Anisotropic pair potential

The crystal structure of solid CO2 has been discussed as arising from minimization of the interactions between molecular quadrupole moments (Buckingham, 1959). Solid a-Na is isomorphous with CO and therefore the same arguments hold here. Kuan, Warshel, and Schnepp (1969) showed that the diatomic potential used by these authors can equally well account for the observed structure. It was found that the minima in energy with molecular orientations are caused by the minima in the repulsion between molecular ends. This structure can therefore be accounted for by any anisotropic pair potential which contains end-to-end repulsions and relative end-to-center attraction. 

Price S L and Stone A J 1980 Evaluation of anisotropic model intermolecular pair potentials using an ab initio SCF-CI surface Moi. Phys. 40 805... 

The parameter /r tunes the stiffness of the potential. It is chosen such that the repulsive part of the Leimard-Jones potential makes a crossing of bonds highly improbable (e.g., k= 30). This off-lattice model has a rather realistic equation of state and reproduces many experimental features of polymer solutions. Due to the attractive interactions the model exhibits a liquid-vapour coexistence, and an isolated chain undergoes a transition from a self-avoiding walk at high temperatures to a collapsed globule at low temperatures. Since all interactions are continuous, the model is tractable by Monte Carlo simulations as well as by molecular dynamics. Generalizations of the Leimard-Jones potential to anisotropic pair interactions are available e.g., the Gay-Beme potential [29]. This latter potential has been employed to study non-spherical particles that possibly fomi liquid crystalline phases. 

Table 3 Anisotropic Atom—Atom Models for the Intermolecular Pair Potential, Which Can Be Quantified Using Ab Initio Monomer Properties, and Lead References... 

Various methods of statistical mechanics are applied to the calculation of surface orientation of asymmetric molecules, by introducing an angular dependence to the intermolecular potential function. The Boltzmann distribution can also be used to estimate the orientational distribution of molecules. The pair potential V(r) may be written as V(r, 6) if it depends on the mutual orientation of two anisotropic molecules, and then we can write for the angular distribution of two molecules at a fixed distance, r, apart... 

Strongly associating fluids will typically have a very low fraction of monomers. With the fraction of monomers low, the probability that one monomer will encounter another monomer in the correct orientation and separation to form a bonded dimer is extremely small. Association interactions are modeled with short-range, highly anisotropic interaction potentials [8], so that a pair of monomers must be found in the correct molecular separation and orientation before bonding can occur. 

The anisotropic part of the pair potential is conveniently taken in the form... 

Consider the simplest case, namely, the nematic phase consisting of uniaxial rodlike molecules. Generally, the intermolecular interaction again consists of the repulsive and attractive parts but both of them become anisotropic. The potential of pair molecular interaction can be written in the following general form ... 

The very fact that (Pyx) is non zero indicates structural distortion of the fluid which manifests by an anisotropic pair distribution function g(r,y) compressed in the xy plane along axes at angles (p = 124° (0 angle between the projection of r/r onto the xy plane and the ex axis) and most elongated along 0 34°. At the value of the dipole moment yx = 1.955 considered no strong orientational order is observed and the distortion of the fluid structure results mainly from the LJ part of the potential and is only weakly influenced by the dipolar contribution. 

In the molecular dynamics and Monte Carlo simulations which are the source of the results described herein, the overall potential energy surface was assumed to be a pairwise sum of two-body interactions. The Ar-Ar and Kr-Kr pair potentials were represented by the accurate functions determined by Aziz and Slaman, while the Ar-SF5 and Kr-SFg interactions were represented by the detailed anisotropic functions determined by Pack et alJ Although three-body forces may not be negligible in these systems, we doubt that their effects are larger than those of the uncertainties in our knowledge of the Rg-SF two-body potentials. 

The starting point for a theory of the anisotropic intermolecular interaction in liquid crystals is the Maierand Saupe theory [114,115,116,118].This theory is based on the assumption that the intermolecular interaction potential in nematic liquid crystals is determined primarily by Lx)ndon dispersion forces. The effective anisotropic potential U of a molecule C in the anisotropic dispersion field generated by its oriented neighbors s is calculated by averaging the pair potential between two molecules C and s over all orientations of the solvent molecules s and over all... 

The interactions between rod-like molecules in nematics are highly anisotropic. The forces between molecules depend not only on their separation but also on their relative orientations. Unfortunately, the precise form of the pair potential is not known. However, it is possible to proceed with a perfectly general pair potential U12. Three Eulerian angles (0,0,-0) are needed to specify the orientation of a rigid particle. In particular, if the particle is a rigid cylindrical rod, the angle is unimportant. Thus, the pair potential depends on five coordinates ... 

In thermodynamic perturbation theory the properties of the real system, in which the pair potential is u° g, are expanded about the values for a reference system, in which the pair potential is ug3. Here we take the reference potential to be the (n,6) potential, so that the anisotropic parts of the potential in Eq. (1) are the perturbation. Expanding the Helmholtz free energy A in powers of the perturbing potential about the value Aq for the reference system gives... 

Only very recently it became feasible to carry out classical trajectory calculations of transport properties of nitrogen (Heck Dickinson 1994 Heck et al. 1994). In these calculations the nitrogen molecule has been treated as a rigid rotor, and the intermolec-ular pair potential was based on the anisotropic ab initio potential energy surface of van der Avoird et al. (1986). The details of these computations as well as the uncertainties of their results are described in the given ptq)ers and will not be repeated here. 

Two additional Gaussian families of quantum-effective pair potentials v fR) arise from the variational principle for the free energy. These are the isotropic (ISVP) and the anisotropic (ASVP) self-consistent variational potentials. They were proposed by Giachetti and Tognetti [124] and, independently, by Feynman and Kleinert [125]. These potentials can be derived via an extensive use of the Fourier decomposition of the particle paths in modes characterized by the Matsubara frequencies = 2nn PHn 1), the zero frequency mode being the intercentroid... 

In all molecular statistical calculations, the choice of a proper interaction potential is of crucial importance. Hard-core simulations [242-244,274,278,279,330] assume only repulsive forces between ellipsoidal particles, and fail to reflect realistic properties of the nematic phase [264, 269, 280, 281]. Therefore, attractive intermolecular potentials must be added. In most cases, an additive superposition of pair potentials is assumed in the calculations. Hybrid models use hard core repulsive potentials plus some attractive anisotropic potentials, for example, modified van der Waals models (e.g. 

In principle, a combination of short range repulsive forces and anisotropic dispersive interactions between mesomorphic molecules is sufficient to understand their selforganization in mesophases [1]. This basic approach is used for theoretical descriptions of mesomorphic states, especially for their simulations by molecular dynamics or Monte Carlo calculations of systems composed of idealized particles. Even on this simplified level, realistic pair potentials constitute one of the crucial problems. 

There are several levels of approximation possible in the consideration of the NA transition. First there is the self-consistent mean field formulation due to Kobayashi and McMillan [8-10]. This is an extension to the smectic-A phase of the self-consistent mean-field formulation for nematics ( Maier-Saupe theory [11]). Kobayashi-McMillan (K-M) theory takes into account the coupling between the nematic order parameter magnitude S with a mean-field smectic order parameter. In Maier-Saupe theory, the key feature of the nematic phase - the spontaneously broken orientational symmetry - is put in by hand by making the pair potential anisotropic. In the same spirit, the K-M formulation puts in by hand a sinusoidal density modulation as well as the nematic-smectic coupling. 

The corrected rate constants provide a basic description of hopping between states, but it is necessary to determine self-diffusivities from these to enable comparison with experimental measurements. In the case of potential minima within a zeolite pore, the lattice of sorption sites is often anisotropic. The probability of a molecule residing in a certain site is dependent on the type of site, and the rate constants, k,n may be different for each ij pair. A Monte Carlo algorithm, based on a first-order description of the hopping process, is usually used to determine the diffusivities. 

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