Metallic clusters energy expression

Quantum chemical methods may be divided into two classes wave function-based techniques and functionals of the density and its derivatives. In the former, a simple Hamiltonian describes the interactions while a hierarchy of wave functions of increasing complexity is used to improve the calculation. With this approach it is in principle possible to come arbitrarily close to the correct solution, but at the expense of interpretability of the wave function the molecular orbital concept loses meaning for correlated wave functions. In DFT on the other hand, the complexity is built into the energy expression, rather than in the wave function which can still be written similar to a simple single-determinant Hartree-Fock wave function. We can thus still interpret our results in terms of a simple molecular orbital picture when using a cluster model of the metal substrate, i.e., the surface represented by a suitable number of metal atoms. 

It should be noted that the above TDLDA picture a priori involves two touchy approxmations. The first one consists in using the LDA which basically relies on the assumption of weakly varying (in space) electron density. This LDX approximation has been widely used in metal clusters arid does not raise problems with respect to the observables we arc interested in. The second approximation is to use in a dynamical context a functional which has been tuned to static problems. The extension of LDA to TDLDA is thus a further approximation which can he considered as adiabatic , in the sense that we are using, at each instant, the energy density as expressed... 

It was shown in Ref. [22] that the Q30 model correctly reproduces all magic numbers observed for alkali metal clusters up to N = 1500, using for the model parameters the values r = 0.038, A = 0.39. But there, these values were chosen more or less by trial and error, so as to fit the experimental data. With the renormalization of the energy expression described in the previous section, we now have a natural way for determining the deformation parameter r, which is connected with the cluster stability. 

This expression arises by adding in the metal-drop energy of Eq. (15), a term accounting for the coulomb energy of a charged cluster [70,74], and neglecting the curvature term. 

Several studies have focused on extensive MD simulations of Pt nanoparticles adsorbed on carbon in the presence or absence of ionomers [109-113]. Lamas and Balbuena performed classical molecular dynamics simulations on a simple model for the interface between graphite-supported Pt nanoparticles and hydrated Nation [113]. In MD studies of CLs, the equilibrium shape and structure of Pt clusters are usually simulated using the embedded atom method (EAM). Semi-empirical potentials such as the many-body Sutton-Chen potential (SC) [114] are popular choices for the close-packed metal clusters. Such potential models include the effect of the local electron density to account for many-body terms. The SC potential for Pt-Pt and Pt-C interactions provides a reasonable description of the properties of small Pt clusters. The potential energy in the SC potential is expressed by... 

Increasing the dose (the deposited energy into the irradiated medium, expressed in Grays, 1 Gy = 1 J.kg" ) leads to more reduced metal ions and then to more clusters with higher nuclearities. 

Examples of other polynuclear clusters that have more metals, dissimilar metals and/or higher spin metals will be of greater complexity than the simple example above, as they will have a larger number of energy levels to consider and it may be more laborious to specify the correct Van Vleck coefficients for each. The resulting expressions may then involve multiple g values, many exponential terms in the denominator and numerator and appear superficially complex however, the principles for the derivation of such equations are the same as those presented here. Those interested in the methods required to treat the general case are referred to the literature. ... 

---