Polarizability electrostatic interaction model

An electrostatic interaction model has been presented for the calculation of the static electronic polarizability of hydrocarbons, which, contrary to previous models, can describe aliphatic, olefinic, and aromatic systems. It is based on the representation of the C and H atoms by induced electric charges and dipoles, where the actual values of the charges and dipoles are those that minimize the electrochemical energy of the molecule. The electrostatic interactions are described in terms of normalized propagators, which improves both the consistency and the numerical stability of the technique. The calibration of the model is sought by reproducing the molecular polarizabilities obtained by current density functional theory for a set of 48 reference structures. An excellent agreement with the reference data has been obtained as evidenced by the relative errors on the mean molecular polarizabilities of 0.5, 1.4, and 1.9% for alkanes, alkenes, and aromatic molecules, respectively. 

In what follows, we first discuss the basic aspects of DFT in Section 6.2 and the electrostatic interaction model for polarizability in Section 6.3. The density-based theoretical formalism for electric response of clusters and suitable coarse graining forms the subject matter of Section 6.4. Miscellaneous aspects of polarizability calculation of metal clusters are discussed in Section 6.5 which is followed by presentation of concluding remaiks in Section 6.6. 

In a simple electrostatic interaction model proposed originally by Silberstein [37] and Applequist [20], the response of an aggregate of atoms (molecule or cluster) to an external electric field can be determined through the atomic dipoles induced at each individual atom as a response to not only the external field buf also an effective field due to the other induced dipoles, which depend on the atomic polarizability parameters. The basic equation of this interaction approach to obtain the polarizability of the aggregate is given by the expression for the induced dipole p at the a-th atomic site, given by... 

The alkali metals tend to ionize thus, their modeling is dominated by electrostatic interactions. They can be described well by ah initio calculations, provided that diffuse, polarized basis sets are used. This allows the calculation to describe the very polarizable electron density distribution. Core potentials are used for ah initio calculations on the heavier elements. 

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],... 

For instance, in structure 12-e, the C-X and C-0 dipole moments are additive, leading to a destabilization of the molecule by increasing the energy. In structure 12-a, offset of the C-X and C-0 dipole moments minimizes electrostatic interactions, thus leading to a more stable conformation. This electrostatic model was supported by the observed increase of the percentage of the equatorial conformation of 2-methoxy tetrahydropyran (14) when moving from a non-polar to a polar solvent (Table 3).12 In this model, the polar groups are not polarizable and lead to dipole/dipole (hard/hard) interactions. 

Inductive effects on dipole moments and the effects of intervening atoms on electrostatic interaction energies are represented by polarizability centers In conjunction with bond centered dipoles. Solvation energies are estimated by means of a continuum dlpole-quadrupole electrostatic model. Calculated energies of a number of conformations of meso and racemic 2,4-dichloropentane and the iso, syndio, and hetero forms of 2,4,6-triehloroheptane give satisfactory representations of isomer and conformer populations. Electrostatic effects are found to be quite important. 

In a simple model a neutral molecule can be described through two properties related to its electron distribution, the permanent dipole moment i and the average polarizability oc. There are therefore four electrostatic interactions between a solute molecule and the surrounding solvent molecules, as shown in Table 3.1. 

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. 

This conclusion is further strengthened considerably by the theoretical calculation of CBE originally performed by Pearson and Gray (102) and later on somewhat modified by Pearson and Mawby (8). Values of CBE are calculated according to three models, viz. the hard sphere model, the polarizable ion model and the localized molecular orbital model. Only the last one, treating the bonds as covalent, is able to account in a satisfactory way for the values found experimentally for such halides as HgCl2 and CdCl2. For LiCl and NaCl, on the other hand, an acceptable fit with the experimental values is obtained already by the hard sphere model, which certainly indicates a predominantly electrostatic interaction. 

To the extent that the polarization of physical atoms results in dipole moments of finite length, it can be argued that the shell model is more physically realistic (the section on Applications will examine this argument in more detail). Of course, both models include additional approximations that may be even more severe than ignoring the finite electronic displacement upon polarization. Among these approximations are (1) the representation of the electronic charge density with point charges and/or dipoles, (2) the assumption of an isotropic electrostatic polarizability, and (3) the assumption that the electrostatic interactions can be terminated after the dipole-dipole term. 

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. 

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