The Periodic Model

Two significant aspects of the symmetry observed in the analysis of periodicity are the inverted electronic energy levels and the approach of Z/N — 1 for all nuclides. The inversion is explained by the computational observation that electronic sub-levels respond differently to compression of an atom. 

The most revealing result was obtained by Goldman and Joslin (1992) who reported the sublevel sequence 

This conclusion immediately leads to an explanation of the Z/N ratio that approaches unity. The most common nuclide with unit ratio is the a-particle, He +. The fusion of a-particles in an equilibrium process under extreme pressure would produce atomic nuclides with uniform Z/N 1, which is a testable hypothesis. 

The simplest case of nucleogenesis by the equilibrium fusion of a-particles must, not only produce nuclides with Z/N = 1, but also with mass numbers A = 4m. Should these nuclides be released into interstellar space many of them would be unstable in a low-pressure environment and decay by positron emission or electron capture. Because of the inherent instability of nuclei with an odd number of both protons and neutrons such a decay must consist of at least two steps, such that 

The reciprocal relationship between matter and the curvature of space implies that the occurrence of different periodic functions at Z/N = 1, r and 0.58 arises from a variation of the electronic configuration of atoms with 

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An adsorption of silver dimer on a rutile (110) surface has been studied using a DFT model within both cluster and periodic approaches. The calculations show that the interaction of silver dimers can occur both with bridging chain of oxygen atoms or with atoms located in the hollows between chains. The bonding of Ag2 in the hollow is characterized by the positive adsorption energy according to the periodic model. On the other hand, the geometry optimization of similar structures within the cluster model leads to desorption or dissociation of silver dimer. The periodic model is shown more appropriate for this system. 

A three-layer slab structure was chosen to simulate rutile (110) surface within the periodic model. Unit cells were 5.91 and 6.49 A along the [001] and [-100] directions, respectively. The slabs were shared out with a 20 A gap. Taking into account the experimental data, geometric parameters of two top layers was allowed to relax during the geometry optimization. Calculations were performed within the DFT model with PBE96 exchange-correlation functional [4]. The Troullier-Martins pseudopotentials were chosen to describe Ti and Ag atoms, and the Hamann pseudopotential was used for O atoms. 

Structures stu(fred within the periodic model are similar to the structures of the cluster model (Fig. 2). The results of calculations are summarized in Table 2. 

Table 2. Binding eneigies ( 0 ), bond distances between the nearest atoms of silver and oxygen r(Ag-0) and betweai two silver-atoms r(Ag-Ag) for Aga/TiOi systems calculated within the periodic model.

The DFT study of adsorption of silver dimer on rutile (110) surface within the cluster and periodic models shows that the interaction occurs both with chain oxygen atoms of the surface and with atoms located between the chains of 0(2c) atoms. Positive binding energy of Ag2 with rutile surface during adsorption between the oxygen chains was obtained only for the periodic model. The latter is concluded to be the preferable for theoretical study of Ag /Ti02 systems. 

Transition state calculations were also run with the periodic force field model for moving ions out of site N. Interestingly the barrier increases for the periodic model over the defect calculation, this contradicts the failure by the DFT to find a site N which would implies a negligible barrier to migration. 

Periodic quantum chemical calculations can be carried out either with the atom-centered basis set or planewave basis set. First calculations of the electronic structure of periodic models of zeolites were carried out with atom-centered basis sets (e.g., Ref. [28-30]). First applications of the periodic model of zeolites employing a planewave basis set appeared just a year later [31-33], Majority of applications use the planewave basis sets at present. 

Fig. 1. Part (a) graphical representation of the periodic model used in this study. The shell-3 clusters has been embedded into the crystallographic T5 position. Part (b) the so obtained DOS for the silicalite (dashed line) and for TS-1 (full line). The inset reports a magnification of the of the two curves in the gap between the symmetric and antisymmetric stretching modes of the T-O-T units, highlighting the appearance of the 960 cm band for TS-1, otherwise hardly visible.

The proposed (Manuel et al., 2006) nuclear cycle that powers the cosmos has many elements in common with some of our arguments. Not unlike the periodic model of stable nuclides and the notion of cosmic self-similarity these authors suggest that stars are subject to the same types of interaction that occur in radioactive nuclides, which depend on the relative amounts of nucleons defined by the numbers A, Z and N. Because of chemical layering an accumulation of neutrons that resembles a neutron star develops at the core of an ordinary star. This core is left behind as the remains of a supernova. 

Me20 The geometries of the CaOCa species are shown in Tables 20.4 and 20.5. Comparing the Ca-O bond lengths between the optimized cluster and periodic models we can confirm that the Ca-O distances are shortened in the periodic model (Table 20.4). A trend was revealed, i.e., a denser attachment of the CaOxCa species to the framework conserves for all X in the periodic model. 

Similarly, we did not get any tri-dentate carbonate species via the periodic study for the interaction between CO2 and Me02Me homologues. With respect to the periodic model of CO2 adsorption in CaOsCaCMOR), the latter confirms the results of the cluster computations. The periodic model also led to the formation of an O2 fragment separated from CaCOsCaCMOR) with carbonate bands at 1517 and 1226 cm (Fig. 20.20b). Assigning the periodic value of 1226-1347 cm of the cluster one gets a re-scaled value of 1669 cm This value has to be compared to 1649 cm for the 8R cluster of MOR or to 1664 cm for the (6R-I-4R) cluster ofFAU (Table 20.13). 

Regarding the cluster and periodic studies of the AE cationic Me20x moieties, Me=Mg and Ca, we observed two important consequences from their comparison. Both approaches showed more or less similar results with respect to the heat AU of oxidation reactions (20.7) by triplet oxygen. Different evaluations of the singlet-triplet energy differences were obtained with the periodic (AUst) or cluster (AUst ") methods. The fact that the absolute AUst " values decrease with the expansion of the cluster model at both the B3LYP and MP2 levels supports the results of the periodic models. Both these results show the possibility of the... 

Conceptually, the method is as follows. The local characteristics of a crystal electronic structure are calculated within the periodic model by the Hartree-Fock method... 

The results for embedded-cluster calculations are different from that for the periodic models. Adsorption energies of a water molecule in the cluster case were much more favorable for the associative mechanism (Table 11.13), which is in better agreement with experimental observations. The predicted adsorption energies for the asso-... 

The fully solvated model cannot easily be used with the dipole-correction or applied field methods discussed above. The absence of a vacuum layer in the fully solvated model makes dipole moment evaluation and field application impossible within the periodic model. Hybrid approaches, in which the dipole is measured in an unsolvated model and solvation approximated in the fully solvated model, might be considered. However, the advantage of the fully solvated model is its integration within the direct charging of the electrochemical interface. 

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