Metal clusters Hamiltonian model

In this review we shall first establish the theoretical foundations of the semi-classical theory that eventually lead to the formulation of the Breit-Pauli Hamiltonian. The latter is an approximation suited to make the connection to phenomenological model Hamiltonians like the Heisenberg Hamiltonian for the description of electronic spin-spin interactions. The complete derivations have been given in detail in Ref. (21), but turn out to be very involved and are thus scattered over many pages in Ref. (21). For this reason, we aim here at a summary that is as brief and concise as possible so that all relevant connections between different levels of approximation are evident. This allows us to connect present-day quantum chemical methods to phenomenological Hamiltonians and hence to establish and review the current status of these first-principles methods applied to transition-metal clusters. 

Clusters are studied in several forms. A study of the ionization energy and electron affinity of a metal cluster in the stabilized jellium model was recently performed by Sidl et al. [83]. A strictly variational procedure for cluster embedding, based on the extended subspace approach, has been presented by Gutdeutsch, Birkenheuer, and R6sch[84]. Initially used with the tight-binding model Hamiltonians, it has the potential to be extended to real Hamiltonians. 

In Section 2 of this paper a brief account of the 3D q -HO is given, while in Section 3 the appUcation of the periodic orbit theory of Balian and Bloch to metal clusters is briefly described. The predictions of the 3D g-HO model axe compared to the restrictions imposed by the theory of Balian and Bloch in Section 4, while in Section 5 a modified Hamiltonian for the 3D q-HO is introduced, allowing for full agreement with the theory of Balian and... 

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. 

This highly unorthodox model of the conducting solid cannot explain the conductivity, although it is not incompatible with it since the wave operator fl would build ionic components from the projected (neutral-only) wavefunc-tion. From the principles, and as shown by the previously mentioned success of Heisenberg Hamiltonians on the most metallic chemical systems (aromatic molecules), almost any system of which the lowest VB determinants are neutral may be treated either by the independent-particle approach followed by a treatment of electronic correlation which reduces the ionic compo-nents or by a Heisenberg-type effective Hamiltonian. This statement seems to be true whatever the p/U ratio. Malrieu et also noticed that the many-body effects (for instance four-body cyclic contributions), which are so important on small molecules and clusters, play a much less important role in the solid. 

We shall see in section 6.2 that this extremely simple model provides a qualitatively correct picture of high-energy spectra of light lanthanide solids. Similar approaches to excitation spectra of lanthanides have been introduced for clusters (Fujimori 1983, Fujimori and Weaver 1985) and have been considered for solids (Kotani et al. 1985). It must be emphazised, however, that the use of a few molecular orbitals in these simplified models of the Anderson impurity Hamiltonian leads to spectra where the excitations appear necessarily as discrete lines. This approach to a solid is missing irremediably the continuum aspects of the band states interacting with the f state. For example, one unrealistic consequence of a cluster model is the fact that the lowest excitation energies are typically of the order of the hybridization energy for both, metals and insulators. This implies zero specific heat and too small susceptibilities for metallic systems (Fujimori et al. 1984). Nevertheless, such models... 

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