1 Introduction

Helium nanodroplets have been used as nanocryostats to isolate atoms or molecules at a temperature of 0.4 K, or to form new weakly bound aggregates [1,2,3]. They offer a unique quantum solvent for studying small and large dopant clusters, neutral or ionic, which can in turn be investigated by electronic or infrared spectroscopy [4]. They also minimize matrix-induced spectral shifts and broadening [5,6,7]. Alkali metal atoms reside on the surface of He droplets due to the short-range Pauli repulsion between their s valence electrons and surrounding helium atoms counteracting the van der Waals attractive forces [8, 9]. However, alkali metal clusters are drawn into He droplets above a critical cluster size [10,11,12]. By increasing the size of the alkali metal cluster, its van der Waals attraction to the helium matrix grows faster than the repulsive energies, and above a certain size it becomes favorable for clusters to submerge in the superfluid HND.

Alkali metal dimers have been investigated both experimentally [13,14,15,16,17,18,19] and theoretically [20,21,22,23,24,25,26,27,28,29] to provide a detailed description of the molecular electronic structure. Gabbanini produced translationally cold Rb dimers in the lowest triplet and singlet states, which were detected by resonance-enhanced photoionization [30]. Lee and Mahan have photoionized the vapor of rubidium in the discrete region of the atomic spectrum [31]. The ions produced were identified by ion mobility measurements, and a lower limit of 0.73 eV (5900 cm−1) was obtained for the dissociation energy (De) of Rb2+. Olson has estimated De = 0.75 eV from the velocity dependence of the Rb–Rb+ charge exchange cross section [32]. Wagner and Isenor have studied multiphoton ionization and dissociation of Rb dimers in a molecular beam; they deduced De = 0.75 ± 0.10 eV [18].

From the theoretical viewpoint, Bellomonte et al. [33] used Hellman-type model potentials to calculate the ground-state properties of Na2+, K2+, Rb2+ and Cs2+. They reported a ground-state energy of 5.04 eV, De = 0.86 eV, equilibrium distance (Re) of 4.45 Å, and harmonic constant (ωe) of 41 cm−1 for Rb2+. Jraij et al. [34] performed ab initio calculations involving non-empirical pseudopotentials, core polarization potentials and semiempirical spin–orbit pseudopotentials to calculate the ground state as well as 25 excited states of Rb2+ for a large range of internuclear distances. Aymar et al. [35] computed the adiabatic potential energy curves associated with dimer cations Rb2+, Cs2+ and RbCs+ for the lowest eight 2Σg,u states using either the Klapisch model potential [36] or an approach based on pseudopotentials, with the aim of analyzing the abilities and limits of the Klapisch model potentials for handling heavy molecular systems with one valence electron. More recently, Smialkowski et al. [37] applied the coupled cluster method to ionic metal dimers and trimers. They investigated the ground-state electronic structure of singly-charged molecular ions, including Rb2+. They obtained Re = 4.82 Å, De = 6151 cm−1 and ωe = 46.0 cm−1. Several other theoretical investigations of Rb2+ have been published [38,39,40,41].

Limited experimental information pertaining to electronically excited states of Rb2+ has been obtained from studying the effect of cooling light on the lifetime of Rb2+ in a magneto-optical trap [42], and from photoabsorption spectra of dense rubidium vapor [43]. Also relevant is a report by Helm et al. who have studied vibrational levels in the 12Πu state of Cs2+ by monitoring Cs+ photofragments resulting from absorption of laser radiation by the 2Σg+ state [44]. They deduced a SO splitting of 280 ± 20 cm−1 in the 12Πu state. We have recently studied Cs2+ complexed with up to 12 He atoms by depletion spectroscopy [45]. Three absorption bands were observed due to transitions from the ground state into the 12Σu+, 12Πu and 22Πu states. The bound 2Πu states in HeCs2+ were SO split by about 315 and 340 cm−1, respectively.

Here, we report depletion spectra of Rb2+ complexed with up to ten He atoms. The ions were formed by doping helium nanodroplets in a pickup cell filled with low-density Rb vapor and subsequent electron ionization. The weak binding between He and Rb2+ ensures that the target ions are in their vibrational ground states. Two absorption bands are observed between 920 and 250 nm, due to transitions into the 12Σu+ and 12Πu states. The transitions are blue- and redshifted, respectively, when the number of He atoms is increased. Spectroscopic constants and the spin–orbit (SO) splitting are deduced for the bound 12Πu state. All experimental findings are supported by ab initio calculations.

2 Experiment

Helium nanodroplets (HNDs) were formed by supersonic expansion through a 5-μm-diameter nozzle into vacuum. The nozzle temperature ranged from 9.6 to 9.85 K for the various spectral scans. The HNDs were doped in a pickup cell filled with Rb at a temperature of 329 K which corresponds to a vapor pressure of 0.0005 Pa [46]. The doped HNDs were ionized by electrons at about 100 eV and an emission current ranging from 300 to 600 μA. HenRbm+ ions ejected from the large multiply charged droplets were deflected by 90° via electrostatic lenses. The ion beam was guided into the extraction region of a high-resolution time-of-flight mass spectrometer (TOFMS) where it was merged with a laser beam from a tunable light source (EKSPLA NT242, line width 5 cm−1, pulse duration 3–6 ns, repetition rate 1 kHz). The laser was calibrated with a spectrometer using an echelle diffraction grating (model SHR, Solar Laser Systems); the wavelength accuracy in the relevant range was 0.05 nm. The TOFMS was operated at 10 kHz, enabling simultaneous measurement of laser-on and laser-off mass spectra. Depletion spectra were extracted from the stored data for Hen85Rb87Rb+ by setting a digital mass filter to ± 0.10 u within the expected mass, for each size 1 ≤ n ≤ 10. Data for homonuclear species Hen85Rb2+ and Hen87Rb2+ could not be extracted because their masses coincide with those of Hen–187Rb2+ and Hen+185Rb2+, respectively. Representative mass spectra are provided in the Electronic Supplementary Information (ESI). Depletion spectra are corrected for the wavelength-dependent output power of the laser. Conversion from wavelengths to energies takes into account the refractive index of air (n = 1.000293). A detailed description of the experiment has been given in the ESI of Ref. [45].

3 Calculations

For modeling the ground electronic state, we used the coupled clusters singles doubles (CCSD) method. For electronically excited states, the equation of motion CCSD (EOMCCSD) [47,48,49] and multireference configuration interaction (MRCI) [50, 51] were employed. We used various basis sets from the def2 series (def2TZVP, def2QZVP, def2QZVPPD) as well as the Stuttgart basis set ECP28MDF [52] for modeling spin–orbit coupling; basis set benchmarking is available as Table S1 in the ESI. In all used basis sets, 17 electrons of Rb2+ are treated explicitly; the remaining ones are included in the effective core potential (ECP). For optimization of the HenRb2+ clusters, tight optimization criteria were used to describe the very floppy potential energy surface. Zero-point energy correction was accounted for in all reported thermochemical values.

Within the (EOM)CCSD calculations, we used either one explicitly correlated electron of Rb2+ (corresponding to the frozen core, FC, approximation) or we correlated all Rb electrons; all He electrons were always correlated. The frozen core approximation is obviously advantageous for reaching the goal of the present study, i.e., modeling weakly bound HenRb2+ clusters, n ≤ 10, leading to a decrease in the computational time. The difference between both approaches is most visible for the Rb–Rb bond length, predicting 5.185 and 4.883 Å with frozen core approximation and when correlating all electrons, respectively, within the CCSD/def2QZVPPD method. Photochemical properties are somewhat less influenced, with 1.57 and 1.52 eV for excitation into the 2Σu+ state and 1.83 and 1.98 eV into the 2Πu state within the EOMCCSD/def2QZVPPD approach. Most importantly, both approaches predict the same trends with respect to shifts induced by helium solvation, which is the main concern of the present publication, and we thus use the frozen core approximation for treating clusters with larger numbers of helium atoms.

For Rb2+, the MRCI calculation was performed with an active space of 17 electrons in 13 orbitals, further denoted as (17,13). For HeRb2+ and He2Rb2+, active spaces of (15,11) and (17,12) were used, respectively. The spin–orbit coupling was calculated using the state-interacting method as implemented in Molpro [53]. The SO splitting of the 12Πu state in Rb2+ does not change considerably (< 2 cm−1) when switching the active space from (1,15) to (1,4) within the MRCI method. Using (17,13) active space, however, the SO splitting increases by about 15 cm−1. Using the all-electron Sapporo-QZP-2012 basis set [54] shifts the splitting by less than 1% within the MRCI(17,13) method. Benchmark calculations on the Rb atom show that for the MRCI(9,12)/ECP28MDF method, the calculated SO splitting of the 12P state (198 cm−1) is underestimated by about 16% with respect to the experimental value (238 cm−1) [55]. The reported SO splitting for Rb2+ could thus be underestimated. When calculating the energy of the 0–0 transition and modeling spectra of the 12Πu state in the HeRb2+ and He2Rb2+ clusters, only the Rb2+ vibration was included into the zero-point energy correction.

Path integral molecular dynamics (PIMD) calculations were performed on the MP2(FC)/def2QZVP(Rb),def2TZVP(He) potential energy surface; this approach was selected as a method that can be used for PIMD simulations of Rb2+ solvated by more He atoms in the future. Sixteen random walkers, time step of 30 a.u. and Nosé–Hoover thermostat with four chains were used. In total, 15,500 steps were calculated, with 5000 initial steps used as a thermalization period. A temperature of 5 K was used for efficient sampling. Out of the trajectory, 1730 structures were used for spectrum modeling within the reflection principle [56,57,58], and calculated points were convoluted with Gaussian functions with full width at half maximum (FWHM) of 0.03 eV.

(EOM)CCSD and density functional theory (DFT) calculations were performed in the Gaussian software [59], and MRCI calculations in the Molpro program. PIMD simulations were performed in the Abin program [60]; vibrationally resolved electronic spectra were modeled within the PGOPHER software [61].

4 Results and discussion

4.1 Rb2 +

Before discussing Rb2+ solvated by helium atoms, let us turn our attention to the properties of Rb2+ itself (Table 1). For the minimum-energy structure in the ground electronic state of 12Σg+, our calculations predict the bond length of 4.883 Å, the vibrational frequency within harmonic approximation of 44.7 cm−1 and dissociation energy of 5992 cm−1 (CCSD/def2QZVPPD), close to previous experimental and computational values. In the Franck–Condon region, there are nine excited states lying within 35,000 cm−1 (Fig. 1). Only transitions to states of 2Σu+ and 2Πu character are bright ones; other transitions are forbidden. However, only the first two electronic states, 12Σu+ and 12Πu, have a considerable oscillator strength. Compared to the case of Cs2+ [45], the 22Πu state has a lower transition dipole moment and is therefore not observed in the present experiment (see also below).

Table 1 Properties of Rb2+, HeRb2+ and He2Rb2+ in the 12Σg+ electronic ground state, 12Σu+ state and spin–orbit (SO) split 12Πu: Equilibrium Rb–Rb internuclear distance, dissociation energies (De or D0), harmonic constant ωe for the Rb–Rb stretch and transition energies (Te and T0 for 12Σu+ and 12Πu, respectively)
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Fig. 1
figure 1

Electronic states in Rb2+ calculated at the (EOM)CCSD/def2QZVPPD level. Bright and forbidden states are shown as full and dashed lines, respectively. Oscillator strengths for bright states are given in parentheses as calculated at the bond length of 4.883 Å

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Focusing on the two lowest electronic states, the 12Σu+ state is dissociative due to the excitation of σ–σ* character (Fig. 2) and converges in the dissociation limit with 12Σg+. We predict the excitation energy from the minimum of the ground state as 12,290 cm−1, close to the previously reported value of 12,690 cm−1 [34]. Note that the 12Σu+ potential is steep (Fig. 1) and already for a bond length shorter by 0.05 Å with respect to the equilibrium value, the excitation energy increases by 180 cm−1. The transition is very intense, with the oscillator strength of 0.37.

Fig. 2
figure 2

Molecular orbitals in which the sole electron is localized in three lowest electronic states of Rb2+, HeRb2+, and two isomers of He10Rb2+. Calculated at the CAM-B3LYP/def2QZVPPD level of theory. Note that a low contour value was chosen as to display the relatively small influence of helium on the electronic structure of Rb2+

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The 12Πu state is bound and has the equilibrium bond length of 5.455 Å (EOMCCSD/def2QZVPPD), about 0.6 Å longer than in the electronic ground state, with the calculated vibrational frequency of 25.1 cm−1. The potential energy surface is shallow, and a small change in the basis set can shift the equilibrium position considerably: When the def2QZVP basis set is used, the bond length of 5.697 Å is retrieved (see Table S1). Due to the same reason, the calculated bond prolongation with respect to the ground state is about twice as large as predicted previously [34] (see also below). In this state, the odd electron is localized in a π orbital (Fig. 2). The vertical excitation energy is calculated as 15,990 cm−1; the energy difference between 12Σg+ and 12Πu minima including the zero-point correction is 15820 cm−1. This is virtually the same value as reported previously, 15,870 cm−1 [34]. Again, the transition is considerably intense, with the oscillator strength of 0.29 for each branch of the 12Πu state.

While there is only a negligible shift due to spin–orbit interaction for the 12Σg+ state (< 0.1 cm−1), the 12Πu state splits considerably, with the gap between the resulting Ω = 1/2 and Ω = 3/2 states calculated as 111 cm−1 for the minimum structure (MRCI(17,13)/ECP28MDF), in reasonable agreement with 142 cm−1 predicted previously [34].

Two other bright states, 22Σu+ and 22Πu, are predicted to lie at about 32,000 cm−1 (Fig. 1). Both are dissociative within the considered Rb–Rb length range and have oscillator strengths below 0.015. At the same time, they have not been observed experimentally so far. Therefore, they are not discussed here further.

4.2 HeRb2 + and He2Rb2 +

Measured depletion spectra of 85Rb87Rb+ complexed with n = 1, 2, 3, 6 and 9 He atoms are displayed in Fig. 3. Data from two separate scans have been combined, one ranging from 600 to 700 nm in 0.1 nm increments and the other from 700 to 920 nm in 0.2 nm increments. The dwell time was 60 s per step. The data shown were averaged over 10 adjacent points in order to reduce statistical scatter.

Fig. 3
figure 3

Low-resolution depletion spectra of 85Rb87Rb+ complexed with n = 1, 2, 3, 6 and 9 He atoms. The solid (red) lines represent fits to the absorption bands that are due to transitions into the 12Σu+ and 12Πu states. The dashed (olive) line is the calculated spectrum modeled at the EOMCCSD(FC)/def2QZVPPD level on structures sampled from the PIMD on the MP2(FC)/def2QZVP(Rb),def2TZVP(He) potential energy surface (see text for details)

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Two absorption bands are shown in Fig. 3 for HeRb2+, a broad one at about 12,700 cm−1 and a narrow one at about 15,700 cm−1. For He2Rb2+, the transitions are slightly blue- and redshifted, respectively. Based on previous theoretical calculations on Rb2+ and potential energy curves in Fig. 1, we assign these states as transitions from the 12Σg+ ground state into the 12Σu+ and 12Πu states, respectively. Their centroids have been determined by fitting Gaussians (for the 12Σu+ state) or square functions (for the 12Πu state); the results are represented in Fig. 3 by solid (red) lines.

When one helium atom is attached to the Rb2+ ion, a linear structure is obtained (Fig. 4), with the Rb–He distance of 3.435 Å. The Rb–Rb bond is mildly affected by the presence of the helium atom, shifting by less than 0.01 Å (CCSD/def2QZVPPD). The adsorption energy of the helium atom is predicted as 26 cm−1 at this level of theory; the helium atom can roam freely around the Rb2+ core starting with 23 cm−1 of kinetic energy. (See the transition state structure in Fig. 4.) In the 12Πu state, the linear structure is slightly distorted, the Rb–Rb bond distance increases considerably to 5.452 Å, and the Rb–He distance shrinks to 3.138 Å. This change can be understood based on the Pauli repulsion (Fig. 2). In the ground state, the helium atom is slightly repelled by the σ orbital. In the excited state, He can position itself closer to Rb2+ due to the shape of the π orbital. Structures calculated for He2Rb2+ show the same trends as for HeRb2+. However, two isomers are retrieved here (Fig. 4b), with both helium atoms on the one side (IIa) or distributed equally (IIb).

Fig. 4
figure 4

a Structures of HeRb2+ for the equilibrium in the 12Σg+ and 12Πu states and the transition state for helium moving to the other side of Rb2+ in the 12Σg+ state, b structures of two isomers of He2Rb2+ for the equilibrium in the 12Σg+ and 12Πu states. Relative energies of isomers and bond lengths (in Å) are given. Calculated at the (EOM)CCSD/def2QZVPPD level of theory

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The structureless band observed for the 12Σu+ state in Fig. 3 reflects the dissociative nature of this electronic state. In HeRb2+, it has an experimental FWHM of 1200 cm−1. The spectrum modeled from the ground-state density sampled by path integral molecular dynamics on the MP2(FC)/def2QZVP(Rb),def2TZP(He) potential energy surface at the temperature of 5 K is also shown. Here, the FWHM is estimated as 900 cm−1, slightly underestimating the experimental width, but still reproducing the measured spectrum well. The agreement in the spectral position is fortuitous given approximations included.

The transition into the bound 12Πu state reveals vibrational structure. Depletion spectra recorded with 0.1 nm increments and a dwell time of 180 s per data point are displayed in Fig. 5 for HeRb2+ and He2Rb2+. For HeRb2+, the envelope of the curve suggests that we are seeing the superposition of two vibrational progressions, due to transitions into the Ω = 1/2 and 3/2 components of the SO split 2Πu state. The vertical lines in Fig. 5a mark possible candidates for the (v’, v’’) = (0,0) transitions at 15,522 cm−1 and 15,705 cm−1. They would imply a SO splitting of 183 cm−1, compared with a theoretical value of 116 cm−1 for HeRb2+ (this work) and of 142 cm−1 for Rb2+ [34]. As discussed in the Calculations section, our results probably underestimate the real value.

Fig. 5
figure 5

a, b High-resolution depletion spectra of heteronuclear Rb2+ complexed with one or two He atoms. The data reveal two overlapping vibrational progressions for transitions into the spin–orbit split 12Πu state. The red solid lines represent fits of two sets of equally spaced Lorentzians to the experimental data. c Modeled depletion spectra of Rb2+ for different equilibrium distances in the 12Πu state (without SO coupling), with other parameters calculated at the (EOM)CCSD/def2QZVPPD level. The harmonic vibrational frequency was kept constant as calculated in the minimum (25.1 cm−1); Lorentzian broadening of 10 cm−1 was employed. The vertical line indicates the (0,0) transition. d Modeled depletion spectrum of HeRb2+ calculated at the (EOM)CCSD/def2QZVPPD level

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The data in Fig. 5a were analyzed by fitting two sets of equidistant Lorentzians with spacing ω1 and ω2, respectively. Results are compiled in Table 1. The fitted vibrational constants, 25.9 and 25.7 cm−1 for the 1/2 and 3/2 states, respectively, are in agreement with 25.0 cm−1 for HeRb2+ (this work) and 28.59 cm−1 for Rb2+ [34] computed for the 12Πu state in the absence of SO coupling.

The depletion spectrum of He2Rb2+ (Fig. 5b) is qualitatively similar to that of HeRb2+. A single progression of 15 equally spaced Lorentzians was fit to the data, resulting in a harmonic constant ωe = 26.03 ± 0.02 cm−1 (Fig. 5b). It is not possible to identify the (0,0) origin of the progression, but the envelope is clearly shifted to lower energies with respect to HeRb2+.

There is no evidence for anharmonicity in the observed vibrational progressions. This is not surprising because the ratio of the anharmonic to the harmonic constants, ωexe/ωe (calculated for the ground state whose values of Re and De are quite similar to those of the 2Πu state), is only 0.2% [39]. Perhaps, more surprising is the absence of a phase shift between the two overlapping progressions, which must be accidental.

As shown in Fig. 1, the calculated equilibrium distance in the 12Πu state exceeds that in the ground state, and the (0,0) transition may well be unobservable. On the other hand, the equilibrium bond length in the 12Πu state varies greatly at different levels of theory due to a floppy potential. In Fig. 5c, we show depletion spectra of Rb2+ simulated for three different Rb–Rb distances in the 12Πu state in the absence of SO coupling. (Note that the x-axis has been shifted in order to facilitate a comparison with the experimental data.) For the Rb–Rb distance of 5.2 Å, there are only five clear bands predicted. The (0,0) transition (marked by a vertical line) has a large Franck–Condon factor, and the (0,2) transition is the most intense one. For Re = 5.455 Å (the equilibrium distance calculated at the (EOM)CCSD/def2QZVPPD level), a larger number of intense transitions are observed, with the most intense one being (0,5). Finally, for Re = 5.6 Å, the maximum shifts to (0,8) and even more bands are seen. In Fig. 5d, the computed spectrum for the equilibrium structure of HeRb2+ is shown, which is very similar to the one for Rb2+. Compared to the experimental data, we can conclude that the modeled spectrum for the Rb–Rb distance of ~ 5.45 Å reproduces the data reasonably well, although one cannot exclude that the Rb–Rb distance for the 12Πu state is somewhat longer. This would imply that the (0,0) transition is not observed in the experiment, and the energies of the transitions into the SO split 12Πu (v’ = 0) state are smaller than the upper limits listed in Table 1 by a few vibrational quanta.

Finally, we analyze spectral shifts induced by the presence of a helium atom. As can be seen in Table 1, the 12Σg+–12Σu+ transition is predicted to shift to a higher energy by 210 cm−1 and the 12Σg+–12Πu transition shifts down by either 60 cm−1 (for vertical excitation energies in the respective minima) or 90 cm−1 (for the (0,0) transition) between Rb2+ and HeRb2+. These shifts can be qualitatively understood by inspection of molecular orbitals depicted in Fig. 2 (see also Ref. [45]): The 12Σg+ ground state is slightly destabilized by the presence of the helium atom; this destabilization, however, increases in the 12Σu+ state. In the 12Πu state, on the other hand, the Pauli repulsion with helium is reduced as the molecular orbital does not extend along the Rb–Rb axis. Thus, there is a considerable shift to higher excitation energies for the 12Σg+–12Σu+ transition due to the destabilization of the target state. For the 12Σg+–12Πu transition, the destabilization in the 12Σg+ state prevails, leading to a smaller negative shift.

For the difference between HeRb2+ and He2Rb2+, we obtain similar spectral shifts of 230 cm−1 and –80 cm−1 for the 12Σg+–12Σu+ and 12Σg+–12Πu transitions, respectively. Here, we can directly compare these values with the experimental ones of 127 ± 26 cm−1 and –58 ± 5 cm−1 (Table 1). The considerable discrepancy for the shift of the 12Σg+–12Σu+ state can be most probably traced to the floppy potential energy surface with respect to the helium position. For example, an increase of 0.3 Å in the Rb–He distance leads to the change in the ground-state energy by 7 and 13 cm−1 for HeRb2+ and He2Rb2+, respectively. However, the spectral shift of the 12Σg+–12Σu+ transition decreases to 151 cm−1 for these structures, i.e., to about 60% of the one for minimum energy structures. The modeled shifts can thus be easily influenced by small changes in the methodology (e.g., inclusion of triples within the CC formalism) as well as by inclusion of dynamic effects and cannot be interpreted quantitatively.

4.3 HenRb2 +, n = 3–10

Let us now turn to clusters solvated with several helium atoms. A vibrational progression is still discernible for He3Rb2+ (see Fig. S3 of the ESI); its envelope is redshifted even further. Vibrationally unresolved depletion spectra were recorded for even larger HenRb2+, n ≤ 10; square functions were fitted to the absorption bands in order to deduce the spectral shift.

The midpoints of the absorption bands versus the number n of attached He atoms are compiled in Fig. 6. Data for He485Rb87Rb+ could not be deduced because its mass coincides with that of impurity ions, H2O85Rb2+ and O85Rb87Rb+. The transition to the 12Σu+ state is blueshifted with increasing size n by + 98 ± 6 cm−1 per added He atom (the dashed line represents the result of a fit to the data). The transition to the 12Πu state is redshifted, on average, by − 38 ± 3 cm−1 for the first five He atoms but gradually tapers off. The total redshift between n = 1 and n = 10 equals about –210 cm−1.

Fig. 6
figure 6

The centroids of the absorption bands in Fig. 3 versus the number n of He atoms attached to the heteronuclear Rb2+ dimer. Transitions into the 12Σu+ state (right scale) are blueshifted with increasing n, while transitions into the 12Πu state (left scale) are redshifted

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Selected calculated structures of HenRb2+, n = 2–10, are shown in Fig. 7 (see the ESI for all isomers); their properties are collected in Table 2. Note that for these larger clusters, we used the frozen core approximation, leading among other effects to prediction of linear isomer IIb. The a, b, c… nomenclature follows the distribution of He atoms on both sides of Rb2+, from the most asymmetric one to more equal ones. Although the calculated structures necessarily represent only a subset of all possible local minima, several trends are clearly visible. Most importantly, isomers with all helium atoms on the one side (a isomers) are usually the least stable ones and exhibit the lowest absolute value of spectral shifts. This can be documented on the spectral shift for the 12Σg+–12Πu transition where even positive shifts are observed (Xa). Structures with a similar number of helium atoms on each side, on the other hand, exhibit higher stability and more considerable spectral shifts. Again, this can be understood based on the molecular orbital picture (Fig. 2). For more helium atoms on the one side (Xa), the molecular orbital might deform toward the other side to compensate the Pauli repulsion. For equal distribution of helium atoms on both sides (Xf), the orbital is more constricted, leading to a more considerable shift. Within the CCSD(FC) method, the Rb–Rb bond length decreases by 0.006 Å when passing from Rb2+ to HenRb2+, n = 6, and by another 0.002 Å between n = 6 and n = 10 (for the most stable isomers in Table 2).

Fig. 7
figure 7

Selected structures of HenRb2+, n ≤ 10, clusters viewed from the side (above) and along the Rb–Rb axis (below). Optimized at the CCSD(FC)/def2QZVPPD(Rb),def2TZVP(He) level. See Fig. S4 for all optimized structures

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Table 2 Properties of HenRb2+, n ≤ 10, clusters with different solvation patterns
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The calculated shifts for the 12Σg+–12Σu+ transition in Table 2 reproduce the experimental trend (Fig. 6) quantitatively. However, this is probably due to error compensation as correlation of all electrons leads to a shift that is considerably larger. (See Table 1 and the discussion above.) The shifts for the 12Σg+–12Πu transition, on the other hand, are considerably underestimated. Here, however, the values presented in Table 2 were calculated for the vertical transition from the minimum. If we account for relaxation in the target state, higher absolute shifts may be expected.

Overall, both thermochemical and photochemical data show that for the Rb2+ solvated by helium atoms, symmetric solvation is the most probable one, as already observed for a low number of helium atoms in lighter alkali dimer ions [62]. This trend can be expected to continue until the first solvation layer is filled.

5 Conclusions

In Rb2+ solvated by helium atoms, only two electronic transitions are observed experimentally, 12Σg+–12Σu+ and 12Σg+–12Πu. The 12Σu+ state is repulsive and shifts to higher energies with helium solvation because helium atoms deform the target σ* orbital more considerably compared to the σ orbital in the ground state. In the 12Πu state, a bound state is formed, split due to the spin–orbit coupling. The shape of the vibrationally resolved spectra emphasizes considerable prolongation of the Rb–Rb bond in the excited state. The 12Πu state shifts to slightly lower energies with helium solvation as helium atoms destabilize the ground state more than the final state. While quantum chemical calculations are able to reproduce the spectral shifts qualitatively, predicted values depend considerably on the computational level. For structures with a higher number of adsorbed helium atoms, we predict that He atoms are distributed symmetrically on both sides of the Rb2+ ion.