Validity of the Approximations

In the preceding sections, we have assumed that an absorption line has a Lorentzian shape. If this is not true, then the linewidth cannot be defined as the full width at half maximum intensity. Transitions from the ground state of a neutral molecule to an ionization continuum often have appreciable oscillator strength, in marked contrast to the situation for ground state to dissociative continuum transitions. The absorption cross-section near the peak of an auto-ionized line can be significantly affected by interference between two processes (1) direct ionization or dissociation, and (2) indirect ionization (autoionization) or indirect dissociation (predissociation). The line profile must be described by the Beutler-Fano formula (Fano, 1961)  

= (E — Er)/ T/2), where Er and T are the energy and full width of the resonance state (and e must not be confused with the kinetic energy of the electron). 

In the preceding sections, the case of one resonance isolated from the others and interacting with a single continuum of a specific nature has been discussed. A brief treatment of more general situations follows. 

If the resonance state interacts with several continua, Eq. (8.9.1) remains valid provided that one defines q by summing over each of these continua in Eq. (8.9.2). In the case where multiple continua are involved - for example, those corresponding to all vibrational levels of a specific electronic state of the ion -it can be shown (Smith, 1970) that the q and T parameters become independent of the vibrational quantum number of the Rydberg state. Note that the lineshape asymmetry parameter, q, does not depend on the quantum number n in contrast, the width T depends on (n ) 3. 

This generalized form of Eq. (8.9.1) can be applied only when a single discrete state interacts with one or several continua that do not interact with each other. This situation is typical for a predissociated state. Unfortunately, it does not often occur for an autoionized Rydberg state. When there are several resonances that have large widths relative to their separations, Fano s formula must be replaced by the formula given by Mies (1968)  

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Note that the relations (23) are valid also if (22) is questionable. Brown [19] refined the approximation (23) by introducing the gn factor, describing the deviation of the mean values for Lj and fi om integers. Validity of the approximation (23) has been checked by means of explicit ab initio calculations, for example, in [20,21]. 

Within the Slater-Koster appro.ximation, we can easily test the validity of the approximations made in eqn (7) based on the graphene model. In Fig. 5 we depict the band gaps using the empirical tight-binding method for nanotube radii less than 1.5 nm. The non-metallic nanotubes n m) are shown in the... 

Thus, for an almost empty surface, the rate assumes its maximum ivith equal amounts of reactants, at the limit of zero conversion. Again, we need to assess the validity of the approximations under the conditions employed. Nevertheless, the above procedure for determining the reaction rate as a function of mole fraction can be quite useful in the exploration of reaction mechanisms. 

Due to the complexity of a full quantum mechanical treatment of electron impact ionization, or even a partial wave approximation, for all but relatively simple systems, a large number of semiempirical and semiclassical formulae have been developed. These often make basic assumptions which can limit their range of validity to fairly small classes of atomic or molecular systems. The more successful approaches apply to broad classes of systems and can be very useful for generating cross sections in the absence of good experimental results. The success of such calculations to reproduce experimentally determined cross sections can also give insight into the validity of the approximations and assumptions on which the methods are based. 

The Feynman-Hibbs and QFH models perform quite well in free energy calculations as long as the quantum corrections are modest. The conditions for validity of the approximations are given above. 

Here we assume simply that some reaction steps remain in thermodynamic chemical equilibrium throughout the process. The validity of the approximation rehes on the fact that both the forward and the reverse reaction steps for the reaction assumed to be in equilibrium are very fast compared to others. 

Despite the fact that Bohr s stopping power theory is useful for heavy charged particles such as fission fragments, Rutherford s collision cross section on which it is based is not accurate unless both the incident particle velocity and that of the ejected electron are much greater than that of the atomic electrons. The quantum mechanical theory of Bethe, with energy and momentum transfers as kinematic variables, is based on the first Born approximation and certain other approximations [1,2]. This theory also requires high incident velocity. At relatively moderate velocities certain modifications, shell corrections, can be made to extend the validity of the approximation. Other corrections for relativistic effects and polarization screening (density effects) are easily made. Nevertheless, the Bethe-Born approximation... 

The approximation that the concentrations of HA and A remain unchanged breaks down for dilute solutions or at extremes of pH. We will test the validity of the approximation at the end of this chapter. 

However, since the QSSA has been used to elucidate most reaction mechanisms and to determine most rate coefficients of elementary processes, a fundamental answer to the question of the validity of the approximation seems desirable. The true mathematical significance of QSSA was elucidated for the first time by Bowen et al. [163] (see also refs. 164 and 165 for history and other references) by means of the theory of singular perturbations, but only in the case of very simple reaction mechanisms. The singular perturbation theory has been applied by Come to reaction mechanisms of any complexity with isothermal CFSTR [118] and batch or plug flow reactors [148, 149]. The main conclusions arrived at for a free radical straight chain reaction (with only quadratic terminations) carried out in an isothermal reactor can be summarized as follows. 

Recently Jain et al. [41] have examined the validity of the approximation Pt p used in deriving Eq. (3.42). They calculated the hole profile in a polymer sample at two different applied voltages. The profiles are shown in Fig. 3.8. Fig. 3.8 shows that near... 

It is clear that the validity of the approximation deaeases, the closer the ion comes toward the dipole, i.e., as r deaeases. 

How valid is the approximation that [HF] = 1.00 M Because this question will arise often in connection with acid-base equilibrium calculations, we will consider it carefully. The validity of the approximation depends on how much accuracy we demand for the calculated value of [H+], Typically, the Ka values for acids are known to an accuracy of only about 5%. Therefore, it is reasonable to apply this figure when determining the validity of the approximation... 

Note that in using this equation, we have assumed that the equilibrium concentrations of A- and HA are equal to their initial concentrations. That is, we are assuming the validity of the approximations... 

The way in which approximation 5 is used will be explained in Section 3.1.4, and the validity of the approximations will be discussed after the solution is obtained. 

When the steady-state approximation of Section B.2.5.2 is applied to nonisothermal systems involving molecular transport, it is sometimes referred to as the extended steady-state approximation. The simplifications that have just been indicated to follow from the application of this approximation underscore the importance of having methods for ascertaining the validity of the approximation. Criteria for the applicability of the extended steady-state approximation have been developed by Giddings and Hirsch-felder [64] and improved by Millan and Da Riva [89]. The discussion... 

The partial-equilibrium approximation differs from the steady-state approximation in that it refers to a particular reaction instead of to a particular species. The mechanism must include the forward and backward steps of any reaction that maintains partial equilibrium, and the approximation for a reaction k is then expressed by setting = 0 in equation (11). It is not always proper to conclude from this that when equations (6), (10), and (11) are employed in equation (14), the terms may be set equal to zero for each k that maintains partial equilibrium partial equilibria occur when the forward and backward rates are both large, and a small fractional difference of these two large quantities may contribute significantly to dcjdt. The criterion for validity of the approximation is that be small compared with the forward or backward rate. 

The 3s momentum distribution confirms the validity of the approximations. The experiment confirms the 3pi Hartree—Fock momentum distribution and eliminates the distribution for the sum over magnetic substates that would apply to target atoms that are not oriented. With perfect... 

Compare these results with (3-27) to verify the validity of the approximation. ////... 

When more than one conjugate add-base pair is in equilibrium with water, the exact mathematical relations for calculation of [H" "] become complex, especially if a single equation is to represent all possible initial conditions. The derivation of such equations can clarify the nature of approximations made in practical applications. In many cases, however, simplification may be achieved at the outset by using approximate calculations to estimate the concentrations of the major species concerned and then testing the validity of the approximations. If the concentration levels or equilibrium constants for a system are so unusual that the simple equations are not valid, exact equations can be used. " ... 

The hyperbolic tangent functions are in good agreement with numerical results, as shown in Fig. 14. To confirm the validity of the approximation, we reproduced... 

It must be stressed that in our analysis, the ft or Yh value used corresponds to a liquid, although the molecules adsorbed at infinite dilution can hardly be compared to an adsorbed liquid film. Therefore, in order to check the validity of the approximation contained in this intrepretation, the results were compared with those obtained using either an analysis developed by Gray (15-171 or die wetting method. 

The push to highlight performance on GPUs has meant that not one of the currently published papers on GPU implementations of MD actually provide any validation of the approximations made in terms of statistical mechanical properties. For example, one could include showing that converged simulations run on a GPU and CPU give identical radial distribution functions, order parameters, and residue dipolar couples to name but a few possible tests. 

The description of the van der Waals interaction based on the Lifshitz approach is now sufficiently advanced to provide accurate predictions for the complete interaction energy. For the geometry of two half-spaces, the exact theory is available in a formulation suited for computational purposes. " In parallel with work on planar systems, there has been a focus on the interaction between spheres. " These developed theories have been used as the exact solutions in the validation of the approximate predictions using the Hamaker approach. The significant contribution of the continuum approach to our understanding of the van der Waals interaction lies in the reliable prediction of the Hamaker constant. The interaction energy for two half-spaces and two spheres is summarized below. 

In the extrinsic case, the expression of overall charge neutrality should take into account the existence of immobile positive centers of donors that lost electrons and/or negative centers of acceptors that gained electrons. Also, the validity of the approximations (4.131) may some times become questionable. We will not dwell on the details of these calculations but it should be clear that they have now been reduced to merely technical issues. 

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