Jellium Model Research Articles

Optical response of C60 fullerene from a time dependent Thomas Fermi approach

We study the collective electron dynamics in C60 clusters within the time dependent Thomas Fermi method in the frame of jellium model. The results regarding the optical spectrum are in good agreement with the experimental data, our simulations being able to reproduce both resonances from and . We compare also, the results with those from other theoretical approaches and investigate the implications of quantum effects including exchange-correlation corrections, or gradient corrections from a Weizsacker term. The nature of the second resonance is studied using transition densities.

First-principles study on stability and magnetism of MAln (n = 2–12; M = Cr, Mn, Fe, Co, Ni, Cu, Zn) clusters

By using the Amsterdam Density Functional (ADF) with TZ2P basis set in conjunction with self-consistent-field (SCF), we have studied the geometric features, stabilities and magnetic properties of MAln (M=Cr, Mn, Fe, Co, Ni, Cu, Zn; n=2–12) clusters. All the calculations have been made in the framework of the generalized gradient approximation (GGA) by means of the Becke–Perdew functional. The pronounced stabilities and magnetism of some 3d metal atom doped Al clusters can be explained by the spherical jellium model (SJM). From atom Mn to Zn, only the MAln (M=Cr, Mn, Fe; n=2–12) clusters exhibit the large magnetic moments, especially for the MnAln (n=2–12) clusters. And for the other MAln (M=Co, Ni, Cu, Zn; n=2–12) clusters, only a few clusters has a total magnetic moment of less than or equal to 3μB.

A correction for the Hartree-Fock density of states for jellium without screening.

We revisit the Hartree-Fock (HF) calculation for the uniform electron gas, or jellium model, whose predictions--divergent derivative of the energy dispersion relation and vanishing density of states (DOS) at the Fermi level--are in qualitative disagreement with experimental evidence for simple metals. Currently, this qualitative failure is attributed to the lack of screening in the HF equations. Employing Slater's hyper-Hartree-Fock (HHF) equations, derived variationally, to study the ground state and the excited states of jellium, we find that the divergent derivative of the energy dispersion relation and the zero in the DOS are still present, but shifted from the Fermi wavevector and energy of jellium to the boundary between the set of variationally optimised and unoptimised HHF orbitals. The location of this boundary is not fixed, but it can be chosen to lie at arbitrarily high values of wavevector and energy, well clear from the Fermi level of jellium. We conclude that, rather than the lack of screening in the HF equations, the well-known qualitative failure of the ground-state HF approximation is an artifact of its nonlocal exchange operator. Other similar artifacts of the HF nonlocal exchange operator, not associated with the lack of electronic correlation, are known in the literature.

Ag21{S2P(OiPr)2}12]+: an eight-electron superatom.

A novel discrete [Ag21{S2P(OiPr)2}12](PF6) nanocluster has been synthesized and characterized by single-crystal X-ray diffraction and also NMR spectroscopy ((1)H, (31)P), ESI mass spectrometry, and other analytic techniques (XPS, EDS, UV/Vis spectroscopy). The Ag21 skeleton has an unprecedented silver-centered icosahedron that is capped by eight additional metal atoms. The whole framework is protected by twelve dithiophosphate ligands. According to the spherical Jellium model, the stability of monocationic nanocluster can be described by an 8-electron superatom with 1S(2) 1P(6) configuration, as confirmed by DFT calculations.

Total energy equation leading to exchange-correlation functional

By solving the total energy equation, we obtain the formula of exchange-correlation functional for the first time. This functional is usually determined by fitting experimental data or the numerical results of models. In the uniform electron gas limit, our exchange-correlation functional can exactly reproduce the results of Perdew-Zunger parameterization from the jellium model. By making use of a particular solution, our exchange-correlation functional could take into account the case of non-uniform electron density, and its validity can be confirmed through comparisons of the band structure, equilibrium lattice constant, and bulk modulus of aluminum and silicon. The absence of mechanical prescriptions for the systematic improvement of exchange-correlation functional hinders further development of density-functional theory (DFT), and the formula of exchange-correlation functional given in this study might provide a new perspective to help DFT out of this awkward situation.

Current-induced spin polarization at the surface of metallic films: A theorem and anab initiocalculation

The broken inversion symmetry at the surface of a metallic film (or, more generally, at the interface between a metallic film and a different metallic or insulating material) greatly amplifies the influence of the spin-orbit interaction on the surface properties. The best known manifestation of this effect is the momentum-dependent splitting of the surface state energies (Rashba effect). Here we show that the same interaction also generates a spin-polarization of the bulk states when an electric current is driven through the bulk of the film. For a semi-infinite jellium model, which is representative of metals with a closed Fermi surface, we prove as a theorem that, regardless of the shape of the confinement potential, the induced surface spin density at each surface is given by ${\bf S} =-\gamma \hbar {\bf \hat z}\times {\bf j}$, where ${\bf j}$ is the particle current density in the bulk, ${\bf \hat z}$ the unit vector normal to the surface, and $\gamma=\frac{\hbar}{4mc^2}$ contains only fundamental constants. For a general metallic solid $\gamma$ becomes a material-specific parameter that controls the strength of the interfacial spin-orbit coupling. Our theorem, combined with an {\it ab initio} calculation of the spin polarization of the current-carrying film, enables a determination of $\gamma$, which should be useful in modeling the spin-dependent scattering of quasiparticles at the interface.

Shape-Dependent Energy of an Elliptical Jellium Background

The jellium model is commonly used in condensed matter physics to study the properties of a two-dimensional electron gas system. Within this approximation, one assumes that electrons move in the presence of a neutralizing background consisting of uniformly spread positive charge. When properties of bulk systems (of infinite size) are studied, shape of the jellium domain is irrelevant. However, the same cannot be said when one is dealing with finite systems of electrons confined in a finite two-dimensional region of space. In such a case, geometry and shape of the jellium background play a role on the overall properties of the system. In this work, we assume that the region where the electrons are confined is represented by a jellium background charge with an elliptical shape. It is shown that, in this case, the Coulomb self-energy of the elliptically shaped region can be exactly calculated in closed analytical form by using suitable mathematical transformations. The results obtained reveal the external influence of geometry/shape on the properties of two-dimensional systems of few electrons confined to a small finite region of space.

Momentum-dependent lifetime broadening of electron energy loss spectra: Sum rule constraints and an asymmetric rectangle toy model

Electron energy loss spectra enable a detailed quantification of the electronic loss mechanisms in a target solid, particularly in the low-energy region dominated by plasmon excitations. Models of the electronic response in condensed-matter systems are usually derived from free-electron gas or jellium models, which commonly neglect to account for the lifetime broadening of individual plasmon and single-electron excitations in a constrained, physical manner. This can lead to potentially significant errors in electron energy loss spectra and electron inelastic mean-free-path (IMFP) calculations. We develop a toy model of plasmon and single-electron excitations that incorporates lifetime broadening for each excitation in an energy- and momentum-dependent fashion. The model is physically constrained using optical and electronic sum rules. We demonstrate the necessity of asymmetric excitation broadening, and show that causally permitted variations in the broadening function can have a significant impact on the dielectric response of the material. Our developments are applied to molybdenum, and compared with previous modeling and high-precision experimental results for the IMFP at electron energies below 120 eV.

Metallic Clusters: Theoretical Background, Properties and Synthesis in Microemulsions

Reducing the size from the bulk material to nanoparticles produces a scaling behavior in physical properties in the later ones, due to the large surface-to-volume fraction. By further size reduction, entering into the subnanometric cluster region, physical properties are largely affected by strong quantum confinement. These quantum size effects (HOMO-LUMO gap), the small size and the specific geometry award subnanometric clusters with totally new and fascinating properties, including cluster photoluminescence, enhanced catalytic activity, etc. In this review, we report an introduction to the physical properties of clusters based on the jellium model; the controlled synthesis by microemulsion methods and the catalytic properties in different areas as heterogeneous catalysis, photocatalysis or electrocatalysis among others.

Ab initionanoplasmonics: The impact of atomic structure

We present an ab initio study of the hybridization of localized surface plasmons in a metal nanoparticle dimer. The atomic structure, which is often neglected in theoretical studies of quantum nanoplasmonics, has a strong impact on the optical absorption properties when subnanometric gaps between the nanoparticles are considered. We demonstrate that this influences the hybridization of optical resonances of the dimer, and leads to significantly smaller electric field enhancements as compared to the standard jellium model. In addition, we show that the corrugation of the metal surface at a microscopic scale becomes as important as other well-known quantum corrections to the plasmonic response, implying that the atomic structure has to be taken into account to obtain quantitative predictions for realistic nanoplasmonic devices.

Special and general superatoms.

Bridging the gap between atoms and macroscopic matter, clusters continue to be a subject of increasing research interest. Among the realm of cluster investigations, an exciting development is the realization that chosen stable clusters can mimic the chemical behavior of an atom or a group of the periodic table of elements. This major finding known as a superatom concept was originated experimentally from the study of aluminum cluster reactivity conducted in 1989 by noting a dramatic size dependence of the reactivity where cluster anions containing a certain number of Al atoms were unreactive toward oxygen while the other species were etched away. This observation was well interpreted by shell closings on the basis of the jellium model, and the related concept (originally termed "unified atom") spawned a wide range of pioneering studies in the 1990s pertaining to the understanding of factors governing the properties of clusters. Under the inspiration of a superatom concept, advances in cluster science in finding stable species not only shed light on magic clusters (i.e., superatomic noble gas) but also enlightened the exploration of stable clusters to mimic the chemical behavior of atoms leading to the discovery of superhalogens, alkaline-earth metals, superalkalis, etc. Among them, certain clusters could enable isovalent isomorphism of precious metals, indicating application potential for inexpensive superatoms for industrial catalysis, while a few superalkalis were found to validate the interesting "harpoon mechanism" involved in the superatomic cluster reactivity; recently also found were the magnetic superatoms of which the cluster-assembled materials could be used in spin electronics. Up to now, extensive studies in cluster science have allowed the stability of superatomic clusters to be understood within a few models, including the jellium model, also aromaticity and Wade-Mingos rules depending on the geometry and metallicity of the cluster. However, the scope of application of the jellium model and modification of the theory to account for nonspherical symmetry and nonmetal-doped metal clusters are still illusive to be further developed. It is still worth mentioning that a superatom concept has also been introduced in ligand-stabilized metal clusters which could also follow the major shell-closing electron count for a spherical, square-well potential. By proposing a new concept named as special and general superatoms, herein we try to summarize all these investigations in series, expecting to provide an overview of this field with a primary focus on the joint undertakings which have given rise to the superatom concept. To be specific, for special superatoms, we limit to clusters under a strict jellium model and simply classify them into groups based on their valence electron counts. While for general superatoms we emphasize on nonmetal-doped metal clusters and ligand-stabilized metal clusters, as well as a few isovalent cluster systems. Hopefully this summary of special and general superatoms benefits the further development of cluster-related theory, and lights up the prospect of using them as building blocks of new materials with tailored properties, such as inexpensive isovalent systems for industrial catalysis, semiconductive superatoms for transistors, and magnetic superatoms for spin electronics.

On the positron in a metal

The self-consistent calculations of the spatial distributions of electrons and potentials in vacancies of metals with a localized positron, the spatial distributions of positrons localized in surface states, and the binding energies of positrons and their lifetimes have been performed in terms of the Kohn-Sham method and the stabilized jellium model. The presence of a localized positron in a vacancy leads to the effect that the vacancy is weakly distinguishable for electron waves: the positron weakens the potential field in the vicinity of the vacancy and leads to a phase shift of the scattering electron wave functions. The calculation of the phase shifts of the wave functions for quasi-free positrons scattered by unperturbed vacancies and the representation of a system of vacancies as a “superlattice” in a metal have made it possible to find the shift of the positron work function and the vacancy contribution to the positron effective mass.

Ions in liquid metal clusters

From the results of computer simulations, especially of 13-atom metal clusters by molecular dynamics, we derive a theory of the solid–liquid phase transition in metal clusters that connects to the jellium model. We analyze the nature of the ion’s behavior in liquid clusters. As a result, we find a dynamical coexistence among the cluster’s configurational states create the liquid aggregate state of this cluster. It follows from the entropy analysis that motions of the ions in liquid metal clusters explore a selective set of ion configurations, rather than following independent motions in a self-consistent cluster field.

Doped aluminum cluster anions: size matters.

The global minima of the cluster anions with the generic chemical formula (XAl₁₂)²⁻, where X = Be, Mg, Ca, Sr, Ba, and Zn, are determined by an extensive search of their potential energy surfaces using the Gradient Embedded Genetic Algorithm (GEGA). All the characterized global minima have an icosahedral-like structure, resembling that of the Al₁₃⁻ cluster. These cages comprise closed-shell electronic configurations with 40 electrons, therefore, in accordance to the jellium model, they are predicted to be highly stable and amenable to experimental detection. The two preferred sites for the dopant species, at the center and at surface of the icosahedral cage, are stabilized depending on the atomic radius of X. Thus, while the small dopants (X = Be, Zn) sit preferably at the center of the cage, the preferred site for X = Mg, Ca, Sr, and Ba is at the surface. Since these dianions are not stable towards electron detachment, one Li cation is added in order to yield stable systems. Our computations show that in the global minimum form of Li(XAl₁₂)⁻, the lithium cation, ionically bonded to the Al atoms, does not change the structure of the (XAl12)²⁻ core.

Superoxide formation on isolated cationic gold clusters.

Gold nanoparticles are known to be highly versatile oxidation catalysts utilizing molecular oxygen as a feedstock, but the mechanism and species responsible for activating oxygen remain unclear. The reaction between unsupported cationic gold clusters and molecular oxygen has been investigated. The resulting complexes were characterized in the gas phase using IR spectroscopy. A strong red-shift in the observed ν(O-O) stretching frequency indicates the formation of superoxo (O2(-)) moieties. These moieties are seen to form spontaneously in systems, which upon electron transfer attain a closed shell within the spherical jellium model (Au10(+) and Au22(+)), whereas an oxygen induced self-promotion in the activation is observed for other systems (Au4(+), Au12(+), Au21(+)).

Truly self-consistent solution of Kohn-Sham equations for extended systems with inhomogeneous electron gas

The density functional approach in the Kohn-Sham approximation is widely used to study properties of many-electron systems. Due to the nonlinearity of the Kohn-Sham equations, the general self-consistent solution method for infinite systems involves iterations with alternate solutions of the Poisson and Schrödinger equations. One of problems with such an approach is that the charge distribution, updated by solving the Schrodinger equation, may be incompatible with the boundary conditions of the Poisson equation for Coulomb potential. The resulting instability or divergence manifests itself most appreciably in the case of infinitely extended systems because the corresponding boundary-value problem becomes singular. In this work the stable iterative scheme for solving the Kohn-Sham equations for infinite systems with inhomogeneous electron gas is described based on eliminating the long-range character of the Coulomb interaction, which causes the tight coupling of the charge distribution with the boundary conditions. This algorithm has been previously successfully implemented in the calculation of work function and surface energy of simple metals in the jellium model. Here it is used to calculate the energy spectrum of quasi-two-dimensional electron gas in the accumulation layer at the semiconductor surface n-InAs. The electrons in such a structure occupy states that belong to both discrete and continuous parts of the energy spectrum. This causes the problems of convergence in the usually used approaches, which do not exist in our case. Because of the narrow bandgap of InAs, it is necessary to take the nonparabolicity of the conduction band into account; this is done by means of a new effective mass method. The calculated quasi-two-dimensional energy bands correspond well to experimental data measured by the angle resolved photoelectron spectroscopy technique.

Linear response of a variational average atom in plasma: Semi-classical model

The frequency-dependent linear response of a plasma is studied in the finite-temperature Thomas–Fermi approximation, with electron dynamics described using Bloch hydrodynamics. The variational framework of average-atoms in a plasma is used. Extinction cross-sections are calculated for several plasma conditions. Comparisons with a previously studied Thomas–Fermi Impurity in Jellium model are presented. An Ehrenfest-type sum rule, originally proposed in a full quantum approach is derived in the present formalism and checked numerically. This sum rule is used to define Bremsstrahlung and collective contributions to the extinction cross-section. It is shown that none of these is negligible. Each can constitute the main contribution to the cross-section, depending on the frequency region and plasma conditions. This result obtained in the Thomas–Fermi–Bloch case stresses the importance of the self-consistent approach to the linear response in general. Some of the methods used in this study can be extended to the linear response in the quantum case.

On the vacancy in a metal

The self-consistent calculations of spatial distributions of electrons and potentials in an isolated spherical cavity (with atomic radius RWS) of the Wigner-Seitz cell, the energy of the vacancy formation, and the vacancy contribution to the electrical resistance have been performed in terms of the modified Kohn-Sham method and the stabilized jellium model. The energy shift of the ground state of electrons in the Wigner-Seitz cell with radius Rv ≫ RWS (where Rv is the average distance between vacancies), the contribution of vacancies to the electron work function, and the effective mass of electrons have been found using the calculated phase shifts of the scattering wave functions and the representation of a system of vacancies as a “superlattice” in a metal.

A Novel Concept for the Synthesis of Multiply Doped Gold Clusters [(M@AunM′m)Lk]q+

Heterometal-doped gold clusters are poorly accessible through wet-chemical approaches and main-group-metal- or early-transition-metal-doped gold clusters are rare. Compounds [M(AuPMe3 )11 (AuCl)](3+) (M=Pt, Pd, Ni) (1-3), [Ni(AuPPh3 )(8-2n) (AuCl)3 (AlCp*)n ] (n=1, 2) (4, 5), and [Mo(AuPMe3 )8 (GaCl2 )3 (GaCl)](+) (6) were selectively obtained by the transmetalation of [M(M'Cp*)n ] (M=Mo, E=Ga, n=6; M=Pt, Pd, Ni, M'=Ga, Al, n=4) with [ClAuPR3 ] (R=Me, Ph) and characterized by single-crystal X-ray diffraction and ESI mass spectrometry. DFT calculations were used to analyze the bonding situation. The transmetalation proved to be a powerful tool for the synthesis of heterometal-doped gold clusters with a design rule based on the 18 valence electron count for the central metal atom M and in agreement with the unified superatom concept based on the jellium model.

On the Work Function and Schottky Barrier Heights of Metal Nanofilms in a Dielectric Environment

We suggest a method for self-consistent calculations of the characteristics of metal films in a dielectric environment. Within a modified Kohn–Sham method and the stabilized jellium model, the most interesting case of asymmetric metal-dielectric sandwiches is considered, for which the dielectric media are different on the two sides of the film. We calculate the spectrum, electron work function, and surface energy of polycrystalline and crystalline films of Na, Al, and Pb placed into passive isolators. It is found that a dielectric environment generally leads to a decrease of both the electron work function and the surface energy. It is revealed that the change of the work function is determined only by the average of dielectric constants on both sides of the film. We introduced the position of a conduction band in the dielectric as a parameter in the self-consistency procedure. The calculations with the use of the image potential for an aluminum film with ideal interfaces vacuum/Al(111)/SiO2 and vacuum/Al(111)/Al2O3 and the sandwich SiO2/Al(111)/Al2O3 are performed. As a result, the effective potential profiles and the Schottky barrier heights are calculated.