Unphysical effect

Figure 1.3 shows how the concentration and rate vary in time for typical values of a0 and kq. Note that the time has to be extrapolated to — oo. This is a particular twist in the behaviour of simple autocatalytic rate laws. If no autocatalyst is present initially, bQ = 0 and hence the rate of production of B is also zero from eqn (1.10). The reaction thus takes an infinite time to get going. This unphysical effect is removed either by including a non-zero initial concentration of B (no matter how small) or by invoking an extra uncatalysed reaction (no matter how slow) converting A directly to B as discussed in the previous section. In the former case, with b0 0, the resulting integrated forms become ... 

The above equation is very conveniently used as the computation of the total energy is the standard quantum-chemical procedure. However, a purely theoretical problem arises when using monomer-centered basis set for evaluation of EA and EB according to (20.1) The intermolecular interaction energy will suffer from what is known as basis set superposition error (BSSE) [3], In order to overcome this unphysical effect which usually manifests itself in too negative interaction energies, one frequently applies the so-called counterpoise correction [4],... 

H + C2H4 this unphysical effect mimics quantum mechanical... 

Figure A2.5.7. Constant temperature isothenns of reduced Helmlioltz free energy A versus reduced volume V. The two-phase region is defined by the line simultaneously tangent to two points on the curve. The dashed parts of the smooth curve are metastable one-phase extensions while the dotted curves are unstable regions. (The isothenns are calculated for an unphysical r = 0.1, the only effect of which is to separate the isothenns...

Note that the Kolmogorov power spectrum is unphysical at low frequencies— the variance is infinite at k = 0. In fact the turbulence is only homogeneous within a finite range—the inertial subrange. The modified von Karman spectral model includes effects of finite inner and outer scales. 

At short distances, approximately equal to the excluded volume diameter, effective pair forces obtained from force matching exhibit unphysically large fluctuations. This is largely due to inadequate sampling of configurations at short distances in... 

From this point of view it is of interest to examine the consequences of full ther-malization of the classical Drude oscillators on the properties of the system. This is particularly important given the fact that any classical fluctuations of the Drude oscillators are a priori unphysical according to the Bom-Oppenheimer approximation upon which electronic induction models are based. It has been shown [12] that under the influence of thermalized (hot) fluctuating Drude oscillators the corrected effective energy of the system, truncated to two-body interactions is... 

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>. 

The existence of many ionic structures in MCVB wave functions has often been criticized by some workers as being unphysical. This has been the case particularly when a covalent bond between like atoms is being represented. Nevertheless, we have seen in Chapter 2 that ionic structures contribute to electron delocalization in the H2 molecule and would be expected to do likewise in all cases. Later in this chapter we will see that they can also be interpreted as contributions from ionic states of the constituent atoms. When the bond is between unlike atoms, it is to be expected that ionic stmctures in the wave function will also contribute to various electric moments, the dipole moment being the simplest. The amounts of these ionic structures in the wave functions will be determined by a sort of balancing act in the variation principle between the diagonal effects of the ionic state energies and the off-diagonal effect of the delocalization. 

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. 

The method of choice has to be carefully evaluated prior to a simulation study to assess whether a chosen functional is appropriate for a particular system. Ionic solvates are strongly dominated by Coulombic interactions and significant polarization effects are observed resulting from the presence of the charged solute. The unphysical self-interaction inherent to DFT is a striking disadvantage in these cases. Furthermore, the parametrization... 

The Aharonov-Bohm effect is self-inconsistent in U(l) electrodynamics because [44] the effect depends on the interaction of a vector potential A with an electron, but the magnetic field defined by = V x A is zero at the point of interaction [44]. This argument can always be used in U(l) electrodynamics to counter the view that the classical potential A is physical, and adherents of the received view can always assert in U(l) electrodynamics that the potential must be unphysical by gauge freedom. If, however, the Aharonov-Bohm effect is seen as an effect of 0(3) electrodynamics, or of SU(2) electrodynamics [44], it is easily demonstrated that the effect is due to the physical inhomogeneous term appearing in Eq. (25). This argument is developed further in Section VI. 

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