Hamiltonian isolated cluster

Here, A ii is the energy of JT stabilization connected with odd cluster vibrations q the fifth term in the Hamiltonian, (27), describes the intercluster interaction through the phonon field. The Hamiltonian of an isolated cluster Hc C = A, B) after transformation contains phonon variables, because the odd coordinate q mixes the exchange-resonance multiplets with the same spin. Then, the crystal is subdivided into two equivalent sublattices A and 5 the clusters of these sublattices are numbered by letters n and m. The interaction between the nearest neighbors is only taken into account the interaction Hamiltonian is the following ... 

Summarising, the valence-only spin-orbit CGWB-AIMP embedded cluster Hamiltonian, which results from choosing the CGWB-AIMP for the isolated cluster, reads ... 

In the canonical or constant number of electrons mode, an isolated cluster or slab is placed in an external electric field. At the level of the Schrodinger equation, the electric field is simply added as an extra term to the Hamiltonian. As a result of the external field, the cluster/slab becomes polarized, with a positive surface charge on one... 

The isolated cluster Hamiltonian can be either the non-relativistic many-electron Hamiltonian or a suitable relativistic choice, both in their all-electron versions or in any effective core potential version. We will discuss later. The embedding potential acting on the cluster electrons reads ... 

Our choices of relativistic isolated cluster Hamiltonian allow us to write the embedded-cluster Hamiltonian of equation 9.1 as a sum of spin-free and spin-orbit coupling Hamiltonians,... 

Up to this point the paramagnetic species have been considered as isolated entities on the surface or in the bulk phase. It is clear, however, that many supported catalysts are actually clusters of ions. If these ions are paramagnetic, such as Cr203 at temperatures above about 30°, then another term must be included in the spin Hamiltonian to account for the exchange energy. This term is written as... 

Suppose we have an isolated system with a single unpaired electron and no hyper-fine interaction. Mononuclear low-spin Fe111 and many iron-sulfur clusters fall in this category (cf. Table 4.2). The only relevant interaction is the electronic Zeeman term, so the spin Hamiltonian is... 

Hmi is the i-th cluster hamiltonian referred to its space fixed frame with axes parallel to the laboratory frame. Kcmi is the i-th cluster kinetic energy operator with total mass Mj. The existence of electronic wave functions for each system is taken for granted as their spectra can be determined in isolation. 

Investigations of the linear and nonlinear optical properties of molecules, polymers, and clusters generally adopt the semi-classical approach. In this approach, the particles are treated quantum mechanically while a classical treatment is applied to the radiation so that the Hamiltonian is written as the sum of two types of terms, one representing the isolated system (Hq) and one being the radiation-molecule interaction term (Hi). For sufficiently large wavelengths with respect to the system dimensions. Hi can be expressed under the form of a multipole expansion ... 

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