Potential electrostatic interactions

It has been found useful to represent the interaction potential for a dimer of homonuclear diatomic molecules [4,5,46,58] as a spherical harmonic expansion, separating radial and angular dependencies. The radial coefficients include different types of contributions to the interaction potential (electrostatic, dispersion, repulsion due to overlap, induction, spin-spin coupling). For the three dimers of atmospheric relevance, we provided compact expansions, where the angular dependence is represented by spherical harmonics and truncating the series to a small number of physically motivated terms. The number of terms in the series are six for the N2-O2 systems, corresponding to the number of configurations of the dimer (for N2-N2 and O2-O2 this number of terms is reduced to five and four, respectively). 

In the derivation of J. D. Ferry s formula. Equation (lb), it was assumed that the solute concentration within the accessible part of the membrane pore is uniform and equal to C2,f- Obviously this assumption cannot hold if we are to acknowledge the presence of Interactive forces between the solute and the pore wall (membrane). Here we will concentrate only on the effect of Van der Waals forces but analogous treatments could be developed for other interactive potentials (electrostatic, etc.). 

Dykstra C E 1993. Electrostatic Interaction Potentials in Molecular Force Fields. Chemical Review 93 2339-2353. 

Here 0p and 0 correspond to the terms in r" and respectively in Equation (1.8) as already pointed out, these contributions are always present, whereas the electrostatic energies 0, and may or may not be present according to the nature of the adsorbent and the adsorptive. In principle. Equation (1.16) could be used to calculate the numerical value of the interaction potential as a function of the distance z of any given molecule from the surface of a chosen solid. In practice, however, the scope has to be limited to systems composed of a simple type of gas molecule and... 

Forces Molecules are attracted to surfaces as the result of two types of forces dispersion-repulsion forces (also called London or van der Waals forces) such as described by the Lennard-Jones potential for molecule-molecule interactions and electrostatic forces, which exist as the result of a molecule or surface group having a permanent electric dipole or quadrupole moment or net electric charge. 

The simplest way to treat the solvent molecules of an electrolyte explicitly is to represent them as hard spheres, whereas the electrostatic contribution of the solvent is expressed implicitly by a uniform dielectric medium in which charged hard-sphere ions interact. A schematic representation is shown in Figure 2(a) for the case of an idealized situation in which the cations, anions, and solvent have the same diameters. This is the solvent primitive model (SPM), first named by Davis and coworkers [15,16] but appearing earlier in other studies [17]. As shown in Figure 2(b), the interaction potential of a pair of particles (ions or solvent molecule), i and j, in the SPM are ... 

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],... 

The QM/MM interactions (Eqm/mm) are taken to include bonded and non-bonded interactions. For the non-bonded interactions, the subsystems interact with each other through Lennard-Jones and point charge interaction potentials. When the electronic structure is determined for the QM subsystem, the charges in the MM subsystem are included as a collection of fixed point charges in an effective Hamiltonian, which describes the QM subsystem. That is, in the calculation of the QM subsystem we determine the contributions from the QM subsystem (Eqm) and the electrostatic contributions from the interaction between the QM and MM subsystems as explained by Zhang et al. [13],... 

Conceptually, the self-consistent reaction field (SCRF) model is the simplest method for inclusion of environment implicitly in the semi-empirical Hamiltonian24, and has been the subject of several detailed reviews24,25,66. In SCRF calculations, the QM system of interest (solute) is placed into a cavity within a polarizable medium of dielectric constant e (Fig. 2.2). For ease of computation, the cavity is assumed to be spherical and have a radius ro, although expressions similar to those outlined below have been developed for ellipsoidal cavities67. Using ideas from classical electrostatics, we can show that the interaction potential can be expressed as a function of the charge and multipole moments of the solute. For ease... 

Schematic forms of the curves of interaction energies (electrostatic repulsion Vr, van der Waals attraction Va, and total (net) interaction Vj) as a function of the distance of surface separation. Summing up repulsive (conventionally considered positive) and attractive energies (considered negative) gives the total energy of interaction. Electrolyte concentration cs is smaller than cj. At very small distances a repulsion between the electronic clouds (Born repulsion) becomes effective. Thus, at the distance of closest approach, a deep potential energy minimum reflecting particle aggregation occurs. A shallow so-called secondary minimum may cause a kind of aggregation that is easily counteracted by stirring.

The above forms for the Lennard-Jones surface-water interaction potential have been used as models of hydrophobic surfaces such as pyrophyl1ite, graphite, or paraffin. If the intention of the study, however, is to understand interfacial processes at mineral surfaces representative of smectites or mica, explicit electrostatic interactions betweeen water molecules and localized charges at the surface become important. 

Two methods for including explicit electrostatic interactions are proposed. In the first, and more difficult approach, one would need to conduct extensive quantum mechanical calculations of the potential energy variation between a model surface and one adjacent water molecule using thousands of different geometrical orientations. This approach has been used in a limited fashion to study the interaction potential between water and surface Si-OH groups on aluminosilicates, silicates and zeolites (37-39). 

Rigid-geometry ab initio MO calculations of 86 torsional isomers of the dimethylphosphate anion (CH30)2P02 led to the determination of parameters for the Lennard-Jones type of nonbonded interaction, two- and three-fold torsional, and electrostatic interaction potential functions (215). Extension of this approach to full relaxation ab initio and MM schemes will be extremely useful, not only for phosphorus but also for other heteroatoms. 

The interaction between nucleosomes plays an important role for the stability of the 30 nm fiber recent experiments on liquid crystals of mononucleosomes [44-47] and also less concentrated mononucleosome solutions [48,49] show an attractive interaction that can be parameterized by an anisotropic Leonard-Jones type potential [50]. Also, an electrostatic interaction potential has been computed using the crystallographic structure of the nucleosome [51]. The influence of these potentials on the structure of the fiber is discussed below together with the corresponding models. 

Less is known about the interaction of the nucleosomes between themselves or with free DNA. The nucleosome-nucleosome interaction has recently been parameterized by using the surface charge density of the known crystal structure [39] in a point-charge model [51]. While in that work only electrostatic interactions were considered and the quantitative influence of the histone tails on the interaction potential still remains obscure, simulations based on this potential allowed to predict an ionic-strength dependent structural transition of a 50-nucleosome chromatin fragment that occurred at a salt concentration compatible with known experimental data (Ref. [65], see below). 

FIG. 13.13 Interaction between polymer-coated particles. Overlap of adsorbed polymer layers on close approach of dispersed solid particles (parts a and b). The figure also illustrates the repulsive interaction energy due to the overlap of the polymer layers (dark line in part c). Depending on the nature of the particles, a strong van der Waals attraction and perhaps electrostatic repulsion may exist between the particles in the absence of polymer layers (dashed line in part c), and the steric repulsion stabilizes the dispersion against coagulation in the primary minimum in the interaction potential. 

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