Bethe lattice approximation

A MODEL FOR THREE-DIMENSIONAL CHEMISORPTION. BETHE LATTICE APPROXIMATION THE CONCEPT OF GROUP ORBITALS. 

This is a drastic simplification compared to the real solid. As a consequence, the local density of states of the Bethe lattice shows very little structure compared to the local density of states computed for real solid structures. On the other hand it can be readily refined to the real solid situation, as we will show later. It appears that the first and second moments of the local density of states are properly calculated within the Bethe lattice approximation. The Green s function for atom 1 is easily found by using the techniques discussed in section 2.4.1 ... 

The Bethe lattice approximation can be readily extended to study chemisorption of an atom. We will discuss the situation of an adsorbate atom coordinated to several surface atoms, three in the example represented in Fig.(2.46). 

In the next chapter results of accurate quantum-chemical calculations on transition metal surfaces will be presented. Here we will develop a method that enables an estimate of elementary quantum-chemical features, without the need for extensive quantum-chemical calculations. The result is of limited and only qualitative value and we will verify it in the next chapter with the aid of more sophisticated procedures. The method is of use to us because it permits the development and illustration of concepts that will facilitate our analysis of more advanced calculations. In this section we will analyze the electronic structure of a face-centered cubic (f.cx.) transition metal. We will use the method to discuss the valence electron structure of the (lll)i (100) and (110) surfaces of f.c.c. crystals. Specifically we are interested to know whether subbands can be identified, whose bandwidths can be estimated by means of the Bethe lattice approximation and how their respective local densities of states behave as a function of valence electron band filling. 

The simulation of catalyst deactivation by coke formation using a 3-dimensional site-bond-site network model is highly attractive, especially for zeolites, as the processes occurring in cavities (also referred to as voids or intersections) and in channels (also referred as necks, capillaries or arcs) can be readily distinguished. This model is flexible and the cormectivity of pores as well as the local homogeneity of the catalyst can be readily altered. Further, a percolation theory is available for site-bond-site models. In the particular case of Bethe lattices, approximated analytical solutions for the percolation probabilities have been derived[7]. 

Although there are probably other universality classes, this transition was successfully modeled by bond percolation [6]. Generally, bond percolation on a lattice has each bond (line connecting two neighboring lattice sites) present randomly with probability p and absent with probability 1-p. Clusters are groups of sites connected by present bonds. For p > Pc zn infinite cluster is formed. Percolation theory (in a Bethe lattice approximation) was invented by Flory (1941) to describe gelation for three-functional polymers. 

Figure 16 illustrates this by presenting the calculated bondstrengths of an hydrogen-type adsorbate to the (111) face of the f.c.c. s-band model metal as a function of the number of valenceband electrons (N i). The same Bethe lattice approximation as discussed earlier has been used. As expected, three-coordinated hydrogen bonds more strongly than mono- or dicoordinated hydrogen atoms at low valence-electron... 

Figure 19 shows the LDOS of different grouporbitals of the s-electron band for the (11 l)-face of a f.c.c. crystal calculated in the Bethe lattice approximation. 

Fig. 19. LDOS ( j(E) of different group orbitals of the electron band for a (111) face of a f.c.c. crystal calculated in the Bethe lattice approximation [51] i = G three coordination i = 2 cr two coordination = 3 a atop coordination i = 4 tt two coordination...

The original degeneracy is lifted by the presence of the metal surface and the resulting surface symmetry electron density of states is sketched in Fig. 33 c (the figure is the result of a Bethe lattice approximation calculation). It is essential to consider this lifting of degeneracy and the creation of asymmetric orbital combinations, since some authors, e.g., Banholzer et al. [100], have erroneously ignored this. As a result,... 

These are results of Bethe lattice approximation to the extended Huckel method calculations... 

In general, percolation is one of the principal tools to analyze disordered media. It has been used extensively to study, for example, random electrical networks, diffusion in disordered media, or phase transitions. Percolation models usually require approximate solution methods such as Monte Carlo simulations, series expansions, and phenomenological renormalization [16]. While some exact results are known (for the Bethe lattice, for instance), they are very rare because of the complexity of the problem. Monte Carlo simulations are very versatile but lack the accuracy of the other methods. The above solution methods were employed in determining the critical exponents given in the following section. 

Linke et al. (1983) and Franz (1984, 1986) developed a different version of the lattice-gas concept. These models emphasize the experimental observation (Secs. 3.4 and 4.4) that thermal expansion of liquids is mainly achieved by a reduction of the average near-neighbor coordination number. A given structure such as the artificial, but mathematically convenient Cayley Tree or Bethe lattices can, when partially populated, be viewed as a crystalline alloy of atoms and vacancies. Tight-binding methods then permit calculation of the electronic structure, in particular the density of electronic states. Franz made use of quantum percolation theory to model the DC conductivity. A more recent model (Tara-zona et al., 1996) employs a body-centered-cubic lattice which, when fully occupied, provides a reasonable approximation to the local structure of liquid metals near the melting point. 

Equations (5.43) with (5.51) give the activities as functions of the densities for this model in the Bethe-Guggenheim approximation. If the occupancies of the cells neighbouring any one cell were indeed independent of each other, these relations would be exact. They are therefore exact for the class of lattices called trees, on which there are no closed paths between any two neighbours of one site there is no path other than via that site, so that, once the state of occupancy of the cell at that site is specified, the cells at the neighbouring sites are decoupled from, and independent of, each other. Because the Bethe-Guggenheim approximation is thus exact for one class of models, it is necessarily therm ynami-cally consistent and, indeed, it may be verified by explicit calculation that (5.43) with (5.51) satisfies (5.44). 

The one-component lattice gas of 5.3 may also be treated in the Bethe-Guggenheim approximation, which is a generalization and improvement upon, the simple mean-field theory. The latter follows from the former in Uie limit of large c and small e. The resulting mean-field theory is then necessarily thermodynamically consistent, because the Bethe-Guggenheim approximation is consistent for all c and e. In the present two-component model, in which the only interactions are infinitely strong repulsions, there is no simplification we can make beyond the Bethe-Guggenheim approximation and still retain thermodynamic consistency there is no parameter e, and, while the coordination number c is at our disposal, there is no limit to whidi we can usefully take h. 

In general, different lattices with the same dimensionality and coordination number, e.g. the hexagonal and Voronoi polygon, exhibit similar behavior [40]. More importantly, the effective diffusion coefficient on these lattice structures can be closely approximated by selecting a Bethe lattice with an appropriate effective coordination number For example, three-dimensional cubic and Voronoi polyhedron lattices, with coordination numbers of 6 and 16, have the same effective diffusion coefficient behavior as Bethe lattices with coordination number of 5 and 7 [44]. Therefore, the effective diffusion coefficient and tortuosity trends shown in Figure 6 are applicable to percolation lattices with widely different geometries. Prediction of the effective diffusivity of a given real lattice follows directly from selection of an effective Bethe coordination number. 

It is obvious that Fig. 2 represents a drastic simplification of reality for complicated molecules The merit of this theory is that it gives a good qualitative picture of real gelation, the first indication for the universality principle that complicated molecular details are not very relevant for the main results. Physicists call Fig. 2 a Bethe lattice, and this gelation process is then called percolation on a Bethe lattice . Hie macromolecules are also designated as Cayley trees since, like trees in a forest, they have no cyclic links between their branches. Many other problems of theoretical physics, besides percolation, have been studied on Bethe lattices. When critical exponents were found they usually agreed with those obtained by using simple approximations for real lattices like mean field (or molecular field) approximation, Landau ansatz for phase transitions, van der Waals equation, etc. We will thus also denote them as mean field approximations. 

A warning for experts A non-negligible fraction of monomers is located on the surface of the Bethe lattice, even within the thermodynamic limit, as one can see already from Fig. 2. We are interested here in the Bethe lattice as an approximation for real three-dimensional space and thus neglect the complications arising from this surface This is achieved by considering the central site of Fig. 2 as a representation of all monomers in the real system. 

This model was solved analytically for the Bethe lattice and with the simple approximation p = 1 - exp(— consW) one can determine its behavior on the simple cubic lattice from Ref. 65. These Monte Carlo results are shown in Fig. 7. They agree qualitatively with the phase diagram in the Bethe lattice and also with the experimental data of Tanaka et al. using a gelatin-methanol water system. As already found for correlated... 

However, one can proceed beyond this zeroth approximation, and this was done independently by Guggenheim (1935) with his quasi-chemicaT approximation for simple mixtures and by Bethe (1935) for the order-disorder solid. These two approximations, which turned out to be identical, yield some enliancement to the probability of finding like or unlike pairs, depending on the sign of and on the coordmation number z of the lattice. (For the unphysical limit of z equal to infinity, they reduce to the mean-field results.)... 

In between these extremes lie a large number of CVM treatments which use combinations of different cluster sizes. The early treatment of Bethe (1935) used a pair approximation (i.e., a two-atom cluster), but this cluster size is insufficient to deal with fhistration effects or when next-nearest neighbours play a significant role (Inden and Pitsch 1991). A four-atom (tetrahedral) cluster is theoretically the minimum requirement for an f.c.c. lattice, but clusters of 13-14 atoms have been used by de Fontaine (1979, 1994) (Fig. 7.2b). However, since a comprehensive treatment for an [n]-member cluster should include the effect of all the component smaller (n — 1, n — 2...) units, there is a marked increase in computing time with cluster size. Several approximations have been made in order to circumvent this problem. 

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