Non-central forces

Commentary on The statistical mechanics of systems with non-central force fields,... 

THE STATISTICAL MECHANICS OF SYSTEMS WITH NON-CENTRAL FORCE FIELDS... 

The development of a general theory of systems with non-central force fields can be divided into two parts. First the many types of directional interaction that may occur have to be classified within a general mathematical framework and then approximate methods of evaluating the partition function have to be devised. This paper summarizes some of the results of a method developed by the author 2 with particular reference to its application to the properties of liquid mixtures. 

With the aim of coarse-graining DPD, the fluid particle method (FPM), a non-central force has been introduced that is proportional to the difference between the velocities of the particles (Espan ol 1998). This force exeits additional drag, which produces rotational... 

Hence, a further advancement in studies of dislocations and interfaces in TiAl requires introduction of non-central forces into atomistic calculations. This character of bonding is, of course, included in ab-initio electronic structure calculations but such calculations are still not feasible on the scale needed when investigating dislocations and interfaces. Empirical attempts to include directionality of bonding have been made (Panova and Farkas 1995) but only employment of queintum mechemics based potentials can reveal the effects of non-central forces unambiguously. Such potentials have been emerging recently but have not yet been employed extensively in studies of lattice defects (Aoki and Pettifor, 1994 Memh et al., 1995 Pettlfor et al., 1995). Incorporation of... 

Although certain of the above-mentioned theories are moderately successful in representing the experimental data of CF4 -t- CH and other fluorocarbon + hydrocarbon mixtures, experimental values of and x are required. At present there is no satisfactory method of obtaining these parameters a priori. Scott, in his 1958 review, considered the various possible factors that could lead to a weakening of the unlike interactions in such mixtures. He concluded that the three most significant were the presence of non-central forces, differences in ionization potential, and differences in size of the two component molecules. The use of the Kihara potential together with the Hudson and McCoubrey rule takes account of all these effects and thus the undoubted success of the Knobler treatment is not surprising. Criticisms could be levelled at his use of a spherically symmetric potential for substances such as n-hexane but the use of a more realistic potential such as the Kihara line-core potential is hardly justified until reliable experimental values for the ionization potentials of the fluorocarbons become available. 

Wang, S. S., C. G. Gray, P. A. Egelstaff, and K. E. Gubbins. 1973. Monte-Carlo study of pair correlation-function for a liquid with non-central forces. Chemical Physics Letters. 21,123. 

An example of a pure liquid in which there are highly non-central forces and for which a pairwise interaction potential would be completely inappropriate is afforded by liquid Se. Comparatively little work of a structural nature has been done on this liquid. The neutron investigation by Scott (1968, unpublished) is interesting because it predicts an interference... 

It is therefore instructive to examine carefully the damping of a(q) about the asymptotic value of unity. For simple liquids, a measure of the interpenetration of the atomic cores may be obtained. On the other hand, the absence of appreciable damping indicates that non-central forces may be playing an important role. 

The first term on the r.h.s. corresponds to a torque and it vanishes for central forces. In the presence of non-central forces the time average of the torque is assumed to vanish. Also the time change of the angular momentum is assumed to approach zero when averaged over long times. Then (1) leads to... 

Buckingham and Pople gave a more complete treatment than Stockmayer s of the interaction between two molecules having non-central force fields. Their treatment allows not only for dipole-dipole interaction but also for dipole-quadrupole, quadrupole-quadrupole, dipole-induced dipole, steric, anisotropic, and other effects. An important feature of the Buckingham-Pople procedure is that the various integrals needed can be calculated from a master table that they give. Buckingham and Pople applied their method to four polar compounds for which values of the dipole moment and the polarizability were known, and used B(T) to... 

Such a procedure is quite arbitrary when applied to globular substances. Non-central forces decrease very rapidly with increasii distance, a fact which does not appear in (13.2.5). Moreover, as the function g is not specified, formula (13.2.5) does not teadi us anything about the nature of these orientational forces. It can only be used empirically to fit certain experimental data. 

Corrections to the central-field approximation. Two important terms are omitted in the central-field approximation. The first is the residual electrostatic interactions represented by JC (equation (3.68)). This gives rise to non-central forces which cause the motion of the electrons to be correlated with one another, rather than completely independent as in the central-field approximation. The second terra is the magnetic spin-orbit interaction which was not included in equation (3.65) ... 

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