Nuclear shell model applied to metallic clusters

We apply the nuclear shell model to jellium clusters of up to twenty-one Na atoms. Binding energies, ionization potentials, and photoabsorption cross sections are calculated and compared with mean-field results.     

Simplified semiclassical theory for the electronic shell structure of metallic clusters

We present a simplified theory for the electronic shell structure of large metallic clusters by extending the semiclassical analysis by Gutzwiller, Balian and Bloch. Thus analytical results are given for the shell effect in the Fermi energy and cohesive energy. In particular, results for the temperature dependence of the shell structure are given. New results are presented for the shell structure in the ionization potential and photoyield and for the existence of electronic- versus atomic shell structure as a function of cluster size and temperature. We obtain good quantitative agreement with previous numerical calculations and with experiments on Na clusters.     

Semi-empirical model for the fission of multiply charged metal clusters

A semi-empirical model is proposed to calculate the fission barrier height for large highly-charged simplemetal clusters. The fusion barrier is obtained making use of a bonding potential, and the classical metallic drop model is employed for calculating the heats of fission and evaporation. We have found a good agreement between calculated critical sizes and experimental results for the appearance of charged clusters in mass spectra. Within the same model, we predict the critical numbers for spontaneous fission.     

Electronic shell structure and metal clusters: the self-consistent spheroidal jellium model

The electronic properties of small metal particles within the recently proposed self-consistent spheroidal jellium model [1] are further explored and compared to recent experimental data. Physical properties investigated include ionization potentials, electron affinities and the binding energy of neutral monomers to cationic clusters. The formalism is applied within the size-range 2≦N≦41, but could easily be extended beyondN=41. Finally, we discuss briefly the implications for the study of the dynamical response of open-shell clusters. In sharp contrast to earlier studies the functional is now corrected for self-interaction error, in a way first proposed by Pedew and Zunger [2]. This enables us to calculate reliable values for the electron affinities within ajellium-based model. This has the advantage, that we can calculate the affinities for Cu for all particle numbers for which experimental data are available. In all cases investigated we obtain excellent agreement with experiment, with pronounced shell-effects both for the electron affinities and for the binding energies, confirming in this way that the abundances map the relative stability of (Me) _N clusters, with Me being a sp-metal atom (Na, K, Li, Cu, Ag, Au etc.).     

Force fields for metallic clusters and nanoparticles

Atomic force fields for simulating copper, silver, and gold clusters and nanoparticles are developed. Potential energy functions are obtained for both monatomic and binary metallic systems using an embedded atom method. Many cluster configurations of varying size and shape are used to constrain the parametrization for each system. Binding energies for these training clusters were computed using density functional theory (DFT) with the Perdew-Wang exchange-correlation functional in the generalized gradients approximation. Extensive testing shows that the many-body potentials are able to reproduce the DFT energies for most of the structures that were included in the training set. The force fields were used to calculate surface energies, bulk structures, and thermodynamic properties. The results are in good agreement with the DFT values and consistent with the available experimental data.     

Description of metallic and covalent clusters with icosahedral symmetry: the polytope model

Compact and tetravalent clusters with icosahedral local or global symmetries are generated by mapping from ideal structures in curved space onto a tangent euclidean 3D space. The observed elastic energy of the clusters can thus be interpreted as an intrinsic curvature associated to a frustrated local order. It is then proposed a kind of classification of the very rich family of possible clusters using a limited set of parameters.     

The Hubbard model for small clusters

The Hubbard model with nearest neighbor hopping, and one type of orbital is applied to clusters of 4 and 5 sites. Results are reported for the ground state as a function of the number of electrons. A rather complicated dependence of the spin of the ground state on occupation number, geometry, and model parameters is found. Ground state energies are compared. The excitation spectrum is illustrated in a particular example.     

The structure-averaged jellium model for metal clusters

We present an extension of the jellium model for alkali clusters which allows a selfconsistent determination of the jellium background in addition to the Kohn-Sham solution of the electron dynamics. It includes the volume and the surface energy of the ionic distribution. We discuss applications to the shape of clusters and to the position and width of the plasmon resonances.     

Ionic vibrational breathing mode of metallic clusters

The energy of the vibrational mode with spherical symmetry, in which the ionic cores oscillate in the radial direction around the equilibrium geometry (ionic breathing mode) is calculated for trivalent (Al_N, 2≤N≤50) and monovalent (Na_N, 2≤N≤73; Cs_N, 2≤N≤74) metallic clusters. The ground-state total energy is calculated using density functional theory, with a spherically averaged pseudopotential to describe the ion–electron interaction and optimizing the geometry by the simulated annealing technique. The energy of the ionic mode is calculated by diagonalization of the dynamical matrix including the electronic relaxation in the linear response approximation. The compressibility and bulk modulus of the metallic cluster are obtained from the energies of the monopole oscillations. These energies present a linear behavior on the inverse of the cluster radius, which is analyzed using a semiclassical liquid drop mass formula for the total energy of the clusters and a scaling model. The values of the vibrational frequencies present electronic shell closing effects for the three metals.©1997 John Wiley & Sons, Inc.     

Molecular clusters as models of metallic catalysts

A critical view of the use of molecular clusters as models of metallic catalysts and as means of modelling chemistry at the surface is presented. The major problems that still exist in the identification of surface adsorbed species and the bonding modes they adopt on reaction with the variety of different metal faces available on the surface of close-packed metal arrays are identified. Limitations have been placed on any direct comparisons because of the relatively small size of clusters under examination and the non-planar characteristics. The use of molecular clusters in this way has led to a much clearer understanding of the bonding mode adopted by a variety of chemisorbed species and can provide answers to long-standing problems.     

Communication: Delayed asymmetric Coulomb fission of molecular clusters: application of a dielectric liquid-drop model

Delayed asymmetric Coulomb fission in size-selected molecular dication clusters has been recorded for the first time. Observations on (NH(3))(n)(2+) clusters show that fragmentation accompanied by charge separation can occur on a microsecond time scale, exhibits considerable asymmetry, and involves a kinetic energy release of ～0.9 eV. The fission process has been modeled by representing the fragments as charged dielectric spheres and the calculated maximum in the electrostatic interaction energy between the fragments gives a good account of the measured kinetic energy release. A simple kinetic model shows that instrumental factors may contribute to the observation of asymmetric fragmentation.     

Imaging nanometer-size metallic clusters with the atomic force microscope

The atomic force microscope has been used in the attractive (non-contact) force mode to produce images of individual nanometer-size clusters pre-formed in the gas phase and deposited on a wide variety of atomically-flat substrates. Using this technique, it is now possible to reliably image pre-formed clusters in their as-deposited positions. Studies of nanometer-size Au clusters supported on highly oriented pyrolitic graphite clearly show how the clusters are distributed across the scanned region. Cluster coverages inferred from atomic force studies are compared to those obtained from TEM studies of amorphous carbon grids simultaneously exposed to the same cluster beam.     

Distribution of interatomic distances in large metallic clusters

Spherically averaged pseudopotential (SAPS) calculations have been done for Mg _n clusters, withn up to 250 within the framework of density functional theory. The electronic structure is computed resorting to the Thomas-Fermi-Dirac-Weizsäcker (TFDW) approximation for the kinetic energy. The equilibrium geometries have been obtained by minimizing the total cluster energy with respect to the atomic positions using the steepest-descent method. The ground state geometries obtained in this way are formed by spherical atomic shells, the number of them increasing with cluster size, up to a number of four for the biggest sizes considered here. An analysis of the distribution of the interatomic distances shows that the more internal is the shell, the more contracted are the interatomic distances. This effect diminishes progressively with increasing cluster size. For the purpose of comparison, similar calculations have been done with Cs _n clusters in the same size range, allowing us to reproduce previous results obtained using a more elaborated density functional technique (Kohn-Sham method). The inhomogeneous contraction of interatomic distances then appears as a general fact for simple metallic clusters and not only for alkaline ones.     

Large fullerenes stabilized by encapsulation of metallic clusters

Structures are proposed for six endohedral metallofullerenes with large carbon cages (from C(92) to C(100)) on the basis of sizeable (LUMO-4)-(LUMO-3) gap and the formal transfer of six electrons to the cages.     

On the shell structure and geometry of monovalent metal clusters

The Hückel model is used to study the electronic structure of monovalent metal clusters. In an fcc cluster the Hückel model gives an estimate to the electronic structure of a free electron cluster. It is shown that the surface faceting of the fcc cluster can destroy the electronic shell structure already when the cluster has about 100 electrons. In the Hückel model the icosahedral structure has smaller total energy than the fcc structures, from which the Wulff construction has the smallest energy already when the cluster has 600 atoms.     

Fission of metal clusters: A comparison of jellium model calculations and shell correction method calculations

We investigate the fission process Ag²⁺₂₃ → Ag⁺₁₂ + Ag⁺₁₁ in order to compare total energies that calculated by the shell correction method and jellium models. A cavity potential and a Woods-Saxon-type potential are used as the shell potential for the shell correction method, with which the single-particle energy levels are calculated. Shell corrections are obtained by using the Balian-Bloch formula and by smoothing the discrete energy levels in the shell potentials. The jellium model calculations are performed in the framework of the local spin density functional approximation. The conventional jellium model and stabilized jellium model are used. Although the qualitative agreement between the shell correction method calculations and the stabilized jellium calculations is very good, an improvement of the liquid drop energy will be necessary for the satisfactory quantitative agreement. © 1996 John Wiley & Sons, Inc.     

Shell correction study of fission of doubly charged silver clusters

Fission of doubly charged silver clusters is investigated by the method of shell corrections. The following fission events are considered: Ag ₂₂ ²⁺ → Ag _n ⁺ + Ag _{22 ?n} ⁺ , (n=11, 10, 9, 8); Ag ₂₁ ²⁺ → Ag _n ⁺ + Ag _{21 ?n} ⁺ , (n=10, 9, 8, 7); Ag ₁₈ ²⁺ → Ag _n ⁺ + Ag _{18 ?n} ⁺ , (n=9, 8, 7, 6). It is found that the shell correction energy is comparable to or larger than the deformation energy of the liquid drop. Threshold energies for the fission events are calculated and compared with the experimental abundance spectra obtained by Katakuse et al. (1990). Correspondence between the calculated threshold energies with the shell corrections and the experimental abundance is very good, showing products from lower threshold fission channels yield more abundance. The threshold energies without the shell corrections are almost constant irrespective of the fission channels and cannot explain the experimental abundance. Abundance of some products are too small to be accounted for only by the threshold energies. The low abundance of those products may be explained by the presence of competing fission channels that have similar minimal energy paths. It is found in fission of Ag ₁₈ ²⁺ that the shell correction overwhelms the Coulomb energy and the fission channel to Ag₈ + Ag ₁₀ ²⁺ is preferred over the fission channel to Ag ₈ ⁺ + Ag ₁₀ ⁺ .     

Effects of surface roughness on the electronic structure of metallic clusters

We study the electronic level density in spherical clusters. Due to the granularity of the ionic background the surface is irregular at the microscopical level. We show that this affects the shell structure and that the level statistics display from the bottom to the top of the spectrum a transition from a poissonian behaviour to one consistent with the predictions of random matrices theory.     

On the electronic shell structure of metallic spheres and circular discs

The electronic shell structure of spherical and circular potential wells of finite depth is examined as an extension of the work of Balian and Bloch. We obtain analytical expressions for the electronic density of states, which are in good agreement with numerical results. The position of the maxima and minima in the density of states depends strongly on the depth of the potential via the quantummechanical scattering phases. For circular discs the oscillating structure in the density of states shows a different supershell pattern than for spheres. This is a consequence of the difference in the effective quantummechanical multiplicity of closed paths, which determine the oscillating structure.