Quasi-bound resonances in electron scattering by small metal clusters

A quantal calculation to evaluate theoretically the elastic cross section of electrons scattered by metal clusters is presented. The method is applied to the elastic dispersion of low energy electrons (up to 5 eV) from the spherical sodium clusters Na₈ and Na₂₀. Strong resonances in the total cross sections are found for some energies which are closely related to the existence of quasi-bound states, i.e. states completely bound in the classical limit.     

Quantum mechanical study on plasmon resonances in small ring clusters

The collectivity of the electronic motion in small sodium clusters with ring structure is studied by time‐dependent density functional theory. The formation and development of collective resonances in the absorption spectra were obtained as a function of the ring radius. In small ring clusters, besides the lower‐energy mode and the higher‐energy mode, there is another plasmon resonance mode, that is, the reverse two‐dipole mode. For the reverse two‐dipole mode, the formations of these two dipoles are due to the external field inducement and the shielding effect, although the resonant excitation is mainly due to the coupling effect of the electrons of these two dipoles. © 2012 Wiley Periodicals, Inc.     

The electronic structure of small sodium clusters

Ab initio molecular orbital calculations using the STO3-21G basis set has been carried out for the cluster series Na _n ⁺ , Na _n , and Na _n ^– (wheren=2–7). The basis set is shown to be reliable compared with more extensive basis sets at the Hartree-Fock level. Thirty-one optimized structures are reported and discussed, many of which (especially for the anions) have not been considered. The STO3-21G//STO3-21G calculations suggest that for most of the species the optimum geometries are planar. In particular, the optimized structures for the anionic species should provide a starting point for more sophisticated configuration interaction calculations.     

Optical response of small closed-shell sodium clusters

Absorption spectra of closed-shell Na(2), Na(3) (+), Na(4), Na(5) (+), Na(6), Na(7) (+), and Na(8) clusters are calculated using a complex Bethe-Salpeter equation derived using a conserving linear response method. In the framework of a quasiparticle approach, we determine electron-hole correlations in the presence of an external field. The calculated results are in excellent agreement with experimental spectra, and some possible cluster geometries that occur in experiments are analyzed. The position and the broadening of the resonances in the spectra arise from a consistent treatment of the scattering and dephasing contributions in the linear response calculation. Comparison between the experimental and the theoretical results yields information about the cluster geometry, which is not accessible experimentally.     

Isomerisation and phase transitions in small sodium clusters

Using a distance-dependent tight-binding hamiltonian, we have studied the influence of the temperature on the geometries of small alkali clusters (Na₄, Na₈, and Na₂₀). We have applied a Monte-Carlo thermodynamical method which consists in performing canonical samplings for various temperatures, these samplings being reexpressed in the microcanonical ensemble. This method provides thermodynamical values such as the entropy and the specific heat. Their behaviour shows one phase transition in the case of Na₄ and Na₈, and two phase transitions for Na₂₀. As concerns Na₄ and Na₈, the transition occurs at 200 K, between a solid-like phase and a phase for which the geometry of these clusters oscillates between numerous shapes. In the case of Na₂₀, the two observed phase transitions can be described as a melting of the surface atoms (at 200 K) preliminarily to the fluctuation of an inner icosahedron seed (at 300 K).     

First-principles investigation of finite-temperature behavior in small sodium clusters

A systematic and detailed investigation of the finite-temperature behavior of small sodium clusters, Na(n), in the size range of n=8-50 are carried out. The simulations are performed using density-functional molecular dynamics with ultrasoft pseudopotentials. A number of thermodynamic indicators such as specific heat, caloric curve, root-mean-square bond-length fluctuation, deviation energy, etc., are calculated for each of the clusters. Size dependence of these indicators reveals several interesting features. The smallest clusters with n=8 and 10 do not show any signature of melting transition. With the increase in size, broad peak in the specific heat is developed, which alternately for larger clusters evolves into a sharper one, indicating a solidlike to liquidlike transition. The melting temperatures show an irregular pattern similar to the experimentally observed one for larger clusters [Schmidt et al., Nature (London) 393, 238 (1998)]. The present calculations also reveal a remarkable size-sensitive effect in the size range of n=40-55. While Na(40) and Na(55) show well-developed peaks in the specific-heat curve, Na(50) cluster exhibits a rather broad peak, indicating a poorly defined melting transition. Such a feature has been experimentally observed for gallium and aluminum clusters [Breaux et al., J. Am. Chem. Soc. 126, 8628 (2004); Breaux et al., Phys. Rev. Lett. 94, 173401 (2005)].     

Shape resonances in molecular clusters: the 2t(2g) shape resonances in S 2p-excited sulfur hexafluoride clusters

Sulfur hexafluoride clusters are investigated in the S 2p excitation regime. For the first time special emphasis is put on high-energy resolution spectroscopy of shape resonances in free clusters. We have investigated the 2t(2g)-shape resonance occurring above the S 2p threshold as one typical example to study size effects that are related to shape resonances. A small redshift of this resonance of 30 +/- 5 meV occurs in clusters relative to the free molecule and changes in line shape are observed. A double-barrier optical potential model is applied for the analysis of intra- and intermolecular effects occurring in SF6 clusters, which is suitable for rationalizing the experimentally observed spectral changes. The experimental and theoretical results are briefly discussed in comparison to previous work on core-excited van der Waals clusters containing diatomic molecules.     

Spin dynamics in sodium clusters

We investigate the dynamics of the electron cloud in a metal cluster from the regime of small oscillations up to high excitations in the multi-plasmon regime. Particular attention is paid to the effect of spin modes. Test cases are a spin saturated cluster, Na₉ ⁺, and a strongly spin polarized cluster, the spherical isomer of Na₆ ⁺. The spectral distributions are dominated by the Mie plasmon resonance and a strong collective spin mode in all cases considered. Nonlinear effects as cross talk between multi-plasmon states or chaotic pattern cannot be observed in the investigated energy range.     

Transition of the electronic response from molecular-like to jellium-like in cold,small sodium clusters

Absolute photoabsorption cross sections for Na _n ⁺ (2≤n≤21) were measured in the visible energy range. The cluster ions were produced in a gas aggregation source and thus have a canonical distribution of internal energy corresponding to a temperature of ～ 105 K. The spectra for n≤9 and 11 exhibit between two and six absorption lines, and are in qualitative agreement with ab inito quantum chemical calculations. For n=15 and 21, the position of the resonances can be explained as excitations of a nearly free electron gas in a spheroidal container. An evolution is thus observed from molecular-like transitions to a giant collective resonance of the electron cloud. The integrated oscillator strength is 0.95 per 3s-electron in the energy range covered for n≥4, showing that the main excitations of the valence electrons have been found.     

Ionization of sodium in water clusters

Structures of Na(H₂O)_n and Na⁺(H₂O)_n clusters with n = 1?6, 19, and 28 are determined in the second order of the Møller-Plesset perturbation theory with the use of extended atomic basis set 6–31++G^**. It is found that when the number of molecules is sufficient for the formation of two solvation shells around sodium, a continuous hydrogen-bond network is formed in both neutral and charged clusters, and the orientation of each molecule is determined by the balance between interactions with the neighboring water molecules and with the field of the central particle. In the cations, this field is stronger, and up to the third solvation shell, molecules have a predominant orientation with respect to sodium. In the neutral clusters, with an increase in the number of water molecules, the maximum of the electron density distribution of the highest occupied molecular orbital becomes more distant from the sodium nucleus, being shifted toward the cluster surface. The energy of this orbital accordingly decreases in absolute value approaching 22 kcal/mol inmicroparticles. In the charged clusters, the distribution of the positive charge generally correlates with the character of the highest occupied orbital in the neutral systems, so that with an increase in the number of molecules, the atomic charge of sodium decreases and tends to zero as n → ∞. The ionization potential of sodium changes in inverse proportion to the linear size of the cluster, and should not exceed 1.1 eV in watermicroparticles.     

Response properties of sodium clusters within a jellium-like model with finite surface thickness

The usual spherical jellium model of alkaline clusters is generalized to allow for a finite surface thickness in the positive background distribution. Static (relative stability, ionization potentials) and response properties have been calculated and the results show a better agreement with experiments.     

Roughness induced surface acoustic resonances

The vibrational and electronic spectra of a semi-infinite crystal with a planar surface is modified in presence of surface inhomogeneities or roughness such as ridges or grooves, quantum wires or tips… Using a Green's function formalism, we present an exact numerical method for obtaining the variation of the density of states associated with the adsorption of a ridge on a flat surface or with a groove cut into an otherwise planar surface. This general method is applied to the determination of the acoustic resonances of shear horizontal polarization associated with such deterministic surface protuberances or indentations. The positions and widths of the peaks in the total or local densities of states give the frequencies and lifetimes of the resonances, which may be more or less pronounced features depending on the relative parameters of the substrate and ridge materials. We also investigate the modifications of these acoustic surface shape resonances due to the interaction between two such defects. This calculation can also be transposed to the study of electronic structure of a wire near a flat surface, in the framework of an effective mass model.     

Vibrational frequencies of sodium clusters

The vibrational frequencies of Na_N clusters (2 ? N ? 72) are calculated by direct diagonalization of the dynamical matrix. Density functional theory with a spherically averaged pseudopotential is used to compute the total energy. The geometry is optimized by the simulated annealing technique. Contributions to the Hessian matrix due to electron relaxation following the ionic displacements are calculated in linear response theory. The frequencies are in the range 0-220 cm^?1 and the electron relaxation strongly modifies those of the modes dominated by radial oscillations, particularly the breathing mode frequencies that are proportional to N?1/3 . The filling of atomic shells produces a stepwise behavior of the highest frequencies. The giant dipole resonance energies are obtained as a byproduct of the calculation. © 1995 John Wiley & Sons, Inc.     

Triaxial shapes of sodium clusters

A modified Nilsson-Clemenger model is combined with Strutinsky's shell correction method. For spherical clusters, the model potential is fitted to the single-particle spectra obtained from selfconsistent Kohn-Sham calculations. The deformation energy surfaces of sodium clusters with sizes of up toN=270 atoms are calculated for a combination of triaxial, quadrupole and hexadecapole deformations. The ground state shapes and energies are determined by simultaneous minimization with respect to the three shape parameters. A significant fraction of the clusters is predicted to be triaxial. The deviations from the axial shape do not generate any systematic odd-even staggering of the binding energies.     

Spectroscopy of size-selected sodium clusters

The method of angular resolved elastic scattering has been used to achieve a size selection of neutral sodium clusters. After the scattering process the clusters are ionized by single photon ionization with various laser energies and detected by a time-of-flight mass spectrometer. The elastic scattering of Na₂ by Ne serves as a test for the experimental setup. Angular dependence of scattered sodium clusters ionized with 308 nm shows a significant amount of fragmentation of the neutral Na₆ to Na₉ clusters which are observed in the Na ₅ ⁺ and Na ₆ ⁺ ion channels.     

Ring currents in polycyclic sodium clusters

In the recent work by Khatua et al. (Khatua, S.; Roy, D. R.; Bultinck, P.; Bhattacharjee, M.; Chattaraj, P. K. Phys. Chem. Chem. Phys.2008, 10, 2461-2474) the synthesis and structure of a fac-trioxo molybdenum metalloligand and its sodium complex containing 1D hexagonal chains of sodium ions was reported. In the same paper, the aromaticity of hexagonal Na clusters was quantified by means of the nucleus-independent chemical shift and electronic multicenter indices. It was shown that the aromaticity of hexagonal Na-clusters is of the same order as the aromaticity of analogous benzenoid hydrocarbons. In the present study current density maps are used to rationalize the aromaticity of polycyclic Na clusters. It is shown that although polycyclic Na systems sustain a diatropic ring current, the induced current density is several times weaker than in analogous benzenoid hydrocarbons. A detailed analysis indicates that the current density in hexagonal Na systems is almost completely determined by four HOMO σ electrons.     

Evaporation rates of hot sodium clusters

A scheme based in density functional theory with pseudopotentials is used to obtain the normal modes of vibration of Na _n clusters (4 ≤n ≤ 22). The monomer and dimer evaporation rates from thermally excited clusters are obtained in this harmonic approximation. The time evolution of the abundance spectra from an initial uniform mass distribution of hot clusters is studied and its influence in the experimentally observed spectra is discussed.