Electron lattice models

We have assumed so far, implicitly, that the interactions are strictly local between neighboring atoms and that long-ranged forces are unimportant. Of course the atom-atom interaction is based on quantum mechanics and is mediated by the electron as a Fermi particle. Therefore the assumption of short-range interaction is in principle a simplification. For many relevant questions on crystal growth it turns out to be a good and reasonable approximation but nevertheless it is not always permissible. For example, the surface of a crystal shows a superstructure which cannot be explained with our simple lattice models. 

The SSH model (Eq. (3.2)) is, essentially, the model used by Peierls for his discussion of the electron-lattice instability [33]. Its ground state is characterized by a non-zero expectation value of the operator. 

We discuss a lattice model where spin-up, spin-down electrons move on a one-dimensional lattice A of size A = r, so that dim = 2 . An annihilator for a spin-up electron on lattice site e A is denoted by a, and that for a spin-down electron by b. An arbitrary operator on can be written as a polynomial in the 2r annihilation and 2r creation operators. 

The sp-valent metals such as sodium, magnesium and aluminium constitute the simplest form of condensed matter. They are archetypal of the textbook metallic bond in which the outer shell of electrons form a gas of free particles that are only very weakly perturbed by the underlying ionic lattice. The classical free-electron gas model of Drude accounted very well for the electrical and thermal conductivities of metals, linking their ratio in the very simple form of the Wiedemann-Franz law. However, we shall now see that a proper quantum mechanical treatment is required in order to explain not only the binding properties of a free-electron gas at zero temperature but also the observed linear temperature dependence of its heat capacity. According to classical mechanics the heat capacity should be temperature-independent, taking the constant value of kB per free particle. 

Electron in the selftrapping dominated region is trapped by the phonons-1 but due to the interactions mediated by phonons-2 the electron can fluctuate to the higher level. Due to the reflection symmetry of the phonons-2 continuum oscillations of the electrons at simultaneous emission and absorption of phonons-1 occurs. These oscillations couple the levels and so the electrons into pairs. This mechanism was described in a recent paper [10] for a lattice model. 

While the electronic structure calculations addressed in the preceding Section could in principle be used to construct the potential surfaces that are a prerequisite for dynamical calculations, such a procedure is in practice out of reach for large, extended systems like polymer junctions. At most, semiempirical calculations can be carried out as a function of selected relevant coordinates, see, e.g., the recent analysis of Ref. [44]. To proceed, we therefore resort to a different strategy, by constructing a suitably parametrized electron-phonon Hamiltonian model. This electron-phonon Hamiltonian underlies the two- and three-state diabatic models that are employed below (Secs. 4 and 5). The key ingredients are a lattice model formulated in the basis of localized Wannier functions and localized phonon modes (Sec. 3.1) and the construction of an associated diabatic Hamiltonian in a normal-mode representation (Sec. 3.2) [61]. 

Regarding the parametrization of the Hamiltonian Eq. (7), the present approach relies on the parameters of the underlying lattice model Eq. (5). However, one could envisage an alternative approach, similar to the one described in Refs. [66-69] for small molecular systems, where a systematic diabatiza-tion is carried out based on supermolecular electronic structure calculations as described in Sec. 2.2. 

Fig. 11 is a drawing of a two-dimensional analogue of the electron-domain model of ethane. Large circles represent valence-shell electron-domains (superimposed on them are the valence strokes of classical structural theory). Plus signs represent protons of the "C—H bonds. The nuclei of the two carbon atoms are represented by small dots in the trigonal interstices of the electron-pair lattice. While these nuclei would not necessarily be in the centers of their interstices, exactly, it can be asserted that an (alchemical) insertion of the two protons on the... 

The electron gas model adequately describes the conduction of electrons in metals however, it has a problem, that is, the electrons with energy near the Fermi level have wavelength values comparable to the lattice parameters of the crystal. Consequently, strong diffraction effects must be present (see below the diffraction condition (Equation 1.47). A more realistic description of the state of the electrons inside solids is necessary. This more accurate description is carried out with the help of the Bloch and Wilson band model [18],... 

We attempt below to put the results in the context of a phase separation [4]. The decomposition of l/63l/ i(T, x) into two terms, as it will be discussed below in more details, manifests itself in a broad temperature interval above Tc. It is limited from above by a T that depends on the concentration, x. We consider T defined in this way as a temperature of a 1st order phase transition, which, however, cannot complete itself in spatial coexistence of two phases because of the electroneutrality condition [5]. It was already argued in [4] that such a frustrated 1st order phase transition may actually bear a dynamical character. The fact that a single resonant frequency for the 63Cu nuclear spin is observed in the NMR experiments, confirms this suggestion. Although in what follows, we use the notions of the lattice model [4, 5], even purely electronic models [6-9] for cuprates may reveal a tendency to phase separation. 

Another method is to calculate the molecular electronic electrostatic potential by replacing p(r ) in Eq. 19 by its multipole formulation (Eq. 8). The quantity obtained represents the electrostatic potential of a molecule removed from the crystal lattice. First calculations have been performed by the Pittsburgh group (Stewart, Craven, He, and co-workers) [43] electrostatic potential calculations were also derived from the Hansen Coppens [lib] electron density model [41,44], The atomic total electrostatic potential including nuclear contribution may be calculated as ... 

Two complementary models can be developed to describe the features mentioned above. Both models consider the electron-electron interaction and the exciton-lattice interaction. Although both models have different starting points, they yield satisfying interpretations of the experimental findings. The one model is the model of Wannier excitons which ensues directly from the one-electron band model discussed above. The other one starts from the many electron states of the [Pt(CN)4]2 ion and takes into account the coupling between neighbouring complex ions. 

Table 1 Total energy values per QHj unit for the five different polyacetylene models obtained by correct truncation of the two-electron lattice sumsb...

There are two extreme views in modeling zeolitic catalysts. One is based on the observation that the catalytic activity is intimately related to the local properties of the zeolite s active sites and therefore requires a relatively small molecular model, including just a few atoms of the zeolite framework, in direct contact with the substrate molecule, i.e. a molecular cluster is sufficient to describe the essential features of reactivity. The other, opposing view emphasizes that zeolites are (micro)crystalline solids, corresponding to periodic lattices. While molecular clusters are best described by quantum chemical methods, based on the LCAO approximation, which develops the electronic wave function on a set of localized (usually Gaussian) basis functions, the methods developed out of solid state physics using plane wave basis sets, are much better adapted for the periodic lattice models. 

The overwhelming majority of the theoretical studies were performed on cluster models of the catalytic site, hi spite of the fact that the role of space confinement and the secondary interactions with the framework atoms is well-known, there are only a few electronic structure calculations on lattice models involving hydrocarbons, using either periodic DFT calculations, or embedding methods. In this brief account of the subject we attempt to overview some of the recent computational results of the literature and present some new data obtained from ab initio DFT pseudopotential plane wave calculations on Cl - C4 alkanes in the chabazite framework. 

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