A First Example

Here c ) (ri) denotes the electrostatic potential generated by charge i as it is felt at the location of qi. This calculation of the repulsion energy is neither difficult nor time-consuming, but it must not be forgotten that the total number of interaction terms in this naive approach scales like O(M ) If we want to evaluate all interaction energies between two sets of M particles qu qiai - i Ma and 1/7, 2, . we end up with a number of pair-wise interaction 

This is where the strength of the multipole expansion comes into play. As we will derive later, the potential arising from an arbitrary charge can be expanded in terms of monopole ( ), dipole (D), quadrupole (Q), and higher order multipole interactions  

The expansion can be truncated after a finite number of, say, L terms. For the spherical multipole expansion, L = 15-21 is known to provide accuracies on the order of 10 a.u. and better in the total energy.Instead of taking into account all M field terms of the individual charges 7, the total field is then given by the L terms of the multipole expansion of the total charge distribution Q, as 

In the same manner, the total interaction energy of a point charge qi with all other charges qj is replaced with the net interaction 

instead of having 0 M) interaction terms for each point charge qi, we end up with a sum over the different orders of the multipoles of the charge dis-tribution(s) Cl. Because L is constant for a certain level of accuracy, we can thus calculate the interaction energy between 7, and one composite charge with complexity that is independent of the number of particles belonging to Cl. 

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A first example of application of microtomography is taken from life sciences. Here X-ray microscopy and microtomography allows to reconstruct the internal three-dimensional microstructure without any preparation and sometimes even of living objects. Fig. la shows an X-ray transmission microscopical image of bone (femoral head). Several reconstructed cross-sections are shown in Fig.lb. Fig.lc shows the three-dimensional reconstruction of this bone. 

In this chapter we illustrate, using a variety of examples, how the designer selects materials so that they provide him or her with the properties needed. As a first example, consider the selection of materials for a... 

Examples of Values of L and AF°. As a first example we may evaluate both L and AF° for a moderately soluble salt in aqueous solution. At 25° a saturated solution of potassium perchlorate has a concentration of 0.148 mole of KCIO4 in a 1000 grams of water that is to say, y+ = y = 0.148/55.5. The activity coefficient in the saturated solution has been taken1 to be 0.70 + 0.05. Using this value, we can estimate the work required to take a pair of ions from the crystal surface to mutually distant points, when the crystal is in contact with pure solvent at 25°C ... 

Battery systems of complex design and structure using—at least for one electrode—expensive materials are (for economic reasons) mainly conceived as storage batteries. Primary (and "reserve") versions of the zinc/silver oxide battery [(-) Zn/KOH/AgO (+)] — as a first example—are only used in particular cases where the question of cost is not crucial, e.g., for marine [26-28] and space applications [29]. 

The appropriate expression for the operator H in the above equations is that appearing in Eq. (8-160). Eor a first example, consider an ideal gas without interactions. Assuming that the one-particle wave functions used in the population density operators are the energy eigenfunctions, then the matrix H0llA is diagonal, and we can write... 

As a first example of applying the techniques described in section 2 let us look at the chain motion of linear polyethylene (LPE). A detailed study of a perdeuterated sample, isothermally crystallized from the melt, has been carried out in our laboratory24,25,44). Since all of this work is published and, in fact, has been reviewed extensively17 we can restrict ourselves to stating the main conclusions here ... 

The last topic to be treated is unequal reactivity by substitution effects. As a first example, the effect of an infinitely negative substitution effect in C due to a reaction with an h group (so I CD Kqj = 0) is compared with the case of equal (random) reactivity of the two functional groups in C for formulation F40. This is suggested as an example of polyesterification with an anhydride and a carboxylic acid, respectively. Figure 15 gives the dramatic effect on... 

As a first example we consider a system bounded periodically in two coordinates and by thermal walls in the other coordinate. The two thermal walls are at rest and maintained at the same temperature, T. The system is subjected to an acceleration field which gives rise to a net flow in the direction of one of the periodic coordinates. For this system, the hydrodynamic equations yield solutions of quadratic form for the velocity and quartic for the temperature. 

Recently, a first example of an organocatalytic asymmetric domino Knoevenagel/ Diels-Alder reaction was reported by Barbas and coworkers (Scheme 2.182) [409]. Spiro[5,5]undecane-l,5,9-triones of type 2-818/2-819 were obtained from comma-daily available 4-substituted-3-butene-2-ones 2-813, aldehydes 2-814, and Mel-drum s acid (2-801) in the presence of 20 mol% of the amino acid 2-815 with 80-95% ee and diastereoselectivity of dr> 12 1. 

Grigg and coworkers developed bimetallic domino reactions such as the electro-chemically driven Pd/Cr Nozaki-Hiyama-Kishi reaction [69], the Pd/In Barbier-type allylation [70], Heck/Tsuji-Trost reaction/1,3 dipolar cycloaddition [71], the Heck reaction/metathesis [72], and several other processes [73-75]. A first example for an anion capture approach, which was performed on solid phase, is the reaction of 6/1-134 and 6/1-135 in the presence of CO and piperidine to give 6/1-136. Liberation from solid phase was achieved with HF, leading to 6/1-137 (Scheme 6/1.30) [76]. 

A first example of a combination of a Rh-catalyzed allylic substitution and a Pau-son-Khand annulation reaction has also been developed by the same group [222]. Thus, [RhCl(CO)dppp]2 is able to catalyze both transformations at different reaction temperatures. Treatment of the allylic carbonate 6-106 with the alkyne derivative 6-107 led to a diastereomeric mixture of 6-108 and 6-109 in 63-84% yield, with 6-108 as the main product (Scheme 6/2.23). 

Gauffre, F. and Roux, D. (1999) Studying a new type of surfactant aggregate ( Spherulites ) as chemical microreactors. A first example Copper ion entrapping and particle synthesis. Langmuir, 15, 3738-3747. 

Whether [6]radialenes have a potential as novel ligands for metal complexes remains to be seen. A first example of a successful complexation is provided by 150, which reacts with tris(acetonitrile)tricarbonylchromium in dioxane at room temperature to give the ortho-xylylene chromium complex 175 in excellent yield (83%) (equation 23)105. 

As a first example of a CFD model for fine-particle production, we will consider a turbulent reacting flow that can be described by a species concentration vector c. The microscopic transport equation for the concentrations is assumed to have the standard form as follows ... 

We consider as a first example sorption of selenate (SeO ), as predicted by the reaction K(, reaction Freundlich, and Langmuir approaches (Sections 9.1-9.3). Alemi et al (1991) observed the partitioning of selenate in batch experiments between 10 g of a loamy soil and 20 ml of a pH 7.5 solution containing small amounts of Na2Se04 their results are shown in Figure 9.1. 

In a first example of how minerals can buffer a fluid s chemistry, we consider how a hypothetical groundwater that is initially in equilibrium with calcite (CaCC F) at 25 °C might respond to the addition of an acid. In REACT, we enter the commands... 

We choose as a first example the evaporation of spring water from the Sierra Nevada mountains of California and Nevada, USA, as modeled by Garrels and Mackenzie (1967). Their hand calculation, the first reaction path traced in geochemistry (see Chapter 1), provided the inspiration for Helgeson s (1968 and later) development of computerized methods for reaction modeling. 

As a first example, we consider the diagenesis of clastic sandstones in the Gippsland basin, southeastern Australia, basing our model on the work of Harrison (1990). The Gippsland basin is the major offshore petroleum province in Australia. Oil production is from the Latrobe group, a fluvial to shallow marine sequence of Late Cretaceous to early Eocence age that partly fills a Mesozoic rift valley. 

As a first example of developments of the last Section, we now consider the electromagnetic field satisfying Dirichlet boundary conditions on parallel planes (metallic plates), normal to the -direction, at finite temperature. In this case, vl(a) is given by Eq. (25) with = — 1 (corresponding to the choice a = (/ , 0,0, i2L)) and Eq. (28) reduces to... 

As a first example, let us consider a metallic thermistor inserted in fig. 3, whose resistance is, in a first approximation, expressed as R(T)=Ro(l+aT). R(T) is the resistance of a PTC thermistor at a given temperature T, Ro is the resistance at To, and I represents a suitable DC (or AC current), while A is the constant gain of a low noise amplifier, operating in a suitable bandwidth. Let us suppose that the injected current I does not induce, through the heating process, a detectable change of the resistance value. 

As a first example, the transient case with Henry isotherm can be considered. Expressions developed in Section 2.3 apply with D replacing Dm,ct m replacing cM (including the substitution of c v M by < M and cfsM by c f ) and Ku (defined as r/cM(r0,t) in both cases, i.e. with or without the presence of L) by AT i / (1 + Kc ). Other cases with analytical solutions arise from the steady-state situation. The supply flux under semi-infinite steady-state diffusion is [57] ... 

For the central violene rc-system, we chose benzene with acetylene spacers for decreasing steric hindrance between the large cationic centers. Therefore, the tetracation 254 salt was designed as a first example for a hybrid with a cyanine unit at the both termini (27). As illustrated in Figure 22, tetracation 254 exemplifies a new hybrid system, which produces another cyanine substructure 252 by two-electron transfer. In this case, overall, four electron transfer should afford a fully neutralized species such as a neutral diradical. 

The systems V-Mo and Cs-Rb presented in Fig. 2.1 are a first example of phase diagrams. 

As a first example we discuss a mechanism in which the formation of the enzyme-intermediate complex, El, is sensitive to hydrogen isotopic substitution, while the next step characterized by rate constants ks and k6 exhibits a carbon kinetic isotope effect. Expressions for the three kinetic isotope effects that can be determined experimentally are ... 

As a first example, let us consider the time-dependent mean position of a normal mode xj of the system. In a mixed quantum-classical calculation, this observable is directly given by the quasiclassical average over the nuclear trajectories x t), that is. 

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