Electronic transitions in Rb2+ dimers solvated in helium

We have measured depletion spectra of the heteronuclear (85Rb87Rb+) dimer cation complexed with up to 10 He atoms. Two absorption bands are observed between 920 and 250 nm. The transition into the repulsive 12Σu+ state of HeRb2+ gives rise to a broad feature at 790 nm (12,650 cm−1); it exhibits a blueshift of 98 cm−1 per added He atom. The transition into the bound 12Πu state of HeRb2+ reveals vibrational structure with a band head at ≤ 15,522 cm−1, a harmonic constant of 26 cm−1, and a spin–orbit splitting of ≤ 183 cm−1. The band experiences an average redshift of − 38 cm−1 per added He atom. Ab initio calculations rationalize the shape of the spectra and spectral shifts with respect to the number of helium atoms attached. For a higher number of solvating helium atoms, symmetric solvation on both ends of the Rb2+ ion is predicted.


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.
From the theoretical viewpoint, Bellomonte et al. [33] used Hellman-type model potentials to calculate the groundstate properties of Na 2 + , K 2 + , Rb 2 + and Cs 2 + . They reported a ground-state energy of 5.04 eV, D e = 0.86 eV, equilibrium 1 3 29 Page 2 of 12 distance (R e ) of 4.45 Å, and harmonic constant (ω e ) of 41 cm −1 for Rb 2 + . 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 Rb 2 + for a large range of internuclear distances. Aymar et al. [35] computed the adiabatic potential energy curves associated with dimer cations Rb 2 + , Cs 2 + and RbCs + for the lowest eight 2 Σ g,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 Rb 2 + . They obtained R e = 4.82 Å, D e = 6151 cm −1 and ω e = 46.0 cm −1 . Several other theoretical investigations of Rb 2 + have been published [38][39][40][41]. Limited experimental information pertaining to electronically excited states of Rb 2 + has been obtained from studying the effect of cooling light on the lifetime of Rb 2 + 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 1 2 Π u state of Cs 2 + by monitoring Cs + photofragments resulting from absorption of laser radiation by the 2 Σ g + state [44]. They deduced a SO splitting of 280 ± 20 cm −1 in the 1 2 Π u state. We have recently studied Cs 2 + 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 1 2 Σ u + , 1 2 Π u and 2 2 Π u states. The bound 2 Π u states in HeCs 2 + were SO split by about 315 and 340 cm −1 , respectively.
Here, we report depletion spectra of Rb 2 + 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 Rb 2 + 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 1 2 Σ u + and 1 2 Π u 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 1 2 Π u state. All experimental findings are supported by ab initio calculations.

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. He n Rb m + 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 He n 85 Rb 87 Rb + by setting a digital mass filter to ± 0.10 u within the expected mass, for each size 1 ≤ n ≤ 10. Data for homonuclear species He n 85 Rb 2 + and He n 87 Rb 2 + could not be extracted because their masses coincide with those of He n-1 87 Rb 2 + and He n+1 85 Rb 2 + , 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].

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 (EOMC-CSD) [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 Rb 2 + are treated explicitly; the remaining ones are included in the effective core potential (ECP). For optimization of the He n Rb 2 + clusters, tight optimization criteria were used to describe the very floppy potential energy surface. Zeropoint energy correction was accounted for in all reported thermochemical values.
Within the (EOM)CCSD calculations, we used either one explicitly correlated electron of Rb 2 + (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 He n Rb 2 + 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 2 Σ u + state and 1.83 and 1.98 eV into the 2 Π u 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 Rb 2 + , the MRCI calculation was performed with an active space of 17 electrons in 13 orbitals, further denoted as (17,13). For HeRb 2 + and He 2 Rb 2 + , 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 1 2 Π u state in Rb 2 + 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)/ECP-28MDF method, the calculated SO splitting of the 1 2 P state (198 cm −1 ) is underestimated by about 16% with respect to the experimental value (238 cm −1 ) [55]. The reported SO splitting for Rb 2 + could thus be underestimated. When calculating the energy of the 0-0 transition and modeling spectra of the 1 2 Π u state in the HeRb 2 + and He 2 Rb 2 + clusters, only the Rb 2 + vibration was included into the zero-point energy correction.
Path integral molecular dynamics (PIMD) calculations were per for med on t he MP2(FC)/ def2QZVP(Rb),def2TZVP(He) potential energy surface; this approach was selected as a method that can be used for PIMD simulations of Rb 2 + 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 PGO-PHER software [61].

Rb 2 +
Before discussing Rb 2 + solvated by helium atoms, let us turn our attention to the properties of Rb 2 + itself (Table 1). For the minimum-energy structure in the ground electronic state of 1 2 Σ g + , 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 2 Σ u + and 2 Π u character are bright ones; other transitions are forbidden. However, only the first two electronic states, 1 2 Σ u + and 1 2 Π u , have a considerable oscillator strength. Compared to the case of Cs 2 + [45], the 2 2 Π u state has a lower transition dipole moment and is therefore not observed in the present experiment (see also below).
Focusing on the two lowest electronic states, the 1 2 Σ u + state is dissociative due to the excitation of σ-σ* character (Fig. 2) and converges in the dissociation limit with 1 2 Σ g + . 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 1 2 Σ u + 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.
The 1 2 Π u 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 1 2 Σ g + and 1 2 Π u 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 1 2 Π u state.
While there is only a negligible shift due to spin-orbit interaction for the 1 2 Σ g + state (< 0.1 cm −1 ), the 1 2 Π u 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, 2 2 Σ u + and 2 2 Π u , 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.

HeRb 2 + and He 2 Rb 2 +
Measured depletion spectra of 85 Rb 87 Rb + 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. Two absorption bands are shown in Fig. 3 for HeRb 2 + , a broad one at about 12,700 cm −1 and a narrow one at about 15,700 cm −1 . For He 2 Rb 2 + , the transitions are slightly blueand redshifted, respectively. Based on previous theoretical calculations on Rb 2 + and potential energy curves in Fig. 1, we assign these states as transitions from the 1 2 Σ g + ground state into the 1 2 Σ u + and 1 2 Π u states, respectively. Their centroids have been determined by fitting Gaussians (for the 1 2 Σ u + state) or square functions (for the 1 2 Π u state); the results are represented in Fig. 3 by solid (red) lines.
When one helium atom is attached to the Rb 2 + ion, a linear structure is obtained (Fig. 4), with the Rb-He  Fig. 4.) In the 1 2 Π u 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 Rb 2 + due to the shape of the π orbital. Structures calculated for He 2 Rb 2 + show the same trends as for HeRb 2 + . However, two isomers are retrieved here (Fig. 4b), with both helium atoms on the one side (IIa) or distributed equally (IIb).
The structureless band observed for the 1 2 Σ u + state in Fig. 3 reflects the dissociative nature of this electronic state. In HeRb 2 + , it has an experimental FWHM of 1200 cm −1 . The spectrum modeled from the groundstate 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 1 2 Π u 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 HeRb 2 + and He 2 Rb 2 + . For HeRb 2 + , 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 2 Π u state. The vertical lines in Fig. 5a  . Calculated at the CAM-B3LYP/def2QZVPPD level of the-ory. Note that a low contour value was chosen as to display the relatively small influence of helium on the electronic structure of Rb 2 + for HeRb 2 + (this work) and of 142 cm −1 for Rb 2 + [34]. As discussed in the Calculations section, our results probably underestimate the real value.
The data in Fig. 5a were analyzed by fitting two sets of equidistant Lorentzians with spacing ω 1 and ω 2 , 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 HeRb 2 + (this work) and 28.59 cm −1 for Rb 2 + [34] computed for the 1 2 Π u state in the absence of SO coupling.
The depletion spectrum of He 2 Rb 2 + (Fig. 5b) is qualitatively similar to that of HeRb 2 + . A single progression of 15 equally spaced Lorentzians was fit to the data, resulting in a harmonic constant ω e = 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 HeRb 2 + . 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, ω e x e /ω e (calculated for the ground state whose values of R e and D e are quite similar to those of the 2 Π u 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 1 2 Π u state exceeds that in the ground state, and the (0,0) transition may well be unobservable. On the other hand, the for the equilibrium in the 1 2 Σ g + and 1 2 Π u states and the transition state for helium moving to the other side of Rb 2 + in the 1 2 Σ g + state, b structures of two isomers of He 2 Rb 2 + for the equilibrium in the 1 2 Σ g + and 1 2 Π u states. Relative energies of isomers and bond lengths (in Å) are given. Calculated at the (EOM)CCSD/def2QZVPPD level of theory equilibrium bond length in the 1 2 Π u state varies greatly at different levels of theory due to a floppy potential. In Fig. 5c, we show depletion spectra of Rb 2 + simulated for three different Rb-Rb distances in the 1 2 Π u 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 R e = 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 R e = 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 HeRb 2 + is shown, which is very similar to the one for Rb 2 + . 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 1 2 Π u 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 1 2 Π u (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 1 2 Σ g + -1 2 Σ u + transition is predicted to shift to a higher energy by 210 cm −1 and the 1 2 Σ g + -1 2 Π u 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 Rb 2 + and HeRb 2 + . These shifts can be qualitatively understood by inspection of molecular orbitals depicted in Fig. 2 (see also Ref. [45]): The 1 2 Σ g + ground state is slightly destabilized by the presence of the helium atom; this destabilization, however, increases in the 1 2 Σ u + state. In the 1 2 Π u 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 1 2 Σ g + -1 2 Σ u + transition due to the destabilization of the target state. For the 1 2 Σ g + -1 2 Π u transition, the destabilization in the 1 2 Σ g + state prevails, leading to a smaller negative shift.
For the difference between HeRb 2 + and He 2 Rb 2 + , we obtain similar spectral shifts of 230 cm −1 and -80 cm −1 for the 1 2 Σ g + -1 2 Σ u + and 1 2 Σ g + -1 2 Π u 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 1 2 Σ g + -1 2 Σ u + 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 HeRb 2 + and He 2 Rb 2 + , respectively. However, the spectral shift of the 1 2 Σ g + -1 2 Σ u + 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

He n Rb 2 + , n = 3-10
Let us now turn to clusters solvated with several helium atoms. A vibrational progression is still discernible for He 3 Rb 2 + (see Fig. S3 of the ESI); its envelope is redshifted even further. Vibrationally unresolved depletion spectra were recorded for even larger He n Rb 2 + , 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 He 4 85 Rb 87 Rb + could not be deduced because its mass coincides with that of impurity ions, H 2 O 85 Rb 2 + and O 85 Rb 87 Rb + . The transition to the 1 2 Σ u + 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 1 2 Π u 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 .
Selected calculated structures of He n Rb 2 + , 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 Rb 2 + , 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 1 2 Σ g + -1 2 Π u 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 Rb 2 + to He n Rb 2 + , n = 6, and by another 0.002 Å between n = 6 and n = 10 (for the most stable isomers in Table 2).
The calculated shifts for the 1 2 Σ g + -1 2 Σ u + 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 1 2 Σ g + -1 2 Π u 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 Rb 2 + 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.

Conclusions
In Rb 2 + solvated by helium atoms, only two electronic transitions are observed experimentally, 1 2 Σ g + -1 2 Σ u + and 1 2 Σ g + -1 2 Π u . The 1 2 Σ u + 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 1 2 Π u 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 Fig. 6 The centroids of the absorption bands in Fig. 3   , 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 number of adsorbed helium atoms, we predict that He atoms are distributed symmetrically on both sides of the Rb 2 + ion. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creat iveco mmons .org/licen ses/by/4.0/. , n ≤ 10, clusters with different solvation patterns (x/y notation denotes x He atoms on the one side of Rb 2 + and y on the other.) Solvation energy per helium atom (E solv ) and shifts of the vertical excitation energies of the 1 2 Σ u + and 1 2 Π u states with respect to Rb 2 + . Note that the shifts are calculated as a difference of vertical excitation energies. Calculated at the EOMCCSD(FC)/ def2QZVPPD//CCSD(FC)/def2QZVPPD(Rb),def2TZVP(He) level. Structures marked with "*" exhibit one small imaginary frequency (< 1 cm −1 ) that could not be removed by following the respective normal mode