Confined model systems constant potential

The state symbols 1Fb, xLa, lBb and xBa shown in Figure 4.27 were introduced in 1949 by Platt.297,298 He assumed a free-electron model, similar to the electron-in-a-box, in which the 7t-electrons of a cyclic system are confined to a one-dimensional loop of constant potential (a circular wire). The eigenvalues of a single electron in a perimeter of length / are given by Equation 4.37. 

The specific structure of the states for Hp was described in detail in [79], where it is mentioned as a well-known physical effect. For example, it was noted in the theory of disordered semiconductors that a similar "ladder" structure of states is realized for the system where the Coulomb potential is modified within a sphere as a constant potential (see [86,87] for a qualitative discussion and analytical solution of the problem). For quantum chemistry, the situation is interesting, as was shown in a series of publications of Connerade, Dolmatov and others (see e.g. [19,88-91] note that the series of publications on confined many-electron systems by these authors is much wider). The picture described is realized to some extent for the effective potential of inner electrons in multi-electron atoms, as it is defined by orbital densities with a number of maximal points. The existence of a number of extrema generates a system of the type described above [89]. This situation was modeled and described for the one-electron atom in [88] it is similar to that one described in Sections 5.2 and 5.3. 

The commonly used DFT-based methods for characterizing activated carbon assume that the pores are geometrically slits with smooth, unterminated graphitic walls of constant wall potential. The experimental data are then modeled as a system of homogeneous, confined slits of varying width. The energetic heterogeneity of the material is therefore completely expressed in terms of its distribution of pore widths. 

In the case of QW heterostructures, the electron and hole of an exciton are well confined within the layer since the band gap discontinuity is quite large, especially for III-V heterostructuies. On the other hand, dielectric and lattice properties of group III (II) and group V (VI) semiconductors (dielectric permittivities, lattice constants, elastic moduli) are in close proximity in their values [7]. Therefore, we model our system by the localized quasi-2D IS-exciton interacting with bulk-like phonon modes in the QW with infinite potential barriers. We describe such an excitonic state by quasi-2D wave fimction... 

Given a molecular or supra-molecular system embedded in a solvent charge distribution, the solute-solvent interaction can be modelled using a mean field (MF) approach [21-24], which treats the solvent as a continuum fully defined by a dielectric constant (s) and by a shape function, uniquely identifying the space regions where the solute and the solvent are placed. The boundary between the two domains is a compact cavity S, which in Fig. 17.2 has been represented as a spherical boundary surface including the explicit molecules. Then, the major issue related to such a scheme is how to model the interactions between the continuum and the explicit molecules placed inside the cavity. In a static picture, such interactions are of two types, electrostatic and non-electrostatic, whereas when such a model is used in MD simulations, an additional potential is included, in order to ensure that all the molecules remain confined inside the cavity during the simulation with a correct density up to the boundary [21]. 

Later, in a model where each cylindrical polymer rod was confined to a concentric, cylindrical, electroneutral shell whose volume represents the mean volume available to the macromolecule, the concept was extended to the macroscopic system itself which was considered as an assembly of electroneutral shells at whose periphery the gradient of potential goes to zero and the potential itself has a constant value. Closed analytical expressions which represent exact solutions of the Poisson-Boltzmann equation can be given for the infinite cylinder model. These solutions, moreover, were seen to describe the essence of the problem. The potential field close in to the chain was found to be the determining factor and under most practical circumstances a sizable fraction of the counter-ions was trapped and held closely paired to the chain, in the Bjerrum sense, by the potential. The counter-ions thus behave as though distributed between two phases, a condensed phase near in and a free phase further out. The fraction which is free behaves as though subject to the Debye-Hiickel potential in the ordinary way, the fraction condensed as though bound . 

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