Exact treatment

The exact treatment of capillary rise must take into account the deviation of the meniscus from sphericity, that is, the curvature must correspond to the AP = Ap gy at each point on the meniscus, where y is the elevation of that point above the flat liquid surface. The formal statement of the condition is obtained by writing the Young-Laplace equation for a general point (x, y) on the meniscus, with R and R2 replaced by the expressions from analytical geometry given in... 

The full quantum mechanical study of nuclear dynamics in molecules has received considerable attention in recent years. An important example of such developments is the work carried out on the prototypical systems H3 [1-5] and its isotopic variant HD2 [5-8], Li3 [9-12], Na3 [13,14], and HO2 [15-18], In particular, for the alkali metal trimers, the possibility of a conical intersection between the two lowest doublet potential energy surfaces introduces a complication that makes their theoretical study fairly challenging. Thus, alkali metal trimers have recently emerged as ideal systems to study molecular vibronic dynamics, especially the so-called geometric phase (GP) effect [13,19,20] (often referred to as the molecular Aharonov-Bohm effect [19] or Berry s phase effect [21]) for further discussion on this topic see [22-25], and references cited therein. The same features also turn out to be present in the case of HO2, and their exact treatment assumes even further complexity [18],... 

Any chemical treatment that reduces general corrosion rates associated with oxygen corrosion will decrease tuberculation. The exact treatment that is best is system dependent. Water chemistries and operating practices may differ widely even among similar industries... 

The exact treatment of the scattering of a metal electron by an impurity traces back to the pioneering work of Friedel [1]. Compared to the complete matrix element... 

In spite of its simplicity this approach, supplemented with Blatt s correction [.3] for lattice distortion, was applied successfully for decades [4, 5] in studies of systematics in the residual resistivity. Its power was the exact treatment of the scattering and the use of the Friedel sum rule [1] as a self-consistency condition ensuring a correct valency difference between impurity and host atom. 

Gonis, A., Sowa, E.C., and Sterne, P.A., 1991, Exact treatment of Poisson s equation in solids with space-filling cells, Phys. Rev. Lett. 66 2207. 

The equilibrium between a pure solid and a gaseous mixture is one of very few classes of solution for which an exact treatment can be made by the methods of statistical mechanics. The earliest work on the theory of such solutions was based on empirical equations, such as those of van der Waals,45 of Keyes,44 and of Beattie and Bridgemann.3 However, the only equation of state of a gas mixture that can be derived rigorously is the virial expansion,46 66... 

The idea of constructing a good wave function of a many-particle system by means of an exact treatment of the two-particle correlation is also underlying the methods recently developed by Brueck-ner and his collaborators for studying nuclei and free-electron systems. The effective two-particle reaction operator and the self-consistency conditions introduced in this connection may be considered as generalizations of the Hartree-Fock scheme. 

The preceding oversimplified mathematical treatment really amounts to an evaluation of the absorption effect (6.1). The exponential term in Equation 6-4 is obviously a product of two exponential terms, each deriving from Beerks Law. One term governs the attenuation of the beam incident upon the volume element in question, and the other governs the attenuation of the characteristic line emerging frcJm this element. The films are so thin that the use of one value each for 6 and for 02 over the entire film thickness is justified. Finally, one must assume that the intensity measured by the detector remains proportional to the intensity of the source. An exact treatment of the problem would be so complicated that one is justified in seeing what can be done with the simple relationships obtained above. 

A further result of Sadler s 2D-simulation was a relation between the step density and growth rate on the one hand and the inclination of the surface with respect to the principal axes on the other. From this relation crystal shapes were derived which show considerable curvature. This result of an exact treatment stands in contrast to Frank s mean-field curvature expression which gives essentially flat profiles. We will return to the discussion of curved edges in Sect. 3.6.3. 

In Sections 42 and 43 we shall describe the accurate and reliable wave-mechanical treatments which have been given the hydrogen molecule-ion and hydrogen molecule. These treatments are necessarily rather complicated. In order to throw further light on the interactions involved in the formation of these molecules, we shall preface the accurate treatments by a discussion of various less exact treatments. The helium molecule-ion, He , will be treated in Section 44, followed in Section 45 by a general discussion of the properties of the one-electron bond, the electron-pair bond, and the three-electron bond. 

The following treatment starts with the complete quantal equations and introduces an eikonal representation which allows for a formally exact treatment. It shows how a time-dependent eikonal treatment can be combined with TDHF... 

Exact Treatment for the Freely Jointed Chain (or Equivalent Chain).4 >5— Consider one of the bonds of a freely jointed chain acted upon by a tensile force r in the x direction. Letting xpi represent the angle between the bond and the o -axis, its component on the x-axis is Xi = l cos pi. The orientation energy of the bond is —rXi, and the probability that its x component has a value between Xi and Xi- -dxi therefore is proportional to... 

The usual Tafel evaluation yielded a transfer coefficient a = 0.52 and a rate constant k of 4x 10 cm s at the standard potential of the MV /MV couple. This k value corresponds to a moderately fast electrochemical reaction. In this electrode-kinetic treatment the changes in the rate of electron transfer with pH were attributed only to the changes in the overpotential. A more exact treatment should also take into account the electrostatic effect on the rate of reaction which also changes with pH. 

Transported PDF methods combine an exact treatment of chemical reactions with a closure for the turbulence field. (Transported PDF methods can also be combined with LES.) They do so by solving a balance equation for the joint one-point, velocity, composition PDF wherein the chemical-reaction terms are in closed form. In this respect, transported PDF methods are similar to micromixing models. 

An exact treatment of the function Cqx = and evaluation of the gradient (dcQjdt)x=Q is complicated. Therefore, an approximate and a simplified method is proposed and presented below. 

It should be noted that the expression of the cavity term in Eq. (22) differs fiom that given by Halicio o and Sinano u (79/, 792) who presented a more exacting treatment of the thermodynamics of cavity formation. However, the difference between the energy calculated by the rigorous formulation and by the iq>proximation in Eq. (22) is only a few percent and seldom exceeds 0.4 kcal/mel. 

In contrast, the exact treatment of Merzhanov (Ref 14) does not give even approximate explosion times unless Eqs 4 are. >vuake/ ko-solved numerically — /itgi57... 

The correlation between the lowest computed barriers (irrespective of mechanism) and the logarithm of the experimental rate constants, k, is shown in Fig. 34. Given that the LFMM parameterization is based solely on ground state properties and that not all possible pathways have been considered, the agreement is remarkable and suggests that LFMM should provide a good basis for more exact treatments of reaction pathways and their attendant TSs. 

The exact treatment yields expressions which have the same form as the expressions given above only the numerical factors are different. The more detailed theory for the diffusion-convection problem between plane walls was developed by Furry, Jones, and Onsager (F10) and that for the column constructed from two concentric cylinders by Furry and Jones (Fll). Recently more attention has been given to the r61e of the temperature dependence of the transport coefficients in column operation (B9, S15). 

We now consider problems of a quantitative analysis of multiphonon transitions. Here an exact treatment seems hopeless at the present time, and to make headway at all a fair number of approximations are required. We shall give an overview of the general difficulties, discuss some (unfortunate) confusion on Born-Oppenheimer terminology, and then illustrate some quantitative problems using the adiabatic formulation (see below). The present discussion will also be used as a basis for subdividing the various papers, to be discussed in Section lOd, into various (perhaps somewhat arbitrary) categories. 

The last term is to be read as a principal value. The justification of this procedure is provided by the more exact treatment in 2. Performing the integration in (4.4) one finds, apart from the frequency shift (2.24),... 

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