis on non-equilibrium and asymptotic points. Apart from the Born–Oppenheimer energy converged to the accuracy better than 0.02 cm⁻¹, the adiabatic and the leading relativistic corrections are computed at each geometry. Possible choices of nuclear masses simulating the non-adiabatic effects in solving the nuclear Schrödinger equation had been analyzed. A small set of theoretically predicted rovibrational transitions were confronted with experimental data [6, 7] in the 10,700–13,700 cm⁻¹ window of the spectrum.

What influences the accuracy of the rovibrational spectrum? The ab initio energy points of the clamped nuclei electronic structure calculations are accurate up to ΔE ≈ 0.02 cm⁻¹, the accuracy of the adiabatic corrections is ΔE ≈ 0.01 cm⁻¹, and errors in the relativistic contributions are of the order ΔE ≈ 0.01 cm⁻¹. The values of the different contributions (BCJK fit [4]) vary with internuclear distances (relative to the value at the equilibrium distances and for R(H₂–H) <6 bohr): 0–58,000 cm⁻¹ (electronic energy), −22 to 55 cm⁻¹ (diagonal adiabatic correction, the value at the equilibrium geometry is ≈115 cm⁻¹), and −2 to 1 cm⁻¹ (relativistic contribution). The largest error originates from the fit of all contributions to the potential energy surface and depends on the polynomial fit; for the region around the minimum the error was mostly not smaller than ΔE ≈ 0.05 cm⁻¹, i.e., for polynomials we have used in recent works [4, 8–11]. Explicit non-adiabatic corrections are still missing, and are simulated by different masses for different nuclear motions, i.e., vibration and rotation. Additionally, the coupling to the electronically bound triplet state is assumed to be small and has not been taken into account. The effects of Lamb-shift have also been neglected.

The main motivation and goal of the recent work [4] was to extend the most accurate CRJK potential [5] to the regions which enable effective rovibrational analysis for transitions to states lying above the barrier to linearity. Beyond the barrier of linearity, which is ≈10,000 cm⁻¹ above the zero point energy, the density of energy states increases considerably. This leads to a stronger mixing (small energy denominators) of rovibrational energy levels with the same “good” quantum numbers. In addition, the treatment of nuclear motion in linear configurations can lead to singular behaviour (depending on the internal coordinates used) in the kinetic energy hamiltonian, which is inversely proportional to the moment of inertia, i.e., results achieved with different coordinate systems can deviate.

We have applied the same methodology, sketched shortly in the following section, as in Ref. [5] maintaining the convergence of the BO energy and computing the adiabatic and relativistic corrections for points located in the non-equilibrium and asymptotic regions of the PES.

In the meantime, data of new experiments [13, 14] are available, in which (a) using the dual-beam double-modulation technique (Morong et al. [13]), 143 transitions in the range of 10,300–13,700 cm⁻¹ are observed, and (b) applying the method of action spectroscopy (Kreckel et al. [14]), developed by Schlemmer and Gerlich [15], 23 transitions in the range of 11,228–13,333 cm⁻¹ have been found.

Whereas, we can compare our transition frequency calculations with both experimental transitions; IR intensities are given only in the first case. We will analyze the present status of agreement and disagreement between theory and experiments, and especially discuss how the inclusion of non-adiabatic effects will modify the results.

2 Methods

2.1 Contributions from the solution of the electronic Schrödinger equation

The potential energy E for the movement of the nuclei is composed of three contributions: the Born–Oppenheimer (clamped nuclei) energy E _BO, the adiabatic correction E _ad, and the relativistic correction E _rel

$$ E=E_{\text{BO}}+E_{\text{ad}}+E_{\text{rel}}. $$

(1)

The BO energy E _BO was obtained by variationally solving the Schrödinger equation with the non-relativistic clamped nuclei Hamiltonian. The adiabatic correction E _ad and the relativistic correction E _rel, as significantly smaller than E _BO, were computed as the leading-order perturbative corrections to E _BO. The Ansatz for the electronic wave function uses two-electron basis functions known as Gaussian geminals (GG) [12].

The adiabatic correction was computed by means of the Born–Handy method [16] in which E _ad is evaluated as an expectation value of the nuclear kinetic energy operator expressed in laboratory coordinates

$$ E_{\text{ad}}=\int\Uppsi(\user2{r}_1,\user2{r}_2)\left(-\sum_{I=1}^3\frac{\nabla_I^2} {2M_I}\right) \Uppsi(\user2{r}_1,\user2{r}_2)\; {\text{d}}\user2{r}_1{\text{d}}\user2{r}_2.$$

(2)

In the above equation, M _I are the nuclear masses and Ψ is the electronic wave function dependent parametrically on the nuclear coordinates. This approach [17] can be considered superior to the classic approach based on the separation of the center of mass motion [18], because it is simpler (cartesian derivatives, and not molecule dependent coordinate derivatives) and includes all contributions (from relative vibrational, rotational and center of nuclear mass motion).

The leading relativistic correction had been calculated in the frames of the Direct Perturbation Theory (DPT) [19–23]. The lowest order correction can be obtained from the following expression [19, 24, 25]

$$E_{\text{rel}} = \langle\hat H_{\rm BP}\rangle + \Updelta_{\rm DPT}, \quad \Updelta_{\rm DPT} = \frac{1}{2} c^{-2} \langle \hat{T} \left(\hat{H} -E_{\text{BO}} \right) \rangle$$

(3)

and $\langle\hat{H}_{\rm BP}\rangle$ is the expectation value of the Breit-Pauli (BP) Hamiltonian computed with the non-relativistic wave function ($\hat{T}$ is the kinetic energy operator and $\hat{H}$ is the non-relativistic Hamiltonian).

The analytical fit to the computed energy points can be performed in different ways getting a global or local description. We choose a form where the electronic energies, relativistic, and adiabatic contributions were added up for each point and a power series expansion in Morse-type symmetry adapted deformation coordinates was performed for the 16th order polynomial fit of the potential V. The weighted root mean square deviation (RMS) for the fit is 1.16 cm⁻¹ and is based on 2,723 points limited to distances shorter than R(H₂ − H) = 6a ₀. An increase in the range of the points to the dissociation limit leads to a poorer fit. Compared to former fits with an RMS of ≈0.05 cm⁻¹ based only on 69 points, the new fit seems to be a change to the worse. If we would have used the original fit based on 69 points, the new energy points would be fitted in the following way: all points up to 20,000 cm⁻¹ would have an RMS value of 3.69 cm⁻¹ and all points surrounding the given 69 points (using Jacobi coordinates: with shifts of ±0.003 bohr for r, ±0.25 bohr for R, and ±3° for the angle θ, as described in Ref. [4]) have an RMS value of 0.60 cm⁻¹ and all new points together used in the present fit would have an RMS of 154.03 cm⁻¹. The present fit shows the following details: all points up to 20,000 cm⁻¹ would have an RMS of 1.17 cm⁻¹, but a slightly worse RMS for all points surrounding the given 69 points of 0.86 cm⁻¹. Whereas the electronic potential is smoothly changing for increasing R(H₂ − H), the adiabatic corrections show a steady increase at r(H ₂) for larger R(H₂ − H) values, i.e. for R > 6 bohr. All the discussed details are a good reason not to rely on older fits based only on 69 points as mostly used in the past. More detailed information about the energy points is given in Ref. [4].

2.2 Solution of the nuclear Schrödinger equation

Perturbation theory is not reliable for the precise solution of vibrational and rotational problems of floppy molecules like H ⁺₃ . In such difficult cases, variational methods have to be used, although, compared to perturbational calculations, their use is usually computationally more expensive [26].

In this paper, we report results of quantum mechanical variational calculations performed for the rovibrational states of H ⁺₃ , using (a) the method of Sutcliffe and Tennyson for triatomic molecules (Jacobi coordinates), implemented in the program package DVR3DRJ [27], (b) using hyperspherical coordinates in conjunction with hyperspherical harmonics [28, 29], and (c) using a filter-diagonalization method [30] with a Fourier-DVR (discrete variable representation) grid for Jacobi coordinates [31–33]. We will not discuss the contributions coming from the different methods, but present results based on the adiabatic approximation and improvements related to non-adiabatic contributions. Details in respect to program suites, basis sets, etc. are given in Ref. [4].

The selection rules for the transitions can be summarized in the following way: the two possible vibrations are the totally symmetric mode ν ₁ and the double degenerate mode ν ₂. The degenerate mode ν ₂ gives rise to an vibrational angular momentum l. The vibrational angular momentum is strongly coupled to the overall rotation of the molecule, and thus k (the projection of the total rotation vector J) nor l are good quantum numbers. It became useful to define $G=\vert k-l\vert$ as a more ‘robust’ quantum number. For H ⁺₃ with three spin 1/2 nuclei one state with total spin I = 3/2 (ortho H ⁺₃ ) and two states with spin I = 1/2 (para H ⁺₃ ) are possible; the ortho/para statistical weights are 4/2. The nuclear spin wave function of ortho H ⁺₃ is totally symmetric with respect to exchange of the particles, while that of para H ⁺₃ is degenerate, i.e., partially antisymmetric. If one ignores the very small hyperfine interaction, three quantum numbers can be regarded as good: the total angular momentum J, the parity ± and the total nuclear spin angular momentum I. Rigorous selection rules are ΔJ = 0 or ± 1; +, ↔, −, and ΔI = 0 for single photon electric dipole transitions. The last selection rule can be interpreted in the sense that ortho (G = 3n) to para (G = 3n ± 1) transitions are forbidden (n is integer). The parity change is related to Δk = ±1 (with parity = (−1)^k). Due to the symmetry restrictions some eigenstates do not exist—most notably the vibrationless states for J even and G = 0. The physical ground state of para H ⁺₃ is (J = 1, K = 1) (with K = |k|), but for ortho H ⁺₃ it is (J = 1, K = 0) [34]. Because the molecular symmetry group for H ⁺₃ is D _3h , transitions only between A ^′₂ ↔ A ^″₂ (ortho) and E′ ↔ E″ (para) states are allowed. For further details about symmetry, selection rules, assignments, and forbidden rotational transitions, see Lindsay and McCall [35].

The nuclear dynamics calculations can be performed with different choices of mass (e.g., nuclear or atomic masses). Depending on the level of theory (see [5] and [9]) used to generate the electronic potential, i.e., BO approximation, BO plus adiabatic contributions, etc., different masses are used within the nuclear Schrödinger equation. The reason for this technique is to simulate the experiment or to remove the deviations to complete theory [8, 36–38]. In the current study, we have used the nuclear mass NU = 1.0072764 amu. The influence of non-adiabatic effects can be simulated by considering the vibrational and rotational motion with different effective masses (VR) [10]. For the case of diatomic molecules, the idea of two different masses has been introduced by Bunker and Moss [36, 37] and rigorous formulas for the internuclear distance dependence of the masses have been derived in Refs. [39, 40, 42]. The analysis of H₂ has shown that adiabatic and non-adiabatic contributions to the frequencies are of the same order of magnitude. This means that if one aims at obtaining high accuracy data for spectroscopic properties or reaction dynamics, the concept of a typical, i.e., E _BO + E _ad, potential energy surface can no longer be justified; in our calculations for the rovibrational spectrum of H ⁺₃ we have reached this point. A promising approach for such a case is the theory of non-adiabatic PES [40–42]. These approaches have shown that in the case of H ⁺₃ , the appropriate masses would be the “atomic mass” ${\bf NU23} (\hbox{NU23}=\hbox{NU} + \frac{2} {3} m_{\rm e})$ for vibration and “nuclear mass” NU for rotation. Such a choice is meaningful if adiabatic corrections have been taken into account at the level of ab initio calculation and non-adiabatic contributions have to be simulated. Numerical calculations at low transition frequencies have already shown in the past quantitative differences (compared to ‘best adapted’ masses) of 0–2 cm⁻¹ when using these different masses. Improvements are expected in the near future.

In the present work, the rovibrational eigenstates and the transitions were calculated in two different ways: (1) using NU for rotation and vibration, and (2) using different effective masses for vibrational and rotational motion [10, 43] (μ_V = 1.007642100 amu, μ_R = 1.0072764551 amu). The latter variant is referred to as VR for vibrational and rotational mass. Polyansky and Tennyson (PT) [43] found that the kinetic energy expression had to be modified by an additional term if vibrational and rotational masses were not the same. This extra term is not large, but was found to be important especially for H₂D⁺ making the residues compared to experimental values systematically smoother and smaller.

3 Results and discussion

Rovibrational eigenstates of H ⁺₃ and its various isotopologues have been analyzed in recent years by several groups [4, 8–10, 29, 43–52, 54]. Most of these calculations were based on less accurate PESs. The highly accurate CI-R12 (configuration interaction with an explicit linear r ₁₂ term in the wave function [8]) and GG PESs (including adiabatic and relativistic effects) used for H ⁺₃ , H₂D ⁺, D₂H⁺, and D ⁺₃ have improved the situation [2]; these potentials are termed RKJK [8] and CRJK [5]. In the most recent work of Schiffels et al. [29, 53] (using our GG PES [9]), band origins up to 13,000 cm⁻¹ have been investigated, termed SAH(CRJK). They could show that non-adiabatic effects can be modeled by empirical corrections based on calculations using only the nuclear masses (termed SAHc), and that these empirical corrections had been proven very helpful for experimental groups to assign their new data [6, 7, 13, 14].

In the present paper, the rovibrational analysis is performed with the latest fitted potential, described in Ref. [4]. Three different numerical procedures for the calculation of the rovibrational energies and intensities have been used: (a) DVR-J: DVR with Jacobi coordinates [27] (using C _2v -symmetry; less accurate for energies above the barrier to linearity, i.e., this might result from a singularity in the kinetic energy operator for linear geometries; for further discussion see Ref. [4]), (b) HYP: hyperspherical coordinates with hyperspherical harmonics [28] (keeping D _3h -symmetry; no numerical problems for energies higher than the barrier to linearity), and (c) FD-J: filter-diagonalization with Jacobi coordinates (C _2v -symmetry; used for cross check of the DVR-J code; results are in close agreement with HYP calculations).

New results of transitions above the barrier to linearity in the frequency range of 10,000–14,000 cm⁻¹ will be presented. The results (see Tables 1, 2, 3, 4) are compared to the most recent experimental results of Gottfried [6, 7], Morong et al. [13], and Kreckel et al. [14]. Whereas for most of the experimental transitions, the uncertainty is claimed to be in the range of ≈0.01 cm⁻¹, nothing is specified for the quality of experimental relative intensities. The calculated rovibrational energies are based on HYP calculations (using the NU masses), and the data were selected with respect to intensity calculations (using DVR-J) in the given frequency range with relative intensities mostly larger than 1%. In addition, a list of transitions (with relative intensities >5%) not seen experimentally, are given in Table 6. The energy corrections E(VR) − E(NU) (in the range of 0–2 cm⁻¹), which take into account the influence of non-adiabaticity using VR masses are calculated with the DVR-J code, because the necessary change of the kinetic energy hamiltonian is implemented only within DVR-J and the calculated eigenstates are in most cases not too different from the numerically correct ones, i.e., HYP calculations. In summary, E(NU) are HYP calculations and E(VR) = E(NU, HYP) + (E(VR, DVRJ) − E(NU, DVRJ)). So, the frequencies are ‘mainly’ based on HYP calculations. Since the HYP calculations provide the correct symmetry for the eigenfunctions, the selection of symmetry allowed transitions is based only on these calculations, whereas the Einstein coefficients for intensities are calculated with the DVR-J code. Less converged DVR-J calculations can lead to suboptimal intensity values, and therefore mismatches compared to experimental intensities might be related to that.

Table 1 Comparison between experimental (Morong et al. [13]) and theoretical transitions (in cm⁻¹) with final and initial assignment of the rovibrational eigenstates using the masses VR

Table 2 Comparison between experimental (Kreckel et al. [14]) and theoretical transitions (in cm⁻¹) with final and initial assignment of the rovibrational eigenstates using the masses VR

Table 3 Comparison between experimental (Morong et al. [13]) and theoretical transitions (in cm⁻¹) using nuclear mass NU

Table 4 Comparison between experimental (Kreckel et al. [14]) and theoretical transitions (in cm⁻¹) using nuclear mass NU

For the calculation of the spectrum we assumed a temperature of T = 600 K (as claimed in the work of Morong et al. [13]; T = 300 K calculations show larger deviations compared to experiment). The intensities of the transitions, computed with the DVR-J suite, are based on the dipole moment functions calculated on CI-R12 level [8]. In the present study, we have used the assignment, especially of the final state, already given in the literature [13, 14], because in several cases we could not clearly assign the transitions to particular vibrational quantum numbers of the final state. The effect of non-adiabaticity, i.e., a comparison of VR and NU results, is plotted in Figs. 1, 2, and 3. With NU masses the theoretical frequencies are slightly too large Δν _exp-calc ∈ 〈−0.5,  −1.5〉 cm⁻¹, whereas the correction with VR is roughly 1 cm⁻¹ towards lower frequencies. The choice of the VR masses requires improvement based on rigorous theory considerations (see e.g., [42]). Some frequencies do not fit into this set of data. The intensity of the transitions deviates stronger from the experimental ones for the frequency range of 10,000–14,000 cm⁻¹ than for lower frequency ranges. The reason for this deviation is not yet clear, but might be related to some poorly converged DVR-J calculations (used for intensities) for energies above the barrier to linearity.

Fig. 1

The differences between the experimentally observed (Morong et al. [13]) and theoretically predicted frequencies (in cm⁻¹). In addition to the results obtained within the current study (VR and NU masses), the literature data are also shown: NMT [46], SAH0 [53], SAH4 and SAH2. All data, except those of the present study (VR, NU), are taken from Table 3 in Ref. [13]

In Table 5 and Figs. 1, 2, we compare our results with earlier calculations [46, 49, 53]. These were mostly based on a potential in which the fit was not sufficiently supported by ab initio data in the energy region above the barrier to linearity (see the discussion of RMS values in Sect. 2.1). Comparisons with calculations based on lower quality ab initio data have been neglected. The best-suited comparison with other theoretical investigations is the one with SAH(CRJK) [29, 53] (see Table 5 and Figs. 1, 2). In SAH the same program (HYP), NU mass and fit (CRJK: [5, 9]) were used.

Table 5 Comparison among observed (Morong et al. [13] and Kreckel et al. [14]) and theoretical frequencies (obs. calc.) (in cm⁻¹)

Fig. 2

The differences between the experimentally observed (Kreckel et al. [14]) and theoretically predicted frequencies (in cm⁻¹). In addition to the results obtained within the current study (VR, NU), the literature data are also shown: NMT [46], unadjusted SAH [53] and adjusted SAHA [14]. All data, except present study, are taken from Table 2 in Ref. [14]

The differences between our new and SAH results refer mainly to the changes in the representation of the BCJK- [4] versus CRJK- [5] potential. This leads to an increase or decrease in the transition frequencies compared to experiment, because non-adiabatic effects have not been included rigorously. As a result, our NU calculations (Table 5) show a slight increase in the average error and RMS value (individual comparisons with the data of Ref. [13] and [14] are termed MO and KR) (MO −1.026, 1.087; KR −1.183, 1.200 cm⁻¹) compared to SAH (MO −0.736, 0.807; KR −0.679, 0.719 cm⁻¹), whereas the standard deviation (MO 0.360; KR 0.206 cm⁻¹) is very similar to the average error in SAH (MO 0.332; KR 0.240 cm⁻¹). For the VR calculations, the standard deviation is further reduced (MO 0.603; KR 0.549 cm⁻¹). In the former work [4] we have used, in addition, the empirical shifts provided by Schiffels et al. [29], which improved the comparison to the experimental data. The investigation of these empirical shifts provides some hint which are the main contributions to the non-adiabatic corrections. Another way to estimate the non-adiabatic contributions is proposed by Alijah and Hinze [55] in calculating rovibrational expectation values of the linear term in a Taylor expansion of the potential energy surface with respect to each individual rovibrational motion. These contributions can be fitted to known experimental transition frequencies, so that the fit can be used for the estimation of corrections of only theoretically known transitions. This had been tested so far for H₂. Our aim is to put more effort in calculating the non-adiabatic corrections based on a rigorous theory to represent the rovibrational spectrum of a given energy region with sub-wavenumber accuracy [40, 42].

As one can see from Table 1, the difference in the transition frequencies ν (exp. calc.) is positive for all states except three (average error of about 0.603). Negative deviations by more than 1 cm⁻¹ are not explainable: in some cases the intensities differ by far more than the average (see also Fig. 4).

In Fig. 3, we see a steady increase of deviation between the VR- and NU-mass results, i.e., rovibrational energies experience a larger portion of the potential energy surface where the energy difference between the electronically ground and excited state is experienced.

Fig. 3

The differences between theoretically predicted frequencies (in cm⁻¹) using VR and NU masses for those frequencies measured by Morong et al. (MO) [13] and Kreckel et al. (KR) [14]

In Fig. 4, we present the ratio of experimental and calculated (exp. calc.) relative intensities, together with deviations in transition frequencies using the VR-mass in the calculation. For 11 out of 143 transitions, the deviations for the ratio in intensities are larger by a factor of 2 (most of them are in the range of 0.8–1.2; the two intensities (exp. calc.) are given in Table 1). It is not clear why there is such a mismatch in 11 cases, except that the different experimental setups have been used or some intensity calculations are poorly converged. In one case (last line in Table 1), the mismatch is by four orders of magnitude and the transition deviates by ≈4 cm⁻¹. Probably the transition is not related to H ⁺₃ .

Fig. 4

Experiment (Morong et al. [13]) versus calculation: ratio of the relative intensities (top) and difference in the transition frequencies (in cm⁻¹) (bottom) using VR masses

In Fig. 5, we propose the spectrum for 600 K in the range of 10,000–14,000 cm⁻¹. Compared to the known experimental transitions (143 from Ref. [13] and 23 from Ref. [14]) more than 10,000 transitions with relative intensities between 1 and 10⁻⁶ are plotted. The most strongest intensities in the given energy range of 10,000–14,000 cm⁻¹ that have been not experimentally detected are given in Table 6. These include P, Q, and R transitions; in case of higher rotational states (J > 5), the magnitude of the intensity might be not very reliable. As one can see from Table 2, the detection technique used by Kreckel et al. [14] is able to find transitions at relatively weak intensities. The stronger transitions in the accessible range were already observed by Gottfried et al. [6, 13]. But in both experiments (MO and KR), weak transitions below a relative intensity (with respect to the strongest intensity in the range of 10,000–14,000 cm⁻¹) of ≈2% have not been measured up to now. There is also experimentally a gap of missing transition between 8,200 and 10,300 cm⁻¹.

Fig. 5

Proposed spectrum (top) of H ⁺₃ for 600 K in the range 10,000–14,000 cm⁻¹ (more than 10,000 transitions) in comparison with experiment (bottom: MO + KR; ca. 160 transitions). Bottom part for the experimental transitions of KR (Kreckel et al. [14]), the proposed calculated values are shown

Table 6 Proposal of the strongest transitions with relative intensities >5% (T = 600 K) in the region of 10,000–14,000 cm⁻¹, yet not measured experimentally

4 Summary

In this paper, we report on new results of ab initio calculations of selected rovibrational transitions of H ⁺₃ in the range of 10,000–14,000 cm⁻¹ and compare them with experimental data available in the literature. The deviation of the computed rovibronic frequencies of <1.5cm⁻¹ is related to non-adiabatic effects. By choosing two different masses for rotational and vibrational motion, non-adiabatic effects have been simulated in an empirical way. As a final result, the deviations became only slightly smaller and have changed the sign. This is a strong indication that one needs an effective, coordinate-depending mass rather than constant mass as already discussed in Ref. [40, 42]. Based on the improvements gained for the non-adiabatic contributions to the rovibrational energies of H₂ and H ⁺₂ and its isotopologes we hope that similar work can be done for triatomics like H ⁺₃ . Work is now in progress to include mass effects for the different individual rovibrational energies based on a more rigorous theory.