Metal clusters spherical approximation

In this Section we want to present one of the fingerprints of noble-metal cluster formation, that is the development of a well-defined absorption band in the visible or near UV spectrum which is called the surface plasma resonance (SPR) absorption. SPR is typical of s-type metals like noble and alkali metals and it is due to a collective excitation of the delocalized conduction electrons confined within the cluster volume [15]. The theory developed by G. Mie in 1908 [22], for spherical non-interacting nanoparticles of radius R embedded in a non-absorbing medium with dielectric constant s i (i.e. with a refractive index n = Sm ) gives the extinction cross-section a(o),R) in the dipolar approximation as ... 

The linear photoresponse of metal clusters was successfully calculated for spherical [158-160, 163] as well as for spheroidal clusters [164] within the jellium model [188] using the LDA. The results are improved considerably by the use of self-interaction corrected functionals. In the context of response calculations, self-interaction effects occur at three different levels First of all, the static KS orbitals, which enter the response function, have a self-interaction error if calculated within LDA. This is because the LDA xc potential of finite systems shows an exponential rather than the correct — 1/r behaviour in the asymptotic region. As a consequence, the valence electrons of finite systems are too weakly bound and the effective (ground-state) potential does not support high-lying unoccupied states. Apart from the response function Xs, the xc kernel /xc[ o] no matter which approximation is used for it, also has a self-interaction error. This is because /ic[no] is evaluated at the unperturbed ground-state density no(r), and this density exhibits self-interaction errors if the KS orbitals were calculated in LDA. Finally the ALDA form of /,c itself carries another self-interaction error. 

One conceptually simple approach which has been used to represent temperature effects in metallic clusters is the random matrix model, developed by Akulin et al. [700]. The principles of the random matrix model, developed in the context of nuclear physics by Wigner and others, were outlined in chapter 10. The essential idea is to treat the cluster as a disordered piece of a solid. In the first approximation, the cluster is regarded as a Fermi gas of electrons, moving in an effective, spherically symmetric short range well. Without deformations, one-electron states then obey a Fermi distribution. As the temperature is raised, various scattering processes and perturbations arise, all of which lead to a random coupling between the states of the unperturbed system. One can... 

Within the jellium model for metal clusters [95,53] as described in the introduction, the positive background potential is in a first approximation normally chosen as a spherical shape of the following form... 

A square confining well is necessarily a simplistic approximation. Other potentials which have been used for metallic clusters include a spherical... 

In this work we start with the primitive jellium model, as appropriate for alkaline metals. In the jellium model for metal clusters a fundamental input is the size-dependent ionic density. Fortunately, when one of us started this calculation in 1984 [3], some experimental data about the size dependence of the nearest-neighbor distance were available from EXAFS (extended X-ray absorption fine structure) measurements [19]. Except for fine details the size dependence is very weak. This means that in a first approximation the bulk density of the metal can be used as input for a cluster calculation. A second question is the size dependence of the shape. Since electron micrographs very often show a spherical shape, at least for the larger clusters, a spherical shape will be assumed for all cluster sizes. This means that for monovalent systems the radius R of the jellium cluster is determined by its bulk density... 

Sharp drops after certain sizes in the abundance spectrum indicate enhanced stability of these clusters compared to neighboring sizes. We will try to understand this phenomenon from the behavior of valence electrons in the clusters by invoking simple quantum mechanical models. The simplest model one uses for valence electrons inside a bulk metal is the free-electron theory valence electrons of all the atoms are free to move over the entire volume occupied by the solid [11]. One can use a similar free electron model in case of metal clusters. As the simplest approximation, shape of the cluster can be taken as spherical, and the electrons strictly confined within the sphere. In this hard sphere model, the Schrbdinger equation describing the valence electrons is... 

As will be shown, model systems for cells employing lipids or composed of polymers have been in existence for some time. Model systems for coccolith-type structures are well known on the nanoscale in inorganic and materials chemistry. Indeed, many complex metal oxides crystallize into approximations of spherical networks. Often, though, the spherical motif interpenetrates other spheres making the formation of discrete spheres rare. Inorganic clusters such as quantum dots may appear as microscopic spheres, particularly when visualized by scanning electron microscopy, but they are not hollow, nor do they contain voids that would be of value as sites for molecular recognition. All these examples have the outward appearance of cells but not all function as capsules for host molecules. 

In the earliest implementation applied to molecular problems, K. Johnson [39] used scattered-plane waves as a basis and the exchange-correlation energy was represented by (13). This SW-Xa method employed in addition an (muffin-tin) approximation to the Coulomb potential of (17) in which Vc is replaced by a sum of spherical potentials around each atom. This approximation is well suited for solids for which the SW-Xa method originally was developed [40]. However, it is less appropriate in molecules where the potential around each atom might be far from spherical. The SW-Xa method is computationally expedient compared to standard ab initio techniques and has been used with considerable success [41] to elucidate the electronic structure in complexes and clusters of transition metals. However, the use of the muffin-tin approximation precludes accurate calculations of total energies. The method has for this reason not been successful in studies involving molecular structures and bond energies [42]. 

One can develop a particularly simple scheme by using the assumption of spherical symmetry together with the jellium model of solid state or nuclear physics to compute the effective potential for clusters of different sizes. In this model, the electrons are treated as free particles by analogy with the conduction band of the solid and the ionic structure within the cluster is completely neglected. This obviously results in a great simplification of the problem, especially if the system is spherical, and might be thought too drastic an approximation. In fact, the jellium model only applies to a specific class of clusters (which we call metallic), but was of enormous importance to the history of the field as it revolutionised cluster physics. 

In terms of the quantum-well picture, a small particle of, e.g., an alkali metal, can be regarded in many respects as a giant atom (or molecule). The electrons are confined by the outer surface of the particle, which presents an approximately spherical potential, similar therefore to the spherically symmetric Coulombic potential in the atom arising from the electron-nucleus electrostatic interaction. Thus, the building-up principle of electrons in such a cluster is quite similar to that underlying the periodic system of the elements, with the characteristic shell-structure for the electrons. Indeed, large differences in reactivity have been observed for clusters with filled or unfilled electron shells An attractive feature of clusters in this respect is, evidently, that the number of electrons (atoms) per cluster can surpass by orders of magnitude the number of elements in the periodic system. 

A schematic representation of the emerging discretization of electronic energy levels with decreasing metallic particle size is shown in Fig. 2. The cluster size appropriate for a SIMIT can thus be readily calculated. Assuming the cluster or particle is approximately spherical, it can be shown that the diameter, D, of a particle containing N atoms is given by ... 

It appears that the authors assumed 100% reduction of cobalt, which may lead to an overestimation of the cluster size and consequently, an underestimation of the true metal dispersion. Assuming that 100% of the cobalt was reduced, a real dispersion of 5.8% would result in an average cobalt cluster size, assuming a spherical morphology, of approximately 18 nm. Since a lower extent of reduction would only increase the true dispersion of the metal, we believe it is safe to presume that the cobalt clusters were, on average, small enough to fit within the pores of the silica support studied. 

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