Curvilinear constant

A three-dimensional body limited by two curvilinear surfaces is called a shell if a distance called a thickness of the shell between the afore mentioned surfaces is small enough. We assume that the thickness is the constant 2h > 0. The surface equidistant from the surfaces is called a mid-surface. Thus, a shell can be uniquely defined introducing a mid-surface, a thickness and a boundary contour. 

As in Sect. 2.1, Dj is the curvilinear centre-of-mass diffusion constant of the chain, and is given in terms of the monomeric friction constant by the Einstein relation Dj =kT/Nl. L is as before the length of the primitive path, or tube length of the chain, which is Finally, we need the initial condition on p(s,t), which... 

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). 

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).

Semilogarithmic plots of formation pressure versus reciprocal absolute temperature yield straight lines, over limited temperature ranges, for hydrate formation from either liquid water, or ice. From Equation 4.13 such linear plots either indicate (1) relatively constant values of the three factors (a) heat of formation, AH, (b) compressibility factor, z, (c) stoichiometry ratios of water to guest or (2) cancellation of curvilinear behavior in these three factors. 

For three-dimensional problems the integral formulations previously obtained are also valid and are implemented into two-dimensional elements that cover the domain surface as shown in Fig. 10.15. Here, we use triangular and rectangular elements as used with FEM. Again, depending on the number of nodes per element, we can have constant, linear and quadratic elements. To be able to represent any geometry it is best to use curvilinear isoparametric elements as schematically illustrated in Fig. 10.16. 

If the energy acceptors are in great excess (c Jf), then the condition (3.556) is not too rigid the interval of relatively weak fields where Bs c is rather wide. Within this interval the conventional Stem-Volmer law is valid and its constant is given by Eq. (3.555). This statement relates to the three upper curves shown in Figure 3.68, which are almost linear in c. However, at a much higher density of fluorophores the inequality (3.556) is inverted at small c and the concentration dependence of the quantum yield becomes curvilinear similar... 

It is nice to have a distinctive notation for the curvilinear co-ordinates, which emphasizes their difference from and yet their one-to-one correlation with the Rt co-ordinates. Most authors reporting anharmonic calculations do not in fact make any distinction they denote the curvilinear co-ordinates by the same symbols customarily used to denote the corresponding rectilinear coordinates in harmonic calculations. For many purposes this is satisfactory, particularly since the harmonic force constants are not altered by the change from rectilinear to curvilinear co-ordinates. However, in a general discussion it is important to distinguish the two sets, and so for the remainder of this section we shall follow Hoy et al.12 and write the curvilinear co-ordinates with the symbol Hi. 

The discussion so far may be summarized as follows. There are two reasons for using curvilinear co-ordinates to represent the anharmonic force field of a polyatomic molecule, despite their apparent complexity. The first is that it is only in this way that we obtain cubic and quartic force constants which are independent of isotopic substitution. The second is that in terms of curvilinear bond-stretching and angle-bending co-ordinates we obtain the simplest expression for the force field, in the sense that cubic and quartic interaction terms are minimized. The first reason is compulsive the second reason is not compulsive, but it does make the curvilinear co-ordinates very desirable. 

The quadratic force constants in the normal co-ordinates are, of course, diagonal, i.e. H = eor<5J<5 .) The expansion (49) is then obtained by substituting (42) and (44) into (48). In fact it is possible to write closed formulae for the force constants in terms of the force constants /, as is done in equations (11) of ref. 12. It is an important property of this transformation that a purely quadratic force field in the curvilinear co-ordinates 8 gives rise to quadratic,... 

Symmetry, and the Number of Independent Force Constants.—As in harmonic calculations, the rather general discussion of the preceding section can be simplified in particular cases by making use of symmetry, as discussed by Hoy et a/.12 Thus we may choose the curvilinear co-ordinates Jfin linear combinations that span the irreducible representations of the point group we denote such symmetrized curvilinear co-ordinates by the symbol S, and we define them by means of a U matrix exactly analogous to that used for rectilinear coordinates ... 

Table 8 Force constants in curvilinear internal co-ordinates for HCN ...

Kuchitsu and co-workers5 7 were the first to introduce what is perhaps the simplest and most generally useful model, in which they assume all anharmonic force constants in curvilinear co-ordinates to be zero with the exception of cubic and quartic bond-stretching constants. These may be estimated from the corresponding diatomics, or from a Morse function, or they may be adjusted to give the best fit to selected spectroscopic constants to which they make a major contribution. This is often called the valence-force model. It is clear from the results on general anharmonic force fields quoted above that this model is close to the truth, and in fact summarizes 80 % of all that we have learnt so far about anharmonic force fields. 

In this case, the binding sites and binding constants can be determined from each linear portion of the curvilinear line as shown in Figure 3.30. 

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