Assembling of hydrogenated aluminum clusters

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The electronic and atomic structure of Al13H has been studied using Density Functional Theory. Al13H has closed electronic shells. This makes the cluster very stable and suggests that it could be a candidate to form cluster assembled solids. The
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  Eur. Phys. J. D  16 , 285–288 (2001) T HE  E UROPEAN P HYSICAL  J OURNAL  D c  EDP SciencesSociet`a Italiana di FisicaSpringer-Verlag 2001 Assembling of hydrogenated aluminum clusters F. Duque 1 , L.M. Molina 2 , M.J. L´opez 2 , A. Ma˜nanes 1 , and J.A. Alonso 2 , a 1 Departamento de F´ısica Moderna, Facultad de Ciencias, Universidad de Cantabria, 39005 Santander, Spain 2 Departamento de F´ısica Te´orica, Facultad de Ciencias, Universidad de Valladolid, 47011 Valladolid, SpainReceived 21 November 2000 Abstract.  The electronic and atomic structure of Al 13 H has been studied using Density Functional Theory.Al 13 H has closed electronic shells. This makes the cluster very stable and suggests that it could be acandidate to form cluster assembled solids. The interaction between two Al 13 H clusters was analyzed andwe found that the two units preserve their identities in the dimer. A cubic–like solid phase assembled fromAl 13 H units was then modeled. In that solid the clusters retain much of their identity. Molecular dynamicsruns show that the structure of the assembled solid is stable at least up to 150 K. A favorable relativeorientation of the clusters with respect to their neighbors is critical for the stability of that solid. PACS.  36.40.Cg Electronic and magnetic properties of clusters – 36.40.Mr Spectroscopy and geometricalstructure of clusters – 61.46.+w Nanoscale materials: clusters, nanoparticles, nanotubes, and nanocrystals The high stability of the Al 13 H cluster indicates that itcould be a promising candidate for the synthesis of newcluster assembled materials [1]. The neutral cluster hasa HOMO–LUMO gap of 1 . 4 ± 0 . 2 eV, measured by pho-toelectron spectroscopy (PES) of the Al 13 H − anion [1].The experimental observations are consistent with the pre-dictions of the jellium model for aggregates of   s –  p  ele-ments [2]; in this model the 40 valence electrons of Al 13 Hgive a structure of closed electronic shells. PES experi-ments [3–5] and  ab initio  calculations [6] confirm the va-lidity of the jellium model predictions for pure aluminumclusters.In the present work we use the Density Functional The-ory (DFT) [7] to study the electronic and atomic struc-ture of Al 13 H, investigating the equilibrium location of the hydrogen atom. Then, the interaction between twoAl 13 H clusters is studied as a function of their distanceand relative orientation. Finally an assembled cluster solidis modeled as a cubic lattice built from Al 13 H units andits stability is tested by molecular dynamics simulations.We use DFT, with the local density approximation(LDA) for exchange and correlation [8], to study first thefree Al 13  and Al 13 H clusters . The Kohn–Sham (KS) equa-tions [7] are solved for the valence electrons using the ADFcode [9], treating the Ne–core of each Al atom as frozen.The basis set is formed by  s ,  p , and  d  Slater–type atomicorbitals. For each Al atom, the basis contains three  s , nine  p  and five  d  orbitals, and for H, three  s  and three  p  or-bitals. The basis set is non–orthogonal. The calculationis a spin restricted, nonpolarized one. However, to obtainthe correct binding energies, a spin polarized calculation a e-mail: is performed for the free H and Al atoms. No symmetryrestrictions are imposed to the geometry of the clusters,neither to their electronic states.We confirm that the equilibrium structure of Al 13  is adistorted icosahedron with one central atom, a geometryfound in previous  ab initio  calculations [10]. The Al–Aldistances are given in Table 1. The anionic cluster Al − 13 ,with 40 valence electrons, has a more regular icosahedralstructure, with a slight contraction with respect to theneutral. In contrast, a more distorted icosahedron is ob-tained for the cation Al +13 , with a small expansion withrespect to the neutral. The calculated binding energy of Al 13 , measured with respect to the separated atoms isgiven in Table 2. The binding energy is larger than thevalue of 35.97 eV obtained by local spin–density calcula-tions for the perfect icosahedral structure [11]. When aspin–polarized calculation is performed for the open shellAl 13 , a very small increase in binding energy is obtainedwith respect to the non-polarized result (0.05 eV), with-out noticeable relaxation of the geometry. This indicatesthat the Jahn–Teller distortions allowed for in our calcu-lation have a much larger effect than those due to spinpolarization. The HOMO–LUMO gap of the closed–shellAl − 13  cluster, calculated from the corresponding orbital en-ergies, is  ∆ǫ  = 1 . 81 eV.Several locations for the H atom bound to Al 13  havebeen tried, following the suggestions of previous work [12]:top, bridge and hollow sites, and also positions inside thecage. In the top site, the atom is placed outside the Al 13 cage, in the radial direction passing through one of the sur-face atoms. The bridge position corresponds to the atomabove the middle of the edge connecting two neighbor-ing Al surface atoms. In the hollow position the atom sits  286 The European Physical Journal D Table 1.  Bond lengths (in atomic units) for the equilibriumgeometries. Distances from the central Al to the outer Al atomsare indicated by d ; D are distances between neighbor Al atomson the surface of the cluster. For the hydrogenated aggregates, d H is the distance from H to the central Al atom, and D H areH–Al nearest-neighbor distances for Al at the surface. Sub-scripts  m and  M   indicate minimum and maximum values. d m  d M   D m  D M   D H m  D H M   d HAl +13  5.04 5.20 5.10 6.66Al 13  5.02 5.10 5.20 5.59Al − 13  5.04 5.05 5.26 5.33Al 13 H + 5.00 5.10 5.20 5.51 3.65 3.70 5.73Al 13 H 5.00 5.08 5.20 5.54 3.67 3.67 5.68Al 13 H − 5.01 5.12 5.25 5.69 3.59 3.60 5.40 above the centre of one of the triangular faces. In eachcase the equilibrium structure was obtained by first mov-ing radially the H atom, with the Al cage frozen, till aminimum of the energy is found, and at that stage a com-plete relaxation of the cluster was performed. The lowestenergy equilibrium configuration was found for the hollowsite, with the H atom at equal distances of 3.67 a.u. fromthe three Al atoms of the triangular face. Other relevantdistances are given in Table 1. The H binding energy is3.36 eV. The bridge and top configurations are only saddlepoints. A very small barrier of 0.08 eV separates hollowsites in two adjacent faces, suggesting a high mobility of the H atom over the surface at room temperature. Thecalculated binding energy of an isolated H 2  molecule bythe present method, using the spin–polarized result for theH atom, is 4.95 eV, to be compared with the experimentalvalue 4.75 eV. This means that there is a net energy gainof 1.77 eV when the H 2  molecule dissociates and each of the two H atoms binds to a different Al 13  cluster. We havealso investigated the possibility of placing the H atom in-side the Al 13  cage, but it was no possible to find a stableequilibrium position. The geometries of the charged clus-ters have been calculated starting from the equilibriumconfiguration of the neutrals. Data for the bond lengthsare given in Table 1. In Al 13 H − , the H atom is closer tothe cluster surface than in the neutral, and closer to thecentral Al atom as well, in spite of the small expansion of the Al 13  cluster. For Al 13 H + , the H atom is slightly moredistant from the cluster surface.The HOMO–LUMO gap of Al 13 H, estimated from thecorresponding orbital energies, is  ∆ǫ  = 1 . 77 eV. This canalso be obtained as the energy difference between the high-est occupied molecular orbital and the next occupied or-bital of the anion Al 13 H − , that provides a more convenientestimate because only occupied orbitals are involved. Thecorresponding value,  ∆ǫ  = 1 . 63 eV, is within the errorlimits of the experimental result, 1 . 4 ±  0 . 2 eV [1]. Theadiabatic ionization potential of Al 13 H is  I   = 7 . 05 eV,in good agreement with other DFT calculation [12]. The Table 2.  Energies of the highest occupied orbital  ǫ HOMO ,HOMO–LUMO gap  ∆ǫ , and cluster binding energy BE withrespect to the separated atoms, in eV. ǫ HOMO  ∆ǫ  BEAl +13  − 7 . 43 0.73Al 13  − 3 . 95 39.22Al − 13  − 0 . 87 1.81Al 13 H  − 4 . 11 1.77 42.58(Al 13 H) 2  − 3 . 14 0.70 88.18 05101520253035-6-5-4-3-2-10    D   E   N   S   I   T   Y    O   F   S   T   A   T   E   S   (   A  r   b .   U  n   i   t  s   )  ENERGY (eV) Fig. 1.  Calculated electronic density of states (DOS) forAl 13 H − with the H atom in the hollow position. large value of   I   reflects the high stability of the aggregate.The calculated adiabatic electron affinity is  A  = 1 . 76 eV.Comparison between the calculated electronic densityof states (DOS) of Al 13 H − and the measured PES spec-trum [1] allows to discuss the location of the H atom. Fig. 2.  Two isomeric geometries of (Al 13 H) 2 . The ground stateis isomer (b) and isomer (a) is 1 . 28 eV above in energy.  F. Duque  et al. : Assembling of hydrogenated aluminum clusters 287 b)a) Fig. 3.  Unit cell of the as-sembled solid for a large valueof the lattice constant (a), andsnapshot of the structure nearthe equilibrium volume for a dy-namical simulation at 150 K (b). To obtain the DOS we start with the KS single–particlestates. The eigenvalue of the highest–occupied KS or-bital,  ǫ HOMO , gives the ionization potential,  I  , (that is I   =  − ǫ HOMO ) for the exact exchange–correlation func-tional [13,14]. However, for the usual approximate contin-uum functionals that relation is not fulfilled. Instead, ageneralized Koopman’s theorem is obtained for  I   [15]: I   = − ǫ HOMO  +  v xc ( ∞ ) ,  (1)where  v xc ( ∞ ) is the asymptotic value of the exchange–correlation potential. This equation can be used to calcu-late  v xc ( ∞ ) as the actual energy shift between  I   (obtainedas the difference between the energies of the ionized andneutral species) and  − ǫ HOMO . The value  v xc ( ∞ ) is thenapplied as a rigid shift to all the KS energies to improvethe absolute energy scale of the electronic spectrum [3,15].For the anion Al 13 H − with the H atom at the hollow site,the value  v xc ( ∞ ) = 2 . 36 eV has been used to calculatethe DOS given in Fig. 1, where each KS energy has beenbroadened by a normalized Lorentzian of width compa-rable to the experimental resolution [1,4]. The structureof the spectrum is in good agreement with experiment,in particular the small feature at  − 1 . 8 eV and the dis-tance between this and the next large feature, that canbe identified with the HOMO–LUMO gap of the neutral.In contrast, when the H is at the top location, the DOSshows an energy gap too small compared to experiment.Next we have studied the dimer (Al 13 H) 2 . The clus-ter Al 13 H has a static dipole moment of 0.28 Debye, inthe direction of the H atom. We first consider two clusterswith a fixed relative orientation, chosen so as to maximizethe dipole–dipole interaction. A “head to tail” geometrywas constructed in which the clusters are face to face andone of them is rotated by  π/ 3 about the axis joining thetwo centers, with one H atom between the two clustersand the other on the outer opposite face. The energy wascalculated for several intercluster distances without anyrelaxation of the cluster geometries, and a minimum wasobtained for a distance  R m  = 11 . 81 a.u. between the clus-ter centers. For the geometry at  R m  a steepest descentrelaxation of the structure was then performed, and theresulting equilibrium geometry is given in Fig. 2(a), whichshows that the two clusters preserve their structures inthe dimer. The relaxation had a very small effect on thestructure of the dimer and the calculated binding energywith respect to the two separated clusters, is 1.78 eV.The intracluster interatomic distances experience smallchanges with respect to those in the separated clusters.The smallest intercluster Al–Al distances have values be-tween 5.59 a.u. and 5.72 a.u., and comparison with thosein Table 1 indicates that they are only a little larger thanthe intracluster distances. The structure has an approxi-mate C 3 v  symmetry. Nevertheless, there is a different iso-mer, in fact more stable than the one just discussed. Itsstructure is shown in Fig. 2(b), and the relevant energiesare given in Table 2. The main structural difference isthat the clusters have contact Al–Al edges perpendicularto each other, that is, one cluster is rotated 90 degreeswith respect to the other. The H atom of the bottom clus-ter adopts a bridge position inside the cluster. The otherH atom is located on a top position in the upper clus-ter. The binding energy of the dimer with respect to twoAl 13 H units is 3.03 eV, and the gain of 1.25 eV is due tothe favorable orientation of the units. A similar structurewas found as the ground state for the (Al 13 ) 2  dimer inpair potential calculations [16]. As a consequence of thecluster–cluster interaction, near–degeneracies of electroniclevels split and shells broaden. Consequently the HOMO–LUMO gap is reduced to 0.71 eV. Of course, there is somefreedom for the initial locations of the H atoms, that couldlead to different final positions for those atoms. We havenot investigated this point in detail because, as shown be-low, the H atoms easily find their way to their optimallocation in the simulations of the assembled solid.We have modeled the assembling of Al 13 H clusterswith a Car-Parrinello type code [17] that uses a supercellgeometry, a basis set of plane waves and nonlocal norm–conserving pseudopotentials. The plane wave energy cutoff was set at 25 Ry. The forces on the atoms were calcu-lated by means of the Hellmann-Feynman theorem, andthe molecular dynamics simulations at constant tempera-ture were performed with a Nos´e–Hoover thermostat.The optimal relative cluster–cluster orientation in theground state of the dimer suggests that the structure to  288 The European Physical Journal D −12.0 −10.0 −8.0 −6.0 −4.0 −2.0 0.0 Energy (eV)    D  e  n  s   i   t  y  o   f  s   t  a   t  e  s   (  a  r   b .  u  n   i   t  s   ) Fig. 4.  Electronic DOS of the assembled cluster solid near theend of a molecular dynamics simulation at 150 K. be tried for the assembled solid should be one in whicheach cluster is alternated 90 degrees with respect to all itsnearest clusters, in order to have the closest edges betweencluster neighbors perpendicular to each other. The numberof cluster neighbors is then 6 and this leads to a structureof a simple cubic lattice with 8 clusters per unit cell [18].For simplicity the H atom is placed in a bridge positionat the beginning of the simulation, although this is notcrucial. In a first step the energy of the assembled solidwas calculated as a function of the lattice constant, main-taining the component clusters frozen. Figure 3(a) showsthe structure of that solid for a lattice constant such thatthe clusters are well separated. Two energy minima wereobtained. The outer minimum, that corresponds to a lat-tice constant of 32 a.u., has a binding energy of 0.35 eVper cluster (with respect to the separated clusters). Inthis configuration the Al atoms of neighbor clusters arewell separated. A second, deeper minimum is found, witha large binding energy of 15 eV per cluster and a smallerlattice constant of 24.2 a.u. In this case the Al–Al inter-cluster distances become comparable to the intraclusterdistances. Although a value of the binding energy of 15 eVis large, that number is still small with respect to the in-ternal binding energy of the clusters with respect to theseparated atoms, which is 42.58 eV as given in Table 2.The assembled solid has a gap at the Fermi energy for thevolume corresponding to the small outer minimum but nogap occurs for the main deep minimum.At that point we fully relaxed the atomic coordinates.For that purpose we performed constant temperature dy-namical simulation runs at 150 K, for a cell with a latticeparameter of 24.0 a.u. The time step was 5 fs and the to-tal simulation time was about 3 ps. The results show thatat that temperature the assembled solid is stable againstdeformations that could induce a transition to anotherstructure. Figure 3(b) shows a snapshot near the end of the trajectory in one of the simulations. Although the Al 13 units are connected and a little bit distorted, they retainthe icosahedral structure along the trajectory. The maineffect of the temperature is to allow the H atoms to mi-grate from their initial positions to more favorable inter-stitial places within the cubic arrangementof clusters. Theenergy gain due to this migration is about 0.7 eV per Hatom. In summary, the assembling of the Al 13 H clustersleads to a cluster solid that appears to be stable at leastup to 150 K. The density of states, given in Fig. 4 showsthat the assembled solid has metallic character. We havealso performed simulations for a more compact solid. Anfcc-type lattice of Al 13 H clusters was constructed and theatomic positions were relaxed for various values of thelattice constant. In this case the interaction betweenthe clusters is much larger and they lose completely theirindividual character. The structure obtained cannot beconsidered any more as a solid of clusters. In the fcc lat-tice each icosahedral cluster has twelve neighbor clustersand the favorable orientational requirements are not met.The conclusion is that the relative orientation of the clus-ters, that controls the degree of packing in the lattice,plays an important role, in addition to other obvious re-quirements like a high intrinsic stability of the individualclusters. That stability arises from a combination of elec-tronic (large HOMO–LUMO gap) and structural factors. Work supported by DGESIC of Spain (Grants PB98-0190 andPB98-0345). Computational resources of CESCA and CEPBA,coordinated by C 4 , are acknowledged. References 1. S. Burkart, N. Blessing, B. Klipp, J. M¨uller, G. Gantef¨or,G. Seifert, Chem. Phys. Lett.  301 , 546 (1999).2. W. de Heer, Rev. Mod. Phys.  65 , 611 (1993).3. J. Akola, M. Manninen, H. H¨akkinen, U. Landman, X. Li,L.S. Wang, Phys. Rev. B  60 , 11297 (1999).4. X. Li, H. Wu, X.B. Wang, L.S. Wang, Phys. Rev. Lett.  81 ,1909 (1998).5. G. Gantef¨or, W. Eberhardt, Chem. Phys. Lett.  217 , 600(1994); C.Y. Cha, G. Gantef¨or, W. Eberhardt, J. Chem.Phys.  100 , 995 (1994).6. F. Duque, A. Ma˜nanes, Eur. Phys. J. D  9 , 223 (1999).7. W. Kohn,  Density Functional Theory  , edited by E.K.U.Gross, R.M. Dreizler (Plenum Press, New York, 1995).8. S.H. Vosko, L. Wilk, M. Nusair, Can. J. Phys.  58 , 1200(1980).9. G. te Velde, E.J. Baerends, J. Comput. Chem.  99 , 84(1992); P.M. Boerriger, G. te Velde, E.J. Baerends, Int.J. Quantum Chem.  33 , 87 (1988).10. J.E. Fowler, J.M. Ugalde, Phys. Rev. A  58 , 383 (1998).11. X.G. Gong, V. Kumar, Phys. Rev. Lett.  70 , 2078 (1993).12. S.N. Khanna, P. Jena, Chem. Phys. Lett.  218 , 383 (1994).13. J.P. Perdew, M. Levy, Phys. Rev. B  56 , 16021 (1997).14. C.O. Almbladh, U. von Barth, Phys. Rev. B  31 , 3231(1985).15. D.J. Tozer, N.C. Handy,J. Chem. Phys. 109 , 10180 (1998).16. D.Y. Sun, X.G. Gong, Phys. Rev. B  54 , 17051 (1996).17. M. Bokstedte, A. Kley, J. Neugebauer, M. Scheffler, Com-put. Phys. 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