Discrete atom method

AN INSTABILITY ANALYSIS OF HETEROEPITAXIAL INTERLACES VIA A DISCRETE ATOM METHOD... 

M. J. Bramley and T. Carrington Jr.,/. Chem. Phys., 99, 8519 (1993). A General Discrete Variable Method to Calculate Vibrational Energy Levels of Three- and Four-Atom Molecules. 

By the method of introducing Pt into the DLC, the platinum metal is assumed to be distributed over the carbonaceous material bulk as discrete atoms or clusters [154], Essentially, Pt is not a dopant in the DLC, in the sense that the term is used in semiconductor physics. Nor is the percolation threshold surpassed, since the admixture of Pt (not exceeding 15 at. %) did not affect the a-C H resistivity, as was shown by impedance spectroscopy tests p 105 Q, cm, like that of the undoped DLC (see Table 3). It was thus proposed that the Pt effect is purely catalytic one Pt atoms on the DLC surface are the active sites on which adsorption and/or charge transfer is enhanced [75], (And the contact of the carbon matrix to the Pt clusters is entirely ohmic.) This conclusion was corroborated by the studies of Co tetramethylphenyl-porphyrin reaction kinetics at the DLC Pt electrodes [155] redox reactions involving the Co central ion proceed partly under the adsorption of the porphyrin ring on the electrode. 

Within the density functional theory (DFT), several schemes for generation of pseudopotentials were developed. Some of them construct pseudopotentials for pseudoorbitals derived from atomic calculations [29] - [31], while the others make use [32] - [36] of parameterized analytical pseudopotentials. In a specific implementation of the numerical integration for solving the DFT one-electron equations, named Discrete-Variational Method (DVM) [37]- [41], one does not need to fit pseudoorbitals or pseudopotentials by any analytical functions, because the matrix elements of an effective Hamiltonian can be computed directly with either analytical or numerical basis set (or a mixed one). 

A fundamental postulate of quantum mechanics is that atoms consist of a nucleus surrounded by electrons in discrete atomic orbitals. When atoms bond, their atomic orbitals combine to form molecular orbitals. The redistribution of electrons in the molecular orbitals determines the molecule s physical and chemical properties. QM methods do not employ atom or bond types but derive approximate solutions to the Schrodinger equation to optimize molecular structures and electronic properties. QM calculations demand significantly more computational resources than MM calculations for the same system. In part to address computer-resource constraints, QM calcu-... 

It is well known that the energy of interaction of an atom with the continuous solid is 2-3 times less than with the discrete (atomic) model (cf., e.g., Ref. [38], Figs. 2.2-2.4). Thus, to obtain the same Henry s Law constants with the two models, one has to increase e for the continuous model. This, however, does not discredit the continuous model which is frequently used in adsorption calculations. In particular, we can use the above mentioned results of Ref. [37] to predict the value of e for Ar which would have been obtained if one had carried out Henry s Law constant calculations for Ar in the AO model of Ref. [17] and compared them with experiment. One can multiply the value of e for CH4 obtained from AO model by the ratio of e values for Ar and CH4 in the CM model [36] to obtain tjk = 165A for Ar in the AO model. This is very close to the value of 160 K obtained in Ref. [21, 28] by an independent method in which the value of the LJ parameter e for the Ar - oxide ion interaction was chosen to match the results of computer simulation of the adsorption isotherm on the nonporous heterogeneons surface of Ti02. Considering the independence of the calculations and the different character of the adsorbents (porous and nonporous), the closeness of the values of is remarkable (if it is not accidental). The result seems even more remarkable in the light of discussion presented in Ref. [28]. Another line of research has dealt with the influence of porous structure of the silica gel upon the temperature dependence of the Henry constants [36]. 

Discrete variational method LCAO (Slater orbitals, or numerical atomic orbitals)... 

It is also worth mentioning that numerical solutions of the Schrodinger equation frequently enclose the atom in a spherical box of finite radius for example, discrete variable methods, finite elements methods and variational methods which employ expansions in terms of functions of finite support, such as -splines, all assume that the wave function vanishes for r > R, which is exactly the situation we deal with here. For such solutions to give an accurate description of the unconfined system it is, of course, necessary to choose R sufficiently large that there is negligible difference between the confined and unconfined atoms. 

We present a new effective numerical method to compute resonances of simple but non-integrable quantum systems, based on a combination of complex coordinate rotations with the finite element and the discrete variable method. By using model potentials we were able to compute atomic data for alkali systems. As an example we show some results for the radial Stark and the Stark effect and compare our values with recent published ones. 

Bramley, M.J., Carrington, T. Jr. A general discrete variable method to calculate vibrational-energy levels of 3-atom and 4-atom molecules, J. Chem. Phys. 1993,99,8519 1. 

Figure 11.10 Convergence of amber99/C-PCM calculations for (alanine iooo using SWIG discretization with 110 Lebedev points per atom. Method (a] uses the factorization in Eq. (11.47) with no pre-conditioning method (b) uses a diagonal pre-conditioner and method (c) uses a biock-diagonai pre-conditioner. The convergence threshold was set to a maximum residual of lO"" and electrostatics were computed exactiy (no FMM).

Keywords Atomization Chemical reactions Craiservation equations Constitutive equations Drop breakup Drop deformation Drop collisions Evaporation LES Newtonian fluids RANS Spray modeling Spray PDF Stochastic discrete particle method Source terms Turbulence... 

The discrete-particle methods proposed earher can be used as the components of the problemsolving environment (PSE) based on the conception of multiresolutional wavelets. As shown in Figme 26.38, the whole series of simulations can be performed over three different spatio-temporal levels similarly as it is for wavelets but here the various shapes of wavelets will depend on the model of particle (atom, DPD droplet, FPM drop, and SPH chunk of fluid) and consequently the interactions between particles. In fact, the shapes of short-ranged interaction can be treated as some sort of wavelets. The interactions are short-ranged with compact support and well localized... 

A different approach was developed by Baerends, Ellis, and Ros (1973). In addition to adopting the Slater potential for the exchange, their approach had two distinct features. The first was an efficient numerical integration procedure, the discrete variational method (DVM), which permitted the use of any type of basis function for expansion, not only Slater-type orbitals or Gaussian-type orbitals, but also numerical atomic orbitals. The second feature was an evaluation of the Coulomb potential from... 

To analyze the individual heat transfer kinetics of droplet clusters within the spray of twin-fluid atomizers, the local correlations between the droplet concentration and the heat and flow conditions are evaluated. Numerical simulations of the spray flow analyzed in this paper have been carried out with Large-Eddy-Simulation (LES) models with Lagrangian particle tracking (discrete particle method) for the droplet motion. A synthetic perturbation generator [30] for the inflow conditions for the gas flow and simple perturbations are added to the dispersed phase to induce realistic vortex patterns at the nozzle and in the consequent spray. 

The two relativistic four-component methods most widely used in calculations of superheavy elements are the no-(virtual)pair DF (Coulomb-Breit) coupled cluster technique (RCC) of Eliav, Kaldor, and Ishikawa for atoms (equation 3), and the Dirac-Slater discrete variational method (DS/DVM) by Fricke for atoms and molecules. " Fricke s DS/DVM code uses the Dirac equation (3) approximated by a Slater exchange potential (DFS), numerical relativistic atomic DS wavefunctions, and finite extension of the nuclei. DFS calculations for the superheavy elements from Z = 100 to Z = 173 have been tabulated by Fricke and Soff. A review on various local density functional methods applied in superheavy chemistry has been given by Pershina. ... 

Figure 3 Structural alignments with discrete properties. Methods are based on discrete properties using the DG algorithm (1) or clique-detection (11) as implemented in distance comparisons (DISCO), and Apex-3D. The structure representation, based on discrete properties, resorts to one atomic descriptor (I), usually the atom type, or multiple atomic or site descriptors (II). In the first method (I), the conformational analysis is restricted to the generation of molecular geometries which allow a common arrangement of selected phaimacophoric moieties present in a rigid compound used as template. In the second method (II), the conformational analysis procedure may involve a systematic enumeration of all the possible conformadons for each ligand. The search similarity is directed towards the confirmation of a predefined pharmacophore postulated by the modeler or from some classical SAR in the case of the active analog approach (1), or the automated identification of pharmacophores and bioacdve conformations (II)...

Ryzhkov et al. [49] carried out a study of the electronic structure of neutral endohedral An C28 (An = Th-Md) confirming our results from fully relativistic discrete variational method. The 6d and 5f contributions to the bonding were found to be comparable for the earlier actinides. In addition, the actinides (Th-Md) series stabilize a C40 cage with a noticeable overlap between the 5f, 6d, 7s, 7p orbitals of the central actinide atom and the 2p(C) of the cage [54]. The most stable complex was found to be Pa C4o. 

Molecula.rMecha.nics. Molecular mechanics (MM), or empirical force field methods (EFF), ate so called because they are a model based on equations from Newtonian mechanics. This model assumes that atoms are hard spheres attached by networks of springs, with discrete force constants. 

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