Potential, anisotropic

This Legendre expansion converges rapidly only for weakly anisotropic potentials. Nonetheless, truncated expansions of this sort are used more often than justified because of their computational advantages. 

Pack R T 1978 Anisotropic potentials and the damping of rainbow and diffraction oscillations in differential cross-sections Chem. Phys. Lett. 55 197... 

Wilson M R, Allen M P, Warren M A, Sauron A and Smith W 1997 Replicated data and domain decomposition molecular dynamics techniques for the simulation of anisotropic potentials J. Comput. Chem. 18 478-88... 

The model of non-correlated potential fluctuations is of special interest. First, it can be solved analytically, second, the assumption that subsequent values of orienting field are non-correlated is less constrained from the physical point of view. The theory allows for consideration of a rather general orienting field. When the spherical shape of the cell is distorted and its symmetry becomes axial, the anisotropic potential is characterized by the only given axis e. However, all the spherical harmonics built on this vector contribute to its expansion, not only the term of lowest order... 

Conserving only the first two terms of the anisotropic potential expansion (7.49) and summing as noted in (7.70) over j, we get... 

For simplicity, let us restrict ourselves to the case when fluctuations of the anisotropic potential are connected only with variation of its direction, determined by the Euler angles X. The energy of interaction of a rotator with the directing held depends only on the difference angle... 

Hikosaka presented a chain sliding diffusion theory and formulated the topological nature in nucleation theory [14,15]. We will define chain sliding diffusion as self-diffusion of a polymer chain molecule along its chain axis in some anisotropic potential field as seen within a nucleus, a crystal or the interface between the crystalline and the isotropic phases . The terminology of diffusion derives from the effect of chain sliding diffusion, which could be successfully formulated as a diffusion coefficient in our kinetic theory. 

There have been many attempts to formulate a procedure to avoid it and both a posteriori and a priori schemes are available. The counterpoise approach (CP) (Boys and Bemardi, 1970) and related methods are the most conunon a posteriori procedures. Although this technique represents the most frequently employed a posteriori procedure to estimate this error, several authors have emphasised that the method introduced by Boys and Bemardi does not allow a clear and precise determination of the BSSE. The addition of the partner s functions introduces the "secondary superposition error" a spurious electrostatic contribution due to the modification of the multipole moments and polarizabilities of the monomers. This is particularly important in the case of anisotropic potentials where these errors can contribute to alter the shape of the PES and the resulting physical picture (Xantheas, 1996 and Simon et al., 1996). 

Anisotropic potentials. The anisotropy of the interaction potentials may be taken into account in the computation of spectral moments. For the zeroth and first moments, the anisotropy of the interaction affects the pair distribution function, g(R,Qi,Q2), which thus becomes dependent on the orientations of molecules 1 and 2. A perturbation treatment based on the assumption of small anisotropy was given later for an estimate of the effects of the anisotropy of H2-He and H2-H2 pairs [293], Moments of more strongly anisotropic molecules (N2, CO2) were recently considered [67, 122],... 

The eigenvalue problem for the simple cos y potential of Eq. (4) can be solved easily by matrix diagonalization using a basis of free-rotor wave functions. For practical purposes, however, it is also useful to have approximate analytical expressions for the channel potentials V,(r). The latter can be constructed by suitable interpolation between perturbed free-rotor and perturbed harmonic oscillator eigenvalues in the anisotropic potential for large and small distances r, respectively. Analogous to the weak-field limit of the Stark effect, for linear closed-shell dipoles at large r, one has [7]... 

To consider defect recombination controlled by the anisotropic potential, equation (3.1.4), it was suggested [76] to rewrite the quasi-steady-state equation (4.2.22) in a form... 

The variational procedure was developed in [61, 63] for calculating Tieff for anisotropic potentials. Employing different trial functions, it is shown that for a strong elastic interaction the effective radius is... 

Tang KT, Toennies JP (1978) A simple theoretical model for the Van der Waals potential at intermediate distances. II. Anisotropic potentials of He-H2 and Ne-H2. J Chem Phys 68 5501-5517... 

It has been the merit of Picken (1989, 1990) having modified the Maier-Saupe mean field theory successfully for application to LCPs. He derived the stability of the nematic mesophase from an anisotropic potential, thereby making use of a coupling constant that determines the strength of the orientation potential. He also incorporated influences of concentration and molecular weight in the Maier-Saupe model. Moreover, he used Ciferri s equation to take into account the temperature dependence of the persistence length. In this way he found a relationship between clearing temperature (i.e. the temperature of transition from the nematic to the isotropic phase) and concentration ... 

It is possible [27,49,50] to improve the predictive capability of the atom-atom potential method by introducing anisotropic potentials—in particular when the structure is based on intermolecular contacts to halogen atoms. In this way one takes empirically into account the orbital interactions in nonbonding molecular contacts [51]. 

R. H. Boyd and co-workers have explored the application of anisotropic united-atom potentials in the MD simulation of liquid polyethylene. Their effort was motivated by recent evidence that united-atom CH2 Lennard-Jones nonbonded potentials are inadequate in some important respects. The use of the anisotropic potential gave good agreement between experiment and simulations for the equation of state P,V,T) and heat of vaporization. 

In spite of the single form of these distance fuiK ions and the usual assumption that the angular expansion can be truncated after very few terms (for instance, only the isotropic and die first anisotropic L, Lg 0 terms), the number of parameters is mostly too large and these parameters are too strongly interdependent in affecting the measured properties, for a fully experimental determination of these parameters to be possible. Only for very simple systems such as atom- liatom systems or atom-tetrahedral molecule systems the experimental data could be used to yield a parametrized anisotropic potential of the form (1) and even there it appeared advantageous to extract part of the parameters from ab initio calculations. For other molecular systems only isotropic potentials are knovm mostly in simplified forms sudi as (6) or (7). 

Therefore, if one needs an anisotropic potential one often includes only specific anisotropic contributions, e.g. the molecular quadnipole-quadrupole interaction making the ad hoc assumption that all other anisotropic terms are small, or one invokes model potratials with fewer parameters which are intrinsically anisotropic. Examples of the latter are ... 

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