Multipole interactions, higher-order

It is of special interest for many applications to consider adsorption of fiuids in matrices in the framework of models which include electrostatic forces. These systems are relevant, for example, to colloidal chemistry. On the other hand, electrodes made of specially treated carbon particles and impregnated by electrolyte solutions are very promising devices for practical applications. Only a few attempts have been undertaken to solve models with electrostatic forces, those have been restricted, moreover, to ionic fiuids with Coulomb interactions. We would hke to mention in advance that it is clear, at present, how to obtain the structural properties of ionic fiuids adsorbed in disordered charged matrices. Other systems with higher-order multipole interactions have not been studied so far. Thermodynamics of these systems, and, in particular, peculiarities of phase transitions, is the issue which is practically unsolved, in spite of its great importance. This part of our chapter is based on recent works from our laboratory [37,38]. 

This is where the strength of the multipole expansion comes into play. As we will derive later, the potential arising from an arbitrary charge can be expanded in terms of monopole ( ), dipole (D), quadrupole (Q), and higher order multipole interactions ... 

More realistic treatment of the electrostatic interactions of the solvent can be made. The dipolar hard-sphere model is a simple representation of the polar nature of the solvent and has been adopted in studies of bulk electrolyte and electrolyte interfaces [35-39], Recently, it was found that this model gives rise to phase behavior that does not exist in experiments [40,41] and that the Stockmeyer potential [41,42] with soft cores should be better to avoid artifacts. Representation of higher-order multipoles are given in several popular models of water, namely, the simple point charge (SPC) model [43] and its extension (SPC/E) [44], the transferable interaction potential (T1PS)[45], and other central force models [46-48], Models have also been proposed to treat the polarizability of water [49],... 

A related methodology that makes use of the calculated surface charges at the cavity surface to estimate the interaction with the solvent has been described in Ref. [54] in addition, the reaction field model can be extended to include the effects of higher order multipoles [55], In the present implementation, only dipole effects are considered. 

On a scale of the order of atomic size, molecular multipole fields vary strongly with orientation and separation. As a consequence, one will generally find induced dipole components arising from field gradients of first and higher order which interact with the so-called dipole-multipole polarizability tensor components, such as the A and E tensors. 

In Eq. (18), we recognize the first quantization fcth-order electronic multipole moment operator (r - Ro). An analogous expansion of r - Rm 1 would yield the nuclear multipole moment operator (Rm — Rq). The higher-order multipole moments generally depend on the choice of origin however, to simplify the notation, we omit any explicit reference to this dependence. The partial derivatives in Eq. (18) are elements of the so-called interaction tensors defined as... 

Energy of Induced Multipole Interaction.—Interaction between a Multipolar System and External Fields. We calculated above the potential energy of electrostatic interaction between a multipole system and external fields, or thefirst-orderenergyduetothefirstpowerofthefield. Besides that energy, which took account only of the reoriratation of permanent multipoles, we have to take into consideration contributions due to the drcum-stance that an external electric field induces higher-order multipole moments given by the expressions (72) and (79). 

On neglecting, in equations (87a), (88a), and (89a), the anisotropies of polarizabilities i.e. putting k< = 0) one obtains the Debye-Falkenhagen formulae. Similarly, it is possible to calculate induced energies of interaction between higher-order multipole moments. ... 

The explicit form of the interaction Hamiltonian // ,(() consists of a series of multipolar terms, but for most purposes the electric-dipole (El) approximation is sufficient. Although the results are calculated within this approximation for each molecular center detailed analysis of the coupling provides results equivalent to the inclusion of higher-order multipole terms for the pair. The same assumption underlies the well-known coupled-chromophore model of optical rotation (Kuhn 1930 Boys 1934 Kirkwood 1937). The Hamiltonian for the system may thus be written as... 

A common further approximation assumes that the investigation of a small subvolume containing only one rod and its counterions will suffice to unveil much of the interesting physics. The main justification for this approach is that the subvolume has zero net charge. Moreover, the counterions will also efficiently screen higher order multipoles. Hence the interactions between two such subvolumes, which are neglected when focusing on just one rod, will be fairly weak. This approximation is called the cylindrical cell model, and it provides the framework for this study. 

Higher order multipole terms can also be computed and included in the expression for Hna,m/3 (see, for example, 41, 42 in (13)). An analogous computation of multipole interactions for small-radius excitons can be found in (41). 

The most important component of the water trimer nonadditive energy is the induction interaction of the second order in V. Its simple mechanism is shown in Fig. 33.3 a permanent multipole moment on monomer A induces multipole moments on monomer B which in turn interact with the permanent multipole moments of monomer C. Higher orders involve interactions between induced moments. The nonadditive induction energy is in general the most important nonadditive component for hydrogen-bonded systems. As already mentioned, it is the only term used—and only in the asymptotic approximation, i.e. neglecting charge-overlap effects—in the polarizable empirical potentials. 

Section 1.2 discusses the obstacles encountered in producing a linear-scaling quantum force field and the methods used to overcome them The linear-scaling quantum force field energy is described in Section 1.3. Section 1.4 discusses the consequences of including higher-order multipoles into the model and assess the quality of the mDC method in reproducing nucleobase interactions. The mathematical details used in the mDC model are collected into a series of small appendices at the end of the chapter (Sections 1.6.1-1.6.5) to facilitate the narrative. 

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... 

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. 

A contains the Coulomb and Pauli repulsions, while B represents the attractive contributions from Keesom, Debye and London forces, as well as interactions due to higher order multipoles. Therefore Equation (1) represents general interactions commonly called Van der Waals forces. When the interacting partners possess some special... 

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