Point-multipolar models

The simulations based on the point dipole model do exhibit a dynamic yield stress [173,292,296,315] and its dependence on the voliune fraction and electric field agrees with experiment [173]. However, the magnitude of the yield stress is severely underestimated in comparison with experiment, likely due to the neglect of multipolar and multibody interactions [173,243]. 

One other aspect of nonprimitive electric double layer theories which is particularly relevant to the inner Stern region are the models for the water molecule and the ions. The simplest models for a water molecule and an ion are a hard-sphere point dipole and point charge, respectively. A more realistic model of the hard-sphere water molecule would include quadrupoles and octupoles and also polarizability. However the hard-sphere property is best avoided and replaced, for example, by a Lennard-Jones potential. An alternative to a multipolar water model are three point charge sites associated with the atoms within the water molecule. 

From a historical point of view, rare gases have been fundamental for the development of models. Although the first proposed model turned out not very realistic at a later analysis even for these simple systems, still they provided a framework for many models of everyday usage, such as the LJ or Buckingham potentials. In polyatomic systems, only at very large separations can the interaction be described by multipolar terms located at the center of the distributions. At short to medium distances, a most important range for condensed phases, multipolar multicenter expansion are used, whereby the centers may be located at the position of the nuclei or not. 

From a practical point of view, the number of sites and of functions per site should be kept as small as possible to reduce computational times. This is the main reason behind the success of empirical and semiempirical models based on LJ functions plus electrostatic terms corresponding to multipolar expansions with very few sites. Conversely, potentials derived from accurate ab initio calculations have been less widely used in view of their complexity, not compensated by real advantages, unless they include many-body terms. 

An electrostatic model of crystal-field interactions can be developed in which the crystal-field components A n defined by eq. (32) are expanded in a multipolar series (Hutchings and Ray, 1%3). It is assumed that the lattice consists of a series of points, one for each ion in a lattice, that have multipolar moments associated with them that are defined by... 

The description of the mDC method in the present work is supplemented with mathematical details that we Have used to introduce multipolar densities efficiently into the model. In particular, we describe the mathematics needed to construct atomic multipole expansions from atomic orbitals (AOs) and interact the expansions with point-multipole and Gaussian-multipole functions. With that goal, we present the key elements required to use the spherical tensor gradient operator (STGO) and the real-valued solid harmonics perform multipole translations for use in the Fast Multipole Method (FMM) electrostatically interact point-multipole expansions interact Gaussian-multipoles in a manner suitable for real-space Particle Mesh Ewald (PME) corrections and we list the relevant real-valued spherical harmonic Gaunt coefficients for the expansion of AO product densities into atom-centered multipoles. 

Higher-order multipole moments enhance the forces between particles at short distances and their neglect is extremely questionable, especially if fine effects are looked at, as for instance the ground-state properties of close-packed lattice structures [244,246-251] or the viscosity To go beyond the point dipole approximation Klingenberg and co-workers [ 173,252] developed an empirical force expression for the interaction between two dielectric spheres in a uniform external field from the munerical solution of Laplace s equation [253]. Recently, Yu and co-workers [254,255] proposed a computationally efficient (approximate) dipole-induced-dipole model based on a multiple image method which accounts partially for multipolar interactions. 

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