Molecular Discrete

Fundamentals of a method for developing models of mass transfer of low-molecular substances in non-reconstructing microheterogeneous membranes were formulated. The local properties of membranes differ in sorbability with respect to species and in the probability for a species to jump from one sorption site to another. Because of this, the permeability of a membrane depends on the amounts of different-type sites, their mutual arrangement, mutual influence of adjacent molecules, and the probabilities of jumps between different sites. The probabilities of occupation of different sorption sites are described by kinetic equations, which take into account the interactions between species. The atomic-molecular discrete and continuous models of mass transfer for thin and thick films are constructed. 

Similar situations, involving diagonal matrix algebra, are encountered when molecular discrete n-dimensional MO LCAO spaces and operator representations are studied. The formalism for these cases is discussed elsewhere [49]. 

POMs are characterized by molecular, discrete, nanosized multi-metal oxide poly anionic scaffolds, with a structural motif being at the interface between molecular complexes and extended oxides. Therefore, they show the reactivity of metal oxides with, in addition, the tunability of a molecular species. 

But besides MNP MOFs and Polymer MOFs, reports on in situ assembly of pre-adsorbed precursors leading to molecular (discrete) species inside MOFs are stiU very scarce. This is indeed surprising, given that several precedents exist for the preparation of molecular species encapsulated in zeolites, as commented... 

Hereafter, some examples are presented to illustrate the large variety of molecular (discrete) organic and metallorganic compounds that have been successfully incorporated inside the pores of MOFs. Most often, these systems have been prepared by adding the species directly during the synthesis of the MOF, though some limited examples also exist on the assembly of precursors inside the pores. 

Specific solute-solvent interactions involving the first solvation shell only can be treated in detail by discrete solvent models. The various approaches like point charge models, siipennoleciilar calculations, quantum theories of reactions in solution, and their implementations in Monte Carlo methods and molecular dynamics simulations like the Car-Parrinello method are discussed elsewhere in this encyclopedia. Here only some points will be briefly mentioned that seem of relevance for later sections. 

Morris K F and Johnson C S Jr 1993 Resolution of discrete and continuous molecular size distributions by means of diffusion-ordered 2D NMR spectroscopy J. Am. Chem. See. 115 4291-9... 

Leforestier C and Museth K 1998 Response to Comment on On the direct complex scaling of matrix elements expressed in a discrete variable representation application to molecular resonances J. Chem. Phys. 109 1204... 

The classical microscopic description of molecular processes leads to a mathematical model in terms of Hamiltonian differential equations. In principle, the discretization of such systems permits a simulation of the dynamics. However, as will be worked out below in Section 2, both forward and backward numerical analysis restrict such simulations to only short time spans and to comparatively small discretization steps. Fortunately, most questions of chemical relevance just require the computation of averages of physical observables, of stable conformations or of conformational changes. The computation of averages is usually performed on a statistical physics basis. In the subsequent Section 3 we advocate a new computational approach on the basis of the mathematical theory of dynamical systems we directly solve a... 

Fig. 2. Left Time average (over T = 200ps) of the molecular length of Butane versus discretization stepsize r for the Verlet discretization. Right Zoom of the asymptotic domain (r < 10 fs) and quadratic fit.

G. Ramachandran and T. Schlick. Beyond optimization Simulating the dynamics of supercoiled DNA by a macroscopic model. In P. M. Pardalos, D. Shal-loway, and G. Xue, editors. Global Minimization of Nonconvex Energy Functions Molecular Conformation and Protein Folding, volume 23 of DIM ACS Series in Discrete Mathematics and Theoretical Computer Science, pages 215-231, Providence, Rhode Island, 1996. American Mathematical Society. 

To compute the above expression, short molecular dynamics runs (with a small time step) are calculated and serve as exact trajectories. Using the exact trajectory as an initial guess for path optimization (with a large time step) we optimize a discrete Onsager-Machlup path. The variation of the action with respect to the optimal trajectory is computed and used in the above formula. 

In this paper, we discuss semi-implicit/implicit integration methods for highly oscillatory Hamiltonian systems. Such systems arise, for example, in molecular dynamics [1] and in the finite dimensional truncation of Hamiltonian partial differential equations. Classical discretization methods, such as the Verlet method [19], require step-sizes k smaller than the period e of the fast oscillations. Then these methods find pointwise accurate approximate solutions. But the time-step restriction implies an enormous computational burden. Furthermore, in many cases the high-frequency responses are of little or no interest. Consequently, various researchers have considered the use of scini-implicit/implicit methods, e.g. [6, 11, 9, 16, 18, 12, 13, 8, 17, 3]. 

The standard discretization for the equations (9) in molecular dynamics is the (explicit) Verlet method. Stability considerations imply that the Verlet method must be applied with a step-size restriction k < e = j2jK,. Various methods have been suggested to avoid this step-size barrier. The most popular is to replace the stiff spring by a holonomic constraint, as in (4). For our first model problem, this leads to the equations d... 

Tn general, the. solvent-accessible surface (SAS) represents a specific class of surfaces, including the Connolly surface. Specifically, the SAS stands for a quite discrete model of a surface, which is based on the work of Lee and Richards [182. They were interested in the interactions between protein and solvent molecules that determine the hydrophobicity and the folding of the proteins. In order to obtain the surface of the molecule, which the solvent can access, a probe sphere rolls over the van der Waals surface (equivalent to the Connolly surface). The trace of the center of the probe sphere determines the solvent-accessible surjace, often called the accessible swface or the Lee and Richards surface (Figure 2-120). Simultaneously, the trajectory generated between the probe and the van der Waals surface is defined as the molecular or Connolly surface. 

Many molecular mechanics potentials were developed at a time when it was computationally impractical to add large numbers of discrete water m olecules to ih e calcu la Lion to sim ulate th e effect of ac ueous media. As such, tech n iq ties cam e into place that were intended to Lake into account the effect of solvent in some fashion. These tech niqiieswcre difficult to justify physically but they were used n cvcrth eless. 

It is always possible to convert internal to Cartesian coordinates and vice versa. However, one coordinate system is usually preferred for a given application. Internal coordinates can usefully describe the relationship between the atoms in a single molecule, but Cartesian coordinates may be more appropriate when describing a collection of discrete molecules. Internal coordinates are commonly used as input to quantum mechanics programs, whereas calculations using molecular mechanics are usually done in Cartesian coordinates. The total number of coordinates that must be specified in the internal coordinate system is six fewer... 

Molecular Formula. For compounds consisting of discrete molecules, a formula in accordance with the correct molecular weight of the compound should be used. 

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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