Point multipoles

A common feature of many mesogenic molecules is the presence of polar substituents and aromatic cores [3]. The electrostatic interactions between such groups can be incorporated into a molecular potential with the addition of dipolar and quadrupolar terms, respectively. Rather than represent these permanent electrostatic interactions by using a model in which a charge distribution is scattered over the surface of the molecule, it is very common to use one (or more) point multipoles [2,29]. Thus for an electrostatic Gay-Berne model, the pair potential is given by the sum... 

Giese and York (GY) [68] used the branch-free FMM algorithm of Watson et al. [69] and the recursive bisection ideas of Perez-Jorda and Yang (PJY) [70] to create an adaptive FMM for systems of particles composed of point multipoles, as opposed to the trivial case of point charges (monopoles). GY spent most of their effort in... 

Sharma, R.R., Das, T.P., and Orbach, R. 1966. Zero-field splitting of S-state ions. I. Point-multipole model. Physical Review 149 257-269. 

The asymmetry found in the electron density of these soft ions can be modelled by adding one or more point multipoles to the point charge used to... 

The Coulomb field defined here is not to be confused with the physical Coulomb field calculated using the full electron density. The Coulomb field used here is based on point charges and point multipoles, not on extended electron densities. 

Under the assumption of the existence of ideal ionic crystals, built up from point charges and point multipoles, the NQR spectrum is completely determined by the crystal field of the electric multipoles. The experimental results of NQR can be explained within the frame of this model. Refinements of the model, such as the dependence of multipole polarizabilities upon the crystal field or the influence of overlapping of the electron clouds, are not yet understood quantitatively. 

The point multipole expansion of the Coulombic matrix element governing the first-order electric dipole moment in Eq. (3) gives the selection rules of the ligand polarization model through Eq. (4),... 

The other two components of the octahedral Ai Ti transition, due to the single-orbital promotions, dyz - dy2 j,2 andd x - d2i xJ, give rise analogously to the rotational strength components, Rg and Rg, respectively. The forms of the latter two components are obtained in the point-multipole approximation by cyclic permutation of the coordinates in Eqs. (21) and (22). 

A more refined approach uses the two-centers potential expansion, which considers the interaction between point multipoles located at the centers of the two charge distributions. Since the exchange integral interactions vary as 1/r and the numbers of interactions increase as r, one has to choose spherical shapes for the convergence of the sum to be guaranteed (Ewald sums method)... 

The variation of this approach merges with the PD point multipole models discussed in the next section. It is well known that an atom in a molecule can possibly have a dipole, quadrupole, etc. An atomic dipole, quadrupole, etc. can be simulated by a distributed multipole. For example, an atomic dipole ascribed to a lone pair can be simulated by placing a lone pair charge site near the atom, at the appropriate direction and distance. An atomic quadrupole can be simulated by placing several monopolic sites near the atom. 

All successful water models make use of a distribution of point charges rather than of point multipoles. The main reason is that the directional properties of intermolecular hydrogen bonds can be obtained efficiently with oifly 3 or 4 point charges. Furthermore, Goldman and Backx [39] have shown that model molecules with such distributions of point charges are more effective as solvents (for instance in their ability to dissociate ion pairs) than molecules with equivalent point dipoles and quadrupoles. 

It is natural to contemplate the next extensions to less-simple models. Of course, a fluid of spherical particles carrying point multipoles presents no... 

There have been some debates among people using the COSMO approach [33, 34] there are no formal reasons asking for a constant factor equal for all tesserae anyway, the examination of simple models for which there are analytical solutions (point multipoles within a regular cavity) suggests a constant value of the type (e — l)/(e + 1 ) with k ranging from 0 to 2. In the present implementation we have selected k = 0 and then f = e — l)/e. 

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