Monisha
Rastogi
a,
Christian
Leidlmair
a,
Lukas
An der Lan
a,
Josu
Ortiz de Zárate
b,
Ricardo
Pérez de Tudela
c,
Massimiliano
Bartolomei
b,
Marta I.
Hernández
b,
José
Campos-Martínez
b,
Tomás
González-Lezana
*b,
Javier
Hernández-Rojas
d,
José
Bretón
d,
Paul
Scheier
a and
Michael
Gatchell
*ae
aInstitut für Ionenphysik und Angewandte Physik, Universität Innsbruck, Technikerstr. 25, A-6020 Innsbruck, Austria. E-mail: michael.gatchell@uibk.ac.at
bInstituto de Física Fundamental, IFF-CSIC, Serrano 123, 28006 Madrid, Spain. E-mail: t.gonzalez.lezana@csic.es
cLehrstuhl für Theoretische Chemie, Ruhr-Universität Bochum, 44780 Bochum, Germany
dDepartamento de Física and IUdEA, Universidad de La Laguna, 38205 Tenerife, Spain
eDepartment of Physics, Stockholm University, 106 91 Stockholm, Sweden
First published on 8th August 2018
We report on a combined experimental and theoretical study of Li+ ions solvated by up to 50 He atoms. The experiments show clear enhanced abundances associated with HenLi+ clusters where n = 2, 6, 8, and 14. We find that classical methods, e.g. basin-hopping (BH), give results that qualitatively agree with quantum mechanical methods such as path integral Monte Carlo, diffusion Monte Carlo and quantum free energy, regarding both energies and the solvation structures that are formed. The theory identifies particularly stable structures for n = 4, 6 and 8 which line up with some of the most abundant features in the experiments.
Several theoretical and experimental studies have focused on the solvation structures of He snowballs around alkali metal cations. Reatto et al. employed variational Monte Carlo (MC) simulations with shadow wave functions to probe alkali ion impurities (Li+, Na+, K+, and Cs+) in liquid helium for equilibrium densities at 0 K.7–9 The chemical potential, local order, single particle excitation, and effective ionic mass were determined from the simulation model. A substantial difference in the snowball for corresponding ions could be observed. It was predicted that only Na+ and K+ have a tendency to form a solid snowball whereas the localization is not as prominent for Li+ and Cs+ species. Gianturco et al. later conducted a dedicated study based on the solvation of Li and other alkali metals in helium matrices employing a combination of classical energy minimization techniques and of exact quantum Diffusion Monte Carlo (DMC) methods.10–13 Small HenLi+ clusters with n ≤ 30 were considered for their investigations and they treated the full cluster interaction as a sum of pairwise potentials for Li+–He and He–He. It could be deduced that three particularly stable structures exist at n = 6, 8, and 10 with the most stable structure being found for n = 6. Additionally, evaluation of single particle evaporation energies, employing classical and quantum techniques, shed light on the formation of a rigid layer of helium with approximately 8 atoms being more tightly bound to the central ion. After this first shell, the evaporation energy was mainly governed by He–He interaction and not by interactions with the ionic core. The behavior was found to be similar to Na+ and K+ doped helium clusters, where the initial rigid layer was comprised of 9 and 12 He atoms, respectively. This rigid behavior of a fully developed first solvation shell for n = 8 was also reported in the ground state path integral calculation performed by Paolini et al.14 who found a stable structure of He atoms forming two parallel squares rotated by π/4 with respect to each other repeated in successively larger clusters (n ∼ 70, for example).
Previous investigations found that three-body (3B) contributions are rather insignificant in the stability of these helium clusters doped with alkali ions. Marinetti et al.13 observed some shifts of the radial distributions to slightly larger distances in their study on HenLi+ when the coupling between induced dipoles on the He atoms were taken into account. The ab initio calculations of the potential energy curve of HeLi+ and optimal structures for HenLi+ with n = 1–6 performed by Sebastianelli et al.12 concluded that the overall interactions were in fact governed mainly by diatom-like interactions between the ion and He atoms and that the pairwise approximation turns to be an acceptable description for these systems. The theoretical investigations performed by Issaoui et al.15 to study HenNa+ clusters added a self consistent many-body contribution between induced dipoles to the pairwise diatomic energy curves. The analysis performed on these studies revealed a notable overestimation of the energies predicted by the 2B approximation in larger clusters, which was found to delay the onset of delocalization and snowball features.15
Müller et al. carried out a systematic investigation on the formation and stability of helium snowballs created by employing femtosecond photoionization (PI) and electron impact ionization (EII) of alkali clusters (Na, K, Rb and Cs).16 From PI spectra, it could be deduced that alkali metal ions that originate from fragmented alkali clusters are more likely to constitute snowball complexes than their ionized monomer counterparts. This could be attributed to the fragmentation of clusters into singly charged ions, due to multiple ionization. Additionally, it was concluded that the size of a snowball with respect to the mass of alkali metals is a function of the kinematics of photofragmentation. For Na+ and K+ ions, they only observed the formation of small snowball sizes (up to 3 and 10 He atoms, respectively), which prohibited a direct comparison with predicted first shell closures from theory. However, with the heavier Rb+ and Cs+ ions they observed the formation of snowballs with up to 41 He atoms and identified the closures of the first solvation layers at He14Rb+ and He16Cs+, somewhat smaller than the shell sizes predicted by theory.9,16 Later, An der Lan et al. studied He droplets containing Na and K monomer and dimer cations.17 They reported on snowballs containing up 30 He atoms, with the first shell closures identified after He9Na+ and He12K+. The lightest alkali ion, Li+, was excluded from both of these experimental studies (and others like them) as the small mass and isotopic composition of the Li ion could potentially obstruct the evaluation of mass spectrometric data, corresponding to alkali-helium snowball complexes.
In this present work we report on the solvation of Li+ ions in helium, evaluated with high-resolution mass spectrometry measurements and different theoretical methods. In our experiments, HenLi+ complexes containing several tens of He atoms are identified and anomalies in specific cluster size yields let us probe the ion stabilities of these systems. These results are compared with both classical and quantum mechanical (QM) simulations of a Li+ ion solvated with He atoms. In particular, and in a similar fashion as previous investigations of clusters formed doping coronene molecules with rare gas atoms and H2,18–20 we have carried out basin-hopping (BH), DMC and path integral Monte Carlo (PIMC) calculations. In addition to this, estimations of the quantum free energy (QFE) have been calculated, leading to very similar results to those obtained with QM corrections of the BH results including zero-point energy (ZPE) effects. Geometries and energies of the stable configurations observed for the different HenLi+ clusters have been investigated and, in particular, the behavior as a function of the size of each cluster has been analyzed in an attempt to understand the abundances observed in the experiment for each n.
The structure of the paper is as follows: in Section 2 we present the essential details of the experimental setup, in Section 3 we present the theoretical approaches employed in this work, and in Section 4 results are shown and discussed. Finally in Section 5 the conclusions are listed.
![]() | (1) |
![]() | (2) |
The corresponding parameters for both the He–He and He–Li+ potentials using the ILJ analytical expression are given in Table 1. The effects of 3B terms are investigated by introducing an induced dipole-induced dipole interaction as that employed in previous studies13,25 with damping functions in our PES:
![]() | (3) |
![]() | (4) |
m | r m | ε | β | |
---|---|---|---|---|
He–Li+ | 4 | 1.90 | 81.3 | 4.2 |
He–He | 6 | 2.97 | 0.947 | 8 |
Fα(T) = −kBT![]() ![]() | (5) |
![]() | (6) |
There is an alternative version of this method in which the partition function given in eqn (6) is replaced by its classical expression. This classical free energy tends asymptotically to the BH results when the temperature is decreased to zero, whereas the QFE tends to the QM corrected BH + ZPE values.
![]() | (7) |
The integrated counts from each HenLi+ complex are shown in Fig. 2. The standout features in this spectrum are the local maxima observed for n = 2, 6, 8, and 14, indicating that these are particularly stable systems (compared to their neighbors). At higher masses there are a few dips in the spectrum at n = 21, 24, and 27–28 on top of an underlying distribution that smoothly tapers off towards large cluster sizes.
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Fig. 3 Evaporation energies obtained by means of the BH (green circles), QFE (full red circles), DMC (blue triangles) and PIMC (black squares) methods. |
Second energy differences, defined as Δ2E = En+1 + En−1 − 2En, are also a useful magnitude to search for stable HenLi+ clusters at specific numbers of He atoms. The results obtained by means of the PIMC, BH + ZPE and QFE approaches are shown in Fig. 4. As expected the features observed in the curve of the evaporation energy of Fig. 3 also manifest as peaks when we plot these Δ2E differences. Thus, noticeable maxima are observed also at n = 4, 6, and 8, which, in view of the BH result also included in Fig. 4, seem to have their origin in the minima of the PES. The comparison between the second energy differences obtained by means of the BH and those calculated with the other methods reveals however noticeable discrepancies between classical and QM approaches. The classical result suggests a similar feature at n = 10 as well but the QM calculations do not entirely confirm this regard, in apparent agreement with the experiment. The integrated yield shown in Fig. 2 also exhibits maxima at n = 6 and 8, whereas for n = 4, only a suggested shoulder is seen.
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Fig. 4 Second difference energies (see text for details) obtained with the PIMC (black), QFE (red), BH + ZPE (blue) and BH (green) theoretical methods. |
Two of the most prominent maximum peaks observed in Fig. 4, those for He6Li+ and He8Li+, are also clearly visible in the measured abundances shown in Fig. 2. The equilibrium geometries associated to the minimum energy configurations have been investigated before in previous works: He6Li+ exhibits a symmetrical octahedral configuration with the He atoms coordinating the Li+ impurity, located at the center of the cluster.10,13 He8Li+, on the other hand, has a stable configuration formed by two parallel squares rotated by π/4 to each other surrounding the Li+ ion,14 which was found also as the inner core of larger clusters, thus suggesting that it corresponds to the geometry of the first solvation shell. This hypothesis was confirmed by the integration of the shell performed by Paolini et al.14 with ground state path integral calculations which yielded a value of 8.24 He atoms.
In this work we have performed DMC and PIMC calculations of the probability density functions corresponding to the HenLi+ clusters with n = 4, 6 and 8, those which correspond to special features in the curves as a function of the number of He atoms shown in Fig. 3 and 4. In the top panels of Fig. 5 we show the PIMC distributions obtained using a representation on the Eckart frames for specific snapshots of the quantum beads for each atom and their corresponding average represented as a cloud surrounding the expected location of both the He and Li+ atoms. The choice of a system satisfying Eckart conditions47 to guarantee an optimal separation between rotation and vibration, is made here only for pictorial purposes. Analogously, geometries obtained by averaging the positions of the DMC replicas (after rotation to a common body-fixed frame) are included in the figure. Both methods yield distributions which are not far from the equilibrium structures predicted by classical energy minimization algorithms.10 Thus, the QM approaches find a structure for He4Li+ which contains the ionic impurity caged inside a tetrahedron formed by the four He atoms and, for n = 6 and 8, the above mentioned octahedral and parallel squares structures found in previous investigations are reproduced here by means of the DMC and PIMC calculations. Although the probability density functions (not shown here) for the inter-particle distances, He–He and He–Li+, and the corresponding angles obtained with the DMC approach, certainly exhibit an intrinsic broadening, the maxima are only slightly deviated with respect to the stick values predicted in classical energy minimization studies.
Our calculations also reveal the stability of the structure found for He8Li+. Thus, Fig. 6 shows BH and PIMC results for He10Li+ indicating that both the classical optimized geometry and the QM probability density function consists, in essence, of the core observed at n = 8 with the two extra He atoms located over the center of each parallel square. This result is consistent with previous findings for this particular cluster.10 This trend is maintained even for larger cluster sizes, and the analysis of radial and angular distributions reveals that the inner shell is quite similar to the structure seen for He8Li+ (see ESI†).
The comparison between these theoretical results and the experimental abundances reveals agreement for peaks at n = 6 and 8. However, the prominent maximum seen for n = 2 in the experimental data in Fig. 2 does not have a definitive direct explanation from the theory. This abundance anomaly from n = 2 is also observed in experiments with Na+ and K+ ions17 which suggests that this is a product of the ionization mechanism itself, which is not covered by the simulations. One possible explanation is that a He2* is formed by the initial electron impact which then through associative Penning ionization forms a He2Li+ complex. Furthermore, the high abundance of the He14Li+ complex in the experiments is not reproduced by calculations. This structure could be explained by the nesting of a parallel square structure like He8Li+ in an octahedron like He6Li+, or vice versa, similar to the nested solvation shells observed for the HenAr+48 and Hn−
49 clusters. However, a particularly stable structure with such a geometry is not observed in the present simulations. In fact, further DMC calculations were carried out starting with a geometry where a He6Li+ octahedron is nested inside a cube formed with eight He atoms (a higher energy classical local minimum) but, after the simulations, the cluster rearranged to a structure with a core formed by eight atoms. This final geometry was not particularly stable as compared with their closest neighbors n = 13 and 15.
In an attempt to test the effect of 3B terms on our present results we introduce a conveniently damped induced dipole-induced dipole interaction contributions as in ref. 25 and calculate the corresponding evaporation energies. The comparison of this magnitude as a function of the number of He atoms obtained by means of the present BH and DMC methods is shown in Fig. 7. Some differences are certainly observed between those energies calculated only with the 2B pairwise description and those with the 3B terms included, especially for the smallest clusters n < 8. Beyond that size evaporation energies are practically the same regardlessly the potential interaction employed. However the qualitative trend is the same for both the classical and the QM results in the figure. In addition, the second difference energies (not shown here) calculated with the 3B effects do not exhibit substantially different features in comparison with Fig. 4, and in particular, no new peaks are seen. This suggests that contributions from terms beyond a mere 2B description, being significant in terms of the absolute energies of the clusters, do not improve in essence the comparison between theoretical and experimental results shown in this work.
The theoretical results show that classical approaches such as basin-hopping (BH) predicts qualitatively similar cluster properties as the quantum mechanical approaches of Quantum Free Energy (QFE), Diffusion Monte Carlo (DMC), and Path Integral Monte Carlo (PIMC). The simulations identify three particularly stable He shells surrounding the Li+ ions for n = 4, 6, 8, and a slightly weaker structure at 10. The sizes of n = 6 and n = 8 line up well with the experimental findings, suggesting that these features in the experimental mass spectrum are the results of these clusters stable octahedral and parallel square geometries, respectively. A larger magic structure observed for He14Li+ is observed in the experiments, but not in the simulations, could indicate the formation of multiple rigid and nested solvation shells.
Footnote |
† Electronic supplementary information (ESI) available: Radial distributions of HenLi+ snowballs. See DOI: 10.1039/c8cp04522d |
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