Spherical distribution pair potential

The reliability of results obtained by molecular dynamic simulations strongly depends on the pair-potential functions employed. If molecules are not strictly spherical, the choice of structure models for the molecules becomes an essential factor determining the reliability of results. A brief discussion of various models will be given. Also discussed are the electron distribution within a water molecule and potential functions... 

The formulation of spatially separated a and 7r interactions between a pair of atoms is grossly misleading. Critical point compressibility studies show [71] that N2 has essentially the same spherical shape as Xe. A total wave-mechanical model of a diatomic molecule, in which both nuclei and electrons are treated non-classically, is thought to be consistent with this observation. Clamped-nucleus calculations, to derive interatomic distance, should therefore be interpreted as a one-dimensional section through a spherical whole. Like electrons, wave-mechanical nuclei are not point particles. A wave equation defines a diatomic molecule as a spherical distribution of nuclear and electronic density, with a common quantum potential, and pivoted on a central hub, which contains a pith of valence electrons. This valence density is limited simultaneously by the exclusion principle and the golden ratio. 

Polar fluids, whose molecules bear non-negligible permanent multipole accounting for a significant deviation of the charge distribution from sphericity. A multipole-multipole interaction term is thus added to ttnon-ci to construct an effective pair potential,... 

This atom-atom potential is essentially modelling the interaction between molecules as the sum of the interactions between spherical atomic charge distributions. The 6-exp form is a simple realistic model for the repulsion and dispersion forces, which can give a qualitatively reasonable description of the properties of argon, but lacks the flexibility to represent accurately the argon pair potential (Maitland et al., 1981). 

There are many other indications that the electrostatic effects of non-spherical features of the charge distribution, such as lone pairs and n electrons, can be important in determining molecular crystal structures. At the extreme of homonuclear diatomics (X2), the electrostatic potential outside the molecule arises from the non-spherical distribution of the valence electrons. Just as there are considerable variations in the bonding orbitals in the diatomics, there are also considerable variations in the lowest temperature ordered crystal structure. 

A system of nonspherical particles in two dimensions is, from the computational point of view, an intermediate case between spherical particles (in either two or three dimensions) and nonspherical particles in three dimensions. The pair potential in this case depends on three coordinates (see below), compared with six in the three-dimensional case. Some very useful information on the numerical procedure, on the problem of convergence, and so forth, can thus be gained in a system which is relatively simpler than the three-dimensional case. We shall also present some results on the generalized molecular distribution function, which thus far are available only in two dimensions, yet are of relevance to the case of real liquid water. 

Now using Stone s formalism [3.15], the case when the symmetry of the orientational distribution of the fi2 vector is lower than spherical is considered. In particular, possible contributions to potential of mean torque by electrostatic and dispersion interactions are briefly examined. A general expansion of the pair potential [Eq. (3.20)] as a product of a distance dependent and an angular dependent function is [3.15]... 

Chandrasekhar and Madhusudana have also considered the calculation of the coefficients required in V. The first contribution that these authors examined was the permanent dipole-permanent dipole forces. These were shown to vary as and provided a V dependence to Ui. It was shown however, that this term vanished when the pair potential V12 is averaged over a spherical molecular distribution function. The authors thus discard this term and provide further arguments for its neglect based on the empirical result that permanent dipoles apparently play a minor role in providing the stability of the nematic phase. The second contribution considered was the dispersion forces based on induced dipole-induced dipole interactions and induced dipole-induced quadrupole interactions. As mentioned above, the first of these gives a dependent contribution, while the second provides a contribution depending on The final contribution considered... 

Recent refinements on the atom-atom potential method include the development of accurate anisotropic model intermolecular potentials from ab initio electron distributions of the molecules. The non-spherical features in these charge distributions reflect features of real molecules such as lone pair and 7t-electron density, and therefore are much more effective at representing key interactions such as hydrogen bonding. 

Probably the most notable work on the structure in liquid water based upon experimental data has been that of Soper and co-workers [6,8,10,30,46,55]. He has considered water under both ambient and high temperature and pressure conditions. He has employed both the spherical harmonic reconstruction technique [8,46] and empirical potential structure refinement [6,10] to extract estimates for the pair distribution function for water from site-site radial distribution functions. Both approaches must deal with the fact that the three g p(r) available from neutron scattering experiments provide an incomplete set of information for determining the six-dimensional pair distribution function. Noise in the experimental data introduces further complications, particularly in the former technique. Nonetheless, Soper has been able to extract the principal features in the pair (spatial) distribution function. Of most significance here is the fact that his findings are in qualitative agreement with those discussed above. 

In addition to the repulsive part of the potential given by Eq. (4), a short-range attraction between the macroions may also be present. This attraction is due to the van der Waals forces [17,18], and can be modelled in different ways. The OCF model can be solved for the macroion-macroion pair-distribution function and thermodynamic properties using various statistical-mechanical theories. One of the most popular is the mean spherical approximation (MSA) [40], The OCF model can be applied to the analysis of small-angle scattering data, where the results are obtained in terms of the macroion-macroion structure factor [35], The same approach can also be applied to thermodynamic properties Kalyuzhnyi and coworkers [41] analyzed Donnan pressure measurements for various globular proteins using a modification of this model which permits the protein molecules to form dimers (see Sec. 7). 

We start by considering the grand canonical ensemble characterized by the variables T, V, and fi where p = (/q, p2,..., pc) is the vector comprising the chemical potentials of all the c components of the system. The normalization conditions for the singlet and the pair distribution functions follow directly from their definitions. Here, we use the indices a and [l to denote the species a, jS = 1, 2,..., c. The two normalization conditions are (for particles not necessarily spherical)... 

As a second example of the application of the functional derivatives, we show that the pair distribution function can be obtained as a functional derivative of the configurational partition function. For a system of N spherical particles, with pairwise additive potential, we write... 

In the work-interpretation derivation, the force field due to the pair-correlation density is first determined, and the potential then obtained as the work done to move an electron in this field. The force field and potential due to g (r,r p(r) are the Hartree field t nfr) and potential Wn(r) = Vnfr), respectively, since the spherically symmetric Fermi hole p r,F p(r) does not contribute to the field at the electron position. The field arises only due to the density p(r ) which is a charge distribution that is not spherically symmetric... 

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