Benchmarking Density Functional Based Tight-Binding for Silver and

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Benchmarking DFTB for Silver and Gold Materials: From Small Clusters to Bulk Luiz Fernando Lopes Oliveira, Nathalie Tarrat, Jerome Cuny, Joseph Morillo, Didier Lemoine, Fernand Spiegelman, and Mathias Rapacioli J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.6b09292 • Publication Date (Web): 26 Sep 2016 Downloaded from http://pubs.acs.org on October 2, 2016

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The Journal of Physical Chemistry

Benchmarking DFTB for Silver and Gold Materials: From Small Clusters to Bulk Luiz F. L. Oliveira,† Nathalie Tarrat,¶ Jérôme Cuny,† Joseph Morillo,§ Didier Lemoine,‡ Fernand Spiegelman,† and Mathias Rapacioli∗,† †Laboratoire de Chimie et Physique Quantiques LCPQ/IRSAMC, UMR5626, Université de Toulouse (UPS) and CNRS, 118 Route de Narbonne, F-31062 Toulouse, France ‡Laboratoire de Collisions Agrégats et Réactivité LCAR/IRSAMC UMR5589, Université de Toulouse (UPS) and CNRS, 118 Route de Narbonne, F-31062 Toulouse, France ¶CEMES CNRS UPR 8011, 29 rue Jeanne Marvig, BP 94347, 31055 Toulouse Cedex 4, France §CEMES CNRS UPR 8011, 29 rue Jeanne Marvig, BP 94347, 31055 Toulouse Cedex 4, France and Université de Toulouse(UPS), 118 Route de Narbonne, F-31062 Toulouse Cedex 9, France E-mail: [email protected]

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Abstract We benchmark existing and improved self-consistent-charge density functional based tight-binding (SCC-DFTB) atomic parameters for silver and gold clusters as well as for bulk materials. In the former case, our benchmarks focus on both the structural and energetic properties of small-size AgN and AuN clusters (N from 2 to 13), medium-size clusters with N = 20 and 55, and finally larger nanoparticles with N = 147, 309 and 561. For bulk materials, structural, energetics and elastic properties are discussed. We show that SCC-DFTB is quite satisfactory in reproducing essential differences between silver and gold aggregates, in particular their 2D-3D structural transitions, and their dependency upon cluster charge. SCC-DFTB is also in agreement with DFT and experiments in the medium-size regime regarding the energetic ordering of the different low-energy isomers and allows for an overall satisfactory treatment of bulk properties. A consistent convergence between the cohesive energies of the largest investigated nanoparticles and the bulk’s is obtained. Based on our results for nanoparticles of increasing size, a twoparameter analytical extrapolation of the cohesive energy is proposed. This formula takes into account the reduction of the cohesive energy for undercoordinated surface sites and converges properly to the bulk cohesive energy. Values for the surface sites cohesive energies are also proposed.

Introduction Silver and gold materials, both noble metals of group 11, have been largely studied due to their large range of applications in materials, surface and nano-science. For instance, silver and gold nanoparticles exhibit some unique chemical and physical properties such as photo-absorption, 1–5 fluorescence, 6–8 and catalytic activity, 9–15 which can be applied, among others, for biological labelling, 16–19 and for the development of light emitting sources in optoelectronics. 20–23 Fine tuning of these systems properties - within bulk, surfaces and aggregates - are closely related to their geometrical arrangements and electronic structure details that

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are hardly accessible by experimental measurements only. Consequently, one has to resort to quantum chemical simulations to provide a better understanding of experiments such as vibrational or photoelectron spectroscopy in order to help in ascertaining the structure of silver and gold clusters. This has been achieved essentially either via wave-function based methods for the characterization of small clusters, 24–27 and via density functional theory (DFT) for bulks, surfaces and middle-size clusters. In particular, the structure of neutral, cationic and anionic clusters with N up to ∼20-25 has been the subject of numerous DFT investigations for both silver, 22,27–32 and gold. 33–48 Nevertheless, while DFT can handle small to middle-size aggregates, the resulting properties show significant dependence over the functional, and it is only recently that the use of accurate functionals such as revTPSS, 47 M06, 49,50 SOGGA, 51 or RPA 47 methods seems to have lifted the uncertainties about the structural and electronic features of even small-size clusters. For instance, the 2D-3D transition in gold clusters has been in debate for sometimes and only recent theoretical studies provided results in agreement with the experimental data. 47 This transition was shown to depend on the cluster’s charge and to occur earlier for gold cations (N = 8) than for neutrals (N = 11) or anions (N = 12). Another example is Ag20 for which recent publications still propose different structure orderings. 27,32 Force-fields, generally based on many-body potentials, 52–58 allow for global structural investigations of metal clusters with N as large as a few hundreds particles 59–66 and local minimization investigations of systems up to several thousands/tens of thousands particles. 67–69 However, they do not allow to describe electronic properties such as ionization, electron attachment or Jahn-Teller effects. They have nevertheless been fruitful in providing size-dependant structural transitions, 67,68,70–72 and also for selection of candidate structures to be further optimized at a higher level of theory. 27,73 Despite fast progress in the derivation of accurate functionals and efficient algorithms to be implemented on high performance computers, DFT is not yet efficient enough to be directly incorporated into global and extensive exploration of potential energy surfaces (PES). Density functional based tight-binding (DFTB) is a quantum mechanical method

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that is derived from DFT via several approximations. 74–79 It is a quantum formalism that provides a balance between accuracy, computational efficiency and ability to explicitly take into account the electronic structure. It has proved to be particularly efficient for the description of complex molecular systems. 80–84 Furthermore, DFTB has previously been applied to investigate the structural and dynamical properties of small gold anionic clusters, either free or supported on films and substrates. 84–87 DFTB is a parametrized method and its accuracy may depend significantly on the quality of the parametrization. In particular, it requires the use of parametrized integrals, generally referred to as the Slater-Koster integrals, constructed for each atom pair of the chemical system of interest. These Slater-Koster integrals can be developed using various methodologies and reference DFT functionals, and are not uniquely defined. In the particular case of silver and gold, several SCC-DFTB parameter sets have been proposed in the literature. For gold, Koskinen and co-workers developed a parametrization achieving a near DFT-like accuracy for the description of small anionic gold 85 In a latter study by Fihey et al., another set of Slater-Koster clusters Au− N up to N = 14.

integrals was developed for Au-Au, as well as integrals for all the Au-X (H, C, S, N, O) atomic pairs, in order to describe gold-thiolates hybrid systems. 88 For silver, Szűcs et al. simulated the tunnelling current between a silver tip and a sulfur-passivated GaAs substrate using the SCC-DFTB formalism. 89 The Ag-Ag parametrization used provided satisfactory results, nevertheless and to our knowledge, it has not been further benchmarked for the description of other silver systems. The SCC-DFTB parameters should be, in principle, transferable to various chemical systems. However, it has been demonstrated by Wahiduzzaman et al. in the particular case of carbon that it is not always possible to develop a SCC-DFTB potential that is transferable to chemically different phases of the same element. 90,91 Consequently, the Au-Au and Ag-Ag parametrization mentioned above are not necessarily accurate for the description of both clusters, nanoparticles, surfaces and bulks. The main goal of the present paper is to check and possibly improve the reliability and transferability of the available DFTB parameters for silver and gold, examining the whole domain from small to large clus-

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ters and for the bulk. In the small- and medium-size regime, we will check the ability of DFTB to reproduce, with a unique parametrization, details of the geometrical and energetic properties and electronic structures to account for the important differences between silver and gold aggregates in their neutrals, cations and anions. At the same time we will check its ability to reproduce bulk structural and energetic properties and the convergence of large cluster energetics towards the bulk limit. The paper is organized as follows: in a first section , the methodology and the SCC-DFTB parameters discussed along this paper are presented. In the second section, a comparison between DFT and SCC-DFTB results for small gold and silver clusters (N ≤ 13) is given, testing various SCC-DFTB parametrization sets with a particular attention to the 2D-3D transition. Bulk properties (structure, relative stabilities and elastic constants) are then presented. Finally, medium-size clusters (N = 20 and 55) are investigated and we propose a two-parameter analytical extrapolation of the cohesive energy of large clusters (N = 147, 309 and 561), which converges properly to the bulk cohesive energy. Conclusions and perspectives are provided in the last section.

Methodology and computational details Several reviews on the DFTB and SCC-DFTB methods can be found in the literature. 74–79 SCC-DFTB mostly differs from the Kohn-Sham DFT expressed on a local basis set by the following approximations: -(i)- the DFT energy is expanded up to the second-order with respect to electronic density fluctuations around a given reference density -(ii)- all three centers interaction integrals are neglected -(iii)- the molecular orbitals are expressed in a minimal atomic basis set -(iv)- the short distance repulsive potential E rep is expressed as a function of two body interactions -(v)- the second-order term in the DFT energy expansion is expressed as a function of atomic charges and a Γ matrix which takes into account the

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charge density fluctuations. With these approximations, the total SCC-DFTB energy reads:

E SCC−DFTB =

atoms X

rep Eαβ +

1 2

ˆ 0 |ϕi i ni hϕi |H

i

α,β6=α

+

X

atoms X

Γαβ qα qβ

(1)

α,β

ˆ 0 is the Kohn-Sham operator at the reference density. The non-diagonal elements of where H ˆ 0 matrices in the atomic basis, as well as E rep are interpolated from DFT the overlap and H αβ calculations on atomic dimers and expressed as a function of interatomic distances. The diagonal elements of the Hamiltonian matrix are the orbital energies of the isolated atoms. Finally, Γαβ terms are derived from Hubbard parameters. ni is the occupation number of molecular orbital ϕi and the qα are the atomic charges (Mulliken). In the following, the SCC-DFTB approach will simply be referred to as DFTB. Calculations involving clusters have been performed with the deMonNano code. 92 The strategy followed for the structural study of small clusters in the range N ≤ 13 was two-fold. (i) We achieved a global optimization search using Parallel Tempering canonical Molecular Dynamics (PTMD) followed by quenching. Precisely, the PTMD scheme involved simultaneous non-SCC MD runs for 60 temperatures equally distributed in the range 0-5000 K, allowing configurational exchanges between trajectories. Fragmentation was hindered by using a spherical box of radius 10 bohrs. The MD timestep was set to 3 fs and exchanges were attempted using a Boltzman energy criterion every 10 timesteps. A chain of 5 Nose-Hoover thermostats with a unique frequency 80 cm−1 was used. The lengths of the trajectories were 3.105 fs. From each of the three respective MD runs at 83 K (low T ), 1000 K (medium T ) and 3000 K (high T ) of the PTMD process, 125 configurations equally spaced in time along the simulation were selected and optimized locally with the SCC-DFTB conjugated gradient scheme, providing the results for the final stable structures. This generated a bunch of low energy isomers for each size, and was repeated for neutrals, anions and cations. (ii)

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In addition, we have also checked some extra geometrical configurations found by previous authors and not present in the final yields of our own global optimization scheme. Actually, the global minimum was always found with the above PTMD+quenching scheme and in a few cases, new candidates were found with respect to the literature data. The quality of the DFTB results will be probed by benchmarking the corresponding properties against reference calculations : (i) the evolution of the binding energy per atom as a function of the size defined as:

Eb (N ) = (N E1 − EN )/N

(2)

where EN is the energy of a cluster of N atoms. (ii) the second-order derivative of the total energy: 1 ∆2 E(N ) = (EN +1 + EN −1 − 2EN ) 2

(3)

This quantity is relevant since it allows to evidence particularly stable structures, also known as magic numbers structures. (iii) the size corresponding to the 2D-3D structural transition. For bulk materials, our computational investigations were conducted using the DFTB+ simulation package. 93,94 The periodic simulation cell was a N = 4 atoms f cc Bravais cubic cell. A 40×40×40 Monkhorst-Pack mesh of k-points was used to sample the Brillouin zone. For computing the cohesive energies of the hcp bulk, a periodic rhombohedral simulation cell with N = 2 atoms was used together with a 40×40×40 Monkhorst-Pack mesh of k-points. In order to evaluate the performance of the DFTB parameters regarding the bulk properties, we calculated the equilibrium lattice parameter a0 , the binding energy Eb and the three independent cubic elastic constants B0 , C ′ and C44 of the Au and Ag f cc structures, and finally checked the stability of the f cc structure against the hcp structure. The bulk modulus B0 represents the response of the crystal to an isotropic stress (homogeneous pressure P ) 7



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whereas C ′ and C44 its response to plane shear stresses on the (010), (110) plane in the [100], [1¯10] direction, respectively. Their ratio z =

C44 C′

(the Zener ratio) is thus a measure of the

anisotropy of the cubic crystal. B0 , C ′ and C44 must be positive for the crystal stability against any deformation. A nice agreement between calculated and experimental values of a0 , Eb , B0 , C ′ , C44 and z is thus a good criterion of the ability of the DFTB potential to describe the equilibrium cubic crystalline structure of the material under consideration and of its anisotropy. The f cc and hcp close-packed structures have the same first and second neighbor shells of atoms and are thus very close in energy. A proper description of the relative stability of these two structures is thus a stringent test of the accuracy of any interaction model to describe subtle energy changes in the bulk properties of close packed structures. a0 and B0 at T = 0 K and P = 0 Pa were deduced from a fit of the T = 0 K energy versus volume curve using the Murnaghan equation of state (see supplementary information). 95 C ′ and C44 were obtained through a sixth-order polynomial fit of the elastic energy as a function of the corresponding applied shear strain δ (see supplementary information). The binding energy Eb per atom for both the f cc and hcp structures, defined as the energy gain in the crystal formation from the isolated atoms, was calculated using:

Eb = Eatom −

E0 N

(4)

where Eatom is the energy of the isolated atom in vacuum and E0 is the energy of the bulk unit cell with N atoms. Finally, from the B0 , C ′ and C44 values we deduced the z ratio and the elastic constants C11 and C12 : 3B0 + 4C ′ 3 3B0 − 2C ′ = 3

C11 =

(5)

C12

(6)

Note that, in order to check the validity of our results, the C11 constant has also been computed through a sixth order polynomial fit of the elastic energy as a function of a 8


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uniaxial tensile/compressive strain applied in the [100] direction. In both cluster and bulk calculations, the basis set includes s, p and d shells, as standardly used in the description of noble and transition metal systems which are likely to undergo valence transitions.

Small clusters Gold Clusters !

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Figure 1: Evolution of the DFTB binding energy of gold clusters as a function of the cluster size computed with three different DFTB parameter sets (DFTBα,β,χ , see text). DFT results have been added for comparison: LDA results from 96 and GGA results from. 29 9



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The reference data chosen for AuN clusters in the range N=1-20 are taken from DFT calculations performed with the functionals LDA, 96 GGA, 29 revTPSS and RPA. 47 In the present study, we will benchmark the parameters from Fihey et al. 88 (hereafter refered as the first set of parameters, or DFTBα ) and those from Koskinen et al. 85 (refered as the second set of parameters or DFTBβ ). With these DFTB parameters, we optimized all the structures from Johansson et al. 47 and Fernandez et al., 29 in addition to the structures obtained from global optimization search. As will be discussed in the following, the DFTBα set provides reasonable results for bulk properties but not for clusters in term of 2D-3D transition whereas the opposite is observed for the DFTBβ set. We noticed that a major difference between these two parameters relies on the energy attributed to the p orbital (-0.0279 Ha in the first case, -0.0096 Ha in the second case). Therefore, we introduced a third set of parameters consisting of DFTBα parameters with a significant shift of the p orbital energy to a value of -0.00001 Ha). This will be refered as the third set, or DFTBχ . Figure 1 compares the binding energies obtained for the most stable clusters with different levels of theory. We note that revTPSS and RPA calculations on gold clusters are not presented in this figure as the authors only reported relative energies. 47 Although the values reported at the DFT level are in fair agreement for the dimer, the binding energies obtained for larger sizes with the LDA functional exceed those published using GGA. At the DFTB level, the general trend of the binding energy increase is pretty well reproduced by the three sets of parameters. We notice however that while the first and third sets of parameters provide dimer binding energies in agreement with ab initio data, these are overestimated when using the second set. For the larger sizes, the binding energies of the first and second sets lie in between the two DFT values. Finally, the third set of parameters reproduces pretty well the GGA results. The cluster size corresponding to the 2D-3D transition in gold clusters has been widely investigated at the DFT level, and attributed to relativistic effects. 97 The most accurate calculations at the moment performed at the revTPSS and RPA level indicate that 2D

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and 3D structures are almost isoenergetics for clusters of 11 units. 47 Figure 2 (a) reports the binding energy of the most stable 2D and 3D structures of AuN obtained at the DFTB level. It can be seen that the transition occurs much too early with the first set of parameters (3D structures are clearly more stable for sizes above 6 units). With the DFTBβ parameter set this transition is shifted towards larger sizes in better agreement with DFT. We notice that the correct 2D-3D transition was already reported for these parameters in the context of negative and neutral gold clusters. 85,87 We however note that the 2D structures remain competitive even for sizes of 13 units. Interestingly, with the third set of parameters, the transition seems to occur within the correct size range with a 2D-3D competition for sizes from 8 to 11 units. Clearly, the shift of the p orbital (the only difference between the first and third sets of parameters) to higher energy values is responsible for preserving the planar geometry for large sizes. This increases the gap between the d and p orbitals and results in a larger population of d orbitals with respect to p orbitals, the first ones favoring planar structure and the latter ones 3D geometries. We note that DFT results also correlate the conservation of AuN planar geometries up to larger sizes with a larger d-p gap. 29 Let us also mention that the choice of the atomic configuration to extract p orbitals (virtual) versus d and s orbitals (occupied) remains an open question in DFTB parameterization. In order to further document the validity of the here proposed DFTBχ parametrization, we also report − a study of singly charged clusters Au+ N and AuN (see illustrations and cartesian coordinates

in supplementary information). Figure 2 (b) shows that the 2D-3D transition occurs for N = 8-9 in the case of cations, versus N = 12-13 in the case of anions, which is consistent with recent DFT studies using expectedly accurate functionals, 47 namely the size of transition increases from cations (N = 8) to neutrals (N = 11) and then anions (N = 12). The most stable structures obtained with the DFTBχ parameters are reported for neutrals in Figure 3 and the corresponding binding energies are given in Table 1. Figure 4 reports the second energy differences ∆2 E for DFT and DFTB results. It can be seen that all DFTB parametrizations yield the odd-even alternation of the second

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Figure 2: Evolution of the DFTB binding energy of gold clusters as a function of the cluster size for the most stable 2D and 3D structures. (a) Neutral clusters computed with three different DFTB parameter sets (DFTBα,β,χ , see text). (b) Cationic and anionic clusters computed with the DFTBχ set.

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Table 1: Binding energies, Eb (eV/at) of the most stable 2D and 3D neutral clusters with the DFTBγ (silver) and the DFTBχ parameters (gold). Size 2 3 4 5 6 7 8 9 10 11 12 13

AgN (2D) -1.03 -1.09 -1.37 -1.46 -1.62 -1.59 -1.65 -1.69 -1.78 -1.76 -1.83 -1.81

AgN (3D)

1.42 1.64 1.74 1.86 1.84 1.90 1.92 1.96 1.98

AuN (2D) 1.59 1.70 2.10 2.27 2.55 2.51 2.63 2.63 2.72 2.72 2.80 2.79

AuN (3D)

2.12 2.35 2.49 2.62 2.63 2.72 2.73 2.81 2.82

Figure 3: Lowest-energy geometries of AuN clusters (DFTBχ parameters). First row: 2D structures (N ≤ 7), second and third rows: 2D and 3D structures (8 ≤ N ≤ 11), fourth row: 3D structures (N = 12 and 13) (associated cartesian coordinates are given in supplementary information). 13



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Figure 4: Second energy derivative as a function of cluster size for Au clusters. The ab initio methods are the same as referenced on Figure 1 and DFTB is performed with three sets of parameters (α, β and χ on top, middle and bottom patterns, respectively).

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energy derivative with size, characterizing most simple metal clusters. The hexamer appears as an electronic magic number at the DFT level, as a result of the planar geometry pattern (possibly interpreted in the 2D quantum dot model see Ref 41 ). This feature is well reproduced at the DFTB level with the second and third parameters whereas it is less pronounced with the first set of parameters. This can be correlated with the too early appearance of 3D geometries for the heptamer with the DFTBα .

Silver Clusters !

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Figure 5: Evolution of the DFTB binding energy of silver clusters as a function of the cluster size computed with two different DFTB parameter sets (DFTBφ,γ , see text). DFT (TPSS, 32 PBE, 29 M06, 27 B97, 27 PW91 27 ), CCSD(T) 27 and CI 24 results have been added for comparison.

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The DFTB parameters tested for silver belong to a set originally developed in the context of surface modelling by Szűcs et al. 89 (hybride set from www.dftb.org website). This latter will be refered as DFTBφ . As will be shown in the following, it leads to an overestimation of the silver cluster binding energies and a too early 2D-3D transition. To reach reasonable binding energies, we introduced a slight modification to this parameter consisting in scaling the inter-atomic DFTB Hamiltonian matrix elements by a factor of 0.9. This modified parameter will be referred to as DFTBγ . The results of these two sets will be benchmarked against DFT calculations reported in the literature that were performed using TPSS, 32 PBE, 29 M06, 49 B97 49 and PW91 49 functionals as well as against CI 24 and CCSD(T). 49 The latter can be considered as the highest level of calculation available for these systems. In addition to the structures obtained from global optimization search, the most stable 2D and 3D structures reported by Fernandez al. 29 and Chen. et al. 27 were optimized within the DFTB framework. As can be seen frim Figure 5, TPSS, M06, B97 and PW91 functionals, CI and CCSD(T) calculations give consistent binding energies within the investigated range, whereas the PBE functional, although providing the correct trend for the size evolution, seems to significantly overestimate the binding energy versus the previous methods. Interestingly, the same observation can be made for the DFTB results with the first set of parameters, the binding energy overestimation being even larger. The modified parameters strongly reduce this overestimation, the binding energies being even closer to the CI ones. In the literature, the 2D-3D transition is usually reported to lie in the range of 6-7 units. 27 At the DFTB level, it can be seen from Figure 6 that, with the DFTBφ set of parameters, the transition occurs much too early, namely below 5 units. On the opposite, the transition occurs at 6 units with the modified DFTBγ parameters. The corresponding optimized structures and binding energies are reported in Figure 7 and Table 1. When looking at the second energy derivatives (see Figure 8), all DFT calculations predict particular stability for clusters containing an even number of units. The same is

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Figure 6: Evolution of the DFTB binding energy of silver clusters as a function of the cluster size for the most stable 2D and 3D structures. (a) Neutral clusters computed with two different DFTB parameter sets (DFTBφ,γ , see text). (b) Cationic and anionic clusters computed with the DFTBγ set (see illustrations and cartesian coordinates in supplementary information).

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Figure 7: Lowest-energy geometries for silver clusters (DFTBγ parameters). First row: 2D structures (N ≤ 4), second row: 2D and 3D structures (N = 5 and 6), third and fourth rows: 3D structures (7 ≤ N ≤ 13) (associated cartesian coordinates are given in supplementary information).

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Figure 8: Second energy derivatives as a function of size for AgN clusters. The ab initio methods are the same as referenced in Figure 5. DFTB is performed with two sets of parameters (φ and γ in upper and lower panels, respectively).

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observed at the CI level apart from the heptamer that displays a value of ∆2 E larger than for the hexamer. The latter singularity is not present in the recent CCSD(T) results. At the DFTB level with the first set of parameters the odd-even behavior is respected except in the 58 size range with ∆2 E(5) < ∆2 E(6) < ∆2 E(7) < ∆2 E(7). The modified DFTBγ parameters present the odd-even pattern in agreement with both DFT and CCSD(T) results. We notice finally that the octamer appears as a magic number whatever the DFTB parametrization used. This is in agreement with all ab initio results. This size is characteristic of electronic magic number of spherical systems (with reference to the 3D spherical jellium model, see discussion in 41 ). Table 2: Vertical ionisation potential (VIP) and vertical electron detachement energies (VDE) in eV obtained for silver and gold cluters (with DFTBγ and DF T B χ parameters, respectively) Size 2 3 4 5 6 7 8 9 10 11 12 13

Ag VIP VDE 6.90 1.22 5.46 2.42 6.19 1.70 5.74 2.23 6.42 2.19 5.80 2.28 6.35 1.45 5.01 2.29 5.62 1.95 5.43 2.41 5.77 2.00 5.31 2.57

Au VIP VDE 9.12 1.29 7.91 3.39 7.56 2.63 7.05 3.35 7.42 1.76 6.45 3.03 6.78 1.97 6.58 2.81 6.75 2.97 5.88 2.99 6.50 2.16 6.46 2.84

Vertical ionization potentials and vertical detachment energies Finally, we show in Figure 9 and Table 2 two electronic properties that can be almost directly compared with experimental data, namely the vertical ionization potentials (VIPs, probing the geometries of neutrals) and the vertical detachment affinities (VDEs, probing the geometries of the anions). The optimized geometries of anions and cations (not directly used in the 20


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present subsection) are shown in Supplementary Information. In the case of silver (left), the vertical IPs’s obtained with DFTB show a global underestimation of 0.5 eV with respect to the experimental IPs of Jackschath et al. 98 However, despite of this shift, the size-evolution of the IPs is in excellent agreement with the experimental one, and in particular the odd-even alternation is quite well reproduced by DFTB. An even better convergence is obtained for the VDEs which follow almost perfectly the experimental points of Ho et al. 99 The situation is clearly somewhat deteriorated for the VIPS of gold. While the DFTB results still exhibit an odd-even alternation, this alternation is much smoother than in the experiment. 98 Also the DFTB IPs tend to decrease too fast at the larger sizes. This does not seem to be due to the DFTB geometries of neutrals, which are in general good correspondance (with a few exceptions) with those obtained in DFT/B3PW91 calculations. 100 At the contrary, the evolution of the VDEs is quite satisfactory with respect to the experimental determination of Taylor et al, 101 but for size n=10 which provides an exception to the odd-even alternation of the VDEs. Let us mention here that the geometries obtained for gold anions are generally the same as those derived using DFT (BP86 functional) by Furche et al., 102 except for N =5, 6, 7 and 13. For size N =10, the present geometry is the same as that of Furche et al., who however found a much lower VDE from their second anionic isomer than from the lowest one.

Bulk properties Gold Bulk Experimental and theoretical bulk properties are reported in Table 3. We see that the lattice constant of f cc gold computed with the DFTBβ parameters differs by 8% from the experimental value. By contrast, the lattice constant computed with the DFTBα and the DFTBχ parameters match very nicely (+0.09% and +0.14%, respectively) the experimental one, even better than DFT (+2.9%). We thus exclude the DFTBβ parameters from subsequent tests. 21



10

10

VIP 8

8

VIP

Energy (eV)

Energy (eV)

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6

AgN clusters

4

VDE

2

6

AuN clusters

4 2

0

VDE

0 0

2

4

6

8

10

12

14

0

2

4

n

6

8

10

12

14

n

Figure 9: Size evolution of the vertical ionization potentials (VIP) and vertical detachments energies (VDE) for silver (left) and gold (right) clusters. DFTB is in full line. Symbols with joining lines stand for the experimental IPs of silver and gold clusters, 98 and the experimental VDEs for silver 99 and gold 101 The binding energy of the solid is underestimated by all methods. However, despite being too small, the DFTB computed values remain close to experimental ones (-4.6% for DFTBα and -8.1% for DFTBχ ), which is not the case of DFT (-13.0%). Concerning the elastic constants, one must keep in mind that, even the most accurate DFT quantum-mechanical calculations provide results which are typically off by 15% with respect to the experimental values 103 and this error is often much larger for small elastic constants. The bulk modulus, which experimental value is 170 ± 10(±6%) GPa, is surprisingly rather well reproduced by both DFTBα and DFTBγ models (-5%), even better than with DFT (-19%). Regarding the shear elastic moduli C and C44 both DFTB models ′

give similar poor results. Not only they differ significantly from the experimental ones (respectively ≈ +22% and -70%) but since they underestimate C44 and overestimate C they ′

lead to a highly underestimated Zener ratio (≈ 0.7 compared to 2.84) and thus a very bad description of the crystalline anisotropy. This is not the case of DFT which slightly underestimates both elastic moduli (-21 and -3 %) and thus leads to a Zener ratio (3.50) closer to the experimental one. The performance of the DFTB parameters regarding the relative stability of the f cc and hcp close-packed structures is similar to that of DFT. Concerning the first neighbours distance (respectively d1 (f cc) and a(hcp)) and the 22


c a

ratio, both DFTB models

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present the same tendencies as DFT (a < d1 and (1.633)). Compared to experimental results, this

c a c a

larger than the theoretical compact ratio is overestimated and the first neighbour

variation between f cc and hcp structures is opposite. However, the comparison with the available experimental data 104 is not straightforward, since these values were obtained on nanoparticles. As a partial conclusion, the performance of the DFTBα and DFTBχ parameters are comparable and very good regarding the lattice constant, the Bulk modulus, the cohesive energy and the stability of phases and only acceptable regarding the shear elastic constants and the anisotropy. Table 3: Experimental and DFTB calculated properties of f cc gold crystal. Lattice constant a0 (Å). Binding energy Eb (eV/at). First neighbours f cc distance d1 ′ (Å). Elastic constants (GPa): bulk modulus B0 , shear elastic moduli C and C44 , cubic elastic constants C11 and C12 . Zener ratio z = C44 /C ′ . ∆E = Eb (f cc) − Eb (hcp) (meV/at). ac ratio and a lattice parameter (Å) of the hcp cell. GGA data from the literature were added for comparison. Property a0 d1 Eb B0 ′ C C44 C11 C12 z ∆E hcp( ac ) hcp(a)

Exp. DFTBα DFTBβ DFTBχ 106 4.0782 4.082 4.401 4.084 2.884 106 2.886 3.112 2.889 3.69 107 3.52 3.39 160-180 108–110 162 162 15.98 110 19 (12a ) 20 (11a ) 45.44 110 14 13 201.63 110 172 (187b ) 170 (189b ) 169.67 110 149c 149c 2.84 0.74 0.65 3.4 2.4 1.635 104 1.736 1.736 2.96 104 2.837 2.838 ′ C11 −C12 a calculated with the formula C = 2 ′ b calculated with the formula C11 = 3B0 +4C 3 ′ c calculated with the formula C12 = 3B0 −2C 3

GGA 105 4.197 2.968 3.21 137.6 12.6a 44.1 154.4 129.2 3.50 1.9 1.655 2.952

Silver Bulk Table 4 gathers the different bulk properties computed to evaluate the performance of the DFTBφ and DFTBγ parameters, GGA data from the literature were added for comparison. The lattice constant of f cc silver computed with the DFTBφ parameters differs from 23



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the experimental value by more than 4%. Its performance is non-convincing for both clusters and bulk. The lattice constant obtained with the DFTBγ model is in fair agreement with the experimental one (-0.46%). The binding energy is slightly overestimated by the DFTBγ model (+2.3%) while it is poorly estimated by DFT calculations (≈ 16%). Concerning the bulk modulus, while it is somehow significantly overestimated (+14%) with the DFTBγ model, it is incorrectly estimated by quite large amounts (-23% and +7%) by DFT. Regarding the shear elastic moduli C and C44 , the results of the DFTBγ parametriza′

tion are acceptable. Both the C and the C44 calculated values overestimate the experimental ′

ones (+ 41% and +27%, respectively), leading to a good Zener ratio (2.71 versus 2.99) and thus to a satisfactory description of the anisotropy. These results compare quite well with the DFT results which also overestimate both shear constants but to a lesser extent (respectively 23% and 14%), leading to a similar Zener ratio (2.76). The DFTB computed relative stability of the f cc and hcp close-packed structures is correct. In hcp bulk, the

c a

ratio is very close to experiments (+0.2%), even closer than the DFT one (+1.6%). The first neighbours distance in hcp bulk does not vary significantly from f cc bulk in both DFTB and DFT calculations. The overall performance of the DFTBγ model relative to the description of bulk properties is thus good, nearly comparable to the DFT one.

Larger clusters and convergence to the bulk From the two previous sections, we showed that only DFTBχ for gold and DFTBγ for silver are able to reproduce reasonably both cluster and bulk properties. In the following, only these sets of parameters will be used.

Clusters N =20 and 55 We investigated larger clusters with sizes N = 20 and 55. Those are particularly sensitive tests in the medium-size range since vibrational or photoelectrons spectra were recorded and 24


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Table 4: Experimental and DFTB calculated properties of f cc silver crystal. Lattice constant a0 (Å). First neighbours f cc distance d1 (Å). Cohesive energy Eb (eV/at). Elastic constants (GPa): bulk modulus B0 , shear elastic moduli ′ C and C44 , cubic elastic constants C11 and C12 . Zener ratio z = C44 /C ′ . ∆E = Eb (f cc) − Eb (hcp) (meV/at). ac ratio and a lattice parameter (Å) of the hcp cell. GGA data from the literature were added for comparison. Property Exp. DFTBΦ DFTBγ GGA 106 111 a0 4.0857 4.255 4.067 4.159 -4.069 112 106 d1 2.889 3.009 2.876 2.941 111 -2.877 112 Eb 2.95 113 4.07 3.02 2.49 111 -3.445 112 110 B0 108.72 105 124 83.3 111 -116 112 ′ C 17.08 110 24(30a ) 21.05a C44 51.09 110 65 58.1 114 110 b C11 131.49 167(156 ) 161.2 114 C12 97.33 110 108c 119.1 114 z 2.99 2.71 2.76 ∆E 0.6 3 112 1.63 106 1.633 1.656 112 hcp( ac ) hcp(a) 2.877 2.866 112 ′ a 12 calculated with the formula C = C11 −C 2 3B0 +4C ′ b calculated with the formula C11 = 3 ′ c calculated with the formula C12 = 3B0 −2C 3

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comparison with DFT calculations were carried out. The structures of four Ag20 isomers are reported in Figure 10. The three lowestenergy ones (labelled a, b and c) are partly ordered structures based on pentagonal patterns. Structure c in particular derives from a double 19-atom double icosahedron from which one apex and two atoms on the side have been subtracted. The pyramidal Td structure lies at higher energy (see Table 5). Depending on the conditions of the calculations and in particular the basis set extension and the functionals, DFT calculations did not yield the same results for the structure of Ag20 . 27,32 Alternatively, Au20 was experimentally investigated via vibrational spectroscopy 42 and the resulting data were shown to correlate with DFT vibrational frequencies originating from the pyramidal Td structure. 115 Such finding is also obtained in the present DFTB framework as reported in Table 5 and Figure 11 (the labels a, b, and c correspond to the lowest-energy structures). Moreover, from correlation of photoelectron spectroscopy data and DFT calculations, Au− 20 was also proven to have a Td symmetry. 115 The present DFTB results are thus consistent with the literature. It is also interesting to see that although small differences in the structures are obtained for various charge situations, the isomer ordering is conserved in each case, constantly different in gold and silver, whatever the charge state. This shows that the addition or removal of a single electron at that size, N = 20, does not fully alter the structure of those clusters. We have also examined the larger clusters of 55 units, as well as their anions and cations. The results are shown in Table 6 and in Figure 12. In the present study, we started the DFTB optimizations of the neutral, cationic and anionic clusters with the same − structures tested by Häkkinen et al. 116 in their LDA study of the Ag− 55 and Au55 clusters: the

symetrical cuboctahedral, decahedral and icosahedral structures and three less symmetric structures (Figure 12), called hereafter (G), (M) and (S-C) named as such because the authors achieved LDA gradient optimization starting from structures 117 globally optimized with various analytical potentials, Glue (G), 55 Morse (M) 118,119 and Sutton-Chen (S-C) 53,57,60 potentials.

26


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(a)

(b)

(c)

Td

Figure 10: Low-energy isomers of Ag20 .

(a)

(b)

(c)

Td

Figure 11: Low-energy isomers of Au20 .

27



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Table 5: Energy ordering (in eV) of various isomers of Ag20 and Au20 at different charge states obtained with the modified DFTB parameters (DFTBχ and DFTBγ for gold and silver, respectively). Ionisation potentials (IP) and electron affinities (EA) are reported in eV (all structures are relaxed). Isomer ∆E (a) (b) (c) Td (a) (b) (c) Td (a) (b) (c) Td

IP EA ∆E EA Ag20 Au20 0 5.522 1.837 1.180 0.342 1.766 0.599 1.803 0.972 0 6.421 1.333 Ag+ Au+ 20 20 0 0.452 0.234 0.677 0.487 1.207 1.297 0 Ag− Au− 20 20 0 1.143 0.245 1.253 0.452 1.281 1.043 0

The lowest energy structure of Ag55 whatever its charge is clearly a slightly distorted 116 icosahedron, which is consistent with the Ag− and 55 LDA calculations of Häkkinen et al.

the TPSS results of Johansson (see table 6). The DFTB lowest energy structures of Au55 and Au+ 55 are those derived from the S-C potential whereas in the case of the anion, it is that derived from the Morse potential which is only slightly more stable than the S-C structure (see Table 6). The present DFTB results thus depend somewhat on the cluster charge, however whatever the charge, the icosahedron, cuboctahedron and decahedron have much higher energies than the less ordered structures (see Table 6): the Sutton-Chen, Morse and Glue structures are always at energies lower than 0.5 eV from the ground state structure, while the icosahedron, decahedron and cuboctahedron lie at energies higher than ≈ 1.2 eV. Let us mention that the exact ordering of these low energy isomers (G, M, S-C) is different from that of LDA calculations. Nevertheless, the present DFTB results are globally consistent with the Au− 55 LDA calculations of Häkkinen et al. 116 who showed that, contrarily to other metallic 55-mers, the low-energy structures of Au55 are not the symmetric icosahedral structure, but less symmetric structures. The same

28


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Ico

Cubo

Deca

M

S-C

G

Figure 12: Low-energy isomers of Au55 . Initial geometries from Häkkinen et al. 116 ) authors also determined the theoretical photoelectron spectra corresponding to these various isomers and showed that these less ordered structures yielded spectra in good agreement with experiments.

Convergence of Clusters Properties to the Bulk Several investigations were concerned with convergence to the bulk and structural transitions in noble metal clusters, 67–72,120,121 most of them using various semi-empirical many-body potentials (MBP). Baletto et al. specifically addressed noble metal clusters with size ranging up to ∼2000 atoms focussing on the transitions between non − f cc structures (icosahedron (Ih ) or Marks decahedron (Dh )) towards f cc structures (truncated octahedron (TO)). They showed that an Ih to Dh transition occurs for N ≤ 400 in the case of silver and N < 100 for gold, and the transition from Ih to T O at N≈ 400 for silver and < 100 in gold. 69 The findings that the Ih structures are not favorable even for smaller sizes in the case of gold was confirmed by the LDA calculations of Häkkinen et al. 116 on Au55 (see the discussion above). In the present section, we do not intend to address thoroughly the structural transition question, but the convergence of cluster cohesive energies towards the bulk. We have thus

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Table 6: Energy ordering (in eV) of various isomers of Ag55 and Au55 at different charge states obtained with the modified DFTB parameters (DFTBχ and DFTBγ for gold and silver, respectively). Ionisation potentials (IP) and electron affinities (EA) are reported in eV (all structures are relaxed). Isomer

∆E

cubo deca G M S-C ico

2.605 1.855 1.042 1.795 2.412 0

cubo deca G M S-C ico cubo deca G M S-C ico

2.735 1.777 1.243 1.936 2.239 0 2.742 2.300 1.067 2.045 2.810 0

IP Ag55

EA

5.024 3.057 Ag+ 55

Ag− 55

∆E 1.620 1.313 0.169 0.006 0 1.204 2.086 1.281 0.426 0.376 0 1.448 1.421 1.180 0.220 0 0.342 1.265

30

IP Au55

EA

4.968 2.827 Au+ 55

Au− 55


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achieved calculations of clusters with specific sizes and structures, namely N = 147, 309 and 561, representative of clusters with closed shells of atoms either in the icosahedral or cuboctahedral packing (see Figure 13). In the case of silver, we investigated both Ih and cuboctahedron structures, the latter being compatible with the f cc lattice, even though it may not be the most favorable for a finite size cluster (vs the T O structure for instance). In the case of gold, we only report the cuboctahedron structures. Starting from ideal polyhedra, the structures were relaxed using all-atom DFTB minimization. As expected, the present DFTB calculations provide Ih structures always lower than the Oh ones, whatever the number of shells. In the case of Ag561 , the icosahedron is lower than the cuboctahedron by 9.360 eV. In order to obtain an analytical extrapolation extending the present results to the asymptotic range throughout to the bulk, we have consider the following analytical expression for the cohesive energies of large size clusters:

ǫcoh (N ) = ǫv (Nv +

X

(ci /cv )α Ni )/N

(7)

i

This formula introduces the account of the reduction of the cohesive energies for surface, edge and apex sites according to their coordination ci smaller than the volume coordination, namely cv = 12 associated with a volume cohesive energy ǫv . The approximation for the dependence of the cohesive energy with coordination is inspired by MBP/SMA schemes, assuming that the many-body site energy is the square root of a sum of pair interactions with the other atoms, 54 i.e. related to a power (1/2) of the coordination in the nearest-neighbour approximation. Since the coordination numbers in the external shells of icosahedral and cuboctahedral structures are well determined (see Table 7) and so are the total numbers of individual site atoms Ni for each size, the above formula provides a simple two-parameter extrapolation, fitted here considering four data points (sizes 55, 147, 309 and 561). The energy ǫv then describes the asymptotic energy associated with a given structural family, independently of the explicit bulk calculation carried on with periodic DFTB, and the site energies can also be derived easily as ǫi = (ci /cv )α ǫv . Table 7 provides the various 31



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values of the site energies. Note that in the icosahedron case, all facets corresponds to a (111) surface, while we have distinguished the (111) triangular and (100) square facets in the cuboctahedron case.

Figure 13: Icosahedron and cuboctahedron structures for N = 309. One can see in particular that the volume energies, which can be associated with determinations of the bulk cohesive energies are 3.022 eV for silver and 3.427 eV for gold, extremely close to the cohesive energy values determined using periodic calculation, namely 3.02 and 3.39 eV respectively. The quality of the fits and the fact that the energy in the range N ≤ 561 has not yet reached a pure volumic behaviour can be seen in Figure 14. The present DFTB data predict a crossover for silver from Ih to Oh (f cc) around N = 2057 (6 atomic shells). Detailed exploration of the crossovers between the various plausible polyhedral structurations, including Dh and TO is beyond the scope of the present section focusing on the compared energetics of large finite systems and bulk periodic systems. Moreover, it cannot be excluded that other geometries should be more stable, as it is the case for Au55 . Finally, DFTB allows to calculate the ionization potentials (IPs) and electron affinities (EAs) of large clusters reported in Table 8. In the case of silver, the adiabatic IP (from relaxed icosahedral Ag561 to relaxed Ag+ 561 is found to be 4.54 eV (In the case of silver, this also corresponds to the adiabatic IP). This value can to be compared with experimental work functions of silver metal surfaces namely 4.22, 4.14 and 4.46 eV obtained from photoeletric measurements by Chelvayohan and Mee 122 for (100), (110) and (111) surfaces respectively (4.62, 4.54 and 4.74 eV respectively in the older work of Dweydari et al, 123 and 4.26 eV 32


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Table 7: Site coordination numbers c and cohesive energies ǫ (eV) of icosahedral and cuboctahedral structures of gold and silver clusters derived from the interpolation of large clusters binding energies (see Eq. 7 for definitions of α and ǫv ; and ǫi = ǫv (ci /cv )α ) Element Ag ico

site

ci

ǫi

volume surface(111) edge apex

12 9 8 6

3.012 2.542 2.372 2.004

Ag cubo

α 0.590465

0.522146 volume surface(111) surface(100) edge apex

12 9 8 7 5

3.022 2.600 2.445 2.280 1.913

volume surface(111) surface(100) edge apex

12 9 8 7 5

3.4266 3.254 3.185 3.110 2.926

Au cubo

0.179829

Table 8: Ionization potentials and electron affinities of large gold and silver clusters (all geometries relaxed) Cluster Ag55 (ICO) Ag147 (ICO) Ag309 (ICO) Ag561 (ICO) Au55 (CUBO) Au147 (CUBO) Au309 (CUBO) Au561 (CUBO)

IP (eV) 5.02 4.58 4.66 4.54 4.97 4.36 4.41 4.31

33

EA (eV) 3.06 2.95 3.39 3.48 3.02 2.61 2.87 3.22



Cohesive energy (eV)

3.1 3 2.9 561

2.8

309 2.7 147 2.6 2.5

55

2.4 2.3

0.1

0.2

0.3

N-1/3 Cohesive energy (eV)

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Page 34 of 52

3.4 561 309

3.3

147 3.2

3.1

55

0.1

0.2

N

0.3

-1/3

Figure 14: Size extrapolation of the cohesive energy of large clusters as a function of N−1/3 . Upper panel: icosahedral (red line) and cuboctahedral (blue line) silver clusters. Lower panel: cuboctahedral gold clusters. The squares and triangles are the DFTB calculated values for N = 55, 147, 309 and 561.

34


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in the polycrystaline case 124 ). Note that the icosahedral facets correspond to slightly deformed (111) surfaces. In contrast, the DFTB IPs seem to be underestimated for large gold clusters (4.31 eV for cuboctahedral Au561 ) in regard of various experimental determinations of the work functions 5.22, 5.20/5.12 and 5.30/5.26-5.55 eV for (100), (110) and (111) surfaces, respectively, obtained from photoemission spectroscopy. 125–127 The results of LDA calculations 128 are 5.10/5.32, 4.85/5.05, 4.67/4.83 eV for the unrelaxed surfaces using various planewaves pseudopotentials (i.e. Vanderbilt/Troullier-Martin definitions), relaxation reducing the above values by ≈ 0.02 eV.

Conclusion We have shown that simple modifications of existing DFTB parameters improve the accuracy of the DFTB parametrization for silver and gold clusters. DFTB answers nicely for the structural properties of neutrals, cations and anions in comparison to DFT calculations with recent functionals and also experiments. It is able to document the structural transition in the small cluster regime. We have shown that DFTB copes quite correctly with the differences between gold and silver clusters -(i)- the 2D-3D transitions in the size range N = 8-13 -(ii)- the differences for the most stable structures of the 20-mers -(iii)- the differences between the symmetric structures obtained for the 55-atom clusters. The periodic boundary conditions calculations also show that the bulk structure, energetics and elastic properties are reasonably well reproduced, about as nicely as DFT calculations. Calculations on selected structures of larger clusters show that the convergence between calculations for finite systems and bulk cohesive energies are consistent. Based on DFTB calculations with structure optimisation of selected large nanoparticles, a two-parameter analytical extrapolation of the size evolution of the cohesive energy is proposed, taking into account the reduction of the coordination of surface sites and the deviation from a pure N−1/3 linear dependence of the cohesive energy in the size range of a few hundreds of atoms. Indeed in this range, the

35



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surface/volume ratio is still non vanishing, from 0.62 for N = 147 to 0.45 for N = 561. From this analytical expression, surface site cohesive energies are also proposed. Some electronic properties have also been calculated on the basis of global or local minimization at the DFTB level. The IPs for larger clusters are shown to depend weakly on the clusters shapes. The IPs of large silver clusters seem to converged to reference DFT calculations or experimental work functions. In gold, the present DFTB IPs of the larger nanoparticles seem to be significantly underestimated with respect to DFT calculated work functions and experiment. A possible explanation could be that size convergence to the bulk is not yet reach for the sizes investigated here in the case of gold, but is not the most likely explanation. Another possible cause for such an underestimation could be a too small value of the on-site Γαα parameter, not influencing too seriously the smaller clusters with few atoms, but yielding a global size additive effect on large clusters. The present benchmark tests should nevertheless justify the use of the present DFTB parametrization in various dynamical problems such as global structural search in the medium-size range, or various dynamical molecular dynamics simulations of processes such as fragmentation of gas phase clusters, molecule-surface interaction and collisional processes involving surface reconstruction. Supporting Information We provide as supporting information : (i) additional computational details (bulk modulus and elastic constants), (ii) illustrations of secondary minima for neutral clusters obtained during PES exploration, (iii) illustrations of the lowestenergy structures of cationic and ionic clusters and (iv) cartesian coordinates of all small clusters. Acknowledgments This work was granted access to the HPC resources of CALMIP (Grants p1303 and p0059) and from IDRIS (Grant i2015087375). It was supported by a CNRS-Inphyniti Grant (ATHENA 2015 project) and the NEXT grant ANR-10-LABX-0037 in the framework of the Programme des Investissements dÁvenir (CIM3 project), the French ANR grant ANR-11-INTB-1005 DRAGS and the CNRS/GDR EMIE. We thank Augusto F. Oliveira, G. Seifert and Thomas Heine for fruitful interactions about DFTB parametrization.

36


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Benchmarking Density Functional Based Tight-Binding for Silver and

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