Model generalized pseudopotential theory

Analytic derivatives have been reported for both the LSCF and GHO models, making them attractive options for MD simulations (Amara et al. 2000). Their generalization to ab initio levels of theory through the use of core pseudopotentials (along the lines of the pseudohalogen capping atoms described above) ensures that they will see continued development. 

In the case of two or more valence electrons we have to make a choice which is absent from the single-electron case we must choose a model for the electronic structure. We have to decide if we shall use a single determinant for the pseudo-wavefunction (using an obvious generalisation of the term pseudo-orbital) or a more accurate model containing electron correlation. Obviously the detailed form of the pseudopotential and of the pseudo-wavefu notion will depend on this choice of model and the development will become too complex to be useful. Let us make the opposite choice look at the formal equations independent of model and see if there are some general decisions to be made which will enable us to use the theory developed so far for a single electron. 

Main experimental findings both for the ground state (magic numbers for the stability of clusters [3] and the existence of supershells [4]) and for excited states (the dominance of collective states in the photoabsorption of metal clusters MeA with N > S) were predicted [5] before their experimental confirmation. Recently we were able to explain the temperature dependence of the absorption of small metal clusters as observed by Haber-land s group [6]. If the model is complemented by pseudopotential perturbation theory [7] the results obtained match qualitatively those obtained by demanding quantum-chemical methods (e.g. the photoabsorption spectra of Na6). Further improvement of the model includes the removal of self-interaction effects, the so-called SIC [8-10] (a consequence of using the local density approximation (LDA) to general density functional theory (DFT)). 

Because of the unphysical nature of a pseudopotential that depends on thermodynamic state, one would like an alternative description valid over the whole liquid range. One approach is to generalize the van der Waals mean field model by incorporating a pressure term of the form F" rather than the van der Waals V term. The expression for the cohesive energy in Eq. (3.28) is then U — —constant/F " with m > 1. An additional improvement of the simple van der Waals model is made by using hard sphere theory. The corresponding equation of state is then... 

To formalize these concepts, we first develop the frozen-core approximation, and then examine pseudoorbitals or pseudospinors and pseudopotentials. From this foundation we develop the theory for effective core potentials and ab initio model potentials. The fundamental theory is the same whether the Hamiltonian is relativistic or nonrela-tivistic the form of the one- and two-electron operators is not critical. Likewise with the orbitals the development here will be given in terms of spinors, which may be either nonrelativistic spin-orbitals or relativistic 2-spinors derived from a Foldy-Wouthuysen transformation. We will use the terminology pseudospinors because we wish to be as general as possible, but wherever this term is used, the term pseudoorbitals can be substituted. As in previous chapters, the indices p, q, r, and s will be used for any spinor, t, u, v, and w will be used for valence spinors, and a, b, c, and d will be used for virtual spinors. For core spinors we will use k, I, m, and n, and reserve i and j for electron indices and other summation indices. 

Unfortunately there are no simple theories to predict the cohesive energies of the metals like the coulomb attraction in ionic crystals. More sophisticated quantum mechanical theories using pseudopotential or other modeling techniques are generally required. There are some interesting correlations, however. 

As mentioned above, the results discussed below are obtained using Ab initio methods. Other methods used to study QDs are effective mass theory (EMT) and the pseudopotential techniques. EMT uses a particle-in-a-box model where the electron and hole masses are given by their bulk values. EMT is an intuitive description that explains general trends seen in experiments. The atomistic pseudopotential technique can be applied to large systems, but requires careful parameterization for each material. Ab initio approaches use minimal parameterization and are applicable to most materials. This makes them particularly useful for studying dopants, defects, ligands, core/shell systems and QD synthesis. The Hartree-Fock (HE) method and density functional theory (DFT) have been around for many decades, while time domain (TD) DFT and non-adiabatic molecular dynamics (NAMD) are more recent areas of research. Currently, ab initio TDDFT/NAMD is the only... 

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