Space complementary

Pig. 18.6 Example of a correlation map produced in the case of a simple cubic structure simulated at finite temperature (left). This map is obtained by averaging over the atoms and configurations of the MD trajectory. For each atom, the two first neighbors are selected, which define a projection pltme. Other atoms from the simulation box ate projected perpendicularly onto that plane if they belong to the space complementary to two cones centered on the defining atom and perpendicular to the pltme (rightpanel). By construction, the map is not symmetric (the shortest interatomic distance is located on the positive x axis)... 

In general, multiple-time-step methods increase computational efficiency in a way complementary to multipole methods The latter make use of regularities in space, whereas multiple-time-stepping exploits regularities in time. Figure 2 illustrates the general idea ... 

One of the most efficient algorithms known for evaluating the Ewald sum is the Particle-mesh Ewald (PME) method of Darden et al. [8, 9]. The use of Ewald s trick of splitting the Coulomb sum into real space and Fourier space parts yields two distinct computational problems. The relative amount of work performed in real space vs Fourier space can be adjusted within certain limits via a free parameter in the method, but one is still left with two distinct calculations. PME performs the real-space calculation in the conventional manner, evaluating the complementary error function within a cutoff... 

This quantity is found to be related to the local polarization energy and is complementary to the MEP at the same point in space, making it a potentially very useful descriptor. Reported studies on local ionization potentials have been based on HF ab-initio calculations. However, they could equally well use semi-empirical methods, especially because these are parameterized to give accurate Koopmans theorem ionization potentials. 

Within multiple-output forward converters, it is possible to easily combine the filter chokes of complementary outputs (i.e., +/- 5V, etc.) together on the same core (see Figure 3-2f). This offers several advantages it saves space, vastly improves cross-regulation of those outputs, and exliibits superior ripple voltage levels on both outputs. 

The nondiagonal elements are needed to preserve the complementary character of wave mechanics, e.g., to permit the description of a certain physical situation in ordinary space as well as in momentum space. 

A complementary approach has been reported very recently [43]. hi this case negative charges were introduced into the salen ligand Iq (Fig. 14) with the aim of exchanging it on cationic supports, such as a layered double (Zn, Al) hydroxide. The expansion in the basal spacing indicated intercalation, at least partially, of the Ig-Mn complex between the layers of [Zn2,i5Alo,86(OH)6,o2]- The complex was used in the epoxidation of (i )-limonene with molecular oxygen and pivalaldehyde. The use of N-... 

In Section 29.3 it has been shown that a matrix generates two dual spaces a row-space S" in which the p columns of the matrix are represented as a pattern P , and a column-space S in which the n rows are represented as a pattern P". Separate weighted metrics for row-space and column-space can be defined by the corresponding metric matrices and W. This results into the complementary weighted spaces and S, each of which can be represented by stretched coordinate axes using the stretching factors in -J v and, where the vectors w and Wp contain the main diagonal elements of W and W. ... 

Both theoretical and experimental data (in the solid, liquid, and gas phases) prove that the tendency of halocarbons to work as XB donors decreases in the order I > Br > Cl [66-68]. Clearly, polarizability and not electronegativity plays a key role. 3-Halo-cyanoacetylene works as self-complementary module and the N X distance is beautifully consistent with the scale reported above, being 2.932, 2.978 and 2.984 A in the iodo, bromo and chloro derivatives, respectively [69,70]. The same trend is observed when a phenyl, rather than a triple bond, spaces the donor and acceptor sites. The N Br distance in 4-bromobenzonitrile is longer than in the 4-iodo derivative [71,72] and no XB is present in the chloro and fluoro analogues, wherein molecules are pinned by N H and X- H short contacts [73]. PFCs have a very poor tendency, if any, to work as XB donors [74-77] and no crystal engineering can be based on such tectons. However, F2 is a quite strong XB donor and several adducts have been described in the gas phase [11,18] (see also the chapter by Legon in this volume). 

Fig. 27. Packing relations in the crystal structure of 47 benzene (1 1) 64>. Stereo drawing of complementary stick style and space filling representations of host and guest molecules, respectively (atomic radii of the corresponding guest atoms in the space filling style are set to about half of their common van der Waals values the H atoms of the host molecules are omitted)...

Fig. 34. Stereo drawing of the packing in the 20 DMSO clathrate 851 (complementary space filling and stick style representations of host and guest molecules, respectively O atoms of the host are shaded). Space around guest molecules in the center of the drawing, related by the symmetry center operator, indicates the opportunity for disorder...

In a complementary approach the backbones of dendrimers are jacketed as tightly as possible with dendrons of the largest possible space demand. Here the dendrons serve as space filling parts to implement sufficient strain to the dendrimers backbone in order to stretch it to the extreme. A good example for this is dendrimer 22a with its third generation dendrons [15b]. The molecular dimensions obtained from MD-simulations in vacuo show that (a) it attains an almost stretched conformation, (b) the dendritic layer around the PPP backbone... 

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