Melting of the Au20 Gold Cluster: Does Charge Matter? - The Journal

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A: Molecular Structure, Quantum Chemistry, and General Theory 20

Melting of the Au Gold Cluster : Does Charge Matter? Mathias Rapacioli, Nathalie Tarrat, and Fernand Spiegelman J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.7b12522 • Publication Date (Web): 23 Mar 2018 Downloaded from http://pubs.acs.org on March 25, 2018

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The Journal of Physical Chemistry

Melting Of The Au20 Gold Cluster : Does Charge Matter ? Mathias Rapacioli,∗,† Nathalie Tarrat,∗,‡ and Fernand Spiegelman

†

†Laboratoire de Chimie et Physique Quantiques LCPQ/IRSAMC, UMR5626, Université de Toulouse (UPS) and CNRS, 118 Route de Narbonne, F-31062 Toulouse, France. ‡CEMES, Université de Toulouse, CNRS, 29, rue Jeanne Marvig, 31055 Toulouse, France E-mail: [email protected]; [email protected]

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Abstract We investigate the dependence upon charge of the heat capacities of the magic gold cluster Au20 obtained from density functional based tight binding theory within parallel tempering molecular dynamics and the multiple histogram method. The melting temperatures, determined from heat capacity curves, are found to be 1102 K for − neutral Au20 and only 866 and 826 K for Au+ 20 and Au20 . Both the canonical and the

microcanonical aspects of the transition are discussed. A convex intruder, associated to a negative heat capacity, is evidenced in the microcanonical entropy in the case of the neutral cluster, but absent in the melting processes of the ions. The present work shows that a single charge quantitatively affects the thermal properties of the gold twentymer such as the melting temperature.

Introduction Investigation of the thermodynamical behaviour of small metal clusters has been strongly developed since the early investigations by Pawlow 1 and Buffat and Borel 2 on finite size gold particles, showing that small particles have a lower melting point than bulk materials. After the pioneering nanocalorimetry experiment of Schmidt et al., 3 several experimental techniques 4–6 have made possible the determination of the heat capacity curves of metal clusters as a function of temperature down to selected sizes as small as typically ten atoms, in particular in the region of the finite size equivalent of the solid-liquid transition. Unlike in the bulk, the phase transition in finite systems is not abrupt but extends over a finite temperature interval, as formalized by Berry 7 and documented in several textbooks. 8–10 Both the melting temperatures and the latent heat have been shown to depend strongly on size. 3 Alkali metal clusters have been specially documented, 3,5 however caloric curves of other metal particles, such as tin, 11 aluminum 12–14 and gallium 4,10,15,16 have also been experimentally obtained. Both structural and electronic factors are known to be determining in the description of size effects of metal clusters via the completion of either geometric shells or 2


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electronic shells. The influence of electronic effects, and in particular charge, on the melting of metal clusters was investigated on sodium, 17 aluminum 12,18 and gallium. 19,20 The case of Al36 , a cluster characterized as doubly magic (as simultaneously exhibiting closed electronic and geometrical shells), and its neighbouring sizes was discussed in detail by Aguado and Lopez 18 from Density Functional Theory (DFT) simulations. They showed that the canonical heat capacity peaks of the solid-to-liquid transition in the cation and in the anion were broader than in the neutral species and shifted to lower temperatures. They also reported the occurrence of a negative branch of the microcanonical heat capacity for Al36 , corresponding to a convex intruder in the entropy or equivalently a backbending in the microcanonical temperature curve as a function of the energy. Gold clusters and nanoparticles have been the object of numerous studies, due to their remarkable properties in several application fields such as catalysis, 21 nano-electronics, 22 nano-luminescence 23 or medicine. 24 The structural and static properties of AuN clusters and nanoparticles have been widely investigated, some of the works providing explicit comparison between data from calculated structures (mostly in the DFT framework) and experimental data. 25–44 However although the early experiment of Buffat and Borel 2 was concerned with gold nanoparticles, no experimental size-selected determination of the caloric curves in the cluster regime range (N less than 50) has been published so far to our knowledge. A number of theoretical investigations 45–52 usually achieved with many body potentials or thermodynamical models have examined thermodynamically induced structural conversion and melting of large nanoparticles. Simulations of thermal properties were also carried out in the cluster regime size (from one to a few tens). Several works were concerned by simulations of specific thermal behaviours related to 2D-3D transitions in small clusters 53,54 or to thermodynamical aspects of vibrational heating. 55 The caloric curves of neutral gold clusters in the small and medium size range were quite systematically investigated by Soulé du Bas and coworkers. 56,57 In their comparison between Au19 and Au20 in particular, they showed that strong differences could be induced by a single missing atom, which is a manifestation of finite size effects where each atom and

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defect counts. In the case of Au16 and Au17 , specific features on the heat capacities were also found, 58 associated with isomerization of the cage global minima of those clusters. The shape of neutral Au20 is a highly symmetric Td pyramid that can be viewed as a part of the − f cc lattice. The shapes of the ionized clusters Au+ 20 and Au20 are also essentially pyramidal,

with very small Jahn-Teller deformations from the Td symmetry. Those geometries are now well established from theory 35,59 and also from experiment, namely infrared spectroscopy 35 for the neutral species, Trapped Ion Electron Diffraction 59 or Ion mobility 41 techniques for the cation and the anion. Direct atomic imaging insight was given by Wang and Palmer. 60 The electronic structure of neutral Au20 can be qualitatively modeled as a closed shell system with 1s2 1p6 1d10 2s2 superorbital configuration for the outer delocalized electrons in the simple spherical Jellium model (eventhough 5d electron bonding and atomic 6s − 5d hybridization can certainly not be neglected). This electronic shell closing together with the symmetric f cc tetrahedral packing of Au20 supports a particular large stability and its assignment as a magic cluster. 61 In recent works, we have adapted and benchmarked DFTB parameters 62,63 for gold materials from clusters up to bulk. 64 We additionally checked the convergence of the cohesive energies of larger nanoparticles of a few hundreds of atoms to the bulk values, as well as the structural, elastic and energetical properties of the bulk itself. As a key advantage, DFTB proved able to yield differential and selective results for charged clusters, namely Au+ N and Au− N , in addition to the neutrals, providing fairly consistent ionization potentials and electron affinities. In a subsequent work, we combined DFTB with a global search algorithm based on the Parallel Tempering Molecular Dynamics (PTMD) scheme 65 completed with (0,+,−)

periodic quenching to obtain the lowest energy isomers of Au20

(0,+,−)

and Au55

. One con-

clusion of the work for Au20 was that the isomerization energy gap is related to the specific charge state of the cluster, namely the gap is large for the neutral and much smaller for the anion and the cation. The present work is dedicated to the investigation of the influence of the charge state of the cluster on the heat capacity curves in the temperature region about

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the solid-liquid transition temperatures in the bulk. Apart from this fundamental aspect, consideration of ions is also of interest because they can be more easily size-selected experimentally via mass-spectrometry setups. The nature of the phase changes in the interval 100-1700 K and its dependence upon charge is analysed from the caloric curves and the temperature-evolution of the isomer populations. Finally, while the present simulations are achieved in the canonical ensemble, we also discuss microcanonical aspects.

Computational details The potential energy of the neutral, cationic and anionic clusters were determined using the second-order version of DFTB 66–68 (Self-Consistent Charge SCC) and the parameters for gold introduced in our previous publication 64,69 (derived from the parameters of ref. 62,63 ). Note here that the SCC scheme is relevant since it provides a self-consistent account of electrostatics, especially important at the surface of charged metal clusters. The Potential Energy Surface (PES) at various temperatures was explored using classical Molecular Dynamics with Parallel Tempering scheme, 65 as previously implemented 70 in the deMonNano code. 71 This technique strongly enhances the ergodicity in the simulations. For each case, we used a temperature range going from 50 to 2000 K with 60 replica using an exponential distribution of temperatures. The length of each trajectory was 7.5 ns using a timestep of 1.5 fs to integrate the classical equations of motion. Exchanges were attempted using the Metropolis energy criterion every 1.2 ps. We used a Nosé-Hoover chain of 5 thermostats with frequencies of 80 cm−1 to achieve an exploration in the canonical ensemble. Evaporation did not occur as a crucial problem here, since there is a quite large difference between the solid-liquid and the liquid-gas transition temperatures (respectively 1337 K 72 vs 3243 K 73 for the bulk). Nevertheless in order to avoid any problems in the solid-liquid region of the heat capacity, we enclosed the clusters within a large rigid spherical potential centered on the cluster center of mass defined by V (r) = a(r − r0 )8 with a=0.08 Hartree ˚ A−1 and r0 = 20

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˚ A for r > r0 . Several approaches exist to determine the heat capacity curve. In the present study, as in the work of Krishnamurty al., 57 we applied the multiple histogram method developed by Labastie and Whetten. 74 This approach reduces the statistical noise and allows to extrapolate heat capacities to temperatures not explicitly simulated. In addition to the canonical heat capacity, those simulations provide the microcanonical entropy S(E) (within a constant), the microcanonical temperature 1/T ∗ = ∂S/∂E and the microcanonical heat capacity C(E) = −(∂S/∂E)2 /(∂ 2 S/∂E 2 ) (the star labels the microcanonical temperature vs the canonical one ). To avoid finite difference calculation of the second derivative, the first derivative of the entropy (1/T ∗ ) was fitted by a polynomial function, easily derivable. Finally, the isomer population analysis was done after periodic quenching. For each temperature, 1024 quenches (conjugate gradient optimization), regularly distributed along the trajectory were achieved, allowing to assign the isomer basins (catching areas). Identification of the quenched structures was done combining an energy threshold (difference in energy less than 10−4 Hartree) together with an ordered-distances criterion (the interatomic distances based similarity function introduced by Joswig et al. 75 with a similarity threshold of 0.95). Note that a number of extra quasi-degenerate isomers, similar however not identical, were obtained with respect to our previous work. 69

Results and Discussion The heat capacity curves as a function of the (canonical) temperature are plotted in figure 1. The first noticeable feature is that all three curves are unimodal with a single and well defined peak and that no premelting feature can be inferred from the canonical heat capacities (unlike for instance in Au17 58 ). The temperatures of the peak maxima provide the average melting temperatures Tmelt in the canonical ensemble. The start of the transition in the canonical ensemble is not precisely defined and a stronger definition is given in the microcanonical ensemble, as will be discussed below. Nevertheless

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Au20

7

Au20+

Au20-

6

5 C/N (kB)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


4

3

2

1

0 0

200 400 600 800 1000 1200 1400 1600 T(K)

− Figure 1: Canonical heat capacity curves per atom N for Au20 , Au+ 20 and Au20 . Note that the classical kinetic contribution is not included.

one may tentatively estimate the start as the intersection between the low temperature linear raise and the tangent at the inflexion point of the heat capacity. For the neutral cluster, this estimation is around ∼900 K while it occurs at lower temperature for the cation and the anion, namely ∼640 and ∼550 K. The peak maxima for the neutral, the cation and the anion are located at 1102, 866 and 826 K respectively. Thus the temperature extension ranges of the transition in the canonical picture can be estimated roughly as 400, 450 and 550 K respectively (estimated as the peak bottom width taken as twice the difference between the start and the peak maximum). The present result for the neutral is in qualitative good correspondence with the DFT/LDA work of Krishnamurty et al. 57 who also observed a steep raise in the temperature evolution of the root mean square displacement attributed to a direct solid-liquid transition for Au20 . Note that the peak maximum in the present work is shifted to significantly higher temperatures, likely due to the larger isomerization energies in DFTB vs LDA. Let us mention that the DFT isomerization energies may significantly depend on the functional used. Actually, the present DFTB first gap to isomerization (0.64 eV) is in much better correspondence with the DFT/TPSS gap of Letchken et al. 59 (0.75 eV)

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than with the DFT/LDA results (0.44 eV). The width of the phase change interval (∼400 K) in the present DFTB simulation is nevertheless consistent with the heat capacity data of Krishnamurty et al. 57

Figure 2: Structural excitation energies of the 14th lowest energy isomers of Au20 (green), − Au+ 20 (yellow) and Au20 (blue). Isomers belonging to the same meta-basin are framed using boxes (see supporting information). An original outcome of the present work concerns the comparison with ions and shows that there is a significant reduction of the melting temperature for charged gold clusters with identical size, and even a perceivable difference can be made between the cation and the anion. The DFTB isomers structural excitation energies are reported in figure 2. Note that the quenching procedure used here (achieving quenches from all the MDPT trajectories at all the samples temperatures) generated extra isomers in addition to those of our previous work among the 15 lowest ones for each charge state. As mentioned above, the lowest energy isomer of all three species is a pyramidal-like f cc structure. One may also mention that quasi-degenerate isomers with this lowest energy structure are obtained consisting of very small distortions. In the ions, these correspond to various Jahn-Teller distortions, generating several, almost isoenergetics isomers. The above deformations only affect the T=0 K limit

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of the heat capacities, however they do not create any visible associated feature on canonical the heat capacity curves. Note that the T=0 K limit cannot be properly described in the present work due to the neglect of quantum effects. Let us however mention that several groups of quasi-degenerate isomers with very neighboring topologies can be found below 1 eV. For instance, isomers 3-12 of the neutral are distinct but they only differ by one or two bonds or a small distortion (see Supplementary Information figures S1 and S2). Such sets of isomers may be thought of as defining meta-basins of the PES. The main differences between the neutral, the cations and the anions however concern the first non-pyramidal isomers. In Au20 , the set of isomers 3-12 lie in the range 0.64-0.79 eV above the lowest one and the isomerization energy of the next one (13) is 0.97 eV, not considering the barriers. In the cation, the isomerization energy to isomer 4 is 0.34 eV. In the anion, the isomerization energy to isomer 5 is 0.27 eV. Both are significantly smaller than the corresponding gap for the neutral. A continuous progression above the first gap is observed. Again this decrease of the structural excitation energies above the pyramid meta-basin for ions with respect to the neutral are in reasonable correspondence with the DFT/TPSS values of Letchken et al. 59 Partial insight in the atomistic aspect of melting can be gained in the analysis of the isomer basin populations (figure 3). Note that such analysis only provides energetical information about the PES basins and no actual picture about the mechanisms, since the barriers between the basins are not considered (except for determining the basins catching areas) and moreover the PTMD scheme breaks the true dynamics, preventing the following of the kinetic aspects. In the neutral case, it appears that the decrease of the population of the pyramidal isomer starts around 800 K. The decrease is quite regular, not showing any steps, and depletion is fully achieved around 1300 K. Only a single meta-basin (isomers in the range 0.64-0.76 eV, essentially isomer 5) contributes with a visible population (however minor < 10 percent at its maximum) within the melting temperature range, namely 7001200K and then disappears at higher temperature. The main observation for T ≥ 900 K, is a growing population consisting of a collection of many different higher energy isomers

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(labeled ”others” in figure 3), revealing strong geometrical fluctuations, so that the cluster can be considered as melted. The minor contribution of some meta-basin structures 3-12 thus appears more as a prelude of a direct and continuous solid-to-liquid transition than as a solid-to-solid transition. Almost all the lowest isomers are essentially generated by the migration of a corner atom (possibly two) to a face of the pyramidal cluster, inducing possible distortions. Those results are qualitatively consistent with the conclusion of the DFT/LDA simulation of Krishnamurty et al., 57 despite of the quantitative differences in the melting temperatures between the two calculations. The qualitative evolution of the isomer populations for the ionized species is found similar to the case of neutrals. For the cation case, depletion of the pyramidal lowest energy structure starts as soon as 450-500K while populations of isomers 3, 4, 7 and above somewhat increase up to a maximum. At higher temperature, these populations decrease, while the isomers labeled ”others” become predominant as in the neutral case. In the anion, none of the low-energy isomers is ever significantly populated. Thus as for the neutral clusters, melting appears as a direct and continuous solid-liquid transition, however shifted to lower temperatures due to a lower energy needed for the departure from the pyramid basins. In no cases any obvious onset of a solid-to-solid type premelting feature can be observed. It was nevertheless evoked by Aguado and Lopez 18 that even a unimodal however broad heat capacity peak could hide several simultaneous processes occurring within a same temperature range. Possible isomerization at low temperatures between quasi-degenerate pyramidal − isomers (1 and 2 for Au20 and Au+ 20 or 1-4 for Au20 ) does not significantly cause any specific

shoulder on the canonical heat capacities. Of course, despite the exchange of the PTMD process, one may not exclude incomplete ergodicity at low temperature. The present result are consistent with the simulations of Aguado and Lopez on Al0,+,− 36 who also found that the canonical heat capacities of the ionic species were broader and shifted to smaller temperatures with respect to the neutral cluster. Further insight can be gained when considering the behaviour of the microcanonical

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Pyr (1-2) 3 4 5 6 7

8 9 10 11 others

Isomer population (%)

100 80 60 40 20 0 0

400

800 T(K)

Pyr (1-2) 3 4 5 6 7

1200

1600

8 9 10 11 others

Isomer population (%)

100 80 60 40 20 0 0

400

800 T(K)

Pyr (1-4) 5 6 7 8 9

1200

1600

10 11 12 13 others

100 Isomer population (%)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


80 60 40 20 0 0

400

800 T(K)

1200

1600

Figure 3: Evolution of the isomer population as a function of Temperature for Au20 (top), − Au+ 20 (center) and Au20 (bottom).

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60

Au20

C/N (kb units)

40

20

0

-20

-40

-60 0.05

0.1

0.15

0.2

0.25

0.3

E (Hartree)

Au20+ Au20-

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8 6 4 2 0 0.06 0.08

0.1

0.12 0.14 0.16 0.18

0.2

E (Hartree)

Figure 4: Microcanonical heat capacities curves per atom N for Au20 (top) (infinite values − at E1 =0.144 and E2 =0.167 Hartree) , Au+ 20 and Au20 (bottom).

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− entropy. The microcanonical heat capacities of Au20 , Au+ 20 and Au20 are shown in figure

4. A negative heat capacity range is observed for the neutral cluster between E1 =0.144 and E2 =0.167 Hartree, with infinite values at these limits. These values correspond to the inflexion points of a convex intruder in the microcanonical entropy. We can also examine the probability distributions p(E, T ). In the case of neutral Au20 , a bimodal distribution pT (E) (energy distribution at given T) with respect to the energy is observed within a narrow temperature range around the melting temperature, as shown in figure 5. For the melting temperature Tm =1102 K, the energy distribution is close to equiprobability of two coexisting phases with maxima at E’1 =0.134 and E’2 =0.171 Hartree. Note that the two underlying contributions to the bimodal feature are significantly broader (>2eV) than the distance between their maxima (the Tm =1102 K distribution is weakly asymmetric). E’1 and E’2 correspond to the boundaries of the convex intruder in the microcanonical entropy, as illustrated in figure 5. This feature appears more clearly in the bottom panel of figure 5, which reports the difference between the tangent at E’1 and E’2 and S(E). We have checked that the convex intruder characteristics and the above microcanonical details remain stable when achieving the Multiple Histogram treatment from DFTB-PTMD simulations run with a finer grid (60 temperatures within the 800-1300K transition region). No entropy convex − intruder could be observed in the case of Au+ 20 and Au20 , again in similarity with the results

of Aguado and Lopez 18 for Al+,− 36 . The microcanonical heat capacities of both ions are found to be unimodal, broader and shifted towards smaller energies with respect to the maximum of the negative heat capacity of the neutral, with quite high peak values. No significant premelting feature is observed in any of the three cases. One can only notice that the raise of the microcanonical heat capacity with increasing temperature starts earlier for the anion. Finally, taking the example of the the neutral case, one can see that the PE (T ) distributions at the transition (figure 6) extend over temperatures between ∼700K and ∼1300 K, framing the melting range extension of the canonical heat capacity discussed above.

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1.2 1 PT(E)

0.8 0.6 0.4 0.2 0 S(E) (kb units)

170 160 150 140 130 120

Δ S(E) (kb units)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

0 -0.2 -0.4 -0.6 -0.8 0.10 0.15 0.20 0.25 E(Hartree)

Figure 5: Top : probability energy distributions PT (E) = P (E, T ) for different fixed temperatures in the range T=1023-1154 K (increasing from left to right, all maxima scaled to 1). The distribution at the melting temperature Tm =1102 K is reported with bold line. The peak energies of the bimodal distribution are E’1 =0.134 and E’2 =0.1707 Hartree and 14 reported by vertical dashed linesACS over all the plots. Middle : Entropy as a function of the Paragon Plus Environment energy and the tangent at E’1 and E’2 . Bottom : ∆S difference between the linear fit and the entropy.

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1.2

1

0.8 PE(T)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.6

0.4

0.2

0

600

800

1000 1200 1400 1600 1800 T(K)

Figure 6: Probability temperature distributions PE (T ) at different energies in the range 0.076 to 0.217 Hartree (from left to right, all normalized to 1). The three distributions with bold lines correspond to energies 0.13, 0.14, 0.17 Hartree. The vertical line denotes the melting temperature at 1102 K.

Conclusion Using the DFTB approach and a canonical PTMD exploration scheme, we have achieved − a comparative analysis of the canonical heat capacity curves of Au20 , Au+ 20 and Au20 as a

function of temperature. The melting temperatures are estimated at 1102 K, 866 K and 826 K respectively. The present work shows that the change in the potential energy landscape induced by the cluster charge yields a significant variation in the energetical distribution of the low energy isomers and results in a quantitatively different solid-to-liquid transition temperature. Nevertheless, in all cases the transition is found to correspond to direct melting. This extends the findings of Krishnamurty et al. 57 addressing thermodynamical finite size effects and reveals for gold a behaviour with charge similar to that demonstrated by Aguado and Lopez in aluminum clusters. 16,18 Evidence for a convex intruder in the microcanonical entropy is also found in the present simulations for such a small size as 20 in the case of the neutral cluster, but absent in the melting processes of ions.

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Acknowledgements This work was granted access to the HPC resources of CALMIP (Grants p1303, p0059 and p18009) and from IDRIS (Grant i2015087375). It was supported by a CNRS-Inphyniti Grant (ATHENA project), the CNRS-GDR EMIE and the NEXT grants ANR-10-LABX-0037 in the framework of the Programme des Investissements d0 Avenir (CIM3 and EXTAS projects).

Supporting Information Available Structures of isomers 3 to 14 of Au20 . This material is available free of charge via the Internet at http://pubs.acs.org/.

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Influence of the charge on the melting of Au20 .

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Melting of the Au20 Gold Cluster: Does Charge Matter? - The Journal

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