Discovery of Magnetic Superatoms and Assessment of van der Waals

Article pubs.acs.org/JPCC

Discovery of Magnetic Superatoms and Assessment of van der Waals Dispersion Effects in Csn Clusters Andrés Aguado* Departamento de Física Teórica, Atómica y Ó ptica, Universidad de Valladolid, Valladolid 47071, Spain S Supporting Information *

ABSTRACT: We report the global minimum (GM) structures and electronic properties of Csn clusters with up to 80 atoms, obtained employing a density functional theory method which accounts for van der Waals dispersion interactions (vdW-DFT). The GM structures of Csn are found to differ markedly from those of lighter alkali clusters like Nan. Three main physical factors are invoked to interpret the differences: vdW attraction, sd hybridization, and spin polarization. vdW effects are found to modify the GM structures of many Csn clusters as compared to the predictions of local and semilocal DFT approximations and tend to favor compact structures. sd hybridization accounts for the enhanced stability of strained motifs such as icosahedra or Kasper polyhedra. Finally, spin polarization stabilizes highly symmetric structures with a high orbital degeneracy. The electric dipole moments of most Csn clusters are close to zero, a distinguishing feature of metallicity previously observed for Nan clusters. However, our predictions show that some clusters like Cs10 or Cs26 have sizable electric dipole moments. High spin multiplicities are observed for many different sizes. The spontaneous magnetization of pure Csn clusters is intrinsic and exclusively due to delocalized electrons. This allows us to extend the concept of superatoms (multicenter systems that mimic the electronic behavior of elemental atoms) so that it encompasses also magnetic properties, without the need to introduce localized electrons or extrinsic impurities. The physical reasons for the stability of magnetic superatoms are identified and described.

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INTRODUCTION Geometry and electronic structure are the most fundamental properties of atomic clusters, as they ultimately determine their chemical reactivity as well as magnetic, thermal, optical, and other properties of interest to the field of Nanoscience. Typical nanoparticle properties are strongly size-dependent due to quantum confinement and surface effects as well as different from the properties of isolated molecules and bulk systems.1−6 An understanding of these properties may lead to a large number of practical applications in Nanotechnology, for example, in the field of catalysis.7 Since the seminal experiments of Knight et al.8 on sodium clusters, alkali clusters have been extensively studied as representatives of simple metal clusters with delocalized valence electrons. Several properties like the electron shell structure, single-particle and collective optical response properties, melting-like transition, fission, and Coulomb stability of highly charged clusters, etc.,1,2,5,9,10 have been addressed. The observed experimental abundances of alkali clusters are consistent with the magic numbers predicted by structureless jellium models, suggesting that the electron shell structure is a more important factor than geometric structure in determining alkali cluster stability. In those early experimental measurements, however, clusters were produced in a hot liquid-like phase,4 while geometric shells are expected to contribute more significantly to the stabilities of cold, solid clusters. Currently available experimental techniques only provide indirect probes of cluster structure, and so the global minimum (GM) © 2012 American Chemical Society

structures of most alkali clusters have remained unassigned for a long time. Very recently, an assessment of the structures of Nan and Nan− clusters has been reported11,12 based on an explicit comparison between theoretical predictions and lowtemperature photoemission11 and beam deflection13 experiments. The results show that the magic numbers of sodium clusters at T = 0 K are mainly determined by geometric packing for n ≥ 55, while electronic shell effects are more important for n < 55. Despite the substantial interest brought about the light alkali metal clusters, the number of studies devoted to the heavier Csn clusters is comparatively scarce. Martin and co-workers reported experimental abundances, ionization energies, and photoabsorption spectra of small pure cesium clusters and also of bigger cesium clusters doped with oxygen or with SO2 impurities.14−18 They observed ionization energy maxima when the number of valence electrons Ne equals 8, 18, 34, 58, 92, ..., in agreement with jellium model predictions, although clusters with 20 and 40 electrons did not show any significant feature. The jellium picture is appropriate even when ionic impurities are present, the main effect of each oxygen impurity being simply a reduction in the number of delocalized electrons. A similar picture has been recently shown to be valid also for oxidized sodium clusters.19,20 McHugh et al.21 reported Received: December 11, 2011 Revised: February 28, 2012 Published: March 5, 2012 6841

dx.doi.org/10.1021/jp2119179 | J. Phys. Chem. C 2012, 116, 6841−6851

The Journal of Physical Chemistry C

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photoelectron spectra of Csn− with n = 1−3. Bhaskar et al.22 reported experimental mass abundances of Csn+ cluster ions with n = 2−21 and found that clusters with Ne = 8 and 20 delocalized electrons have an enhanced stability. Large cesium clusters were produced by Aman et al.23 and Gspann,24 but an analysis of relative stabilities was not provided. Regarding theoretical calculations, Moullet et al.25 reported an ab initio study of neutral and ionized cesium dimers. They concluded that the pseudopotential approximation is accurate once relativistic and nonlinear core corrections are included and that the hybridization with both p and d atomic orbitals is important for an accurate description of the electron density. Onwuagba26 reported the binding energies of Csn clusters with up to 21 atoms predicted by the spherical jellium model. Lammers et al.27,28 identified inconsistencies in the jellium model predictions for cesium clusters and reported the structures of Csn up to n = 78 by employing a spherically averaged pseudopotential approximation (SAPS). The SAPS approximation, however, tends to favor structures in which the atoms are arranged into spherical layers about the cluster center. Springborg29 compared the predictions of three different models (jellium, tight-binding, and spherical infinite well), finding that all of them produce the same electron shell closings. Li et al.,30 Lai et al.,31 and Hristova et al.32 reported the structural predictions of an empirical Gupta potential. Ali et al.33 investigated the structure of Csn and Csn+ clusters with n = 2−10 using all-electron ab initio methods. A global optimization was not attempted as the starting geometries for optimization were generated empirically, and relativistic effects were not accounted for. Flórez and Fuentealba34 improved upon the previous ab initio results by properly accounting for relativistic effects and employing a more flexible basis set in a density functional study of cesium clusters with up to 8 atoms. However, the spin multiplicities were constrained to their lowest possible value which might affect the accuracy of the reported structures. Finally, Assadollahzadeh et al.35 have hitherto reported the most exhaustive theoretical study of cesium clusters, covering the size range n = 2−20 and employing a generalized gradient approximation to density functional theory (DFT). The determination of optimal structures involves a preliminary unbiased search with a genetic algorithm (GA),36 employing a small basis set and the local density approximation. Some of the most stable structures found in that initial step are reoptimized at a higher level of theory. It is noteworthy that these authors also focused just on low-spin isomers, and higher spin multiplicities were not considered. The aim of the present work is to provide a more definitive assessment of the structures of Csn in a broad size range n = 2− 80, employing ab initio DFT methods. There are several motivations for such a study. The scarcity of theoretical results for n ≤ 20, and the total absence of results for n > 20, is certainly one of them. More important to us, however, is to assess the impact of van der Waals (vdW) dispersion interactions on the structural properties. Recent works37−39 have shown that dispersion effects are essential to accurately describe the properties of bulk cesium, which was indeed classified as “soft matter”.38 Dispersion effects are selfconsistently incorporated into DFT by employing a nonlocal correlation functional, first proposed by Dion et al.40 The structural freedom is nevertheless restricted in the bulk limit due to the translational symmetry constraint, and incorporation of vdW effects only modifies the equilibrium volume, elastic

constants, and other properties but does not alter the crystalline structure of bulk cesium. Dispersion effects are expected to be more important in systems with reduced dimensionality which have a larger structural freedom. Recent works have already demonstrated that vdW interactions significantly modify the physical properties at interfaces,41 as well as the dimensionality and structures of small gold42 and sodium12 clusters. In this work, we will offer a comparison between the structural predictions of vdW-DFT and standard DFT (the one employing local or semilocal exchange-correlation functionals). A final motivation of our work is to offer a comparison with the previously reported structures of sodium clusters.11,12 Sodium and cesium are similar metallic systems, with delocalized electrons coming from the s-like atomic orbitals, but there are a number of differences which may induce significant disparity in the geometrical properties. First of all, the 6s-orbital of cesium is contracted by relativistic effects, but the 6p-orbital is essentially unaffected by the relativistic correction.25 Thus, while the 6p-orbital is quite diffuse, the spatial extent of the 6s-orbital becomes comparable to that of the 5d-orbital. Concomintantly with this observation, the excitation energies of the electronic configurations 6s06p15d0 and 6s06p05d1 become very similar to each other, and both are lower, by about 0.4 Ry, than the excitation energy of the 3s03p1 configuration of sodium. Hybridization effects, and particularly sd hybridization, will therefore be much more important in cesium than in sodium. In fact, it is known that significant sd hybridization occurs at the Fermi surface of bulk cesium, which is enhanced upon pressurization of the sample.43,44 Typical interatomic distances in metallic clusters are contracted as compared to the bulk limit values, so that hybridization effects are expected to be more relevant in the cluster phase. They can contribute to the stabilization of strained structures, like those based on icosahedra or Kasper polyhedra, because the internal atoms are subjected to a substantial pressure in those structures.45 A second important difference between sodium and cesium concerns the magnitude of the electron shell effects. The ionization energies of alkali clusters show sharp decreases upon the opening of a new electronic shell, but the magnitude of those sharp drops is almost 70% smaller in cesium than in sodium.15 This difference is well captured by jellium models, which show that the confinement potential is much shallower for cesium. A possible implication associated with this observation is that purely geometric packing effects can play a more important role in determining the structures of Cs clusters. Also, if the contribution of electronic shell effects to the cluster energy is small enough, new mechanisms for electronic stabilization, mainly spin polarization, may become relevant. This is why we will not constrain the spin multiplicity in the present paper. A third difference between Na and Cs involves their atomic masses: zero-point nuclear delocalization effects are expected to be much smaller in cesium. The electric dipole moments of small Nan clusters have been measured13 and found to be essentially zero, a distinguishing feature of metallicity. In a recent work,12 we demonstratated that zeropoint nuclear vibrations significantly affect the obtained electric dipole moments and polarizabilities of sodium clusters. It is interesting to check wether the electric dipole moments of Csn clusters continue to be zero, to analyze the generality of the proposal by the de Heer’s group. Finally, we notice that accurate binding energies of Csn clusters are needed to calculate excess or mixing energies in alkali nanoalloys of different sizes,46 which is one of our goals in the near future. 6842



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atom was smaller than 0.01 eV/Å. The initial geometries for structural optimization are obtained from extensive basin hopping optimizations58 based on a Gupta empirical potential30 plus subsequent relaxation of many candidate structures at the LDA-DFT level. The detailed protocol employed to select the candidate structures is explained in our previous works.11 Finally, we reoptimized at the vdW-DFT level all the isomers which, according to the LDA calculations, have total energies up to 0.4 eV above the putative GM. Although vdW-DFT usually changes the relative stabilities of isomers as compared to the LDA, the energy window of 0.4 eV is quite safe for not missing any relevant isomer.

The paper is organized as follows: The theoretical approach for locating the GM structures is described in Section 2. Section 3 describes the structures, stabilities, and also some electronic properties of Csn clusters. Finally, section 4 summarizes the main conclusions of our work.

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COMPUTATIONAL METHODS The strategy employed to locate the GM structures of Csn clusters is the same as the one employed in our recent works on sodium clusters,11,12 so we just provide here a brief summary. The first-principles calculations are performed at the Kohn− Sham47 DFT48 level. We employ the SIESTA code49 with different implementations of the exchange-correlation functional: (i) the well-known spin-polarized local density approximation (LDA functional), as parametrized by Perdew−Zunger50 and based on Ceperley−Alder simulations of the electron gas;51 (ii) the spin-polarized generalized gradient approximation (GGA), as implemented by Perdew, Burke, and Ernzerhof (PBE functional);52 (iii) the fully nonlocal van der Waals functional (vdW-DFT), as proposed by Klimes, Bowler, and Michaelides (KBM functional).53 This functional has been implemented by Román-Pérez and Soler54 in the SIESTA code. The main results are obtained with the KBM functional, while LDA and PBE are used only to assess the effect of vdW interactions on the obtained structures. We employ norm-conserving pseudopotentials55 to describe the effect of core electrons, including scalar relativistic and nonlinear core corrections56 which are essential for cesium.25 The electronic wave function is expanded into a basis of numerical atom-centered orbitals, which includes four, three, and two basis functions of s, p, and d, angular symmetry, respectively. Both the size of the basis set (nine functions per atom) and the spatial extension of the basis functions were numerically optimized to obtain converged total energies for the bulk and dimer limits. The resolution of the real-space grid is defined by a cutoff energy of 100 Ry. Benchmark tests are offered in Table 1. They clearly show that vdW-DFT reproduces the experimental results in both dimer and bulk limits much more accurately than either LDA or PBE. All equilibrium cluster geometries were obtained from unconstrained conjugate-gradients structural relaxation using DFT forces. The structures were relaxed until the force on each

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RESULTS Global Minimum Structures. Figure 1 shows the structures of Csn in the size range n = 10−20. All the structures

Figure 1. Putative GM structures for Csn clusters with n = 10−20 atoms. The size and approximate point group symmetry are provided below each cluster. Several isomers are shown (for example, for n = 14) when they are nearly degenerate (total energy difference smaller than 0.01 eV). To help visualization, atoms in the cluster core are yellow (light), while those at the cluster surface are plotted red (dark). The atomic coordinates of all putative GM structures are available in the Supporting Information.

with n < 10 coincide with those reported by Assadollahzadeh et al.35 and are not shown here explicitly. The GM of Cs10 is obtained by adding two atoms to the D2d structure of Cs8. Cs11, Cs12, and one of the C2 isomers of Cs14 adopt deltahedral structures formed by fusing several pentagonal bipyramids together. Cs13, a perfect icosahedron, is the smallest size with an internal atom. Cs15 and Cs16 are Kasper polyhedra with two and three disclination lines, respectively. The perfect D6d symmetry of the Z14 Kasper polyhedron3 is lowered to D2 in Cs15 by a Jahn−Teller (JT) distortion, resulting in a stabilization of 0.015 eV. For Cs17, we obtain two very similar degenerate isomers (in practice, we define two structures as degenerate when their total energies differ by less than 0.01 eV), which are distortions of the D4d polyhedron observed in Na17−.11 Cs18 has two degenerate structures: the Cs isomer is obtained by adding one atom to Cs17; the C2v isomer is another compact structure, with a single central atom coordinated to 17 surface atoms. Cs19 is a double-icosahedron with D5h symmetry and contains two core atoms, although there is a Cs degenerate structure with a single central atom. For n = 20, the structures are based on the double icosahedron: The C2v isomer has a clear adatom in the equatorial region, while the C2 isomer incorporates the additional atom into the cluster shell. Our results in this size range differ from Assadollahzadeh’s predictions35 for sizes n = 13, 15−17, and 19. For n = 17, the discrepancy is minor, as they report a perfect D4d structure which differs from ours just by a small JT distortion. Regarding n = 15 and 16, Assadollahzadeh et al. do not even report the Kasper polyhedra as possible local minima, so it is probable that

Table 1. Equilibrium Bond Distance d, Cohesive Energy Ecoh, and Vertical Ionization Potential (VIP) of the Cs2 Dimer and Lattice Constant a, Bulk Modulus B, and Cohesive Energy Ecoh of Bulk Cs, As Predicted by Several Exchange-Correlation Functionals, Are Compared to Experimental Resultsa Cs2 LDA PBE KBM exp. Cs (bulk) LDA PBE KBM exp.

d (Å) 4.589 4.757 4.670 4.651 a (Å) 5.762 6.171 5.988 6.039

Ecoh (eV)

VIP (eV)

0.251 0.201 0.216 0.226 Ecoh (eV)

3.95 3.56 3.77 3.7 B (GPa)

0.88 0.69 0.80 0.81

2.65 1.92 2.18 2.1

a

Experimental values are taken from ref 37 for bulk cesium and from refs 57 and 16 for the dimer. 6843



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with a well-defined core and a shell of monatomic thickness. The number of core atoms progressively increases from 9 atoms for Cs45 to 12 atoms in Cs54. Structures in the n = 45−54 size range have no apparent structural motif and so look quite disordered. They typically have elongated shapes and contain a number of disclinations lines, most easily appreciated in the structure of Cs52. Figure 3 shows the structures for sizes n = 55−80. A Mackay icosahedron with a small distortion toward C2h symmetry is

these structures were simply not located in their GA optimizations. We notice that those GA runs were performed at the ab initio DFT level of theory and so were necessarily short. The disagreement is most important for Cs13 because the perfect icosahedron was reported to be 0.112 eV less stable than their suggested GM structure.35 We analyze the reasons for this discrepancy in the next subsection. We also notice that, with the only exceptions of n = 11, 17, and 18, Csn and Nan clusters12 adopt different structures. The main observation is that Csn structures are more compact: for example, an interior atom does not appear in sodium clusters up to n = 17, while it emerges at n = 13 in cesium clusters; similarly, while Cs19 and Cs20 have two atoms in the core, the corresponding sodium clusters have a single central atom. Figure 2 shows a selection of the GM structures for n = 21− 54. In this size range, many structures are based on a

Figure 3. Representative selection of the putative GM structures for clusters with n = 55−80 atoms. The rest of the caption is the same as in Figure 1.

obtained as the GM structure of Cs55. Contrary to the results on sodium clusters, however, the icosahedral structure is degenerate with a Cs isomer in which one of the vertex atoms is shifted to a new position within the cluster shell, creating a vacancy and three rosette-like defects.59 Surface structural excitations have therefore a lower energy as compared to sodium clusters. The extra atom in Cs56 is also incorporated into the surface layer via the formation of rosettes. Cs59 has a remarkable structure with C6v symmetry. The core contains 15 atoms and coincides with the Z14 Kasper polyhedron, i.e., the GM structure of Cs15. These structures dominate the size range n = 57−62 and are therefore more compact than the corresponding structures of Nan clusters. The core of Cs65 is an atom-centered Z16 Kasper polyhedron, containing four disclination lines in tetrahedral arrangement,3 which results in another highly compact structure. For n = 66−69, the core contains 18 atoms and, apart from small distortions, is the same as the GM structure of Cs18. Cs70−72 adopt low-symmetry structures with 19 atoms in the core and quite spherical shapes. For n = 73 and 74, the 20-atom core has a high D3d symmetry, and it is formed by two 13-atom icosahedra fused together so that they share only six atoms (as opposed to seven atoms in a double icosahedron). A complete Mackay overlayer covers the core for n = 74, the whole cluster thus keeping all the point group symmetries of the core. For these two sizes, the GM structures of cesium and sodium clusters coincide. For n = 75− 80, the number of core atoms increases gradually from 21 for Cs76 to 23 for Cs79. In particular, the core of Cs79 contains once more a Z16 unit, with a pentagonal cap added on top of one of the disclination lines. As a general conclusion, we notice that the structures of Csn clusters are, on average, both more compact and more spherical than the corresponding structures of Nan clusters. We also observe an increased preponderance of Z14−Z16 Kasper polyhedra in the structures of Csn clusters, while most of the Nan structures were based on pIh packing. The differences are therefore substantial even if the two metals are similar and point to a more important role of geometric packing effects (as compared to electron shell closing effects) in determining the most stable structures of cesium clusters. As detailed in the next

Figure 2. Representative selection of the putative GM structures for clusters with n = 21−54 atoms. The rest of the caption is the same as in Figure 1.

combination of Cs13 units (polyicosahedral (pIh) packing) and/or Cs15−16 units (packing of Kasper polyhedra as in the Frank−Kasper bulk phases). For example, the GM structure of Cs21 can be seen as a Cs15 unit with a six-atom cap. The other two degenerate isomers represent different ways of inserting two Cs atoms into the shell of Cs19. For n = 25−27, the structures have three atoms in the core and show pIh packing. Cs30, with four core atoms, has two degenerate structures: the Cs isomer contains a Cs21 unit with a nine-atom cap, and the C2v isomer is obtained by removing four atoms from the so-called 5-fold pancake structure which is the GM structure of Na34 and has D5h symmetry.11 Cs34 adopts a C2v structure with a perfectly flat five-atom core. It is based on a three-layer hexagonal stacking, and its global shape is also pancake; however, it is almost 0.3 eV more stable than the D5h pancake, once more a big difference with respect to sodium clusters. For n = 35−37, the core contains six atoms. Cs36, Cs37, and the C1 isomer of Cs35 contain a slightly distorted octahedral core, so they just represent different possibilities for building an overlayer about a Cs6 octahedron. The distortions typically remove the mirror plane symmetries, resulting in chiral structures. The core of the D3 isomer of Cs35 is a slightly twisted trigonal prism. The C2v isomer is quite different, as it is formed by interpenetrating four Cs13 and two Cs15 units. Cs39 has a seven-atom core with tetrahedral shape. Cs40 has two competing structures: the C1 isomer is grown on a seven-atom pentagonal bypiramid core, and the D2 isomer already has an eight-atom core, namely, the Z8 Kasper polyhedron,3 that coincides with the GM structure of Cs8 and which is also the core of Cs42. It can be appreciated that most clusters can be described as endohedral structures 6844



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calculations where the d functions were removed from the basis set. For this particular cluster size, vdW effects are not strong enough to modify the GM structure, although they swap the relative stabilities of the C2 and Cs isomers and modify the magnitudes of the total energy differences. We notice that the Ih isomer is the GM structure also at the PBE level of theory (the one employed by Assadollahzadeh et al.35) if no constraints are applied, but dispersion effects further stabilize the most compact isomer. In general, we have found that dispersion effects can modify the energy differences between local minima by as much as 0.07 eV as compared to LDA and/or PBE results, so the predicted GM structures differ from PBE results for many sizes. sd hybridization effects are seen to be essential to the stability of the icosahedral structure, and they probably represent the most important quantitative difference between cesium and sodium cluster systems: removing the d functions from the basis reinstores both the higher stability and the near degeneracy of the C2 and Cs structures, rendering Cs13 very similar to Na13. However, sd hybridization alone is not enough to fully explain the stability of the icosahedral structure. Neither PBE nor vdW-DFT favors the Ih isomer if the spin multiplicity is constrained to a doublet configuration. The conjoint stabilization effects of sd hybridization and spin polarization are needed to produce the final result. We show the spin-resolved electronic density of states (DOS) of Cs13 in Figure 4, together with the partial contribution of s, p, and d orbitals and the jellium labeling in the bottom panel. The DOS is responsible for the so-called band energy contribution to the total energy. The 1S level is fully described by the s-like atomic orbitals; s and p orbitals

subsection, sd hybridization, spin polarization, and vdW effects all contribute to the stabilization of compact motifs. Analysis of Hybridization, Spin Polarization, and vdW Effects. We present here a detailed analysis of the stability and electronic structure of Cs13. We choose this cluster as a representative example due to the large discrepancy with previous theoretical predictions,35 although we have checked that the conclusions are general and can be applied to other cluster sizes. According to the spherical jellium model,1,2 the electron shell configuration of Cs13 is 1S21P61D5. The cluster has an incomplete outer shell, and according to deformable jellium models, the jellium background will undergo a deviation away from spherical shape. The symmetry breaking tends to open a gap at the Fermi energy by removing the orbital degeneracy and so typically leads to a lower-energy state with minimum spin multiplicity. This JT distortion arises from vibronic coupling between the electronic degrees of freedom and the collective nuclear degrees of freedom which describe the jellium shape. The distortion induces a restructuring of the electronic shells, so that subshell closings emerge at particular sizes providing an additional electronic stabilization. As the outermost shell is exactly half-filled in Cs13, oblate and prolate shape deformations are equally favored. In our previous work on sodium clusters,12 we have indeed located two nearly degenerate spin-doublet structures for Na13: a C2 isomer with oblate shape and a Cs prolate isomer, thus in almost perfect agreement with the jellium predictions. We notice that the GM structure of Cs13 reported by Assadollahzadeh et al. coincides with the C2 isomer of Na13. There is, however, a possible different scenario for Cs13. If the cluster keeps a spherical shape, the superatom picture8 predicts an electronic state with maximum spin polarization: if all the D orbitals are singly occupied, the electron−ion attractive interaction will be optimized, as the electronic and ionic densities will have a congruent shape. If the magnitudes of electron shell and vibronic coupling effects are sufficiently small, the stabilization provided by spin polarization may overcome that provided by shape distortion effects. In fact, our calculations predict that Cs13 is a spin-sextet with a spherical (icosahedral) shape. We notice that only levels with angular momentum l > 2 are split by the Ih point group,60 so at least regarding the electronic structure Cs13 can be viewed as perfectly spherical. To analyze the effects of different contributions, we report in Table 2 the total energy differences between Ih, C2, and Cs isomers of Cs13 at both vdW-DFT and PBE levels of theory. We include results where the total spin multiplicity was constrained to its minimum value, as well as results of Table 2. Total Energy Differences (in eV) between Three Isomers of Cs13, at Different Levels of Theorya isomer

vdW

S=1

sp basis

PBE

S=1

Ih C2 Cs

0.00 0.074 0.100

0.051 0.00 0.026

0.194 0.00 0.002

0.00 0.044 0.029

0.087 0.015 0.00

Figure 4. Spin-resolved electronic DOS for the putative GM structure of Cs13 (thick black curves) is compared to the DOS obtained when the d orbitals are not included in the basis (green or light curves in the upper graph) and with the DOS obtained when the spin multiplicity is constrained to be minimal (green or light curves in the middle graph). The partial contribution of s, p, and d orbitals, together with the jellium labeling of the cluster levels, is shown in the lower graph. A Gaussian broadening of 0.05 eV is employed for each KS level.

a

The spin S is measured here by the number of electrons with unpaired spin, so S = 1 identifies the spin-doublet configuration. vdW and PBE columns show results with unconstrained spin polarization and the full basis set. The (S = 1) columns show results of calculations with fixed spin and the full basis set; in the sp basis column, the d functions were removed from the basis, but the spin was unconstrained. 6845



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gap separating the filled 1D↑ spin shell from the empty 1D↓ spin shell. Being a closed shell cluster, a JT distortion does not appear. A similar situation is found in other clusters like Cs74, which also preserves full D3d symmetry. For most other sizes, however, the JT distortion is clearly appreciated. Cs15, for example, undergoes a JT distortion from a reference perfect Kasper polyhedron with D6d symmetry to the distorted D2 arrangement shown in Figure 1. This cluster has an open shell 1D7 = 1D↑51D↓2 outer electron configuration, and so the maximum spin multiplicity (2S + 1 = 4). In the reference D6d arrangement, the average exchange splitting separating the occupied D↑ and D↓ levels is 0.1 eV. In the JT-distorted D2 configuration, the degeneracy is separately lifted in the two spin channels, but the shifts in the energies of the one-electron levels are all smaller than 0.02 eV. This implies that, even after the JT distortion, the five D↑ levels continue to have a lower energy than the two D↓ levels. Therefore, the JT distortion does not kill the magnetism because the filling of spin orbitals, in the natural order of increasing energy, leads to the maximum spin multiplicity. Similar JT distortions occur in many other highly symmetric clusters, such as Cs55, which distorts from perfect Ih to C2h symmetry. However, in all such cases, the JT effect on the energy levels is 1 order of magnitude smaller than the exchange splitting, so the magnetism survives the distortion. In summary, spin polarization effects stabilize nearly spherical or otherwise highly symmetric atomic arrangements which possess large orbital degeneracies or near-degeneracies. The main reason for the stabilization is always a favorable electron−ion interaction in the high multiplicity state, in agreement with Hund’s rule. This conclusion is expected to be valid for all those nearly free electron clusters for which electron shell and vibronic coupling effects are sufficiently weak, and cesium perfectly matches this condition. It will not be valid, of course, for magnetic clusters of transition or rare-earth metals which are far from the free electron limit and the superatom concept. In these systems, the magnetism has mostly an atomic origin associated to incomplete d- and/or f-localized atomic states. To complete the picture, it is important to emphasize that the magnitude of the exchange splitting stabilization depends on the degree of symmetry breaking. The exchange splitting is large only in clusters with either a high symmetry or a spherical shape, and all other clusters prefer to adopt instead a low-spin configuration. As an example, we have also calculated the energy of the C2 isomer of Cs13 by constraining the spin multiplicity to a sextet state. The sextet is now more than 0.2 eV less stable than the singlet, mainly because of an unfavorable electron−ion interaction in the high spin state. This is also consistent with the electronic structure obtained from the superatom picture: clusters with a strong shape deformation will have the spherical shells split into subshells, with sizable energy gaps separating the several subshells, and so imposing a high spin state will force some electrons to occupy high energy levels. In this case, it is the lowest spin state which has the most favorable electron−ion attraction. To conclude, the strength of the electron−ion interaction is at the same time the main factor that favors a high spin multiplicity in highly symmetric clusters and that quenches magnetization in low-symmetry clusters. This competition between spin-polarized symmetric shapes and deformed shapes with low spin is well documented in the literature dealing with the spin-polarized jellium model62,63 and occurs not only in cesium but also in other alkali clusters. In the case of Na13, for example, we also locate an icosahedral isomer which is a spin sextet; however, because of the stronger electron

almost completely account for the 1P level; and the 1D level has approximate 50%, 25%, and 25% contributions from s, p, and d orbitals, respectively. Removing the d orbitals from the basis set has a major effect on the 1S and 1D levels (see top panel). The D shell is significantly stabilized by sd hybridization, leading to a favorable band energy contribution. Similar plots for the Cs and C2 isomers (not shown explicitly) demonstrate that d orbitals have a smaller relative weight in the total DOS. Also, a local analysis of the DOS shows that the central atom in Cs13 carries the largest d orbital contribution, indicating that sd hybridization mostly contributes to strengthen the bonding between the core atom and the shell. This picture is consistent with the observation that sd hybridization effects increase with pressure because the icosahedron is a strained structure which induces a substantial local stress in the cluster interior. The involvement of d electrons thus accounts for the higher stability of compact strained structures like icosahedra or the Kasper polyhedra observed in bigger clusters. Constraining the spin configuration to a doublet state (middle panel in Figure 4) results in a slight stabilization of the S and P shells (as measured by the energies of the centroids of spin up and spin down channels), but once more the D shell is strongly destabilized, resulting in the end in a band energy term 0.06 eV more stable in the sextet state. All the molecular orbitals of Cs13 are spread over the whole cluster, so the magnetism is clearly due to delocalized electrons. We have performed an analysis of the different contributions to the total energies of the sextet and doublet states, fixing the geometry to that optimal for the sextet state, so that the Coulombic nuclear contribution is the same in both systems. The sextet is more stable than the singlet by 0.124 eV. This energy is 1 order of magnitude larger than typical JT stabilization energies (0.015 eV in Cs15; for example, see previous subsection), demonstrating that the vibronic coupling is weak in cesium clusters. The electron−ion interaction becomes 0.127 eV lower in the sextet, which demonstrates the validity of the superatom picture. The higher congruence between electronic and ionic densities also makes the electronic kinetic energy 0.102 eV lower in the sextet state. Notice that the trend in kinetic energy is opposite to the one typically observed in atoms, which have a single nuclear attractor. In an atom, electrons in singly occupied orbitals are less shielded from the nuclear attraction, so that such orbitals contract, enhancing both the electron−nuclear attraction and the “repulsive” electron kinetic energy.61 Therefore, those two energy contributions compete in isolated atoms. In Cs13, however, both energy terms decrease because the electron density is more evenly distributed over the whole cluster volume in the sextet state, so high spin multiplicities are expected to be even more stable in superatoms than in atoms. The electron−electron interaction (Hartree plus exchangecorrelation energies) is 0.105 eV higher in the sextet. The results thus support the validity of Hund’s rule of maximum multiplicity for the Cs13 superatom, although its now obsolete explanation in terms of a lower electron−electron repulsion in the high spin state is clearly wrong. From this analysis, we can conclude that in highly symmetric structures the stabilization associated to a high spin multiplicity is quantitatively more important than the stabilization associated with a possible JT distortion. However, JT distortions are not necessarily suppressed. For the particular case of Cs13 just discussed, the large exchange splitting equips the cluster with a closed shell electronic structure, with a sizable 6846



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models.1,2 There are only slight departures between the ab initio and jellium predictions regarding the electronic properties: for example, the shell closings at Ne = 34 and 40 are not very marked in the ab initio results. In fact, Ne = 34 is only seen as a local minimum in the VEA, but no appreciable feature is seen in the other two indicators of electronic stability. Similarly, a local maximum in the VIP is seen at Ne = 39 instead of Ne = 40, and the beginning of the drop in the VIP is indeed located at Ne = 36. Our VIP results are in good agreement with the experimental measurements of Limberger and Martin,16 performed up to n = 15. In particular, we reproduce the observation that the VIP of Cs6 is higher than the VIP of Cs8, as well as the absence of significant odd−even alternation effects in the VIP for n = 9−15, which is once more partially connected to the occurrence of high spin multiplicities. Figure 6 shows the spin multiplicities for the putative GM structures of Csn clusters. As explained in the previous

shell effects, the shape deformation stabilization is bigger, and the low-spin C2 isomer is the GM structure. In Cs13, shape deformations have a smaller impact on the stability because of weaker electron shell effects. The combined effect of vdW dispersion interactions and sd hybridization effects (which are much less important in sodium) contribute to stabilize the more compact icosahedral geometry, which is further stabilized by spin polarization. All the effects need to be taken into account as shown by the results in Table 2. Both systems have in common that highly symmetric structures are spin polarized, and low-symmetry structures are not, in agreement with the electronic structure of jellium; however, only in cesium do the spin-polarized structures become the global minima at many cluster sizes. Cluster Stabilities and Electronic Properties. Figure 5 shows the energy difference between the highest occupied

Figure 5. HOMO−LUMO gaps (upper graph), vertical ionization energies (middle graph), and vertical electron affinities (lower graph) for the putative GM structures of Csn clusters. The main electron shell closings (sizes preceding a large drop in the HOMO−LUMO and/or VIP values, or marked minima in the VEA values) are explicitly indicated.

Figure 6. Electron dipole moments (upper graph) and spin multiplicities (lower graph) for the putative GM structures of Csn clusters. In cases of near degeneracy, the different isomers typically have very similar μ and (2S+1) values. Some exceptions are mentioned in the text.

molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO), as well as the vertical ionization potential (VIP) and vertical electron affinity (VEA) of Csn clusters. Odd−even alternation (mostly in εHOMO−LUMO) and nonmonotonic behaviors are observed, superimposed to the average trend predicted from classical charged sphere models. A large drop in the εHOMO−LUMO or VIP values is taken as indicative of the opening of a new electron shell. Also, clusters with a completely filled electron shell are expected to show up as local minima in the VEA curve. Clusters with a particularly stable electronic structure are thus found for Ne = 6, 8, 14, 18, 20, 26, 30, 34, 36−37, 39−40, 50, 58, 64, and 70 electrons. Some of the observed electron shell closings, namely, Ne = 8, 18, 20, 34, 40, 58, and 70, are already predicted by a simple spherical jellium model; some others like Ne = 14, 26, 30, and 50 are predicted by spheroidal jellium or Clemenger−Nilsson

subsection, high-spin states are observed for many of the more spherical and/or high-symmetry geometries, which present a high orbital degeneracy. In particular, we have checked that the occurrence of high-spin states in the size range n = 35−38 is essential to reproduce the enhanced electronic stability of Cs36 as reported in Figure 5. Cs55 is also a spin sextet, like Cs13. At first sight, such a spin configuration can not be easily reconciled with jellium model predictions. The filling sequence predicted by the jellium model for Cs55 is 1S21P61D102S21F142P61G15, with the 2D, 3S, and 1H nearly degenerate shells unoccupied. The three holes in the 1G shell would allow for a maximum spin multiplicity of 2S + 1 = 4, which is in fact the one observed in Na55.11 However, in icosahedral symmetry, the 1Gg shell splits into two subshells with room for 10 and 8 electrons, respectively,60 which increases its effective width. An analysis of the DOS of Cs55 6847



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demonstrates that the conjoint effect of the 1G “crystal-field” splitting and spin polarization is to shift the energy of the 3S↑ shell to a position just between the two 1G↓ subshells, so the correct electronic configuration for Cs 55 is indeed 1S21P61D102S21F142P61G143S1, thus explaining the observed spin multiplicity. The rounded and compact geometries observed for many of the larger clusters are also stabilized by spin polarization effects. In particular, the high electronic stability of Cs64 (see Figure 5) can only be explained in terms of its septet spin configuration, as it dissappears when the spin multiplicity is constrained to be minimal. Cs64 adopts a highly spherical structure, so its DOS conforms to jellium model expectations. The 2D3S shells have very similar energies and can be viewed as a single shell which is exactly half-occupied, so the explanation of the maximum spin multiplicity is the same as for Cs13. Finally, the prolate but otherwise highly symmetric D3d structure of Cs74 adopts a nonet spin configuration, the highest spin multiplicity observed, to the best of our knowledge, in simple metal clusters with delocalized electrons. The splitting of the spherical jellium levels by the D3d point group symmetry can be clearly identified in the DOS of Cs74 up to the 1G level. The splitting pattern of the nearly degenerate 2d and 1h outer shells is more complex and results in a high orbital degeneracy at the Fermi level, which explains the huge spin multiplicity. We comment here that high spin multiplicities have recently been observed in metallic superatoms doped with an endohedral transition metal impurity,64 for example, FeMg8. These clusters contain both localized d electrons and delocalized electrons which distribute in superatom shells. It was shown that the exchange splitting in the transition metal impurity is preserved in the cluster and that hybridization between localized and delocalized shells is the driving force for the spin polarization of the superatom shells. We emphasize that our finding is qualitatively different, as no impurities nor localized d electrons are present in pure Csn clusters. The magnetism observed in Csn clusters does not even depend on the participation of the 5d atomic orbitals of Cs, as exactly the same spin multiplicities are obtained in calculations where the d orbitals are removed from the basis set. The spontaneous magnetic polarization of Csn clusters is intrinsic, involves exclusively delocalized electrons, extends up to large cluster sizes, and results in higher spin multiplicities than those reported in ref 64. Its driving force is simply the conjoint optimization of the electron−ion interaction and electronic kinetic energy, which is possible in superatoms due to the spread, multicenter, nature of the nuclear attraction. The spin polarization stabilization far outweighs the energy gain associated to shape distortions because the vibronic coupling and electron shell effects are weak in cesium, so that Hund’s rule of maximum multiplicity applies. This allows, for the first time, to extend the concept of superatoms (multicenter systems that mimic the electronic behavior of elemental atoms) so that it encompasses also magnetic properties, without the need to introduce localized electrons or explicit impurities. The electric dipole moments are shown in the upper part of Figure 6. Those sizes with nearly degenerate structures typically have very similar μ-values, so we just quote the lowest of them. The most significant exception is Cs6: a planar D3h structure has zero dipole for symmetry reasons, but a degenerate isomer with C5v symmetry has an enormous dipole of 231 mD/atom. The electric dipoles are significantly lower than 10 mD/atom for most sizes, reflecting the ability of Csn clusters to efficiently screen internal electric fields. However, some clusters like Cs10

or Cs26 have significantly larger dipole moments, which would make them ideal candidates for future beam deflection experiments like those performed on sodium clusters.13 In a previous work,12 it was found that zero-point vibrations are an important ingredient to reproduce the tiny dipole moments of sodium clusters. The zero-point energies that we obtain for Csn clusters are about 5 meV/atom, which would amount to approximately 20 K if converted to an internal temperature scale. This energy is about three times smaller than the zeropoint energy of sodium clusters. As the vibrational frequencies are also lower on average for cesium, we have estimated the zero-point effect on the dipole moment of Cs26 explicitly, by employing the same molecular dynamics-based method described in ref 12. The conclusion is that zero-point vibrations can modify the μ-values by at most 2 mD/atom. Therefore, zero-point effects are not expected to modify appreciably the results of Figure 6. Our theoretical results predict that some small Csn clusters may have sizable dipole moments. We hope this prediction can be experimentally tested in the future. Figure 7 shows three different stability measures: the cohesive energy (or binding energy per atom) Ecoh(n) = E1 −

Figure 7. Three different stability measures for Csn clusters. The upper graph shows the cohesive energies, referenced to Efit, which is a fourparameter fit of the form Efit = A0 + A1N1/3 + A2N2/3 + A3N. The middle graph shows the evaporation energies, and the lower graph shows the second-energy differences. Clusters with enhanced stability (magic sizes) are shown for each of the stability indicators.

En/n, the evaporation energy Eevap(n) = (E1 + En−1) − En, and the second energy difference Δ2(n) = En−1 + En+1 − 2En = Eevap(n) − Eevap(n + 1). The cohesive energy quantifies the total internal energy content of a cluster and is therefore a measure of its global (or absolute) stability. Meanwhile, Eevap and Δ2 provide “local” stability measures, by comparing the energy of a cluster of size n to that of clusters with neighboring sizes, and so they are more suitable quantities to interpret the results of abundance mass spectra. The cohesive energies show an oscillating pattern about the average behavior (obtained by a smooth fit to the numerical data). Clusters with n = 8, 17−20, 36−39, and 58−65 atoms have an enhanced stability as compared to that average. The oscillations are in qualitative 6848



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van der Waals dispersion interactions. We have compared our results with those of previous works on cesium clusters with n ≤ 20,35 improving on the description of the GM structures for n = 13, 15−17, and 19. The predictions for n > 20 are new. We have also compared the properties of Csn clusters to those of Nan clusters, which were recently reported.11,12 Although Na and Cs are both alkali (and therefore similar) metals, we have found that the structures of Nan and Csn clusters are significantly different except for a few sizes. As compared to sodium, cesium clusters tend to adopt a more compact geometric packing, with a much more preponderant occurrence of Kasper polyhedral motifs with negative disclination lines. On average, Csn clusters adopt also more spherical shapes as compared to Nan. Most Csn cluster structures can be described as endohedral or core−shell, possessing a well-defined atomic core covered by an overlayer of monatomic thickness. Except for a few sizes, the shell does not contain clear vacancies or adatoms. This implies a reconstruction of the shell almost for each cluster size; i.e., most GM structures can not be obtained by simply adding or removing one atom from the structures of neighboring sizes. The core of Cs59, for instance, is a Z14 atomcentered Kasper polyhedron and thus containes 15 atoms. Cs65 is another geometric shell closing, and its 17-atom core is an atom-centered Z16 polyhedron. The substantial structural differences between Csn and Nan cluster systems have been rationalized in terms of three separate physical effects which conspire to stabilize the more compact motifs seen in Csn clusters: first, vdW dispersion effects are more substantial in cesium due to its high atomic polarizability and tend to stabilize compact structures over more open motifs, simply because dispersion interactions are purely attractive; second, sd hybridization stabilizes compact strained structures like icosahedra and the Kasper polyhedra. The stretched surface bonds in these structures put the cluster core under pressure, which enhances the involvement of the d atomic orbitals of interior Cs atoms and strengthens the bonding between core and shell; finally, high spin multiplicities emerge to further stabilize many spherical or otherwise highly symmetric structures possessing a high orbital degeneracy. The additional stabilization provided by a spontaneous spin polarization overcomes the one due to static Jahn−Teller deformations because electron shell closing and vibronic coupling effects are much weaker in Csn, as compared to Nan clusters. High spin multiplicities are obtained for many Csn clusters in the whole size range considered in this paper. As typical examples, Cs13 and Cs55 are spin sextets; Cs64 is a septet; and Cs74 is a nonet, the highest spin multiplicity observed. The magnetism in Csn clusters is completely due to delocalized electrons, as opposed to previous findings in metal clusters doped with a transition metal impurity.64 The spin polarization observed in Csn clusters does not depend either on the explicit involvement of d orbitals in bonding, as it is observed even when the d atomic orbitals are removed from the basis set. Highly symmetric Csn clusters spontaneously magnetize according to Hund’s rule of maximum multiplicity. The physical factors driving the cluster toward a spin-polarized electronic state are the concurrent decreases in both electronic kinetic energy and electron−ion interaction energy. The simultaneous stabilization of those two energy terms upon magnetization is possible in superatoms because the nuclear attractors are themselves distributed over the whole cluster volume. We have thus argued that spin polarization is expected

agreement with the stabilities predicted by jellium-like models, demonstrating that the electron shell structure is a major factor determining the absolute stability of Csn clusters. There are, however, significant deviations from simple jellium rules for clusters with n > 20: for example, neither Cs34 nor Cs40 has a particularly high stability, and the cohesive energy maximum in this size range occurs in fact for Cs39; similarly, the expected sharp decrease in stability after n = 58 is not observed. Instead, the high stability of Cs58 extends up in size and reaches Cs65 before showing a significant decrease; finally, although cohesive energy drops are observed at sizes n = 68 and n = 70 (electron shell closings according to the jellium model), they occur in a size range of low absolute stability. The observed discrepancies between ab initio and jellium results hint at the possibility that geometric packing, and possibly other effects such as spin polarization, contributes significantly to the stability of Csn clusters. We assign the main magic numbers by combining the global and local stability measures: a highly stable cluster size will be more stable than neighboring sizes and at the same time will have a high cohesive energy. Those clusters with a high local stability but low cohesive energy are considered secondary magic numbers. Therefore, the most stable clusters are n = 8, 17, 20, 36, 39, 58−59, and 65, while secondary magic clusters occur for n = 13, 26, 30, 34, 68, 70, 74, and 79. We notice that Cs17 has a higher local stability than Cs18 even if a local maximum in the cohesive energy occurs for Cs18. This is due to the sharp decrease in the cohesive energies of clusters with n < 17. Cs40 is not a magic cluster according to our ab initio results, although Cs34 appears at least as a secondary magic number. The high stability of Cs36 stems mainly from electronic effects, as it correlates with a significant local maximum in the VIP (Figure 5) which is in turn induced by a high spin multiplicity. Cs39, on the other hand, is a highly compact core−shell structure that also shows a local maximum in the VIP, so geometric packing and electronic effects conjointly explain its high stability. The Δ2 and cohesive energy values show that Cs58 and Cs59 are similarly stable: Cs58 is an electronic shell closing, and Cs59 is a compact geometric shell closing with high symmetry. Cs65 is another geometric shell closing, with a shell of monatomic thickness covering a highly compact Z16 core, and is further stabilized by spin polarization effects as explained above. We have checked that the marked maximum in Δ2 for n = 65 would not be reproduced if a minimum spin multiplicity is enforced. Concerning the secondary magic numbers, some of them (n = 26, 30, 34, 68, 70) are induced by the electron shell structure, while some others (n = 13, 74, 79) coincide with geometric shell closings. Sizes n = 13 and n = 74 are additionally stabilized by high spin multiplicities. In summary, the interpretation of the magic numbers of Csn clusters is complex, as it involves the analysis of different electronic and geometric effects. Although electron shell closings affect the cluster stabilities in the whole size range considered in this paper, they become of secondary importance for sizes n > 58. This conclusion is analogous to the one recently advanced for sodium clusters.11 Geometric packing and other electronic effects like spin polarization are needed to understand the stabilities of cold, solid, alkali clusters.

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SUMMARY In summary, we have reported the GM structures, stabilities, and electronic properties of Csn clusters with n = 2−80 atoms, employing a first-principles DFT method which incorporates 6849



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to stabilize the energy in superatoms even more than in real atoms, so high spin multiplicities are to be expected in nearly free electron clusters as long as the competing electron shell effects are sufficiently weak. These conditions are almost perfectly fulfilled by cesium, so that Csn clusters provide the first example of intrinsically magnetic superatoms. Typical electronic properties like ionization energies have been found to be in good agreement with the predictions of deformable jellium models, with few exceptions. Electron shell closing effects are thus visible up to the biggest sizes considered in this paper. Whether these effects are strong enough to substantially affect the cohesive energies is, however, a different problem. We have found that many of the magic numbers do not agree with electronic shell closings but rather with geometric shell closings. Some others are directly induced by spin polarization effects. In any case, electronic shell closings are found to be only of secondary importance to the stabilities of Csn clusters with n > 58. This is at variance with experimental measurements dealing with hot liquid clusters, where only the electron shell-induced magic numbers survive. Geometric packing and spin polarization are, however, essential to interpret the stabilities of cold, solid Csn clusters. The electric dipole moments of most Csn clusters are close to zero, a result analogous to that recently observed12,13 in Nan clusters and which points to a very efficient electronic screening of internal electric fields. This is one of the distinguishing features of the metallic state of matter, so it is surprising that small clusters with a discrete electronic density of states satisfy this property so well, even when the point-group symmetry of most Csn clusters would in principle allow for a finite dipole moment. In Nan clusters, the electric dipole is highly fluxional and couples to the zero-point vibrations of the ionic skeleton, so the nuclear degrees of freedom appreciably contribute to the screening properties. Here, we have found that a similar effect is less pronounced for Csn clusters. Our calculations predict that some Csn clusters, such as n = 10, 26 or one of the degenerate isomers of Cs6, have sizable dipole moments, making them ideal systems to check the scope and implications of the “metallicity” concept in small atomic clusters. We hope our results may stimulate future beam deflection experiments.

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ASSOCIATED CONTENT

* Supporting Information S

A file including the Cartesian coordinates and point group symmetries of the putative GM structures of Csn. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS I am grateful to B. Assadollahzadeh for sending the coordinates of their GM structures for explicit comparison. I also gratefully acknowledge the support of the Spanish “Ministerio de Ciencia e Innovación”, the European Regional Development Fund, and “Junta de Castilla y León” (Project Nos. FIS2011-22957 and VA104A11-2). 6850



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