Models Monte Carlo calculations

Employing a classical model, Monte Carlo calculations were performed to investigate the A1 distribution in zeolites. " Si and A1 atoms were randomly distributed over the tetrahedral sites, and in each step an attempt was made to interchange each A1 with a nearby Si. Loewenstein s rule, which forbids Al-O-Al bridging, was investigated, and results from the simulation indicate that the A1 distribution in zeolite structures is generally in accordance with this rule. 

Molecular dynamics calculations are more time-consuming than Monte Carlo calculations. This is because energy derivatives must be computed and used to solve the equations of motion. Molecular dynamics simulations are capable of yielding all the same properties as are obtained from Monte Carlo calculations. The advantage of molecular dynamics is that it is capable of modeling time-dependent properties, which can not be computed with Monte Carlo simulations. This is how diffusion coefficients must be computed. It is also possible to use shearing boundaries in order to obtain a viscosity. Molec-... 

Chemistry, like other sciences, progresses through the use of models. Models are the means by which we attempt to understand nature. In this book, we are primarily concerned with models of complex systems, those systems whose behaviors result from the many interactions of a large number of ingredients. In this context, two powerful approaches have been developed in recent years for chemical investigations molecular dynamics and Monte Carlo calculations [4-7]. Both techniques have been made possible by the development of extremely powerful, modern, high-speed computers. 

The essential influence of surface roughening is also present in this model. Grand canonical Monte Carlo calculations were used to generate adatom populations at various temperatures up to Chemical potentials corresponding to those in the bulk LJ crystal were used, and these produced adatom densities that increased with temperature and roughly approximated the values observed in Ising model simulations below T. ... 

As described in Section 9.4, the determination and refinement of molecular conformations comprehends three main methods DG, MD and SA. Other techniques like Monte Carlo calculations have only a limited applicability in the field of structure elucidation. In principle, it is possible to exclusively make use of DG, MD or SA, but normally it is strongly suggested to combine these methods in order to obtain robust and reliable structural models. Only when the results of different methods match a 3D structure should be presented. There are various ways of combining the described techniques and the procedural methods may differ depending on what kind of molecules are investigated. However, with the flowchart in Fig. 9.13 we give an instruction on how to obtain a reliable structural model. 

In the case of being successful in calculating multiple conformations by using time- or ensemble-averaged MD restraints the solved molecular structures are presented as 3D models and can be deposited in an electronic structure database (17). Finally, it is recommended to provide an accurate explanation of the procedures used for the structure elucidation because the application of different methods (NMR, DG, MD, SA, Monte-Carlo calculations. X-ray crystallography) may result in varying conformational models which do not implicitly display the real state of a molecule. This aspect should be always kept in mind when dealing with structure determination methods. 

Third, a further simplification of the Boltzmann equation is the use of the two-term spherical harmonic expansion [231 ] for the EEDF (also known as the Lorentz approximation), both in the calculations and in the analysis in the literature of experimental data. This two-term approximation has also been used by Kurachi and Nakamura [212] to determine the cross section for vibrational excitation of SiHj (see Table II). Due to the magnitude of the vibrational cross section at certain electron energies relative to the elastic cross sections and the steep dependence of the vibrational cross section, the use of this two-term approximation is of variable accuracy [240]. A Monte Carlo calculation is in principle more accurate, because in such a model the spatial and temporal behavior of the EEDF can be included. However, a Monte Carlo calculation has its own problems, such as the large computational effort needed to reduce statistical fluctuations. 

Mahoney MW, Jorgensen WL (2001) Rapid estimation of electronic degrees of freedom in Monte Carlo calculations for polarizable models of liquid water. J Chem Phys 114(21) 9337-9349... 

Monte Carlo calculations have been carried out to simulate the spin transition behaviour in both mono- and dinuclear systems [197]. The stepwise transition in [Fe(2-pic)3]Cl2-EtOH as well as its modification by metal dilution and application of pressure have been similarly modelled by considering short- and long-range interactions [52, 198, 199]. An additional study of the effect of metal dilution was successfully simulated with the Monte Carlo treatment considering direct and indirect inter-molecular interactions [200]. A very recent report deals with the application of the Monte Carlo method to mimic short- and long-range interactions in cooperative photo-induced LS—>HS conversion phenomena in two- and three-dimensional systems [201],... 

During the last year two Monte Carlo calculations and one molecular dynamics study 276> on ion hydration were published. Watts et al.274> extended previous Monte Carlo calculations 162> to the Li+F- ion pair, which was surrounded in this model calculation by 50 water molecules. In the more recent work of this group Fromm et at.275 ) increased the number of solvent molecules to 200 and coordination numbers for the ions at room temperature were evaluated ( li+ 6,... 

We review Monte Carlo calculations of phase transitions and ordering behavior in lattice gas models of adsorbed layers on surfaces. The technical aspects of Monte Carlo methods are briefly summarized and results for a wide variety of models are described. Included are calculations of internal energies and order parameters for these models as a function of temperature and coverage along with adsorption isotherms and dynamic quantities such as self-diffusion constants. We also show results which are applicable to the interpretation of experimental data on physical systems such as H on Pd(lOO) and H on Fe(110). Other studies which are presented address fundamental theoretical questions about the nature of phase transitions in a two-dimensional geometry such as the existence of Kosterlitz-Thouless transitions or the nature of dynamic critical exponents. Lastly, we briefly mention multilayer adsorption and wetting phenomena and touch on the kinetics of domain growth at surfaces. 

Fig. 30. Phase diagram of a model for Si/W(110) in the temperature versus 9 plane. Experimentally determined interactions J Jj,are used. Full dots are from Monte Carlo calculations, while triangles are based on transfer matrix finite size scaling using strip widths of 8 and 12. The point labelled L indicates approximate location of Lifshitz point. The dotted line indicates the transition region between the (5 x l)and(6 x 1) phases. (From...

In order to reproduce the temporal behavior of water decomposition products, two theoretical approaches based on spur diffusion model and Monte Carlo calculations have been developed. 

While the study of diarylpentanes is helpful in understanding the conformational behavior of aryl vinyl polymers, a simple weighting of the properties of the model compounds by the tacticity of the polymer does not yield the properties of the polymer. For example, the presence of dl dyads surrounding a meso dyad will suppress the tt conformer in the meso dyad 14fl). Thus, in order to obtain the fraction of tt meso conformers within an atactic P2VN sample, it is necessary to resort to a Monte Carlo calculation utilizing an extended product of statistical weight matrices, 26). 

Verdier,P.H., Stockmayer, W.H. Monte Carlo calculations on the dynamics of polymers in dilute solution. J. Chem. Phys. 36, 227-235 (1962). See also Verdier,P.H. Monte Carlo studies of lattice-model polymer chains. 1. Correlation functions in the statistical-bead model. J. Chem. Phys. 45,2118-2121 (1966). 

Very recently Monnerie31 has described Monte Carlo calculations for a quite realistic lattice model nonintersecting chains confined to tetrahedral lattices performing local stochastic processes involving the simultaneous motion of three or four bonds. Without volume exclusion and with no correlations in the orientation probabilities of neighboring bonds, the model has also been treated analytically,32 with application to the fluorescence depolarization experiment. It is easy to show that this model also leads to the long-time Rouse spectrum. 

Collision Model. Figure 3 shows an instructive, efficient, and convenient method for treating the case of mixed surface and particle-diffusion control. It is instructive because it is easy to visualize. It is efficient because although it is a molecular approach, it does not require Monte Carlo calculations, which are expensive and should be avoided whenever possible. It is convenient because it leads to a set of differential equations... 

Figure 9. X-ray spectra at several times for Model 5LM evaluated at the dates shown from Monte Carlo calculations by Pinto and Woosley (1987). Horizontal error bars show the GINGA sensitivity at various energies.

Despite its bad reputation as an analytical tool, XRF is potentially a traceable method according to the CCQM definition and could be a primary method although it was not selected as such, and won t be for a long time. In fact, it is the only microanalytical method which can at present be considered as a candidate for accurate microscopic elemental analysis. Proof of this statement follows from Monte Carlo calculations in which experimental XRF spectra can be accurately modelled starting from first principles [23], This is not an easy approach but with computing power now available it is feasible, though not worth the effort for bulk chemical analysis where other alternatives are available. 

Mansfield 79 performed Monte Carlo calculations on model dendrimers and determined that as a result of the unique architecture of the branches, even when similar chemically, they are well segregated. Further, he concluded that dendrimers are fractal (D ranges from 2.4 to 2.8) and self-similar only over a rather narrow scale of lengths. 

To calculate the above-indicated expected values, we carry out Monte Carlo calculations on 20 x 20-site lattices. As mentioned, the ground-state wave function of the model depends on 14 parameters and, of course, cannot possibly be analyzed completely. We confine the numerical calculations to the case in which the spinor 4>A/" P depends on one parameter a ... 

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