Unphysical behavior

The reason for this unphysical behavior lies in the assumption that v remains constant. A large value of N should result in very rapid shear stress relaxation which, in turn, reduces v (Figure 7.5). 

Figure 3.51 also contains a dissection of the total energy ( totai) into Lewis (ii(L)) and non-Lewis (ElSL>) components. The localized Lewis component E" corresponds to more than 99.3% of the full electron density, and so incorporates steric and classical electrostatic effects in nearly exact fashion. Yet, as shown in Fig. 3.51, this component predicts local minima (at 70° and 180°) and maxima (at = 0° and 130 ) that are opposite to those of the full potential. In contrast, the non-Lewis component E (NL) exhibits a stronger torsional dependence that is able to cancel out the unphysical behavior predicted by (L), leading to minima correctly located near 0° and 120°. Thus, the hyperconjugative interactions incorporated in E(SL> clearly provide the surprising stabilization of 0° and 120° conformers that counter the expected steric and electrostatic effects contained in ElL>. 

Likewise, if (F) (0) lies well below the reaction zone, the IEM model will collapse all points outside the reaction zone towards the mean values without passing through the reaction zone, i.e., the flame will be quenched even when the local reaction rate is infinite.66 Such unphysical behavior is avoided with the EMST model (Subramaniam and Pope 1999). 

Here, is the distance between atoms i andj, C(/ is a dispersion coefficient for atoms i andj, which can be calculated directly from tabulated properties of the individual atoms, and /dampF y) is a damping function to avoid unphysical behavior of the dispersion term for small distances. The only empirical parameter in this expression is S, a scaling factor that is applied uniformly to all pairs of atoms. In applications of DFT-D, this scaling factor has been estimated separately for each functional of interest by optimizing its value with respect to collections of molecular complexes in which dispersion interactions are important. There are no fundamental barriers to applying the ideas of DFT-D within plane-wave DFT calculations. In the work by Neumann and Perrin mentioned above, they showed that adding dispersion corrections to forces... 

Even for purely adiabatic reactions, the inadequacies of classical MD simulations are well known. The inability to keep zero-point energy in all of the oscillators of a molecule leads to unphysical behavior of classical trajectories after more than about a picosecond of their time evolution." It also means that some important physical organic phenomena, such as isotope effects, which are easily explained in a TST model, cannot be reproduced with classical molecular dynamics. So it is clear that there is much room for improvement of both the computational and experimental methods currently employed by those of us interested in reaction dynamics of organic molecules. Perhaps some of the readers of this book will be provide some of the solutions to these problems. 

There is a possibility that the PM3 parameter set may actually be the global minimum in parameter space for the Dewar-NDDO functional form. However, it must be kept in mind that even if it is the global minimum, it is a minimum for a particular penalty function, which is itself influenced by the choice of molecules in the data set, and the human weighting of the errors in the various observables included therein (see Section 2.2.7). Thus, PM3 will not necessarily outperform MNDO or AMI for any particular problem or set of problems, although it is likely to be optimal for systems closely resembling molecules found in the training set. As noted in the next section, some features of the PM3 parameter set can lead to very unphysical behaviors that were not assessed by the penalty function, and thus were not avoided. Nevertheless, it is a very robust NDDO model, and continues to be used at least as widely as AMI. 

Indeed, in reality everything in physics should be finite, but we know that we possess only some knowledge on asymptotic low-energy behavior of various physical quantities. Very often applying asymptotics beyond their applicability one goes into unphysical behavior of various results and, sometimes, to divergences. To make divergences finite one has to use a complete description, not its asymptotics, with exact laws instead of their asymptotic forms. 

The origin of the unphysical behavior of the diffusion equation and the reaction-diffusion equation can be understood from three different viewpoints (i) the mathematical viewpoint, (ii) the macroscopic or phenomenological viewpoint, and (iii) the mesoscopic viewpoint. 

As discussed in Sect. 2.2 the diffusion equation has the well-known unrealistic feature that localized disturbances spread infinitely fast, though with heavy attenuation, through the system. In that section we described three approaches to address the unphysical behavior of the diffusion equation and reaction-diffusion equation. Since the Turing instability is a diffusion-driven instability, it is of particular interest to explore how this bifurcation depends on the characteristics of the transport process. In this section, we address the effects of inertia in the dispersal of particles or individuals on the Turing instability. Does the finite speed of propagation of perturbations in such systems affect Turing instabilities We determine the stability properties of the uniform steady state for the three approaches presented in Sect. 2.2. 

At low elongational rates the heat flux m the direction of stretching decreases, whereas that in the transverse direction increases. For larger values of the elongation rate both components increase, ultimately becoming infinite. This unphysical behavior is presumably associated with the unrealistic infinite stretching of the dumbbells. 

Thus, Ak is a fluctuating quantity that obeys an equation similar to the fluctuating linear law, but containing the well-behaved random force. The motivation for our final choice of A is that the random force in Eq. (101) maintains the time reversibility of the equation for A°. The equation for the earlier mentioned A°, Eq. (95), was irreversible, giving rise to unphysical behavior at negative time. We now have... 

FCI is size consistent, but truncated Cl methods are not. This means the energy computed for two noninteracting molecules is not identical to the sum of the energies computed for the individual molecules. This unphysical behavior is a major drawback of any truncated CL So that CISD can be made approximately size consistent, the Davidson correction can be applied in which the contribution of quadruples, A q, is estimated from the correlation energy given at the CISD level, AEqisd and the coefficient of the reference configuration, ao ... 

The mounting criticisms of NDIS-93 prompted Soper et al to re-examine their high temperature NDIS experiments with greater attention to the inelasticity correction [116]. Realistic constraints were also imposed on the hard-core diameters of the pair correlations to avoid unphysical behavior at short range. According to the authors, their new method for analysing the raw data better represents the inelasticity corrections and, consequently, the resulting microstructures from different experiments become consistent with one another. 

In this section, we shall consider an exactly solvable model, that of driven Brownian oscillator (DBO) systems. The exact formulations will be presented in this subsection, while the POP-CS-QDT counterpart will be outlined in the next. We will show in Sec. 5 that the POP-CS-QDT and the CODDE agree with the exact QDT very well however, the COP-CS-QDT results in contrary and unphysical behaviors, for example, on its temperature dependence of the equilibrium properties. 

Both the mass—velocity and Darwin terms show unphysical behavior in variational calculations. To demonstrate this, consider a hydrogen-like atom (ion) with a single electron and a nuclear charge Z as a prototype example With a trial wavefunction of the form,... 

While the CP procedure is useful when examining specific structures, it is sometimes desirable to examine a path across the PES or reaction profile that might include, for example, reactants, products, and a transition state (TS). Generating a CP-corrected potential energy curve or reaction profile can introduce a new set of problems in certain circumstances. As demonstrated by Lendvay and Mayer, the Boys-Bernardi CP procedure (and variations thereof) can actually produce discontinuities in the PES near the TS and even give different TS energies for the forward and reverse reactions.Fortunately, this unphysical behavior is normally limited to regions near the TS. 

The extended pom-pom model sometimes predicts unphysical behavior, as discussed in Inkson, N. J., and T. N. PhillUps, J. Non-Newtonian Fluid Meek, 145, 92 (2007). 

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