Optimal influence function

In words, E rp) is the backward finite Fourier transform of the product of —ifc, the forward finite Fourier transform of the mesh based charge density Pm and the so-called optimal influence function ( opt) given by... 

The most interesting ingredient of the P M method is the optimal influence function from Eq. 22. It is constructed such that the result of the mesh calculation is as close as possible to the solution of the original continuum problem. More precisely, the P M method is derived from the requirement that the resulting Fourier-space contribution to the force minimizes the the following error measure Q ... 

It is important to realize that Hockney and Eastwcwd not only provide a closed expression for the optimal influence function Gopt J5ut also a closed expression for the corresponding optimal error Qopt = Q[Gopt] ... 

Admittedly, Eq. 35 looks rather complicated. Still, in combination with Eq. 34 it gives the rms force error of the P M method or, more precisely, of its Fourier-space contribution. After all, the computation of Qopt and that of Gopt are quite similar. It should be emphasized that the formula Eq. 35 for the optimal Q value, just like the one for the optimal influence function in Eq. 22, is of a very general nature. It also works for different charge assignment functions, reference forces, or any differentiation scheme which can be expressed by an operator D(k). 

In the case of forces employing the mixed distribution most approaches neglect terms describing the dependence on the coefficients. In case of the mixed distribution this term is related to their influence on the nodal surface for iI/q and hard to estimate. Usually the contribution of this term is accepted to be small for an energy optimized wave function. One obtains... 

Since the stereochemical course of a catalytic hydrogenation is dependent on several factors, " an understanding of the mechanism of the reaction can help in the selection of optimal reaction conditions more reliably than mere copying of a published recipe . In the first section the factors which can influence the product stereochemistry will be discussed from a mechanistic viewpoint. In subsequent sections the hydrogenation of various functional groups in the steroid ring system will be considered. In these sections both mechanistic and empirical correlations will be utilized with the primary emphasis being placed on selective and stereospecific reactions. 

Ab initio methods allow the nature of active sites to be elucidated and the influence of supports or solvents on the catalytic kinetics to be predicted. Neurock and coworkers have successfully coupled theory with atomic-scale simulations and have tracked the molecular transformations that occur over different surfaces to assess their catalytic activity and selectivity [95-98]. Relevant examples are the Pt-catalyzed NO decomposition and methanol oxidation. In case of NO decomposition, density functional theory calculations and kinetic Monte Carlo simulations substantially helped to optimize the composition of the nanocatalyst by alloying Pt with Au and creating a specific structure of the PtgAu7 particles. In catalytic methanol decomposition the elementary pathways were identified... 

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