Noncentral forces

The mean square torque is another test of the pair potentials used. The calculated mean square torques are very potential dependent they range from 7 x 10"31 to 36 x 10"28 (dyne-cm)2. The experimental values51 of the mean square torque in solid CO at 68°K and in liquid CO at 77.5°K are 19 x 10 28 and 21 x 10 28 (dyne-cm)2, respectively. Therefore, the Stockmayer potential clearly does not represent the noncentral forces in liquid CO, i.e., this potential is much too weak. On the other hand, the noncentral part of the modified Stockmayer potential is too strong. However, as pointed out previously, this problem can easily be solved by using a smaller quadrupole moment. The mean square torques from the other two potentials agree quite favorably with the experimental values. We conclude from the above that the quadrupole-quadrupole interaction can easily account for observed mean square torques in liquid CO. 

Considering nonlinear effects in the vibrations of a crystal lattice (see, for example, [8]) it is necessary to take into account anharmonicity only in the terms connected to the largest interatomic forces, while the potential energy of weak forces of interlayer (or interchain) interactions, as well as noncentral forces should be considered in the harmonic approach. Therefore in (1) it is possible to neglect the summands, containing correlators of the atom displacements from various layers or chains, i.e. the correlators of... 

As shown in Figure 9, the FPM model represents a generalization not only of DPD but also of the MD technique. It can be used as DPD by setting the noncentral forces to zero, or as MD by dropping the dissipative and Brownian components. These three techniques also can be combined into one three-level hybrid model. As shown in Figure 9, the three-level system consists of three different procedures representing each technique invoked in dependence on the type of particle interactions. We define three types of particles ... 

Fluid particles (FP) are the Tumps of fluid partic les in the bulk solvent, with interaction range <1.5 X. Noncentral forces are included within this framework. 

The term m = 0.74048 Vm°/Vm = 1/6 7rN0cJm/Vm> where Vm° is the close-packed volume, N0 is the Avogadro number, and Vm is the molar volume of the system. V° is a simple function of the temperature (T) (10) with a characteristic value V°° at T = 0 K. The last term in Equation 12 was introduced by Alder et al. (II). Dnm are 24 universal constants common for all substances whose radial and higher distribution functions are the same functions of u/kT and the reduced density p = V°/V. As shown by Chen and Kreglewski (10) and Simnick, Lin, and Chao (12), Equation 12 is the most accurate known equation with four characteristic constants a, V°° (V° at T = 0 K), u°/k, and rj/k (see Equations 13 and 14). They also have shown (10) that in order to obtain agreement with second virial coefficient data of the gas and the internal energy or the enthalpy of the liquid, it is necessary to assume that u(r) is a function of T as required by the theory of noncentral forces between nonspherical molecules (13)... 

In all theories of polymer solutions tiy or always are assumed to be independent of the temperature, apparently contradicting the theory of noncentral forces. Our results show that there is no contradiction and that this assumption is allowed for long straight or circular chains at high densities (liquids below their normal boiling temperatures). 

The frequencies of LO and TO modes of the asymmetric O stretch are given by the central and noncentral force approximations [25] ... 

In this section, we present some examples of GMDF s. We confine ourselves to spherical particles in two dimensions. (In Chapter 6, we present some further examples for particles interacting via noncentral forces.) All the illustrations given in this section were obtained by a Monte Carlo computation on a two-dimensional system, consisting of 36 Lennard-Jones particles, for which the pair potential is presumed to have the form [for more details, see Ben-Naim (1973b)]... 

For noncentral forces or bound states containing more than two constituents, eq. (2.11) is rewritten as [100,101]... 

Thus we have seen that no essential difference exists between cubic and hexagonal cohesive energies as long as a central force is assumed between molecules. An alternative approach is therefore needed noncentral forces must be considered, as was pointed out by Prins et al. and Barron and Domb. An explanation by the present author is the following. 

The new surface modes appear because of the broken translational symmetry at the surface. However, at real surfaces, the force constants within the first layer and between the very first layers will be modified with respect to the bulk force constants. This originates from the modified electronic structure at the surface and the surface relaxation. Furthermore, we considered here for simplicity only central nearest-neighbor forces. For a better description, the range of interaction has to be extended to next-nearest neighbors and even longer distances. Additional noncentral forces as, for example, bond-bending forces have to be included for a proper description of the phonon dispersion in most materials. 

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