Anisotropic potential functions

In order to limit the total number of interactions exp>erienced by each molecule in a computer simulation, the potential energy is usually truncated so that a molecule s interaction range is finite. For example, the ordinary (spherical) Lennard-Jones potential is truncated at about 2.5cr the interactions between all molecules separated by more than this distance are weak enough to be neglected. In order to maintain conservation of energy, an anisotropic potential function should be truncated at an equipotential surface. However, if the potential is not too anisotropic, truncation at a fixed distance leads to only minor effects on energy conservation. ... 

This can be inserted in equation (02.2.3) to give tlie orientational distribution function, and tlius into equation (02.2.6) to deteniiine the orientational order parameters. These are deteniiined self-consistently by variation of tlie interaction strength iin equation (c2.2.7). As pointed out by de Gemies and Frost [20] it is possible to obtain tlie Maier-Saupe potential from a simple variational, maximum entropy metliod based on tlie lowest-order anisotropic distribution function consistent witli a nematic phase. 

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],... 

Theory. The theory of collision-induced absorption profiles of systems with anisotropic interaction [43, 269] is based on Arthurs and Dalgamo s close coupled rigid rotor approximation [10]. Dipole and potential functions are approximated as rigid rotor functions, thus neglecting vibrational and centrifugal stretching effects. Only the H2-He and H2-H2 systems have been considered to date, because these have relatively few channels (i.e., rotational levels of H2 to be accounted for in the calculations). The... 

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]... 

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... 

P.-O Astrand, P. Linse and G. Kalstrdm, Molecular dynamics study of water adopting a potential function with explicit atomic dipole moments and anisotropic polarizabilities, Chem. Phys., 191 (1995) 195-202. 

From the simulations, we conclude that two hydrogen bonding force constants are a basic requirement for reproducing the measured spectrum. If a water-water potential generates sufficiently large force constant differences for the different proton configurations (or the different relative dipole-dipole orientations in water or ice), it should produce the same effect as seen in the LR model. The anisotropic properties of the classic potentials are a result of charge interaction and this anisotropy should increase in the polarisable potentials and hence they produce a broad optic peak. This broad peak indicates that the orientational variation of the potential function has been increased considerably but it may still be less than the critical value of 1.5 as we indicated in the section 6.1. One would, therefore, expect that a better polarisable potential would, eventually, be able to reproduce the split optic peaks in the measured INS spectrum. 

Jacobi and Schnepp (1972) and Raich (1972) were the first to develop a quantum-dynamical model for the large-amplitude librations in a-nitro-gen. Their formalism is essentially described in Section IV,C. They first calculated single-molecule mean field states that may be localized as well as delocalized, depending on the height of the rotation barriers from the anisotropic potential. These states were used to construct a basis of exci-tonlike wave functions for the whole crystal. The final step in their calcu-... 

In spite of the simple form of these distance functions and the usual assumption that the angular expansion can be truncated after very few terms (for instance, only the isotropic and the 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-diatom 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 known , mostly in simplified forms such as (6) or (7). 

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... 

It has become attractive to consider a CG particle as ellipsoid because this treatment can give a reasonable approximation to the anisotropic nature of a cluster of atoms. The Gay-Beme anisotropic potential [55, 56] is actually based on a Gaussian-overlap potential [57], and in this work the Gay-Beme anisotropic potential energy function [/qb is represented by the functional form ... 

The anisotropic potential, U, is now known for other concentrations, temperatures and molecular weights, so Eq. (4) can be used to calculate the order parameter as a function of the temperature. 

Perhaps the most significant difficulty in the computer simulation of polyatomic fluids is the formulation of the intermolecular potential function. Extremely little is known about the details of anisotropic molecular interactions, and the possibilities for modeling are restricted by considerations of practicality for computer applications. In this section we shall discuss several approaches that have been used to model the interactions of anisotropic molecules. 

The combination of a truncated potential function and periodic boundary conditions leads to an important programming simplification. Let L be the length of the molecular dynamics box and let r, be a molecule s interaction range. (In the case of an anisotropic potential, r, is the maximum distance of the... 

Note that an atom-atom potential depends implicitly on the internal coordinates, since these determine the position of the atoms relative to the molecular coordinate frame. The potential parameters may be allowed to depend explicitly on the Internal coordinates, but it is unlikely that experimental data will be good enough to characterize such dependence for some time. Nevertheless it is clear that anisotropic atom-atom potential functions of the type described are capable not only of summarizing our present knowledge but of incorporating future developments. 

Following a previous analysis [7,8,1a], for P atoms (L=l) it is convenient to introduce a parameterization of die electrostatic interaction by two functions of the intemuclear distance, a spherical interaction potential, Vq(R), and an anisotropic potential term, V2(R). They are simply related to the more familiar V CR) and V] (R) potential functions, representing different electronic symmetries ... 

P.-O. Astrand, P. Linse, and G. Karlstrdm, Chem. Phys., 191, 195 (1995). Molecular Dynamic Study of Water Adopting a Potential Function with Explicit Atomic Dipole Moments and Anisotropic Polarizabilities. 

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