Rare-gas nanodroplets, in particular helium droplets, provide a unique environment for studying molecular complexes and chemical reactions. They are used as cold isothermal nanoscopic reactors to trap and cool molecules, free radicals, molecular ions and weakly bound clusters of molecules at very low temperatures, while keeping them isolated as in the gas phase (i.e. the species do not condensate). This allows high-resolution spectroscopy on individual molecules. Furthermore, the nanodroplets can be used to form, trap and study metastable species that would not last long enough in the gas phase. They can also be employed to grow molecular species and nanostructures before deposition on a solid target . Nanodroplets isolation technique opened therefore new possibilities for the study of physical and chemical phenomena and of exotic chemical species (see Ref. , for recent reviews).
Droplets are prepared by expansion of a pre-cooled and high-pressure atomic gas in vacuum and generally contain a few thousand atoms. As the droplets’ formation is a statistical process, they are not monodispersed. In standard isolation experiments, the pure droplets are collided with the species to be isolated, considered as dopants of the droplets (the so-called pick-up technique). However, the number of dopant atoms/molecules which may stick to the droplets as well as their location in the droplets depends on the size of the initial pure droplets. Therefore, determining and controlling the size distribution of nanodroplets is an important issue and represents an active research field (see Ref.  and references therein).
Currently, the structure of nanodroplets is probed using mass spectrometry . Droplets are ionized (usually by electron impact) and undergo fragmentation. However, the fragmentation dynamics is fairly complex and determining the structure of the droplets from the mass spectrum is not straightforward . Several theoretical works have focused on the structure and the dynamics of pure and doped droplets, either in the description of the droplets formation and pick-up processes , or on understanding the fragmentation dynamics ,. An obstacle in modeling the latter is that the number of charges and the distances between them are not known. In the case of multiple ionization the explosion of the droplets depends strongly on the distance between the charges. Models have thus to rely on some kind of statistical averaging over the distances. Even in the case of singly-ionized droplets, simulating the fragmentation dynamics is difficult since important quantum effects are expected. A better control of the number of charges and their location is therefore desirable. One of the goals of this project is to investigate Interatomic Coulombic Decay (ICD) as means to control the charges in nanodroplets.
Interatomic Coulombic decay is an efficient non-radiative electronic de-excitation process for atoms and molecules embedded in a chemical environment. Via this effect, the excess energy of the excited system is transferred, within few hundreds of femtoseconds or less, to a neighboring atom/molecule inducing its ionization (Fig. 1). As ICD occurs between nearest neighbors it selectively creates two charges at neighboring sites in the droplets, which leads to rapid fragmentation of the latter. The distance between the charges can be determined from the energy of the ICD electron. The initial situation in the doubly charged droplets can therefore be extremely well characterized, which should help to infer the initial structure and composition of the droplets.
The ICD process was predicted in the late 90’s by Cederbaum et al. . It was experimentally demonstrated about 10 years ago on the example of neon clusters by Marburger et al.  and Jahnke et al. . Since then, it was shown that ICD is a general process, taking place in a large variety of systems, like hydrogen bonded  or van der Waals clusters . It has been observed mainly after photoionization or photoexcitation but has also been experimentally demonstrated after electron  and ion impact . ICD was first predicted and observed after ionization in the inner-valence shell. Later it was also shown to occur after (i) two-electron processes like ionization-excitation  and two-electron excitation , (ii) Auger  and resonant Auger  cascades. Ref.  contains a list of all ICD publications.
Fig. 1 Interatomic Coulombic decay process in He dimer. The process is initiated by photoexcitation of one of the atoms, followed by ICD and ejection of a low-energy electron from the neighbor; the formation of two positively charged species leads to a Coulomb explosion of the system.
Another reason for the current broad interest in ICD is the possibility of analytical applications. For example, Auger effect is widely used to probe the atomic composition and the environment of the atoms of materials via Auger electron spectroscopy. ICD is clearly superior to Auger effect in its sensitivity to the chemical environment. Indeed, here the decay itself is possible only due to the effect of the environment, even if the interaction of the ionized atom or molecule with the environment is very weak. In this context, it was shown that the surroundings of an atom or a molecule can be probed over extremely large distances. The helium dimer for which ICD occurs at interatomic distances up to 14 Å  is a remarkable example of this property.
The aim of the project is to assess the applicability of ICD as a spectroscopic tool for probing the structure of nanodroplets. To answer this question, the electronic decay and subsequent fragmentation dynamics of helium clusters of different sizes have been simulated from first principles. From these simulations, the kinetic-energy distributions (KER spectra) of the ionic fragments, which are the main observables, are obtained and compared with available experimental data in order to establish a relationship between the observables and the size of the initial droplets. It is expected that this project provides enough detailed information to propose a new spectroscopic tool for characterizing nanodroplets. Moreover, this investigation will be a first step towards the study of doped nanodroplets. ICD spectroscopy may help to determine the number of the dopants as well as their location and thus lead to a better control of the isolation technique.
It should be noted that currently there are only very few, relatively small, systems (e.g. helium and neon dimer) for which the ICD process has been given a full theoretical description. Modeling the ICD process and the dynamics associated with it in large systems is challenging. The project provided, for the first time, a complete theoretical description of much more extended systems. In order to achieve the goal of describing ICD in systems containing thousands of atoms, original and state-of-the-art theoretical methods and numerical tools were developed.
Below you can find a short description of the methods we developed and used to accomplish our research goal, a summary of the main results, as well as a list of the project publications.
The “diatomics-in-molecule” (DIM) approach is a configuration interaction method in a limited basis set constructed on the asymptotic atomic eigenfunctions. In its standard implementation , the Hamiltonian is partitioned into atomic (Hα) and diatomic (Hαβ) parts
where N is the number of atoms. The Hamiltonian matrix elements of the system are then evaluated using the energies of the atoms and all pairs of atoms forming the system. The DIM Hamiltonian matrix is small and scales linearly with the number of atoms within the system. Diagonalization of the Hamiltonian provides the energy of the states and the corresponding wavefunctions in the DIM basis set (see Fig. 2). Using the DIM wavefunctions and the decay rates for all pairs of atoms of the system, the total and partial decay rates of the whole system can be computed.
Fig. 2 Scheme of the DIM approach specifically designed for the description of ICD. The energy of the atomic and diatomic states (EA and EAB, respectively) of the atoms and all pairs of atoms forming the system are employed to compute the DIM Hamiltonian matrix elements. Diagonalization of the Hamiltonian matrix (HDIM) yields the energy of the states (EDIM) of the whole system and the DIM eigenvector matrix (UDIM). The latter can then be used to compute the energy gradients using the matrix Δ made up of the gradients of the energies of all diatomic states (EAB) as well as the total and partial decay rates for the whole system using the decay rates of all pair of atoms (VAB).
The semiclassical method recently developed in Ref.  involves a classical treatment of the nuclear motion while retaining a quantum description of the electron dynamics: the nuclei follow classical trajectories on the potential energy surface of the electronic decaying state. To simulate the electronic decay, the trajectories may jump to the final state and propagate on the corresponding potential energy surface. The size and the kinetic-energy distributions of the fragments may be computed from the trajectories. Results of this method for simple systems compare well with that of a full quantum treatment . Furthermore, this approach has low computational costs and can be generalized to the investigation of large systems. This semiclassical approach is combined with the DIM method proposed for computing the rates and energetics of the droplets.
The code of the MD-DIM method can be downloaded here.
The results of the ICDSpect project demonstrate that ICD can be used to probe the size of helium clusters Hen for n = 2 - 112. For n = 2 - 7 we show that the kinetic energy distribution of the ions produced in the ICD process ions are characteristic of the size of the cluster. For larger clusters, our work shows that the photoelectron spectra are a powerful probe of cluster size. Nanodroplets have been partially studied and they are still under investigation.
Fig. 3 Theoretical coincidence map of kinetic energies of the ions occurring after the shake-up ionization and ICD of Hen (n = 2, 3 upper panels, n = 4, 5 middle panels, n = 6, 7 lower panels).
The kinetic energy of the ions contains information both on the nuclear dynamics in the excited states and during the Coulomb explosion. The coincidence spectra of the small Hen clusters for n = 2 - 7 can be seen on Fig. 3. The upper left spectrum corresponds to the KER of the dimer. It exhibits an intense diagonal feature resulting from a Coulomb explosion where the two ions equally share the kinetic energy. Apart from the diagonal dimerlike feature, the spectra of the remaining clusters exhibit an additional non-diagonal feature, which corresponds to a fast and a slow ion. As can be seen from the coincidence maps, the number of the trajectories resulting in a fast and a slow ion increases with the number of atoms.
Fig. 4 Frustrated Coulomb explosion: the formation of a slow ion and a neutral fast atom following ICD. These products are a result of charge transfer between the fast He+ ion and the surrounding neutral He atoms.
To explain the asymmetry in the kinetic energy between the two ions, we analyzed the classical trajectories and we found an alternative collision mechanism, involving a charge transfer between a (fast) ion and a (slow) neutral helium atom (see Fig. 4). We term this mechanism Frustrated Coulomb explosion.
Comparing the coincidence spectra of the trimer and tetramer, one can see that an additional feature appears at low kinetic energies of both ions, which becomes even more intense in the larger clusters (n > 4). Our analysis shows that this feature corresponds to the events where both ions transfer their charge following the Coulomb explosion. This results in the production of two slow ions, the majority of which have kinetic energies below 2 eV.
Consequently, the total coincidence spectra of the clusters with more than two atoms may be divided in three parts which are almost non overlapping: a diagonal part corresponding to dimerlike trajectories with no charge transfer, a non-diagonal part with one fast and one slow ion, corresponding to trajectories with one charge transfer, and a wide low-energy energy part corresponding to trajectories where both ions transferred their charges.
Fig. 5 Kinetic energy release distributions for Hen clusters of sizes n = 2 - 6.
Figure 5 shows the kinetic energy release spectra for different cluster sizes, which is the sum of the kinetic energy of the two ions. The KER spectrum of the dimer agrees well with the theoretical and experimental KER spectra reported in the literature . It exhibits an intense peak at 9 eV, which corresponds to the energy of two ions placed at the shortest distance accessible to a He2+ pair. As the internuclear distance increases, the ICD lifetime increases, hence the intensity decreases for lower KER.
Compared to the spectrum of the dimer, the He3 KER spectrum exhibits an additional feature around 5 eV, which is a result of the charge transfer discussed above. For He5−7, a shoulder starts to build up on the low kinetic-energy side of the peak at 5 eV. It is a result of the double charge transfer, which becomes almost 1/3 of all charge-transfer events for the He7 cluster. The intensity of the shoulder and of the peak at 5 eV increases with the number of atoms, whereas the intensity of the high kinetic energy peak decreases. Thus, for the He7 cluster, about 90% of all trajectories undergo frustrated Coulomb explosion. Figure 5 clearly shows that the KER spectra depend on the cluster size and consequently, they can be used as a tool to probe the cluster size.
The data used to generate the KER spectra can be downloaded here.
The data used to generate the coincident KER maps can be downloaded here.
Our results have been summarized in the following papers:
- Direct observation of Interatomic Coulombic Decay and subsequent ion-atom scattering in helium nanodroplets F. Wiegandt, F. Trinter, K. Henrichs, D. Metz, M. Pitzer, M. Waitz, E. Jabbour al Maalouf, C. Janke, J. Rist, N. Wechselberger, T. Miteva, S. Kazandjian, M. Schöffler, N. Sisourat, T. Jahnke, and R. Dörner https://arxiv.org/abs/1806.05214
- Frustrated Coulomb explosion of small helium clusters S. Kazandjian, J. Rist, M. Weller, F. Wiegandt, D. Aslitürk, S. Grundmann, M. Kircher, G. Nalin, D. Pitters, I. Vela Pérez, M. Waitz, G. Schiwietz, B. Griffin, J. B. Williams, R. Dörner, M. Schöffler, T. Miteva, F. Trinter, T. Jahnke, and N. Sisourat https://arxiv.org/abs/1806.06276
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This project has received funding from the Research Executive Agency (REA) under the European Union's Horizon 2020 research and innovation program Grant agreement No. 705515 and from Agence Nationale de la Recherche through the program ANR-16-CE29-0016-01.