Grid representation

The flux-flux expression and its extensions have been used to calculate reaction probabilities for several important reactions, including H2+02 H + H2O, by explicit calculation of the action of G in a grid representation with absorbmg potentials. The main power of the flux-flux fomuila over the long mn will be the natural way in which approximations and semi-classical expressions can be inserted into it to treat larger systems. 

Fig. 5. Heat-exchange network (a) flow representation, and (b) grid representation. See text.

Often the actions of the radial parts of the kinetic energy (see Section IIIA) on a wave packet are accomplished with fast Fourier transforms (FFTs) in the case of evenly spaced grid representations [24] or with other types of discrete variable representations (DVRs) [26, 27]. Since four-atom and larger reaction dynamics problems are computationally challenging and can sometimes benefit from implementation within parallel computing environments, it is also worthwhile to consider simpler finite difference (FD) approaches [25, 28, 29], which are more amenable to parallelization. The FD approach we describe here is a relatively simple one developed by us [25]. We were motivated by earlier work by Mazziotti [28] and we note that later work by the same author provides alternative FD methods and a different, more general perspective [29]. 

The parameter 4>, is now the value of the wave function, in the DVR grid representation, at the ith grid point that results from acting with the operator exp(—/7 8f/2/i) on the wave function /. 

The step that has just been outlined in detail is the most difficult step in the propagation of the wave function. The action with the operator exp —iV R,t)6t/2h) is straightforward as this operator is a local operator in the grid representation and we just multiply the grid representation of the wave function at grid point i by the value of the operator at the same grid point. 

The grid representation of the streams is shown in Figure 3.25. The vertical dotted lines represent the pinch and separate the grid into the regions above and below the pinch. 

Heat recovery system (HRS), 23 190 energy demand in, 23 190-192 grid representation of, 23 188-189 Heat recovery system process, 23 786-787 Heat release calorimeters, 22 458 Heat removal, from direct HDC chlorination reaction, 25 638 Heat resistance... 

These selective transitions (1), (7), and (9) may be achieved by proper optimization of the parameters eo and w, as described elsewhere [13, 18, 21]. Extensions to IR femtosecond/picosecond laser-pulse-induced dissociation or predissociation have been derived in Ref. 16, using either the direct or the indirect solutions of the Schrodinger equation (2) the latter requires extensions of the expansion (5) from bound to continuum states [16,31]. (The consistent derivation in Ref. 16 is based on S. Fliigge in Ref. 31). The same techniques can also be used for IR femtosecond/picosecond laser-pulse-induced isomerization as well as for more complex systems that are two dimensional, three dimensional, and so on, at the expense of increasing numerical efforts due to the higher dimensionality grid representations of the wavepackets f/(t) or the corresponding expansions (5) (see, e.g., Refs. 18, 20, and 21). 

Leforestier, C. (1991). Grid representation of rotating triatomics, J. Chem. Phys. 94, 6388-6397. 

Heat Exchanger Network Design Grid Representation... 

In recent years the solution of problems of large amplitude motions (LAM s) has usually been based on grid representations, such as DVR,[11, 12] of the Hamiltonians coupled with solution by sequential diagonalization and truncation (SDT[13, 9]) of the basis or by Lanczos[2] or other iterative nicthods[14]. More recently, filter diagonalization (ED) [5, 4] and spectral transforms of the iterative operator[15] have also been used. There has usually been a trade-off between the use of a compact basis with a dense Hamiltonian matrix, or a simple but very large D R with a sparse H and a fast matrix-vector product. 

Si is the only element exhibiting SM III substitution. Various possibilities to generate silicoaluminophosphate (SAPO) frameworks by substitution of P and Al atoms with Si are shown in Figure 2.11.[19] The substitution is conveniently explained by using a two-dimensional grid representation of T-atom configurations. 

Another solution involves reduced representations of the macromolecules. For this purpose, many simplified models of proteins were developed first for folding processes [19-22] and subsequently for docking problems [17, 23]. The various proposed models differ in their degree of simplification. A residue may be described by only one point per side chain [21, 24, 25] or more [20, 22, 26]. Other researchers prefer to use discrete positions with a grid representation of the molecules [4, 27]. A small interval enables obtaining a nearly atomic representation [18] a larger one yields a sharply reduced model [17, 23]. Many other techniques have been proposed to simplify the systems, e.g. spherical harmonics [28, 29], Connolly surfaces [8, 30, 31] etc. 

Recently we have proposed more efficient relativistic molecular theory by an application of the pseudospectral (PS) approach [135]. In the PS approach [136,137], we use the mixed basis function between a grid representation in the physical space and spectral representation in the function space. 

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