Displacement, curvilinear

To obtain the anharmonic terms in the potential, on the other hand, the choice of coordinates is important 130,131). The reason is that the anharmonic terms can only be obtained from a perturbation expansion on the harmonic results, and the convergence of this expansion differs considerably from one set of coordinates to another. In addition it is usually necessary to assume that some of the anharmonic interaction terms are zero and this is true only for certain classes of internal coordinates. For example, one can define an angle bend in HjO either by a rectilinear displacement of the hydrogen atoms or by a curvilinear displacement. At the harmonic level there is no difference between the two, but one can see that a rectilinear displacement introduces some stretching of the OH bonds whereas the curvilinear displacement does not. The curvilinear coordinate follows more closely the bottom of the potential well (Fig. 12) than the linear displacement and this manifests itself in rather small cubic stretch-bend interaction constants whereas these constants are larger for rectilinear coordinates. A final and important point about the choice of curvilinear coordinates is that they are geometrically defined (i.e. independent of nuclear masses) so that the resulting force constants do not depend on isotopic species. At the anharmonic level this is not true for rectilinear coordinates as it has been shown that the imposition of the Eckart conditions, that the internal coordinates shall introduce no overall translation or rotation of the body, forces them to have a small isotopic dependence 132). 

Curvilinear displacement of a monomer (labelled by a dark circle) along the contour of the tube between two conformations is A.v. Only a short section of the tube is shown. 

Displacements of monomers at the two ends of the tube are unrelated to each other on lime scales shorter than the Rouse time of the chain (r < Tr). These incoherent curvilinear displacements lead to tube length fluctuations [Eq. (9.70)] ... 

What is the mean-square curvilinear displacement of monomers in an entangled melt along the confining tube on time scales between the... 

Moreover, even in view of dynamical calculations, the IRP has a rather severe drawback (a similar situation prevails for the genuine steepest-descent path) according to the Reaction Path Hamiltonian philosophy, initially developed by Miller, Handy and Adams [33] and extended by others [15,18,31,36,37], the curvilinear displacement along the IRP (i.e. the parameters t or T above) should be taken as one of the internal coordinates describing the system This is undoubtedly a clever idea, since the value of that very coordinate would, throughout the dynamical calculations, be a measure of the advancement of the reactive process. The complications stem from the fact that (i) it can be fairly complicated to define the complementary 3N-7 internal coordinates (except for few isomerizations in which they can be considered small-amplitude vibrational coordinates) and (ii) in many cases, multivaluation difficulties cannot be avoided, especially when large-amplitude motions drive the representative point of the system in... 

Otherwise, Jacobi coordinates can also be used to account for isomerization reactions, e.g. HCN CNH and HO2 O2H, in which case the reaction coordinate is rather the bending angle a in Figure 2a. In addition, it is clear from Figure 3 that the connection between a and S, the curvilinear displacement along the IRP is nearly linear, for both systems, whatever the PES in the case of O2H and everywhere throughout the reaction. 

Figure 3. The relation between S (the curvilinear displacement coordinate along the IRP) and the Jacobi bending angle a, is almost linear, for HCN CNH and the Murrell et al PES [54] (a) as well as for HO2 O2H and two different PES by Varandas et al [55,56] (b and c)...

In a melt, polymer chains are Gaussian if a monomer has a curvilinear displacement l(t) it moves over a spatial distance r(t) The Euclidian displacement of monomers by reptation is thus... 

Fig. 8. Polymer segments in a tube are displaced along the randomly coiled contour line resulting in the curvilinear displacement s in the experimental diffusion time (drawn in gray). Field-gradient NMR diffusometry measures displacements Rj,z along the gradient direction (G). Contour loops act as traps for segments the displacements of which are monitored along G...

Figure Al.2.6. Anliannonic stretch nonnal modes of a synnnetric triatoniic. The plot is similar to figure Al.2.5. except the nonnal modes are now anliannonic and can be curvilinear hi the bond displacement coordinates r, r. The antisyimnetric stretch is curved, but the synnnetric stretch is linear because of synnnetry.

The dextral displacement reaches a maximal value leading to a curvilinear Schild regression. 

Local reptation regime For times t > xe we have to consider curvilinear Rouse motion along the spatially fixed tube. The segment displacement described by Eq. (18) (n = m) must now take the curvilinear coordinates s along the tube into consideration. We have to distinguish two different time regimes. For (t < xR), the second part of Eq. (19) dominates - when the Rouse modes... 

Fig. 12. Potential energy surface for C2v structures of HjO. Rectilinear displacements of the hydrogen atoms would occur along straight lines on this surface. The dotted line shows a curvilinear motion keeping the OH bond length constant. The contours (- 10, - 9, - 8, - 7 eV) have been drawn from the analytical potential of Murrell and Sorbie U35).

The magnitude of the accelerating force that acts on a particle in curvilinear motion depends on the particle inertia. The greater the inertia of the particle, the greater will be the displacement. Inertia depends on particle mass and velocity. Heavy particles will be displaced more from the streamlines in which they are traveling than light ones, and increases in velocity will increase displacement for a particle of given mass. 

In the mid 1950s Durwaxd W. J. Cruickshank " noted that atoms in, for example, a rotating molecule, are displaced towards the rotation axis. Rotational oscillations of molecules, such as found in crystalline benzene near its melting point, will cause an apparent displacement of atomic positions from their true positions because the best fit to the electron density should be curvilinear but, with the limitations of present-day techniques, is generally linear (Figure 13.15). If the root-mean-square amplitude of libration about an axis is o> (in radians), then the apparent (but not real) shortening of the bond, d, is ... 

Although equal in value to the rectilinear coordinate Qx, the parameter /s can be treated as a curvilinear coordinate that follows the infinitesimal displacement of a point on the seam along the local tangent vector to the curve, t(/3). This moving frame is completed by the normal vector, n(/3). At the expansion point (origin of the frame fj, = 0), the normal and tangent vectors to the seam are parallel to xi and X3 (unit vectors), respectively. However, away from that point, these vectors are different and combine xi and X3 because the seam is curved (Fig. 5). 

For times shorter than the Rouse time of the chain (tcoherent motion of a chain segment consisting of a/t/to neighbouring monomers. The time-dependent curvilinear coordinate of a monomer along the contour of the tube is s t) (Fig. 9.19). The mean-square monomer displacement along the tube is of the order of the mean-square size of this section in three-dimensional space [Eq. (8.58)] ... 

This curvilinear motion continues up to the reptation time trep where the chain has curvilinearly diffused the complete length of the tube, of order aNjN. At times longer than the reptation time (/ > Trep) the mean-square displacement of a monomer is approximately the same as the centre of mass of the chain and is a simple diffusion with diffusion coefficient D [Eq. (9.12)]. 

Sole effects on affinity (with no change in receptor function) result in surmountable antagonism. The dextral displacement reaches a maximal value leading to a curvilinear Schild regression. 

FIGURE 1 Distinction between rectilinear (far//) and curvilinear (fight) representations of valency bond-angle displacements. 

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