Discontinuously interactions

When the cutoff is sharp, discontinuities in the forces and resultant loss of con servation of energy m molecular dynamics calcnla-tionscan result.To minimi/e edge effects of a cu toff, often theciit-off IS implemented with a switching or shifting function to allow the interactions to go smoothly to /ero. 

IlyperChem avoids th e discon tin nily an d, in isotropy problem of th e implied cutoff by iin posing a sin oothed spherical cn toff within the implied cutoff. When a system is placed in a periodic box, a switched cnLoITis aiitoinatically added. The default outer radius, where the interaction is completely turned off, is the smallest of 1/2 R., 1/2 R.. and 1/2 R, so that the cutoff avoids discontinuities and is isotropic, fh is cutoff may be turned off or modified in the. Molecular Mechanics Options dialog box after solvation and before calcii lation. ... 

IS added to the short-range molecule-molecule interaction. Problems with the reaction ethod may arise from discontinuities in the energy and/or force when the number of les j rvithin the cavity of the molecule i changes. These problems can be avoided by dng a switching function for molecules that are near the reaction field boundary. 

We will be concerned with the interaction of waves with boundaries and with other waves throughout this text. To determine how these interactions take place, it is important to consider that discontinuities in either pressure or particle velocity cannot be sustained in any material. If a discontinuity in either of these variables is created at some point by impact or wave interaction, the resulting motion will be such that the pressure and particle velocity become continuous across the boundary or point of interaction. Unless the material separates at that point, the motion will consist of one or more waves propagating away from the point of the discontinuity. For pressure discontinuities, it is easy to see that waves must propagate by again considering an... 

It is important to note that the state determined by this analysis refers only to the pressure (or normal stress) and particle velocity. The material on either side of the point at which the shock waves collide reach the same pressure-particle velocity state, but other variables may be different from one side to the other. The material on the left-hand side experienced a different loading history than that on the right-hand side. In this example the material on the left-hand side would have a lower final temperature, because the first shock wave was smaller. Such a discontinuity of a variable, other than P or u that arises from a shock wave interaction within a material, is called a contact discontinuity. Contact discontinuities are frequently encountered in the context of inelastic behavior, which will be discussed in Chapter 5. 

Section 1.9 showed that as long as an oxide layer remains adherent and continuous it can be expected to increase in thickness in conformity with one of a number of possible rate laws. This qualification of continuity is most important the direct access of oxidant to the metal by way of pores and cracks inevitably means an increase in oxidation rate, and often in a manner in which the lower rate is not regained. In common with other phase change reactions the volume of the solid phase alters during the course of oxidation it is the manner in which this change is accommodated which frequently determines whether the oxide will develop discontinuities. It is found, for example, that oxidation behaviour depends not only on time and temperature but also on specimen geometry, oxide strength and plasticity or even on specific environmental interactions such as volatilisation or dissolution. 

In such cases as these it is evident that a continuous transition from one extreme structure to another could occur. If, however, the structures have different multiplicities, they cannot be combined with one another (so long as spin-orbit interactions are negligible), so that the transition from one extreme bond type to the other would be effectively discontinuous. 

Similar arguments lead to the prediction that the cross conjugate TMM dication should be more stable than the linear conjugate BD dication. The cyclic orbital interaction is favored by the continuity of orbital phase in the TMM dication, but the orbital phase is discontinuous in the BD dication. 

There is a degree in the continuity and discontinuity of the orbital phase [20]. 2-Oxopropane-l,3-diyl (Scheme 10) is a hetero analog of trimethylenemethane (TMM) where the orbital phase is continuous in the triplet diradical (Sect. 2.1.5) and discontinuous in the singlet diradical (Sect. 2.1.6). The n and orbitals of carbonyl bonds are lower in energy than those of C=C bonds. The lowering strengthens the interaction of the radical orbitals (a, b) with and weakens that... 

For the EAG-snbstimted alkenes, the transition states are non-cycUc E-Il-EAG systems. Polarization of Ft, induced by the delocalization from Ft to EAG and E, determines the regioselectivities. The polarization is analogous to that in the TMM dication (Sect. 2.1.4). The cyclic interaction occurs among the electron-accepting orbital eag ) of the substituent, e, n, and n. The a addition is favored by the orbital phase continuity while the P addition is disfavored by the phase discontinuity (Scheme 16). 

The orbital phase theory can be applied to the thermodynamic stability of the disubstituted benzene isomers. The cyclic orbital interaction in the benzene substituted with two EDGs is shown in Scheme 21. The orbital phase is continuous in the meta isomer and discontinuous in the ortho and para isomers (Scheme 22, cf. Scheme 4). 

Butanes are chosen as the simplest models for the normal and branched isomers. Both branched and normal isomers contain a C-C bond (2 ) interacting with the terminal C-H bonds (2 and 2 ) (Scheme 26a). The cyclic -aj-a2 -a3 a2- interaction (Scheme 26b) occurs in the polarization of the middle C-C a-bond by the interactions with the antiperiplanar C-H a-bonds. The orbital phase is continuous in the branched isomer and discontinuous in the normal isomer (cf Scheme 4). The branched isomer is more stable. The basic rule of the branching effects on the stability of alkanes is ... 

The n orbitals on the two CO molecules interact with the same lobe of a vacant 3p orbital on a metal atom in the model for the acute angle coordination, and with different lobes for the obtuse angle coordination (Scheme 29b). Cychc orbital interaction occurs between the occupied 3s orbital and the vacant 3p orbitals on M and the n orbitals, n, and n, of the CO molecules (Scheme 29c). The phase is continuous for the same lobe interaction and discontinuous for the different lobe interaction (Scheme 29d, cf. Scheme 4). The acute-angle coordination is favored. 

Fig. 6a-c Through-bond interactions in the singlet state of 1,3-diradical, a Mechanism of electron delocalization and polarization of a-spin electrons, b Cyclic orbital interaction, c Orbital phase discontinuity (denoted by the dotted lines)... 

To summarize, the properties of triplet and singlet diradicals are closely related to the effectiveness of through-bond and through-space interactions, which are governed by the orbital phase continuity/discontinuity properties. In the next two sections, we will utilize this simple model to predict the spin preference and intramolecular reactivity for a broad range of diradicals. 

T gap (cf. references collected in Table 1). This correlates well with a disfavored cyclic six-orbital interaction by the phase discontinuity in the triplet state of 7 [29] (shown in Fig. 11). In addition, TME is an important topological unit which appears frequently in many non-Kekule diradicals (as exemplified by 15-17 in Fig. 13). 

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