Wan Xu

Xiao-Di Liu

Miriam Peña-Alvarez

Hua-Chao Jiang

Philip Dalladay-Simpson

Benoit Coasne

Julien Haines

Eugene Gregoryanz

Mario Santoro
email: santoro@lens.unifi.it

High Pressure Insertion of Dense H 2 into a Model Zeolite

Keywords: High pressure, hydrogen, zeolites, Raman spectroscopy, X-ray diffraction, Monte Carlo modelling

Our combined high-pressure synchrotron x-ray diffraction and Monte Carlo modelling studies show super-filling of the zeolite, and computational results suggest an occupancy by a maximum of nearly two inserted H 2 molecules per framework unit, which is about twice more than observed in gas hydrates. Super-filling prevents amorphization of the host material up to at least 60 GPa, which is a record pressure for zeolites and also for any group IV element being in full 4-fold coordination, except for carbon. We find that the inserted H 2 forms an exotic topologically constrained glassylike form, otherwise unattainable in pure hydrogen. Raman spectroscopy on confined H 2 shows that the micro-porosity of the zeolite is retained over the entire investigated pressure range (up to 80 GPa) and that inter-molecular interactions share common aspects with bulk hydrogen while they are also affected by the zeolite framework.

with a broad range of industrial applications. Adsorption studies on zeolites at extreme conditions led some of us and other groups to develop a rich and dynamic research area.

Particularly, the high pressure insertion of simple dense molecular systems as the guest species in zeolite hosts leads to the formation of exotic guest phases and to unique properties for the host framework [START_REF] Hazen | Zeolite molecular sieve 4A: anomalous compressibility and volume discontinuities at high pressure[END_REF][START_REF] Lee | Non-framework cation migration and irreversible pressure-induced hydration in a zeolite[END_REF][START_REF] Lee | Pressure-induced argon insertion into an auxetic small pore zeolite[END_REF][START_REF] Haines | Deactivation of pressure-induced amorphization in silicalite SiO 2 by insertion of guest species[END_REF][START_REF] Lee | Pressure-and Heat-Induced Insertion of CO 2 into an Auxetic Small-Pore Zeolite[END_REF][START_REF] Santoro | Silicon carbonate phase formed from carbon dioxide and silica under pressure[END_REF][START_REF] Niwa | Pressureinduced noble gas insertion into Linde-type A zeolite and its incompressible behaviors at high pressure[END_REF][START_REF] Santoro | High-pressure synthesis of a polyethylene/zeolite nano-composite material[END_REF][START_REF] Seoung | Irreversible xenon insertion into a small-pore zeolite at moderate pressures and temperatures[END_REF][START_REF] Gatta | Zeolites at high pressure: A review[END_REF][START_REF] Vezzalini | High-pressure-induced structural changes, amorphization and molecule penetration in MFI microporous materials: a review[END_REF][START_REF] Santoro | High Pressure Synthesis of All-Transoid Polycarbonyl [-(C=O)-] n in a Zeolite[END_REF][START_REF] Scelta | High pressure polymerization in a confined space: conjugated chain/zeolite nanocomposites[END_REF][START_REF] Santoro | Synthesis of 1D polymer/zeolite nanocomposites under high pressure[END_REF][START_REF] Santoro | Intermolecular Interactions in Highly Disordered, Confined Dense N 2[END_REF][START_REF] Santoro | Insertion and Confinement of H 2 O in Hydrophobic Siliceous Zeolites at High Pressure[END_REF] . Once the latter is supported by inserted guests to form a "molecular spring", it resists the applied pressures. But what would happen if the zeolites are filled by the most penetrating fundamental systems and also the most abundant element in the universe: hydrogen? Indeed, dense H 2 is a "master" and benchmark system in high pressure sciences. [START_REF] Gregoryanz | Everything you always wanted to know about metallic hydrogen but were afraid to ask[END_REF] Dense, sub-nano confined states of H 2 could be investigated and compared to those of bulk hydrogen at extreme conditions, since these states would add to the general view of such a fundamental element. The highly penetrating character of H 2 could be also compared to other larger molecular and atomic systems to allow us to investigate the ultimate capability of filled zeolites to resist the pressure-induced pore collapse and the consequent pressure induced amorphization (PIA). The utmost removal of PIA can then provide information on the local structure of the framework cation, silicon in our case, at record extreme conditions. Indeed, it is of great interest to search to which extent the 4-fold coordination of silicon can be preserved. In bulk silica, the thermodynamic 4-fold-to-6-fold transformation occurs below 10 GPa, while metastable 4-fold coordinated phases can survive at most up to 20-40 GPa. [START_REF] Hemley | High-pressure behavior of silica[END_REF][START_REF] Prokopenko | In situ characterization 16 of phase transitions in cristobalite under high pressure by Raman spectroscopy and X-ray diffraction[END_REF][START_REF] Haines | Crystalline post-quartz phase in silica at high pressure[END_REF][START_REF] Bykova | Metastable silica high pressure polymorphs as structural proxies of deep Earth silicate melts[END_REF] In gas hydrates, similar host-guest systems, the four-connected framework is built up by H-bonded water molecules, instead of being covalently bonded like in zeolites, and it is filled by simple gaseous molecules. Recently, a CH 4 filled ice has been found to be stable up to the record pressure of 150 GPa. [START_REF] Schaack | Observation of methane filled hexagonal ice stable up to 150 GPa[END_REF] On the other hand, cold compression of methane clathrates at GPa has been found to lead to PIA with the amorphous form still being a host-guest gas hydrate. [START_REF] Tulk | Structure and stability of an amorphous water-methane mixture produced by cold compression of methane hydrate[END_REF] We focused our work on a model, pure SiO 2 zeolite, silicalite-1 in order to avoid catalytic effects and to investigate how the pore size and shape affect the topology of the confined molecular form under pressure. Silicalite-1 is characterized by a framework of 4-, 5-, 6-, and 10-membered rings of corner-sharing SiO 4 tetrahedra forming interconnected, mutually orthogonal straight and sinusoidal channels, with ambient pressure diameters of close to 5.5 Å (see Figure 1, at 7 GPa, together with inserted H 2 ). [START_REF] Olson | Crystal structure and structure-related properties of ZSM-5[END_REF][START_REF] Hay | Orthorhombic-monoclinic phase changes in ZSM-5 zeolite/silicalite[END_REF] Silicalite-1 is produced in crystals of several tens of microns, which makes it very suitable for optical spectroscopy studies.

In this work we show the high-pressure insertion of H 2 in silicalite-1 up to 82 GPa. Based on synchrotron X-ray diffraction (XRD) and Monte Carlo (MC) modelling, we show that the insertion leads to an exotic glassy-like form of molecular hydrogen. H 2 -filled silicalite-1 is found to be stable up to at least 60 GPa and significantly less compressible than the same zeolite filled by the bigger molecules; this being a consequence of the more penetrating nature of H 2 . The number of inserted molecules vs. pressure is non-monotonic with a maximum close to two per SiO 2 unit. Raman spectroscopy shows that the framework porosity is retained up to at least 82 GPa and it provides important clues on inter-molecular interactions.

Our experimental and theoretical methods are described in Supplementary Materials 30 together with the relevant references. [START_REF] Mao | Specific volume measurements of Cu, Mo, Pd, and Ag and calibration of the ruby R 1 fluorescence pressure gauge from 0.06 to 1 Mbar[END_REF][START_REF] Akahama | Pressure calibration of diamond anvil Raman gauge to 310 GPa[END_REF][START_REF] Howie | Hydrogen (deuterium) vibron frequency as a pressure comparison gauge at multi-Mbar pressures[END_REF][START_REF] Prescher | DIOPTAS: a program for reduction of two-dimensional X-ray diffraction data and data exploration[END_REF][START_REF] Carvajal | Recent developments of the program FULLPROF, in commission on powder diffraction (IUCr)[END_REF][START_REF] Fei | Toward an internally consistent pressure scale[END_REF][START_REF] Desbiens | Water condensation in hydrophobic silicalite-1 zeolite: a molecular simulation study[END_REF][START_REF] Coasne | Intrusion and retraction of fluids in nanopores: effect of morphological heterogeneity[END_REF][START_REF] Leachman | Fundamental equations of state for parahydrogen, normal hydrogen, and orthohydrogen[END_REF][START_REF] Mccarty | Selected properties of hydrogen (engineering design data)[END_REF][START_REF] Kunz | The GERG-2004 Wide-Range Reference Equation of State for Natural Gases and Other Mixtures GERG TM15[END_REF][START_REF] Darkrim | Monte Carlo simulations of hydrogen adsorption in singlewalled carbon nanotubes[END_REF][START_REF] Desbiens | Water condensation in hydrophobic nanopores[END_REF][START_REF] Garberoglio | Adsorption of gases in metal organic materials: comparison of simulations and experiments[END_REF] Powder XRD patterns of H 2 -filled silicalite-1 were measured upon increasing pressure up to 60 GPa (Figure 2), in order to investigate to which extent the framework is stable at high pressures and also to provide inputs for Monte Carlo modelling aimed to determine the amount of stored hydrogen. The large, nanometer scale, unit cell size of silicalite-1 is related to the micro-porosity of the framework and it gives rise to the 101, 011, 200, 020 and 111 diffraction peaks located at very low 2θ angles, down to 2 • -3 • . The widths of the Bragg peaks increased gradually by a factor of two over the entire pressure range investigated. This can be linked to a gradual increase in deviatoric stress in the solid H 2 pressure medium. We also observed some apparent changes in relative intensity, principally due to the separation of overlapping peaks resulting from changes in the unit cell parameters (Figure SM2 and Table SM1). [START_REF] Leachman | Fundamental equations of state for parahydrogen, normal hydrogen, and orthohydrogen[END_REF][START_REF] Mccarty | Selected properties of hydrogen (engineering design data)[END_REF][START_REF] Kunz | The GERG-2004 Wide-Range Reference Equation of State for Natural Gases and Other Mixtures GERG TM15[END_REF] Importantly, at the highest pressures, in spite of the increase in line-width, the peaks retained much of their initial integrated intensity (typically and crystallographic (bottom) view of H 2 inserted in silicalite-1 at room temperature and P = 7 GPa. In the molecular view, the red and orange sticks correspond to the chemical bonds between silicon and oxygen atoms while the white spheres correspond to the hydrogen atoms in H 2 (the sphere radius roughly corresponds to the van der Waals radius of hydrogen). In the crystallographic view, the blue tetrahedra correspond to the silica tetrahedra in silicalite-1 with the red spheres showing the apical oxygens. The black dumbells represent the inserted H 2 molecules. A 2×2×2 unit cell is considered here for the sake of clarity. [START_REF] Haines | Deactivation of pressure-induced amorphization in silicalite SiO 2 by insertion of guest species[END_REF] Black line: equation of state for silicalite-1, measured in a non-penetrating pressure transmitting medium: silicon oil, where amorphization has been found to occur below 10 GPa. 30-100 %) at low pressure. That suggests that silicalite-1 remaines crystalline up to the highest pressure and, consequently, full 4-fold coordination of silicon by oxygen was retained at record pressures among all known group IV compounds, except carbon. Indeed, the thermodynamic transformation pressure from 4-fold to 6-fold coordination in bulk silica is lower by one order of magnitude. LeBail fitting of the XRD patterns (Figure SM3) was used to obtain the pressure behavior of the unit cell volume of H 2 -filled silicalite-1 (Figure 2) normalized to its ambient pressure value. We can compare this compression curve with those observed for other penetrating simple gaseous systems such as Ar and CO 2 . [START_REF] Haines | Deactivation of pressure-induced amorphization in silicalite SiO 2 by insertion of guest species[END_REF] The most striking result is the much higher relative volumes of up to 24 % at 25 GPa measured for H 2 -filled silicalite-1 than for Ar or CO 2 filled silicalite-1, clearly indicating that H 2 is much more penetrating than the other two larger systems. Also, the compression curve of H 2 -filled silicalite-1 shows an anomaly. Indeed, a quasi-horizontal inflection point appears at around 19 GPa. Above this point, the curve is convex rather than concave, up to about 50 GPa. This anomaly is suggestive of substantial pressure changes in the filling of silicalite-1 by H 2 , a hypothesis that can only be tested on the Grand Canonical ensemble based Monte Carlo (MC) model, whose main outputs are the spatial distribution and the number of inserted molecules per unit cell.

In Figure 1, we report the structure of H 2 filled orthorhombic silicalite-1 at 7.0 GPa, obtained by combining the experimentally determined lattice parameters with Monte Carlo modelling. The zeolite framework is entirely filled by guest H 2 molecules, 127 per unit cell.

The state of inserted hydrogen is remarkable. Indeed, this is a disordered, dense glassy-like form within all the investigated pressure range by XRD and MC, i.e. up to 60 GPa. The calculated number of inserted H 2 molecules per unit cell of silicalite-1 by MC as a function of pressure clearly exhibits a maximum (Figure 3, left panel), which corresponds to the anomaly experimentally observed for the pressure behavior of the unit cell volume. The maximum is equal to about 174 and it is located at around 34 GPa. The very low compressibility of H 2 -filled silicalite-1 in the 15-25 GPa pressure interval is now found to be due to H 2 rapidly entering silicalite-1 to a greater and greater extent upon increasing pressure in this range; here, the filling of H 2 hinders the reduction of the host framework volume by strongly reducing the compressibility of the pores. Then, above 30-40 GPa, some amount of previously inserted H 2 is extruded and the volume of silicalite-1 decreases as a result of the combined effect of guest extrusion and normal compression of the filled framework material. H 2 -filled silicalite-1 is thus a hydrogen rich material, with H 2 being physisorbed by the zeolite, and an overall pressure dependent chemical composition: SiO 2 (H 2 ) x , with x=1.32-1.82. In fact, this gas content is substantially greater, in terms of number of stored guest molecules, than that experimentally observed so far in any gas hydrate phase at any pressure-temperature condition. [START_REF] Loveday | High-pressure gas hydrates[END_REF][START_REF] Mao | Clathrate hydrates under pressure[END_REF] At least two remarkable differences emerge between H 2 -filled silicalite-1 and the filling of this zeolite with larger simple systems such as Ar and CO 2 . [START_REF] Coasne | Enhanced mechanical strength of zeolites by adsorption of guest molecules[END_REF] Firstly, the maximum number of confined H 2 molecules exceeds that for bigger molecules by a factor of 3-4. Secondly, H 2 is so much more penetrating that even the cages with small openings of 1.4-3.0 Å diameter built up by 5-and 6-membered rings are well filled in this case. We found that the number of H 2 molecules in these cages is equal to 20 per unit cell over all the investigated pressure range, which is 11-16 % of the total amount of inserted molecules. Also, we found that multiple filling of some cages occurs, similarly to several gas hydrates. [START_REF] Loveday | High-pressure gas hydrates[END_REF][START_REF] Mao | Clathrate hydrates under pressure[END_REF][START_REF] Kuhs | Cage Occupancy and Compressibility of Deuterated N 2 -Clathrate Hydrate by Neutron Diffraction[END_REF][START_REF] Mao | Hydrogen Clusters in Clathrate Hydrate[END_REF] Interestingly, the non-monotonicity for the pressure behavior of the filling appears to be entirely due to molecules inserted in the channels. All this explains why the volume of the zeolite is so much larger when H 2 , rather than bigger guests, is inserted in the framework.

Importantly, the nearest neighbor's intermolecular distance for confined hydrogen is larger than that for bulk hydrogen (Figure 3, right panel) by a few percent, indicating that the H 2 -silicalite-1 interaction adds a negative term to the local pressure.

The Raman spectroscopy investigation provides direct information on dynamical properties of the confined dense form of hydrogen. In Figure 4, we report waterfalls of selected Raman spectra of the H 2 vibron for H 2 /silicalite-1 samples, measured upon increasing pres- sure on silicalite-1 crystals up to 82.5 GPa. In the spectra, we observe the pure H 2 peak due to bulk hydrogen layers surrounding the crystals and several blue shifted extra peaks which can be easily attributed to confined dense H 2 . The blue shift, which increases with pressure, is the net result of the modified intermolecular interactions in confined hydrogen and the interaction between the guest molecules and the internal walls of the porous host. Spectra measured on larger silicalite-1 crystals show up to 4-5 partially resolved peaks for confined hydrogen (Figure 4, left panel), likely to be ascribed to H 2 guest molecules located on distinct host crystallographic sites of silicalite-1 and, as a consequence, experiencing different interactions. These peaks are much broader than the peak of bulk H 2 and they broaden upon increasing pressure till they merge. This finding is compatible with confined hydrogen being highly disordered around the different crystallographic sites corresponding to the distinct peaks, in full agreement with the combined XRD/MC outcome where confined H 2 is found to be in a glassy-like state

In Figure 5, we report the pressure shift of the H 2 frequency for the most intense peak in confined H 2 and for bulk H 2 measured in this work up to 82.5 GPa, and also for isolated H 2 impurities in three crystalline matrices: Ne, Ar and D 2 . [START_REF] Loubeyre | Raman measurements of the vibrational properties of H 2 as a guest molecule in dense helium, neon, argon, and deuterium systems up to 40 GPa[END_REF] The pressure behavior for the confined H 2 frequency is remarkable: it is systematically higher than that of the Raman frequency for pure H 2 and, more importantly, in both cases we observe a maximum, which is located at around 50 GPa in confined H 2 and 35 GPa in pure H 2 . In addition, in confined hydrogen, a frequency jump occurs around 62 GPa, beyond the pressure range of our XRD investigation, which probably relates to a major structural change in silicalite-1 such as a yet unknown phase transition. The origin of the maximum in the H 2 frequency for confined hydrogen can be easily traced back to the H 2 -H 2 vibrational coupling [START_REF] Loubeyre | Raman measurements of the vibrational properties of H 2 as a guest molecule in dense helium, neon, argon, and deuterium systems up to 40 GPa[END_REF][START_REF] Moshary | Pressure dependence of the vibron in H 2 , HD, and D 2 : Implications for inter-and intramolecular forces[END_REF][START_REF] Hanfland | Synchrotron infrared spectroscopy at megabar pressures: Vibrational dynamics of hydrogen to 180 GPa[END_REF][START_REF] Hanfland | Novel infrared vibron absorption in solid hydrogen at megabar pressures[END_REF] (and references therein. See also SM). The Raman frequency being blue shifted in confined H 2 and the maximum being at higher pressures with respect to bulk H 2 suggest that vibrational coupling is weaker in the confined form. This is due the larger inter-molecular distance (see Figure 3) and to the reduced number of H Full and open dots: this work, values for confined in silicalite-1 and bulk H 2 , respectively (see text for details). Black lines: guides for an eye through these two data sets. Error bars are mainly due to the fitting procedure of peaks for confined H 2 . Other lines: pressure shift for H 2 as a single molecule impurity in Ne (black), D 2 (blue) and Ar (green) matrix, respectively. [START_REF] Loubeyre | Raman measurements of the vibrational properties of H 2 as a guest molecule in dense helium, neon, argon, and deuterium systems up to 40 GPa[END_REF] Orange line: pressure shift for the IR active H 2 frequency in pure hydrogen. [START_REF] Moshary | Pressure dependence of the vibron in H 2 , HD, and D 2 : Implications for inter-and intramolecular forces[END_REF][START_REF] Hanfland | Synchrotron infrared spectroscopy at megabar pressures: Vibrational dynamics of hydrogen to 180 GPa[END_REF][START_REF] Hanfland | Novel infrared vibron absorption in solid hydrogen at megabar pressures[END_REF] molecules in the cages of silicalite-1, where the number of H 2 neighbors drops to 1 or 0 and the vibrational coupling is nearly or entirely switched off. This case likely corresponds to the highest frequency components in the Raman frequency distribution for confined hydrogen (Figure 4). The Raman spectra of the rotational peaks (Figure SM4) [START_REF] Leachman | Fundamental equations of state for parahydrogen, normal hydrogen, and orthohydrogen[END_REF][START_REF] Mccarty | Selected properties of hydrogen (engineering design data)[END_REF][START_REF] Kunz | The GERG-2004 Wide-Range Reference Equation of State for Natural Gases and Other Mixtures GERG TM15[END_REF] show that confined H 2 is a near free rotor, the rotations of which are somewhat more hindered than for pure hydrogen. [START_REF] Peña-Alvarez | Quantitative rotational to librational transition in dense H 2 and D 2[END_REF] Our investigation uncovered an exotic form of dense hydrogen inserted in a zeolite. Which is the Mbar/multi-Mbar fate of H 2 -filled zeolites? Which is the fate of silicon coordination in gas filled framework materials and, more generally, of host-guest systems where the host is a covalent or H-bonded network and the guest is a simple atom or molecule, and which is the potential to synthesize novel H rich systems in this way? All these questions among others build up an entirely unexplored field, initiated here and left to future intriguing studies.

Acknowledgement

Supporting Information Available

The Supporting Information is available free of charge at https://

Figure 1 :

 1 Figure 1: Structure of H 2 -filled orthorhombic silicalite-1. Left: principles of MC modeling in the Grand Canonical ensemble. Like in real experiments, a zeolite material is set in contact with an infinite reservoir of bulk H 2 . The purple shaded area denotes the unit cell volume while the blue dashed line indicates the periodic boundary conditions. Right: molecular (top)and crystallographic (bottom) view of H 2 inserted in silicalite-1 at room temperature and P = 7 GPa. In the molecular view, the red and orange sticks correspond to the chemical bonds between silicon and oxygen atoms while the white spheres correspond to the hydrogen atoms in H 2 (the sphere radius roughly corresponds to the van der Waals radius of hydrogen). In the crystallographic view, the blue tetrahedra correspond to the silica tetrahedra in silicalite-1 with the red spheres showing the apical oxygens. The black dumbells represent the inserted H 2 molecules. A 2×2×2 unit cell is considered here for the sake of clarity.

Figure 2 :

 2 Figure 2: Left panel: Selected powder XRD (λ =0.4828 Å) patterns of H 2 -filled silicalite-1, measured upon increasing pressure. A broad Compton scattering background due to air and diamonds has been subtracted. Right panel: experimental pressure behavior for the relative to ambient pressure unit cell volume of H 2 -filled silicalite-1 (filled circles). Error bars are within the size of the circles. Red line: compression curve for Ar and CO 2 filled silicalite-1. 8 Black line: equation of state for silicalite-1, measured in a non-penetrating pressure transmitting medium: silicon oil, where amorphization has been found to occur below 10 GPa.

Figure 3 :

 3 Figure3: MC calculated pressure behavior of the number of inserted H 2 molecules in silicalite-1 (left) and of the ratio of the nearest neighbor's H 2 -H 2 distance for confined hydrogen to that of bulk hydrogen (right). Error bars are due to the simulation and to the evaluation method of the average distances. Red lines through the points are guides for the eye.

Figure 4 :

 4 Figure 4: Selected Raman spectra of H 2 -filled silicalite-1 crystals in the frequency range of the H 2 vibron, measured upon increasing pressure. Left (right) panel: spectra measured on top of a silicalite-1 crystal of 80×40×40 µm 3 (30×15×15 µm 3 ) initial size. Red dashed lines separate the bulk and the confined H 2 peaks. Insets: silicalite-1 crystals in the gasket holes, at 0.4 GPa and 4.2 GPa, in the left and right panels, respectively, immersed in H 2 as the pressure transmitting medium. Ruby chips are also present for pressure measurements.

Figure 5 :

 5 Figure 5: H 2 frequency vs. pressure for different materials and compounds.Full and open dots: this work, values for confined in silicalite-1 and bulk H 2 , respectively (see text for details). Black lines: guides for an eye through these two data sets. Error bars are mainly due to the fitting procedure of peaks for confined H 2 . Other lines: pressure shift for H 2 as a single molecule impurity in Ne (black), D 2 (blue) and Ar (green) matrix, respectively.[START_REF] Loubeyre | Raman measurements of the vibrational properties of H 2 as a guest molecule in dense helium, neon, argon, and deuterium systems up to 40 GPa[END_REF] Orange line: pressure shift for the IR active H 2 frequency in pure hydrogen.[START_REF] Moshary | Pressure dependence of the vibron in H 2 , HD, and D 2 : Implications for inter-and intramolecular forces[END_REF][START_REF] Hanfland | Synchrotron infrared spectroscopy at megabar pressures: Vibrational dynamics of hydrogen to 180 GPa[END_REF][START_REF] Hanfland | Novel infrared vibron absorption in solid hydrogen at megabar pressures[END_REF]

  The research was supported by the Chinese Academy of Sciences President's International Fellowship Initiative Fund (2019VMA0027), the National Natural Science Foundation of China (Grant Nos.11874361, 51672279, 11774354 and 51727806), the CAS Innovation Grant (CXJJ-19-B08), the Science Challenge Project (No.TZ2016001), and the CASHIPS Director's Fund (Grant No. YZJJ201705). We also acknowledge the PRIN project ZAPPING, number 2015HK93L7, granted by the Italian Ministry of Education, Universities and Research, MIUR. MPA acknowledges the support of the European Research Council (ERC) Grant Hecate Ref. No. 695527 and UKRI Future leaders fellowship Mrc-Mr/T043733/1.

  2 neighbors. An extreme case in this respect is that of

		4400	pure H 2 (IR)	H 2 in D 2 matrix
			H 2 in Ne matrix
	)			H 2 in Ar matrix
	2 vibron frequency (cm -1	4300		confined H 2 (peak frequency)	pure H 2
	H	4200
		0	20	40	60	80
				Pressure (GPa)

• Supplementary GCMC results.

• Supplementary XRD results.

• H 2 -H 2 vibrational coupling.

• Supplementary Raman Spectroscopy results


Florent Balacheff
email: florent.balacheff@math.univ-lille1.fr

Florent

A LOCAL OPTIMAL DIASTOLIC INEQUALITY ON THE TWO-SPHERE

Keywords: 2000 Mathematics Subject Classification. 53C23 Conical singularity, diastole, sphere, systole

Using a ramified cover of the two-sphere by the torus, we prove a local optimal inequality between the diastole and the area on the two-sphere near a singular metric. This singular metric, made of two equilateral triangles glued along their boundary , has been conjectured by E. Calabi to achieve the best ratio area over the square of the length of a shortest closed geodesic. Our diastolic inequality asserts that this conjecture is to some extent locally true.

Introduction

Let g be a smooth Riemannian metric on the two-sphere S 2 . Following I. Babenko [START_REF] Babenko | Topological entropy of geodesic flows on simply connected manifolds, and related problems[END_REF] we call general systole, or simply systole in this context, the least length of a non trivial closed geodesic and denote it by sys 0 . For every Riemannian metric g on S 2 , area(S 2 , g) ≥ 1 32 sys 0 (S 2 , g) 2 .

(1.1) Such an inequality was first proved by C. Croke (see [START_REF] Croke | Area and the length of the shortest closed geodesic[END_REF]), and later improved in [START_REF] Nabutovsky | The length of the shortest closed geodesic on a 2-dimensional sphere[END_REF], [START_REF] Sabourau | Filling radius and short closed geodesic of the sphere[END_REF] and [START_REF] Rotman | The length of a shortest closed geodesic and the area of a 2dimensional sphere[END_REF]. A natural question is to ask for the best constant in inequality (1.1). It amounts to find the global infimum of the systolic area defined as the ratio area/sys 2 0 among smooth metrics. A flat metric with three conical singularities was conjectured by E. Calabi (see [START_REF] Croke | Area and the length of the shortest closed geodesic[END_REF]) to be the global minimum. This metric, denoted by g c , is made of two equilateral triangles of size 1 glued along their boundary. Its systolic area is equal to 1 2 √ 3 .

We may wonder why such a metric has any chance to achieve the minimal systolic area. In order to answer and then give (if needed) strength to Calabi's systolic conjecture we ask the following question : Is such a conjecture locally true ?

Observe that the systolic area is not a continuous function of the metric, even when the space of smooth metrics is topologized by the strong topology (for a definition of strong topology, see [START_REF] Hirsch | Differential Topology[END_REF]p. 34]). In order to prove local results we introduce a quantity that coincides with the systole at g c and that varies continuously with the metric g. This quantity is the diastole (see [START_REF] Balacheff | Diastolic inequalities and isoperimetric inequalities on surfaces[END_REF]). The diastole of a Riemannian closed surface is defined as the value obtained by a minimax process over the space of one-cycles, see section 2.1.

It is denoted by dias. In the case of a smooth metric the diastole is realized as the length of an union of disjoint closed geodesics (see [START_REF] Pitts | Regularity and singularity of one dimensional stationary integral varifolds on manifolds arising from variational methods in the large[END_REF]p. 468]). Observe that the notion of diastole holds for any Riemannian closed surface with a finite number of conical singularities. It is proved in [START_REF] Balacheff | Diastolic inequalities and isoperimetric inequalities on surfaces[END_REF] that for every Riemannian closed surface (M, g) of genus k the diastole bounds from below the area:

area(M, g) ≥ C k + 1 dias(M, g) 2 .
Here C is a positive constant. In the case of a two-sphere and since sys 0 ≤ dias for smooth metrics we thus (re)obtain inequality (1.1) albeit with a worse constant. We call the ratio area/dias 2 the diastolic area and observe that it is continuous on the space of smooth metrics with a finite number of conical singularities for the uniform topology on metric spaces (see [START_REF] Aleksandrov | Intrinsic geometry of surfaces[END_REF]chapter 7] for a definition of uniform topology). The infimum of the diastolic area is not known for any surface, and we do not know if in the case of the two-sphere it is realized by g c and so equals 1/(2 √ 3). It is proved in [START_REF] Balacheff | Sur la systole de la sphère au voisinage de la métrique standard[END_REF] that the round metric g 0 is a critical point of the diastolic area over the space of smooth metrics conformal to g 0 topologized by the strong topology. As area(S 2 , g 0 ) dias(S 2 , g 0 ) 2 = 1 π , the round metric is not the global infimum. So its criticity is very interesting and suggests that it may be a local minimum.

In this paper we prove that the metric g c is a local minimum of the diastolic area over the space of singular metrics C 1 -conformal to g c with respect to the C 1 -topology. More precisely, let

M gc := e 2u • g c | u ∈ C 1 (S 2 , R)
be the space of metrics of class C 1 with three conical singularities of angle 2π/3 on S 2 conformal to g c . The space M gc is naturally in bijection with C 1 (S 2 , R * + ) and we will call C 1 -topology on M gc the topology induced by the C 1 compact-open topology on C 1 (S 2 , R * + ) (see [START_REF] Hirsch | Differential Topology[END_REF]p. 34]). The space M gc is adapted to the local study of the diastolic area near the metric g c . In fact we will observe in subsection (2.2) that, thanks to the Riemann uniformization theorem, any smooth metric can be written (up to an isometry) as a metric which differs from g c by a conformal factor which lies in C 1 (S 2 , R + ). Especially finding the infimum of the diastolic area over M gc amounts to find the infimum of the diastolic area over the space of smooth metrics, see proposition (2.2).

Theorem 1.1. There exists an open neighborhood O of g c in M gc with respect to the C 1 -topology such that for all g ∈ O,

area(S 2 , g) ≥ 1 2 √ 3 dias (S 2 , g) 2
with equality if and only if g = g c .

The proof is based on the study of a degree 3 ramified cover of S 2 by the two-torus T 2 .

Preliminaries and definitions

2.1. Isodiastolic inequality for closed surfaces. Recall from [START_REF] Balacheff | Diastolic inequalities and isoperimetric inequalities on surfaces[END_REF] that the diastole of a closed Riemannian surface (M, g) of class C 1 with a finite number of conical singularities is defined by a minimax process over the space of one-cycles and is denoted by dias (M, g).

More precisely, we denote by Z 1 (M ; Z * ) the space of one-cycles of M , see [START_REF] Balacheff | Diastolic inequalities and isoperimetric inequalities on surfaces[END_REF] for a precise definition. Here Z * denotes Z in the orientable case, and Z 2 in the non orientable one. An isomorphism due to F. Almgren [Al60] between π 1 (Z 1 (M ; Z * ), {0}) and H 2 (M ; Z * ) ≃ Z * permits us to define the diastole over the one-cycle space as dias (M, g) := inf

(zt) sup 0≤t≤1 M ass(z t )
where (z t ) runs over the families of one-cycles inducing a generator of π 1 (Z 1 (M ; Z * ), {0}) and M ass(z t ) represents the mass (or length) of z t . From a result of J. Pitts [Pi74, p. 468] (see also [START_REF] Calabi | Simple closed geodesics on convex surfaces[END_REF]), this minimax principle gives rise to a union of closed geodesics (counted with multiplicity) of total length dias(M ) when the metric g is smooth without conical singularities.

Recall the following result (see [START_REF] Balacheff | Diastolic inequalities and isoperimetric inequalities on surfaces[END_REF]).

Theorem 2.1. There exists a constant C such that every closed surface M of genus k endowed with a Riemannian smooth metric g satisfies

area(M, g) ≥ C k + 1 dias(M, g) 2 .
(2.1)

The constant C can be taken equal to 10 -16 .

The best constant involved in inequality (2.1) is not known for any closed surface. Nevertheless the asymptotic behaviour of the above constant for large genus can not be improved, see [START_REF] Balacheff | Diastolic inequalities and isoperimetric inequalities on surfaces[END_REF]. The inequality (2.1) remains valid for Riemannian metric of class C 1 with a finite number of conical singularities, as these metrics can be obtained as uniform limit (as metric spaces) of Riemannian smooth metrics and both the area and the diastole are continuous for this topology (see [START_REF] Aleksandrov | Intrinsic geometry of surfaces[END_REF]p.224 & 269]).

The singular metric and the Riemann uniformization theorem.

By the Riemann uniformization theorem there exists only one conformal structure on S 2 (up to diffeomorphism). So we set S 2 = C ∪ {∞}. Following [START_REF] Troyanov | Les surfaces euclidiennes à singularités coniques[END_REF] (see also [START_REF] Reshetnyak | On the conformal representation of Alexandrov surfaces[END_REF]), g c can be written as

(|z -a 1 | • |z -a 2 | • |z -a 3 |) -4/3 |dz| 2
up to a scale transformation where (a 1 , a 2 , a 3 ) is any triple of pairewise distinct points of S 2 . We fix in the sequel the triple to be (-1, 0, 1).

Let

M gc := e 2u • g c | u ∈ C 1 (S 2 , R)
be the space of metrics of class C 1 with three conical singularities of angle 2π/3 on S 2 conformal to g c .

Proposition 2.2. The infimum of the diastolic area over M gc equals the infimum of the diastolic area over the space of smooth metrics.

Proof. By the Riemann uniformization theorem, every smooth Riemannian metric g is isometric to a metric of the type e 2v • g 0 where g 0 denotes the round metric and v ∈ C ∞ (S 2 , R). As

g 0 = 2 1 + |z| 2 2 |dz| 2 ,
we see that g is homothetic (up to diffeomorphism) to the metric

e 2v • 2 1 + |z| 2 2 • (|z + 1| • |z| • |z -1|) 4/3 • g c
which differs from g c by a conformal factor which lies in C 1 (S 2 , R + ). So we can approximate the smooth Riemannian metric g by elements of M gc with respect to the C 1 -topology. As the diastolic area is continuous with respect to this topology (the diastolic area is in fact continuous for the uniform topology on metric spaces), we get that the infimum of the diastolic area over M gc is less than the infimum of the diastolic area over the space of smooth metrics. The reverse inequality is obtained by approximating elements of M gc by Riemannian smooth metrics for the uniform topology on metric spaces.

2.3. The ramified cover and the isosystolic inequality on the torus.

Let f : T 2 → S 2
be the covering map of degree 3 ramified over the three points {-1, 0, 1}. If g is in M gc , we denote by g the pull-back metric of g by the map f . The metric g is C 1 everywhere. In fact, near the ramified points, the map looks like the map z → z 3 . So each conical singularity of angle 2π/3 lift to a conical singularity of angle 2π. Observe that area(T 2 , g) = 3 • area(S 2 , g).

The flat metric gc given by the hexagonal lattice and appearing as the pullback metric f * g c is very special. First recall that for a closed Riemannian surface the homotopy systole is defined as the least length of a non-contractible closed curve and is denoted by sys π . Loewner's theorem (unpublished, see [START_REF] Pu | Some inequalities in certain nonorientable Riemannian manifolds[END_REF]) states the first known isosystolic inequality:

Theorem 2.3. For all metric g of class C 1 on T 2 , area(T 2 , g) ≥ √ 3 2 sys π (T 2 , g) 2 ,
with equality if and only if (T 2 , g) is homothetic to the flat torus corresponding to the hexagonal lattice (T 2 , gc ).

The study of such isosystolic inequalities is well developped, see [Be00, p.104] for instance.

The systolic geometries of (S 2 , g c ) and (T 2 , gc ) are intimately related : to each systole of (T 2 , gc ) (that is a closed geodesic realizing the homotopy systole) corresponds a geodesic loop of (S 2 , g c ) whose length equals the general systole (see Figure 1). More precisely,

• Either the systole of (T 2 , gc ) passes trough a point of ramification, and its image by f is a simple geodesic loop with base point a conical singularity, • Or the systole of (T 2 , gc ) never goes trough a point of ramification, and its image is a figure eight geodesic avoiding conical singularities.

(1, 0)

f (1/2, √ 3/2) Figure 1. Projection of the systoles
The systoles of (T 2 , gc ) can be classified into three one-parameter families. If we think at (T 2 , gc ) as the quotient of R 2 by the hexagonal lattice spanned by the vectors (1, 0) and (1/2, √ 3/2), the horizontal lines, the lines parallel to the vector (1/2, √ 3/2) and the lines parallel to the vector (-1/2, √ 3/2) correspond to these three families of systoles. Observe now that each of these families projects onto the same one-parameter family of geodesic loops of (S 2 , g c ). We denote by γ s (t) = f (t, s) this family where (t, s)

∈ [0, 1] × [0, √ 3/2].

Proof of the local diastolic inequality

Let g = e 2u • g c be a metric in M gc where u ∈ C 1 (S 2 , R). By application of the Cauchy-Schwarz's inequality and as dv gc = dt ds,

√ 3/2 0 l g (γ s (t))ds = √ 3/2 0 1 0 e u•f (t,s) dt ds = T 2 e u•f dv gc = 3 • S 2 e u dv gc ≤ 3 S 2 e 2u dv gc 1/2 S 2 dv gc 1/2 ≤ 3 area(S 2 , g) 1/2 area(S 2 , g c ) 1/2 . So √ 3 2 inf s∈[0, √ 3/2] l g (γ s ) ≤ 3 area(S 2 , g) 1/2 area(S 2 , g c ) 1/2 , which gives that inf s∈[0, √ 3/2] l g (γ s ) 2 ≤ 2 √ 3 • area(S 2 , g) as area(S 2 , g c ) = 1 2 √ 3 . Lemma 3.1. There exists an open neighborhood O of g c in M gc such that for all g ∈ O, dias(S 2 , g) ≤ inf s∈[0, √ 3/2] l g (γ s ).
Proof of the Lemma. To each geodesic loop γ s of (S 2 , g c ) we associate a oneparameter family of one-cycles {z α s } where α ∈ [0, 1] such that

• z 1/2 s = γ s , • {z α s }
starts and ends at one-cycles made of one or two points,

• {z α s } induces a generator of π 1 (Z 1 (S 2 ; Z), {0}) ≃ Z, • each z α
s is made of one or two closed curves and have length

l gc (z α s ) = 1 -2 • |α -1/2| where | • | denote the absolute value.
For this we consider two cases.

First case. If s = 0, 1 2 √ 3 or 1 √ 3 , then the geodesic loop γ s goes through a single singularity. Thus γ s bounds two disks D 1 and D 2 each one containing in its interior a conical singularity. For i = 1, 2 we homotope γ s in D i to the singularity lying in its interior through a C 0 -family {z β i,s } β∈[0,1] of geodesic loops (whose base point lies on the edge of the triangle) of length

l gc (z β i,s ) = 1 -β.
Then we set

z α s = z 1-2α 1,s if α ∈ [0, 1 2 ] z 2α-1 2,s if α ∈ [ 1 2 , 1] , see Figure 2, First case. Second case. If s = k 2 √ 3 + s ′ where k = 0, 1, 2 and s ′ ∈]0, 1 2 √
3 [, then γ s consists of a figure eight geodesic which avoids singularities. So γ s decomposes into the concatenation of two simple closed geodesic loops γ 1,s and γ 2,s . Let D i be the disk bounding by γ i,s and containing a single singularity for i = 1, 2. Each γ i,s can be homotoped in D i to the singularity lying in its interior through a family {z β i,s } β∈[0,1] of geodesic loops (whose base point lies on the edge of the triangle) of length l gc (z β i,s ) = (1 -β)l gc (γ i,s ). In a similar way, the piecewise geodesic γs obtained by the concatenation of γ 1,s and γ -1 2,s bounds an open disk containing a single singularity in its interior, and we denote by D its closure. Again γs can be homotoped in D to the singularity lying in the interior of D through a family {z β s } β∈[0,1] of piecewise geodesics of length l gc (z β s ) = (1 -β). Each piecewise geodesic z β s consists of two geodesic arcs z β 1,s and z β 2,s . Then we set For all g = e 2u • g c ∈ M gc and by Stokes' Theorem,

z α s = z 1-2α 1,s + z 1-2α 2,s if α ∈ [0, 1 2 ] z 2α-1 s if α ∈ [ 1 2 ,
l g (γ i,s ) -l g (z α i,s ) = γi,s e u•f dt - zα i,s e u•f dt = ci,s e u•f dt + Ωα i,s de u•f ∧ dt = ci,s dt + ci,s (e u•f -1) dt + Ωα i,s ∂ ∂s e u•f ds ∧ dt ≥ ci,s dt - ci,s e u -1 ∞ dt -sup Ωα i,s ∂ ∂s e u•f • area( Ωα i,s , gc ) ≥ (1 -e u -1 ∞ - 1 2 • tan π 6 • sup Ωα i,s ∂ ∂s e u•f ) (l gc (γ i,s ) -l gc (z α i,s ))
Observe that there exists a positive constant B such that sup for all x ∈ T 2 . But g 0 (df x ( ∂ ∂s ), df x ( ∂ ∂s )) goes to 0 when x goes to a point of ramification. This is clear from the fact that g c (df x ( ∂ ∂s ), df x ( ∂ ∂s )) = 1 outside singularities and from the following expression of g 0 : l g (γ s ).

g 0 = 2 1 + |z| 2
Let us finish the proof of the Theorem 1.1. For all g ∈ O, we have (dias(S 2 , g)) 2 ≤ 2 √ 3 • area(S 2 , g).

Now observe that the equality case imposes to the conformal factor e 2u to be constant. So g is homothetic to g c .

Figure 3 .

 3 Figure 3. Domains.

  f ≤ B • de u 0 where de u 0 = sup{|de u x (v)| | (x, v) ∈ T S 2 with g 0 (v, v) = 1}is the uniform norm on closed one-forms on S 2 associated to the round metric g 0 . In fact,∂ ∂s e u•f (x) = de u f (x) • df x ( ∂ ∂s )

2•

  (|z + 1| • |z| • |z -1|) 4/3 • g c where z ∈ S 2 = C ∪ {∞}. So we can set B = sup x∈T 2 g 0 (df x ( ∂ ∂s ), df x ( ∂ ∂s )). Now define the set O as the reunion of the open sets{e 2u • g c ∈ M gc such that e u -1 ∞ < t and 1 2 tan π 6 B de u 0 < 1 -t} for t ∈]0, 1[. The setO is an open neighborhood of g c in the C 1 -topology and we get l g (γ s ) ≥ l g (z α s ) for all g ∈ O, and for all (s, α).Fixs ′ ∈ [0, √32 ] such that l g (γ s ′ ) = inf s∈[0, √ 3/2] l g (γ s ). For all g ∈ O, we obtain dias(S 2 , g) ≤ sup α∈[0,1] l g (z α s ′ )≤ l g (γ s ′ )

  and ci,s . Note that ci,s is composed of two connected components. The curve ci,s is oriented such that the concatenation of the first connected component of ci,s , zα i,s and the second component of ci,s makes sense and is an arc homotopic to γi,s . We endow Ωα i,s with the orientation such that its boundary is the concatenation of γi,s , the first connected component of -c i,s , -z α i,s and the second component of -c i,s . We can easily compute that

	covers a domain Ω α s whose boundary is the reunion of γ s and z α s . We can lift this domain to a domain Ωα s of T 2 as in Figure 3 which decomposes into two domains Ωα 1,s and Ωα 2,s . One of this domain is reduced to a point if s = 0, 1/(2 √ 3) or 1/ √ 3. Each domain Ωα i,s is bounded by three curves:
	γi,s , zα i,s area( Ωα i,s , gc ) =	1 4	tan	π 6	(l gc (γ i,s ) 2 -l gc (z α i,s ) 2 ).
	Observe that
	see Figure 2, Second case. 1,s zα area( Ωα i,s , gc ) ≤ (l Case α < 1 1 2 tan π 6 2 Case α > 1 1] 2 Ωα 1,s Ωα 1,s γs zα 2,s zα 1,s Ωα 2,s	,	γs	Ωα 2,s	2,s zα
	First case			Second case
	γ s				γ s
	Figure 2. Families of one-cycles.
	Denote by • ∞ the uniform norm on C 1 (S 2 , R). For each α ∈ [0, 1/2[∪]1/2, 1], the family of 1-cycles
	{z t/2+(1-t)α s	| t ∈ [0, 1]}

gc (γ i,s ) -l gc (z α i,s )) as l gc (γ i,s ) + l gc (z α i,s ) ≤ 2.

Funded by grant 20-118014/1 of the Swiss SNF.


P Rousseau

J González-Vázquez

D G Piekarski

J Kopyra

A Domaracka

M Alcamí

L Adoui

B A Huber

S Díaz-Tendero
email: sergio.diaztendero@uam.es

F Martín

Timing of charge migration in betaine by impact of fast atomic ions

The way molecules break after ion bombardment is intimately related to the early electron dynamics generated in the system, in particular, charge (or electron) migration. We exploit the natural positive-negative charge splitting in the zwitterionic molecule betaine to selectively induce double electron removal from its negatively charged side by impact of fast O 6+

ions. The loss of the two electrons in such a localized region of the molecular skeleton triggers a competition between direct Coulomb explosion and charge migration that is examined to obtain temporal information from ion-ion coincident measurements and non-adiabatic molecular dynamics calculations. We find a charge migration time, from one end of the molecule to the other, of approximately 20-40 fs. This migration time is longer than that observed in molecules irradiated by ultrashort light pulses and is the consequence of charge migration being driven by adiabatic nuclear dynamics in the ground state of the molecular dication.

Teaser

Timing adiabatic charge migration in doubly charged betaine ions resulting from collisions with highly charged ions

Introduction

Understanding and controlling molecular damage induced by ionizing radiation, such as extremeultraviolet (XUV) or X-ray light, fast electrons or ions, is one of the current challenges in natural sciences. Indeed, it is well established that molecular damage can lead, among other processes, to random mutagenic miscoding or even lethal nanocoding of the DNA sequence, which is at the origin of several diseases (1)(2)[START_REF]DNA Damage Recognition[END_REF][START_REF] Sinha | [END_REF]. However, when conveniently controlled, ionizing irradiation can be the remedy rather than the problem, as demonstrated by current radio-and hadron-therapies [START_REF]Principles of Radiation Oncology in Cancer[END_REF][START_REF] Durante | [END_REF]. An important source of damage is the environment surrounding the molecule and, in this respect, several processes involving electron transfer between neighboring molecules have been proposed, as electron-transfer-mediated decay (ETMD) (7) or interatomic Coulombic decay (ICD) (8). However, pioneering experimental and theoretical work (9,10) has shown not only that molecular damage is also possible in isolated molecules, but also that the nature of the resulting fragments is intimately related to the charge migration induced by the ionizing radiation at the very early stages, where most of the dynamics is governed by electronic motion. To experimentally investigate these early stages, one must first access the natural time scale of electronic motion, i.e., from several hundreds attoseconds to a few femtoseconds, and then follow the ensuing coupled electron and nuclear dynamics within the next few tens of femtoseconds. This is nowadays possible by using XUV/X-ray attosecond and few fs pulses produced either in the laboratory through high harmonic generation (HHG) or in large-scale X-ray freeelectron laser (FEL) facilities, usually in the framework of pump-probe set ups (11)(12)(13)(14)(15)(16). The necessary time resolution is achieved by using a short enough pump pulse, so that ionization is a "sudden" process in comparison with the subsequent charge migration process, which can then be probed by using a second ultrashort pulse and varying the pump-probe delay with attosecond and few-fs resolution. In these experiments, charge migration results from the coherent superposition of electronic states created by the broadband pump pulse in the remaining molecular cation and, therefore, it precedes nuclear motion (13,16) -although nuclear motion can have a significant effect at longer times (17,[START_REF] Delgado | Faraday Discuss[END_REF]. In the case of X-ray pulses, the sudden ionization of the molecule can be followed by Auger decay, which can thus induce charge migration in the corresponding molecular dication (17,[START_REF] Picón | [END_REF]. One can also use ultrafast molecular dissociation associated to, e.g., proton emission or Coulomb explosion or both, to clock the electron dynamics generated by the pump pulse. With the help of accurate theoretical calculations (see, e.g., (16,20,21) and references therein), the latter methods have been able to provide real-time pictures of electron dynamics in small and medium-size molecules ionized by IR, XUV and X-ray light (22)(23)(24)(25)(26).

Charge migration induced by fast ions and electrons is, in general, very different from that induced by photons, as the latter usually involve less electronic channels as a result of selection rules. Also, contrary to photons (and electrons), ions can selectively deposit energy in a well localized region of a bulky target, as they can efficiently penetrate matter with low rates of diffusion and release most of their energy all at once just before stopping (the so called Bragg peak) (27)(28)(29). So they are ideal, e.g., to treat tumors where extreme spatial localization is essential. Hence, understanding charge migration induced in collisions with atomic ions is not only important per se but also to explore new routes to trigger radiation damage by altering the early electron dynamics.

Such studies require the implementation of specific experimental and theoretical approaches.

In recent pump-probe experiments, ultrashort femtosecond pulses of electrons pioneered by Zewail and collaborators (30) have been successfully combined with ultrashort laser pulses with a time resolution of ⇠150 fs (31), which is comparable to that achieved in state-of-the-art femtochemistry experiments performed with lasers but provide in addition spatial atomic resolution.

In the case of atomic ions, this is not yet possible, mainly due to the difficulty to produce ultrashort pulsed ionic beams in conventional ion accelerators. And this in spite of the fact that an ion of approximately a few keV/amu interacts with a small or medium size molecule for no more than a few fs (e.g., a 3 keV/amu projectile has a velocity of 7.6 Å/fs), which is the optimum scenario to induce a "sudden" ionization of the molecular target, hence to investigate charge migration processes in their natural time scale. To achieve the necessary few-fs time resolution, in this work, we will use ultrafast Coulomb explosion as an internal clock of the electron dynamics generated by the ionizing collision. This is not always an available option, but, as we will see below, it can be achieved by a smart selection of the ionic projectile and molecular target.

Here, we exploit the natural positive-negative charge splitting in the betaine zwitterion (N, N, N trimethylglycine, see Fig. 1c) to selectively induce double ionization on the negatively charged end of the molecule by impact of 3 keV/amu O 6+ and then trigger a competition between direct Coulomb explosion and charge migration involving the two positively charged ends of the doubly ionized molecule. The analysis of the kinetic energy distributions of the resulting positively charged fragments detected in coincidence along with the results of ab initio non-adiabatic molecular dynamics simulations allows us to determine a charge migration time from end to end of approximately 20-40 fs. The calculations also show that charge migration is mainly driven by nuclear motion in the ground state of the molecular dication, thus leading to migration times longer than those observed in charge migration triggered by electronic coherences (13,16,32).

Results and discussion

With a positive charge localized on the quaternary ammonium group ( N + (CH 3 ) 3 ) and a negative charge on the carboxylate group ( CO 2 ) (see Fig. 1c), the (neutral) betaine molecule The missing channels are expected to be barely populated in the collision or lie outside the axes range. Notice that the incoming projectile approaches the molecule from the side of the negatively charged carboxylate group, which is the most favorable orientation for single and double capture (see text). The orange curves in (a) and (b) represent the entrance channel. The electron densities, referred to the ground state electron density of neutral betaine, for the lowest states of singly-and doubly-charged betaine resulting from single-and double-electron capture, respectively, are shown in panel (d). The numbers below the density maps indicate the energies (in eV) of the corresponding electronic states of singly-and doubly-charge betaine with respect to the energy of the ground state of neutral betaine.

Figures 1a,b show the potential energy curves of the [O 6+ B] system (where B stands for betaine) for the case in which the O 6+ projectile lies in the direction of the betaine permanent dipole µ (Fig. 1c). The orientation of the molecule is such that the charge-dipole interaction between the projectile (P) and the target (T) is the most attractive. The potential energy curves and the energy ranges associated with the single-and double-capture channels have been obtained by using the experimental bound-state energies of O 6+ , O 5+ and O 4+ (34), the calculated energies of the autoionizing doubly-excited states of O 4+ (35), the calculated first, second and third vertical ionization potentials of betaine (see Methods for details), and the asymptotic analytical formulas that describe the repulsive charge-charge, q P q T /d, and attractive charge-dipole, q P (µ • d)/d 3 , interactions between the different projectile and target species, where d is the distance between the projectile and the center of mass of the target [START_REF] Karplus | Atoms and molecules an introduction for students of physical chemistry[END_REF]. The latter formulas are expected to accurately represent the actual behavior of the potential energy curves except at very short projectile-target distances, where electron delocalization over the projectile and the target can be important. As can be seen, the potential energy curve of the entrance channel O 6+ B intersects the repulsive curves of the O 4+ (1s 2 3lnl 0 ) B 2+ channels, associated with double electron capture, at projectile-target distances between 5 a.u. and ⇠10 a.u., where non adiabatic transitions from the former to the latter channels should be very efficient at the impact energy considered in this work [START_REF] Macías | [END_REF]. The same applies to the O 5+ (1s 2 3l) B + and O 5+ (1s 2 4l) B + channels, associated with simple capture, which appear in the same energy region as the doublecapture channels. Therefore, one can expect that both single and double electron capture (either direct or mediated by single-capture channels) will be very efficient processes. These are also expected to occur, though to a lesser extent, when the molecule is oriented at 90 o with respect to the incidence direction, as µ • d = 0 and the corresponding potential energy curve mostly coincides with the zero-energy axis. This curve crosses single-and double-electron capture channels in a similar range of projectile-target distances, although involving slightly higher n and l quantum numbers in the projectile side, which are less favorable due to the large number of radial and angular nodes in the corresponding wave functions (Barat-Lichten rules (38)). More importantly, if the orientation of the betaine molecule was reversed with respect to the first one, the potential energy curve of the entrance channel would be the mirror image of that shown in Figs. 1a,b with respect to the zero-energy axis, i.e., it would be entirely repulsive and therefore would not lead to significant single and double capture in the above mentioned channels. In other words, the charge-dipole interaction that governs the approach of the projectile to the target and the orientation selectivity of the double-capture process restrict the range of molecular orientations where doubly charged betaine can be formed. We can safely discard ionization as a possible source of doubly charged betaine because, for the impact energy considered in our work (3 keV/amu), the ionization cross section is several orders of magnitude smaller than that associated with electron capture (see, e.g., references (39,40) for the case of proton-hydrogen and proton-uracil collisions). For the same reason, transfer-ionization and electron-transfermediated decay (ETMD) (7,41), in which a betaine electron directly goes into the continuum (ionization) and another betaine electron is transferred to the projectile (capture) thus leading to a betaine dication, are expected to be less likely than two resonant electron transfers to the projectile (39,40,42). Finally, two-electron processes implying the removal of electrons from inner shells (non resonant electron capture) can also be discarded because target inner vacancy states lie very far in energy from the entrance channel (in fact, these states lie out of scale in Fig. 1).

Figure 1d shows the electron density of the four lowest states of singly-and doubly-ionized betaine referred to the ground sCMtate density of neutral betaine. As can be seen, the removal of one or two electrons always involves the CO 2 side of the zwitterion, meaning that, after double capture (either direct or sequential), this side of the molecule will hold a positive charge while the charge distribution in the rest of the molecule will basically remain unchanged. This fact, combined with the high preference for the molecular orientation shown in Fig. 1c, indicates a high selectivity of the double electron capture process, which basically leads to a doubly charged betaine molecule with a positively charged CO + 2 side and a positively charged N(CH 3 ) + 3 side, well separated from each other. These are the optimal conditions for a clean and rapid Coulomb explosion (CE) of B 2+ , which is expected to compete with internal charge migration (CM), where an electron would move from the N(CH 3 ) + 3 side to the CO + 2 high electron-affinity side, thus neutralizing the positive charge in the latter side. The two competing processes are expected to lead to different molecular fragments, as illustrated in Figure 2a: CE can lead to CO + 2 and other molecular fragments, while CM is not expected to lead to any CO + 2 .

In order to confirm this picture, we have performed ion-ion coincident measurements of the ionic fragments produced in collisions of 3 keV/amu O 6+ with fully deuterated betaine.

We have chosen deuterated betaine in order to unambiguously assign fragments using timeof-flight mass spectrometry (e.g., both N(CH [START_REF] Delgado | Faraday Discuss[END_REF], suggesting also that the neutral CD 3 fragment emitted in the second step has a negligible momentum (see Refs. (43,44) for the connection between the slopes of coincident islands and n-body fragmentation).

The 18 + /66 + island is associated with three body fragmentation of doubly charged betaine . The latter further dissociates by emission of a neutral methyl group CD 3 . Experimentally, this process leads to a coincidence 44 + /66 + in the ion-ion map (c). The slope of the measured island, given by momentum conservation, confirms this scheme (see text). In contrast, after CM, the ammonium side of the betaine is doubly charged, so that the emission of a neutral carboxylate group CO 2 is expected and the ulterior charge separation gives rise to two charged fragments, CD + 3 and (CD 2 )N(CD 3 ) + 2 . Experimentally, one observes a coincidence island 18 + /66 + in the ion-ion map (b) with a positive slope, indicating that fragmentation involves the formation of the intermediate species shown in the upper row of (a) (see text).

after neutralization of the CO 2 side by CM, leading to CO 2 /CD + 3 /(CD 2 )N(CD 3 ) + 2 . In this case, the slope of the coincidence island is positive and cannot be explained as resulting from a twostep process in which Coulomb explosion is followed by emission with negligible momentum of neutral CO 2 from one of the charged fragments, as this process would always lead to an island with negative slope. A positive slope indicates that the two charged fragments are emitted in the same direction (45), while neutral CO 2 is emitted with large momentum in the opposite direction to compensate the momentum of the charged particles. This picture is consistent with the formation of an intermediate doubly-charged transient structure as that shown in the upper panel of Fig. 2a, which, according to our calculations (see below), corresponds to a minimum in the potential energy surface arising from the attractive charge / induced-dipole interaction between (CD 2 )N(CD 3 ) + 3 and a neutral CO 2 fragment in the early stages of dissociation.

To get a deeper insight on the results of these measurements, we have performed fewswitches trajectory surface hopping (FS-TSH) calculations on the lowest potential energy surfaces of the betaine dication obtained within the complete active space perturbation theory (CASPT2) approach (see Methods for details). and the ensuing fragmentation of the remaining dication into CD + 3 /(CD 2 )N(CD 3 ) + 2 . This result is compatible with the observation of a positive slope in the 18 + /66 + island, since the fact that CO 2 leaves first makes the two charged fragments go in the same direction (i.e., opposite to the CO 2 emission direction).

The results given in Fig. 3 show that CM is mostly an adiabatic process taking place in the ground state of the betaine dication. This interpretation is confirmed by the absence of avoided crossings between the ground and the excited electronic states of doubly charged betaine and the conservation of the ground-state electronic character along the minimum energy path leading to emission of neutral CO 2 (see Supplementary Information). So, CM is almost entirely driven by nuclear motion in this ground state, which explains why CM in doubly-charged betaine is 2 fragments detected in coincidence with 66 + fragments (associated with direct CE). (c) and (d) Estimation of the time necessary for the separation of the two charges leading to the corresponding ion pairs 18 + /66 + and 44 + /66 + based on their respective kinetic energy release distribution. substantially slower than charge fluctuations resulting exclusively from electronic coherences induced by ionizing radiation, e.g., by using attosecond and few-fs laser or electron pulses.

An analysis of the kinetic energy release (KER) of the CD + 3 /(CD 2 )N(CD 3 ) + 2 and CO + 2 /(CD 2 )N(CD 3 ) + 2 ion pairs associated with, respectively, the 18 + /66 + and 44 + /66 + peaks observed in the ion-ion coincidence spectrum confirms the general trends resulting from our calculations. The details of this analysis are given in the Methods section and the Supplementary Information. The KERs are shown in Figs. 4a,b, respectively, in the form of histograms.

Both distributions peak at zero kinetic energy and decrease with energy. However, the KER associated with CE (i.e., the CO + 2 /(CD 2 )N(CD 3 ) + 2 coincidence) is significantly narrower, indicating that CE (or, equivalently, emission of CO + 2 ) is slower than the fragmentation into CD + 3 /(CD 2 )N(CD 3 ) + 2 that follows CM (see Fig. 2a). This fact can be more clearly seen in Figs. 4c,d, where the corresponding kinetic energy release (KER) histograms have been converted

into time histograms by assuming that direct CE and CD + 3 emission following CM are practically irreversible when the two charged fragments have travelled a distance of 2 a.u. from their initial positions. As can be seen, for CM, the time histogram peaks at around 50 fs, whereas, for CE, approximately 90% of the events take much longer than 50 fs, in excellent agreement with our theoretical findings, in particular with our estimation of the CM time. Interestingly, the time histogram for direct CE given in Fig. 4d shows an increase of the number of events at around 50 fs, which is more or less the time taken by CM to occur. When direct CE is that fast, one can expect that it efficiently competes with CM, thus highlighting the effect of possible electronic coherences between the CE and CM channels. Since determination of the kinetic energy of the fragments from the measured TOF distribution is not as accurate as that provided by position sensitive detectors, further experimental work allowing for the direct measurement of fragments momentum could shed more light on this issue and probably allow for a more accurate retrieval of the CM time from experiment by focusing on the KER region where interferences between CE and CM channels are expected to occur.

Conclusion

Taking advantage of the natural positive-negative charge separation in the betaine zwitterion,

we have been able to selectively induce double electron removal from the negatively charged end of the molecule in collisions with 3 keV/amu O 6+ ions. This process initiates a competition between direct Coulomb explosion and charge migration that can be used to obtain temporal information on the several stages involved in these processes. We have determined an internal charge migration time, from one end of the molecule to the other, of approximately 20-40 fs and show that this process preferentially takes place in the ground state of the betaine dication resulting from the collision, while direct Coulomb explosion mainly occurs in its excited states. The charge migration time is significantly longer than that previously observed in charge dynamics induced by electronic coherences in molecules of similar complexity. This is the consequence of charge migration being mostly driven by adiabatic nuclear dynamics in the ground state of the betaine dication.

Methods Experiment

The experiments have been performed at ARIBE (46), the low-energy ion beam facility of GANIL (Caen, France). The experimental set-up and methods have been described in detail elsewhere (47). Briefly, the O 6+ ions are produced in an electron cyclotron resonance ion source, accelerated at 48 keV and mass-analyzed by a magnet sector. The beam is pulsed in bunches of 500 ns and transported by ion optics towards the crossed-beam collision device where after focusing it interacts with an effusive beam of neutral betaine molecules produced by evaporation of a commercial sample of fully deuterated betaine (purity 98%, Eurisotop) in a heated oven. Just after the passage of the ion bunch, the cationic products of the interaction are extracted by a pulsed electric field into a modified Wiley-McLaren time-of-flight mass spectrometer (48). Their arrival times are recorded in an event-by-event mode allowing to consider coincidences among them (49,50). The kinetic energy release of an ion pair is obtained after differentiation of the distribution of the time difference between two fragments (51,52).

SIMION simulations of the time-of-flight spectrometer show that the ion transmission is 100%

for cations with a kinetic energy up to 10 eV (53). From the relative velocity of the two frag-ments one can estimate the time required to run a given distance.

Theory

Optimized geometry and frequencies for the betaine molecule were obtained in the ground electronic state using the Moller-Plesset (MP2) method with the cc-pvDZ-jkfit basis (54) as implemented in the BAGEL code (55). Starting from this geometry, a set of 50 initial conditions were obtained using a harmonic Wigner distribution of correlated position and momentum. In order to describe the electronic states of the different ionization states of betaine, single point stateaverage complete active space self-consistent field (SA-CASSCF) calculations were performed for the neutral (singlet), cation (doublet), dication (singlet and triplet) and trication (doublet) at the MP2 geometry using a state average of 4 electronic states, without symmetry considerations, with an active space consisting of 8 orbitals and 8 electrons for the neutral species. These calculations were performed by using the OpenMOLCAS package ( 56) with the same basis set. To simulate the non-adiabatic semiclassical dynamics, a local version of the SHARC (57) method was employed. The electronic structure was represented, including gradients, at the XMS-CASPT2 level (58) with an imaginary shift of 0.2 a.u. by using the BAGEL code (55).

For the calculation of the non-adiabatic couplings, the overlap matrix between two consecutive time steps was employed. For this, the perturbed modified wave function, including configuration interaction (CI) vectors and molecular orbitals, was extracted from BAGEL (55) and imported into the OpenMOLCAS code (56) with overlaps obtained with the RASSI program.

Charges on different molecular sites were evaluated by performing a Mulliken analysis.

S2. EXTRACTION OF THE FRAGMENTS' KINETIC ENERGIES

In our experiment, the neutral betaine molecules are isotropically distributed in space.

However, as explained in the main text, in the present case, double electron capture resulting from the collision can only occur for molecules whose negatively charged side faces the projectile. Hence, the probability to form a betaine dication will be maximum when the negatively charged side of neutral betaine (-CO 2 ) is in front of the projectile and zero when it is in the opposite side. Therefore, after the collision, the spatial distribution of doubly charged betaine molecules will be strongly anisotropic along the direction that connects the center of mass of the molecule with the projectile (see Fig. S2). The shape of the spatial distribution of doubly charged betaine molecules is the same for all projectile positions contained in a circle centered on the molecule with radius equal to the e↵ective impact parameter in the collision (see Fig. S2). All these positions are equally accessible in the collision. To reflect the fact that the CO 2 side faces the projectile, the maximum of the blue rectangular distributions) and, therefore, it is directly related to the kinetic energy of that fragment. The TOF peak is thus made by the stacking of rectangular distributions with di↵erent widths (thus di↵erent kinetic energies, see black rectangles). In this way the KE distribution can be obtained by simply analyzing the shape of the TOF-peak or the KER distribution by the shape of the TOF2-TOF1 peak obtained from the coincidence ion pair. During the treatment of the data some resolution is lost compared to a full momentum measurement with position sensitive detectors, but it is enough for the purpose of understanding the mechanisms described in the manuscript.

Noteworthy, in the present TOF spectrometer, the transmission e ciency of the fragment is close to 100% up to kinetic energies of about 10 eV. Moreover, the Daly-type detector also gives high and uniform detection e ciency over a large mass range (the TOF system was designed for the study of metallic clusters and successfully used for clusters of C 60 molecules).

The observation of well-defined slopes also shows that the coincidences are not fortuitous and that the probability to miss one fragment in the case of 3 charged fragments events is low enough to avoid a significant contribution of these events in the recorded 2-stops events.

For instance, in the case of the island 18+/66+, it is clear that the third fragment is neutral (and with a significant momentum) and not a charged one that was not detected.

S5. ADIABATICITY OF CHARGE MIGRATION

We have performed XMS-CASPT2 calculations to obtain the electronic states of doubly charged betaine along the minimum energy path (MEP) that leads to emission of neutral CO 2 in the ground electronic state (the charge migration channel). We have also calculated As can be seen, along the whole MEP, the energy separation between the ground and the lowest excited state of the betaine dication is never less than 1 eV and this energy separation increases as the CO 2 fragment departs from the molecular dication. Thus, we can conclude that no sharp curve crossing is contributing to the production of neutral CO 2 . We can also discard a change of character in the ground state electronic wave function, since the overlap between ground state wave functions in consecutive points of the MEP is always above 0.96, indicating that the electronic character of the ground state remains roughly the same along the MEP. Therefore, the production of neutral CO 2 , hence charge migration, is mainly an adiabatic process.

Figure 1 :

 1 Figure 1: The O 6+ -betaine system. Molecular energies for (a) single-electron capture and (b) double-electron capture channels as a function of the distance between the O 6+ projectile and the center of mass of the betaine molecule oriented as shown in panel (c). For the sake of clarity, only the energy ranges covered by the single-and double-electron capture channels leading to O 5+ (nl) with n < 5 (a) and O 4+ (2lnl 0 ) with n < 4 and O 4+ (3lnl 0 ) (b) are shown. The missing channels are expected to be barely populated in the collision or lie outside the axes range. Notice that the incoming projectile approaches the molecule from the side of the negatively charged carboxylate group, which is the most favorable orientation for single and double capture (see text). The orange curves in (a) and (b) represent the entrance channel. The electron densities, referred to the ground state electron density of neutral betaine, for the lowest states of singly-and doubly-charged betaine resulting from single-and double-electron capture, respectively, are shown in panel (d). The numbers below the density maps indicate the energies (in eV) of the corresponding electronic states of singly-and doubly-charge betaine with respect to the energy of the ground state of neutral betaine.

Figure 2 :

 2 Figure 2: Fragmentation of fully deuterated doubly charged betaine. (a) As a result of double electron capture by the projectile from the carboxylate group, both ends of the betaine molecule are positively charged. Two scenarios are then possible: direct Coulomb explosion (CE) and charge migration (CM). In CE, charge separation by cleavage of the C carboxylate C ↵ bond leads to the two charged fragments CCMO + 2 and (CD 2 )N(CD 3 ) + 3. The latter further dissociates by emission of a neutral methyl group CD 3 . Experimentally, this process leads to a coincidence 44 + /66 + in the ion-ion map (c). The slope of the measured island, given by momentum conservation, confirms this scheme (see text). In contrast, after CM, the ammonium side of the betaine is doubly charged, so that the emission of a neutral carboxylate group CO 2 is expected and the ulterior charge separation gives rise to two charged fragments, CD + 3 and (CD 2 )N(CD 3 ) + 2 . Experimentally, one observes a coincidence island 18 + /66 + in the ion-ion map (b) with a positive slope, indicating that fragmentation involves the formation of the intermediate species shown in the upper row of (a) (see text).

Figure 3

 3 summarizes the most important results.Panel (a) shows the time evolution of the local charge around the CO 2 group for trajectories starting from the ground and first excited state of the betaine dication. As can be seen, within the limited time interval considered in our calculations (see Supplementary Material for an extension up to 150 fs), trajectories starting in the ground state can only lead to CO 2 charges equal to zero or around +0.75, which are consistent with the charges that should be observed just after CM or in the intact betaine dication, respectively. In contrast, trajectories starting in the first excited state can only lead to CO 2 charges equal to +1 or around +0.75, compatible with CE and the intact betaine dication, respectively. Trajectories starting in the second excited state (not shown in the figure for the sake of clarity, see Supplementary Material) follow a similar pattern: they do not lead to neutral CO 2 . Panel (a) also shows that neutralization of the initial positive

Figure 3 :

 3 Figure 3: Molecular dynamics calculations. (a) Evolution of the charge located at the CO 2 end of doubly-charged betaine for all trajectories starting in the ground (1) and the first excited (2) state. The color palette on the right indicates the potential energy surface in which the trajectory evolves at every time. Notice that a significant number of trajectories starting in the ground state lead to neutralization of the CO 2 end. (b) Evolution of the C carboxylate C ↵ bond distance in doubly-charged betaine for all trajectories starting in the ground and the first excited state. The color palette on the right indicates the charge in the CO 2 end at every time. Notice that the C carboxylate C ↵ bond breaks faster in CM than in CE. (c) Evolution of the N C bond distance in doubly-charged betaine for all trajectories starting in the ground and the first excited state. The color palette on the right indicates the charge in the CD 3 site at every time.

Figure 4 :

 4 Figure 4: Kinetic energy release and estimation of the charge separation time. Kinetic energy release (KER) for (a) CD + 3 fragments detected in coincidence with 66 + fragments (associated with CM), and (b) CO +2 fragments detected in coincidence with 66 + fragments (associated with direct CE). (c) and (d) Estimation of the time necessary for the separation of the two charges leading to the corresponding ion pairs 18 + /66 + and 44 + /66 + based on their respective kinetic energy release distribution.

  probability distribution must always lie in the line connecting the center of the circle with the projectile position. In other words, the probability distribution must rotate in the Y Z plane following the di↵erent positions of the projectile in the dashed green circle in Fig.S2. Therefore, the overall orientation of the doubly charged ions is nearly isotropic in the Y Z plane. This plane contains the extraction direction. The same reasoning applies for projectiles not impinging perpendicularly to the plane shown in the figure.

Fig. S3 shows

  Fig.S3shows collision geometries in which a fragment is emitted in di↵erent directions relative to the extraction field for two di↵erent values of the momentum in each direction (represented by red and blue arrows). We have only considered those orientations that can lead to double charge transfer to the projectile, i.e., we are not considering orientations in which the negatively charge side of the target is on the opposite side of the projectile. We have considered the extreme cases where the momentum of the ejected fragment is parallel, perpendicular and antiparallel to the extraction field of the TOF mass spectrometer (along the Z-axis) and two intermediate cases. The parallel and antiparallel cases correspond to, respectively, the shortest and longest time of flights of the charged fragment. In these cases, the total momentum of the fragment is entirely given by the Z-component (the X-and Y -components are close to zero), so that we can determine the total kinetic energy of the fragment. All other cases result in time of flights that lie in between these two extremes.

  FIG. S4: Extended molecular dynamics calculations. Molecular dynamics calculations extended up to 150 fs. (a) Evolution of the charge located at the CO 2 end of doubly-charged betaine for all trajectories starting in the ground (1), the first (2) and the second (3) excited state. The color palette on the right indicates the potential energy surface in which the trajectory evolves at every time. (b) Evolution of the C carboxylate C ↵ bond distance. The color palette on the right indicates the charge in the CO 2 end at every time. (c) Evolution of the C N C bond distance. The color palette on the right indicates the charge in the CD 3 site at every time.

  FIG. S5: Adiabaticity of charge migration. (a) Electronic energies of the betaine dication (XMS-CASPT2 level) along the minimum energy path (MEP) leading to CO 2 emission in the ground electronic state. Black line: ground state; red line: first excited state; purple line: second excited state; blue line: third excited state. (b) Overlap between electronic ground states obtained in consecutive points of the MEP.

Acknowledgments

All calculations were performed at the Mare Nostrum Supercomputer of the Red Española de Supercomputación (BSC-RES) and the Centro de Computación Científica de la Universi-dad Autónoma de Madrid (CCC-UAM). Work supported by the MICINN projects PID2019-105458RB-I00, PID2019-106732GB-I00 and PID2019-110091GB-I00, the "Severo Ochoa" Programme for Centres of Excellence in R&D (SEV-2016-0686), the "María de Maeztu" Programme for Units of Excellence in R&D (CEX2018-000805-M), and the European COST Actions CA18212 MD-GAS and CA18222 AttoChem. The research was conducted in the framework of the International Associated Laboratory (LIA -CNRS) "Fragmentation DYNAmics of complex MOlecular systems -DYNAMO". J.K. acknowledges support for a visit to GANIL, Caen (France) from the Polish Ministry of Science and Higher Education via statutory activity subsidy (No 25/20/B).

Data and Materials Availability

All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials.

Author Contributions

PR, JK, AD and BH designed and performed the experiments and analyzed the experimental data. JGV, DGP, SDT and FM designed and performed the theoretical calculations and analyzed the theoretical data. All authors discussed the results. PR, SDT and FM wrote the first version of the manuscript and all authors contributed to their improvement. FM coordinated the project.

Competing Interests

The authors declare no competing interests.

Supplementary Information: Timing of charge migration in betaine by impact of fast atomic ions P. Rousseau, 1, ⇤ J. González-Vázquez, 2, 3, ⇤ D.G. Piekarski, 2, † J. Kopyra, 4 A. Domaracka, 1 M. Alcamí, 2, 3, 5 L. Adoui, 1 B. A. Huber, 1 S. Díaz-Tendero, 2, 3, 6, ‡ and F. Martín