Polarizable point dipoles

One method for treating polarizability is to add point inducible dipoles on some or all atomic sites. This polarizable point dipoles (PPD) method has been applied to a wide variety of atomic and molecular systems, ranging from noble gases to water to proteins. The dipole moment, p,-, induced on a site i is 

The energy of the induced dipoles, [/jndj can be split into three terms. 

The energy Ustat is the interaction energy of the N induced dipoles with the permanent, or static, field 

Note that the energy is the dot product of the induced dipole and the static field, not the total field. Without a static field, there are no induced dipoles. Induced dipoles alone do not interact strongly enough to overcome the polarization energy it takes to create them (except when they are close enough to polarize catastrophically). 

Some water models use a shielding function, S r), that changes the contribution to E,- from the charge at 26,30,31,37 

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Because this method avoids iterative calculations to attain the SCF condition, the extended Lagrangian method is a more efficient way of calculating the dipoles at every time step. However, polarizable point dipole methods are still more computationally intensive than nonpolarizable simulations. Evaluating the dipole-dipole interactions in Eqs. (9-7) and (9-20) is several times more expensive than evaluating the Coulombic interactions between point charges in Eq. (9-1). In addition, the requirement for a shorter integration timestep as compared to an additive model increases the computational cost. 

The evaluation of Gcicctrostatic has received a great deal of attention. It is clear that Eqs. (32) and (33), which are for nonpolarizable point charges and point dipoles, cannot reproduce the effect of the medium upon the solute molecule. A major contribution was made by Onsager, who took this molecule to be a polarizable point dipole located at the center of a spherical cavity 20 the resulting expression is,... 

Models to describe frequency shifts have mostly been based on continuum solvation models (see Rao et al. [13] for a brief review). The most important steps were made in the studies of West and Edwards [14], Bauer and Magat [15], Kirkwood [16], Buckingham [17,18], Pullin [19] and Linder [20], all based on the Onsager model [21], which describes the solvated solute as a polarizable point dipole in a spherical cavity immersed in a continuum, infinite, homogeneous and isotropic dielectric medium. In particular, in the study of Bauer and Magat [15] the solvent-induced shift in frequency Av is given as ... 

The OWB model describes the solute as a classical polarizable point dipole located in a spherical or ellipsoidal cavity in an isotropic and homogeneous dielectric medium representing the solvent. In the presence of a macroscopic Maxwell field E, the solute experiences an internal (or local) field E given by a superposition of a cavity field Ec and a reaction field ER. In terms of Fourier components E -n, Ec,n, ER,n of the fields we have... 

In the language of reciprocal space, nonlocal metal response refers to the dependence of the metal dielectric constant on the wavevector k of the various plane waves into which any probing electric fields can be decomposed. Such an effect is often mentioned in reports on SERS, but it is usually neglected. One of the oldest papers addressing the importance of nonlocal effects on the polarizability of an adsorbed molecule is the article by Antoniewicz, who studied the static polarizability of a polarizable point dipole close to a linearized Thomas-Fermi metal [63], The static dielectric constant eTF(k) of such a model metal can be written as ... 

The molecule is often represented as a polarizable point dipole. A few attempts have been performed with finite size models, such as dielectric spheres [64], To the best of our knowledge, the first model that joined a quantum mechanical description of the molecule with a continuum description of the metal was that by Hilton and Oxtoby [72], They considered an hydrogen atom in front of a perfect conductor plate, and they calculated the static polarizability aeff to demonstrate that the effect of the image potential on aeff could not justify SERS enhancement. In recent years, PCM has been extended to systems composed of a molecule, a metal specimen and possibly a solvent or a matrix embedding the metal-molecule system in a molecularly shaped cavity [62,73-78], In particular, the molecule was treated at the Hartree-Fock, DFT or ZINDO level, while for the metal different models have been explored for SERS and luminescence calculations, metal aggregates composed of several spherical particles, characterized by the experimental frequency-dependent dielectric constant. For luminescence, the effects of the surface roughness and the nonlocal response of the metal (at the Lindhard level) for planar metal surfaces have been also explored. The calculation of static and dynamic electrostatic interactions between the molecule, the complex shaped metal body and the solvent or matrix was done by using a BEM coupled, in some versions of the model, with an IEF approach. 

The polarizable point dipole models have been used in molecular dynamics (MD) simulations since the 1970s." For these simulations, the forces, or spatial derivatives of the potential, are needed. From Eq. [18], the force on atomic site k is... 

The polarizable point dipole model has also been used in Monte Carlo simulations with single particle moves.When using the iterative method, a whole new set of dipoles must be computed after each molecule is moved. These updates can be made more efficient by storing the distances between all the particles, since most of them are unchanged, but this requires a lot of memory. The many-body nature of polarization makes it more amenable to molecular dynamics techniques, in which all particles move at once, compared to Monte Carlo methods where typically only one particle moves at a time. For nonpolarizable, pairwise-additive models, MC methods can be efficient because only the interactions involving the moved particle need to be recalculated [while the other (N - 1) x (]V - 1) interactions are unchanged]. For polarizable models, all N x N interactions are, in principle, altered when one particle moves. Consequently, exact polarizable MC calculations can be... 

The dynamic treatment of the charges is quite similar to the extended Lagrangian approach for predicting the values of the polarizable point dipoles, as discussed in the previous section. One noteworthy difference between these approaches, however, is that the positions of the shell charges are ordinary physical degrees of freedom. Thus the Lagrangian does not have to be extended with fictitious masses and kinetic energies to encompass their dynamics. 

The vector q represents the set of qi. The second-order coefficient, Jijirij), depends on the distance between the two atoms i andand at large distances should equal l/r,y. At shorter distances, there may be screening of the interactions, just as for the dipole-dipole interactions in the earlier section on Polarizable Point Dipoles. This screened interaction is typically assumed to arise... 

The polarization energy in the EE models can be compared directly to that in the polarizable point dipole and shell models. Consider the first term in Eq. [43],... 

In most electronegativity equalization models, if the energy is quadratic in the charges (as in Eq. [36]), the minimization condition (Eq. [41]) leads to a coupled set of linear equations for the charges. As with the polarizable point dipole and shell models, solving for the charges can be done by matrix inversion, iteration, or extended Lagrangian methods. 

One important difference between the shell model and polarizable point dipole models is in the former s ability to treat so-called mechanical polarization effects. In this context, mechanical polarization refers to any polarization of the electrostatic charges or dipoles that result from causes other than the electric field of neighboring atoms. In particular, mechanical interactions such as steric overlap with nearby molecules can induce polarization in the shell model, as further described below. These mechanical polarization effects are physically realistic and are quite important in some condensed-phase systems. 

As mentioned earlier, the shell model is closely related to those based on polarizable point dipoles in the limit of vanishingly small shell displacements, they are electrostatically equivalent. Important differences appear, however, when these electrostatic models are coupled to the nonelectrostatic components of a potential function. In particular, these interactions are the nonelectrostatic repulsion and van der Waals interactions—short-range interactions that are modeled collectively with a variety of functional forms. Point dipole-and EE-based models of molecular systems often use the Lennard-Jones potential. On the other hand, shell-based models frequently use the Buckingham or Born-Mayer potentials, especially when ionic systems are being modeled. 

Note that linearly polarizable point dipoles provide only an approximation to the true polarization response in two different ways. Eirst, polarization can include terms that are nonlinear in the electric field. Thus, Eq. [3] represents only the first term in an infinite series. 

In the reaction field model (Onsager, 1936), a solute molecule is considered as a polarizable point dipole located in a spherical or ellipsoidal cavity in the solvent. The solvent itself is considered as an isotropic and homogeneous dielectric continuum. The local field E at the location of the solute molecule is represented by (78) as a superposition of a cavity field E and a reaction field (Boettcher, 1973). 

The simplest, and least realistic, method for including induced dipole interactions is to average the potential for pair-polarizable point dipoles over all relative orientations, in which case the induced energy is the well-known Debye component of the van der Waals potential ... 

The Langevin dipole model (LD) developed by Warshel (Warshel and Levitt, 1976) can be considered as an intermediate step between discrete and continuum models. Solvent (actually water) polarization is described by introducing a grid of polarizable point dipoles, responding to other electic fields according to Langevin s formula ... 

The models of this category are based on the pioneering work by Levine, Bell and Smith in 1969, who modeled the inner layer of a charged interface in the absence of solute molecules as a two-dimensional sheet of polarizable point dipoles situated in vacuo. The dipoles are randomly distributed over the sites of a hexagonal lattice. This model has been extended to describe adsorption of solute molecules on electrode surfaces by Sangaranarayanan, Rangarajan and their col-... 

The adsorbed layer is modeled as a two-dimensional sheet with a hexagonal lattice structure composed of adsorbate and solvent molecules, which behave as polarizable point dipoles. The solvent molecules in the form of either monomers or clusters may be in different polarization states (orientations) treated as independent species. Each adsorbed species i may occupy ri lattice sites. In addition, the thickness of the adsorbed layer may vary upon adsorption. For this reason we denote by h the thickness of the adsorbed layer when it is composed exclusively of the i-th species. 

From these data, we may use our simple dipole model to calculate the density dependence of the interaction energy, the simple case of the non-polarizable point dipole the interaction energy of Equation 3 varies almost linearly with CO2 density as shown is Figure 3. The critical density of CO2 is 10.64 mo1/1. 

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