Molecular Multipole Models

Molecular multipole models represent a water molecule by a single site with a van der Waals sphere and electric multipole tensors of the entire molecule [45] (Fig. 9.4). The electrostatic interaction energy between molecules is obtained by expanding Coulomb s law around a single site rather than by multiple sites with partial charges. A significant advantage of the multipole approach is that 

The multipole expansion of Coulomb s law up to the octupole, which can be found in [48], is more complicated than Eq. 9.3. The advantage is that the complexity in the expression leads to computational efficiency in computer simulations once it is programmed, since only one distance between two interacting water molecules is needed. The downside is that while it becomes more accurate as higher order multipoles are added, it also becomes computationally slower as higher-order multipoles are included since each n-pole involves a (n - 1) rank tensor. A soft-sphere model with a dipole, quadrupole, and octupole (SSDQO], which is exact up to the 1/r term and in addition approximates the 1/r term, has been developed for computational efficiency [48]. However, the recent implementation of a fast multipole method in the molecular dynamics program CHARMM [80] should make this approximation unnecessary specifically, the full multipole expansion up to the 

So far, the multipole approach has been rather limited for modeling liquid water. For rigid, nonpolarizable models, the SSDQOl discussed here uses multipoles up to the octupole that have been optimized for various properties of liquid water at STP [29]. In addition, SCME, a polarizable molecular multipole model using multipoles up to the hexadecapole, has recently been developed [49]. 

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Table 9.1 Multipoles for multisite and molecular multipole models, quantum mechanical calculations, and experiment...

Table 9.2 Ratios of multipoles and first hydration shell order (see text] from QM, multisite, and molecular multipole models...

Distributed multipole models for Nj and HF. (Figure adapted from Stone A j and M Alderton 19S5. ibuted Multipole Analysis Methods and Applications. Molecular Physics 56 3 047-1064.)... 

Many of the mesogenic molecules are stericaUy asymmetric, which is determined by the fractures and bending of the molecular core as well as by the presence of the tail chains of different nature, including the branched, biforked or polyphilic moieties (Fig. 2c-f). In terms of the multipole model of molecular asymmetry introduced by Petrov and Derzhanski [34], we can speak about longitudinal or transverse steric dipoles or multipoles (Fig. 3). 

Williams, D. E. 1991. Net Atomic Charge and Multipole Models for the Ab Initio Molecular Electrostatic Potential. In Reviews in Computational Chemistry. K. B. Lipkowitz and D. B. Boyd, eds. VCH Publisher, New York. 

While exchange- and dispersion-induced dipole components are of a quantum nature, the multipole-induced dipole components can be modeled by classical relationships, if the quantum effects are small. For many systems of practical interest, multipolar induction generates the dominant dipole components. The classical multipole induction approximation has been very successful, except for the weakly polarizable partners (e.g., He atoms) [193]. It models the dipole induced in the collisional partner by polarization in the molecular multipole fields. 

Williams DE. Net atomic charge and multipole models for the ab initio molecular electric potential. In Lipkowitz KB, Boyd DB, eds. Reviews in Computational Chemistry. Vol. 2. New York VCH, 1991 219-271. 

Donald E. Williams, Net Atomic Charge and Multipole Models for the Ab Initio Molecular Electric Potential. 

The X-ray scattering may be expressed directly in terms of a formalism related to the molecular orbitals of a transition metal complex, rather than to the charge distribution on the individual atoms, as is done in the multipole model. In such a formalism the overlap density is directly related to the orbital populations, and does not have to be projected into the one-center functions. 

The distributed multipole model incorporates a nearly exact description of the molecular charge distribution into the evaluation of the electrostatic energy. Is the increase in accuracy gained by representing the effects of lone pair and 7i-electron density worth the extra complexity in the potential model Even if there is a significant enhancement, is it worth using such an elaborate model when only crude models, such as the isotropic atom-atom 6-exp potential, are available for the other contributions ... 

A molecular multipole expansion is poorly converged at distances of chemical interest, for instance, at distances between atoms of molecules in a dimer or crystal. To obtain electrostatic interaction energy between heteronuclear diatomic molecules at close distance it is always better to model observed molecular point dipoles by a distributed dipole model that places net charges on the atoms. In general, a distributed multipole model of a molecule consists of a set of sites with each site having its own multipoles. Obviously distributed multipole models are not uniquely defined. For instance, at which site in a polyatomic ion is the ionic charge (monopole) located Do you spread it around evenly among the sites ... 

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