Isotropic solute polarizability

The model of polarizable dipolar chromophores suggests that the 3D nuclear reaction field of the solvent serves as a driving force for electronic transitions. Even in the case of an isotropic solute polarizability, two projections of the reaction field should be included the longitudinal (parallel to the difference solute dipole) component and the transverse (perpendicular to the difference dipole) component. The 8 function in Eq. [18] eliminates integration over only one of these two field component. The integral still can be taken analytically resulting in a closed-form solution for the Franck-Condon factor... 

To answer this question, let us first consider a neutral molecule that is usually said to be polar if it possesses a dipole moment (the term dipolar would be more appropriate)1 . In solution, the solute-solvent interactions result not only from the permanent dipole moments of solute or solvent molecules, but also from their polarizabilities. Let us recall that the polarizability a of a spherical molecule is defined by means of the dipole m = E induced by an external electric field E in its own direction. Figure 7.1 shows the four major dielectric interactions (dipole-dipole, solute dipole-solvent polarizability, solute polarizability-solvent dipole, polarizability-polarizability). Analytical expressions of the corresponding energy terms can be derived within the simple model of spherical-centered dipoles in isotropically polarizable spheres (Suppan, 1990). These four non-specific dielectric in-... 

Fig. 2.2 Self-Consistent Reaction Field (SCRF) model for the inclusion of solvent effects in semi-empirical calculations. The solvent is represented as an isotropic, polarizable continuum of macroscopic dielectric e. The solute occupies a spherical cavity of radius ru, and has a dipole moment of p,o. The molecular dipole induces an opposing dipole in the solvent medium, the magnitude of which is dependent on e.

Quantitative models of solute-solvent systems are often divided into two broad classes, depending upon whether the solvent is treated as being composed of discrete molecules or as a continuum. Molecular dynamics and Monte Carlo simulations are examples of the former 8"11 the interaction of a solute molecule with each of hundreds or sometimes even thousands of solvent molecules is explicitly taken into account, over a lengthy series of steps. This clearly puts a considerable demand upon computer resources. The different continuum models,11"16 which have evolved from the work of Bom,17 Bell,18 Kirkwood,19 and Onsager20 in the pre-computer era, view the solvent as a continuous, polarizable isotropic medium in which the solute molecule is contained within a cavity. The division into discrete and continuum models is of course not a rigorous one there are many variants that combine elements of both. For example, the solute molecule might be surrounded by a first solvation shell with the constituents of which it interacts explicitly, while beyond this is the continuum solvent.16... 

The continuum model of solvation has evolved from these beginnings. The solvent is treated as a continuous polarizable medium, usually assumed to be homogeneous and isotropic, with a uniform dielectric constant e.11-16 The solute molecule creates and occupies a cavity within this medium. The free energy of solvation is usually considered to be composed of three primary components ... 

Most earlier papers dealt with the mercury electrode because of its unique and convenient features, such as surface cleanness, smoothness, isotropic surface properties, and wide range of ideal polarizability. These properties are gener y uncharacteristic of solid metal electrodes, so the results of the sohd met electrolyte interface studies are not as explicit as they are for mercury and are often more controversial. This has been shown by Bockris and Jeng, who studied adsorption of 19 different organic compounds on polycrystaUine platinum electrodes in 0.0 IM HCl solution using a radiotracer method, eUipsometry, and Fourier Transform Infrared Spectroscopy. The authors have determined and discussed adsorption isotherms and the kinetics of adsorption of the studied compounds. Their results were later critically reviewed by Wieckowski. ... 

Christopher J. Cramer and their co-workers during the last decade [61,100, 55, 56], In SMx, terms responsible for cavity foimation. dispersion, solvent structure and local field polarization are present [51,57], The solvation energy is obtained via the usual approximation that the solute, treated at the quantum mechanical level, is immersed in an isotropic polarizable continuum representing the solvent. Therefore the standard free energy of the solute in solution can be expressed as ... 

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... 

The so-called polarizable continuum model (PCM) offers a unified and well sound framework for the evaluation of all these contributions both for isotropic and anisotropic solutions. In PCM, the solute molecule (possibly supplemented by some strongly bound solvent molecules, to include short-range effects such as hydrogen bonds) is embedded in a cavity formed by the envelope of spheres centered on the solute atoms. The procedures to assign the atomic radii and to form the cavity have been described in detail together with effective classical approaches for evaluating K vand ,... 

The Cartesian indices refer to an arbitrarily chosen laboratory frame. For certain NLO processes intrinsic permutation symmetry can be used to reduce further the number of independent components. In the case of the Kerr susceptibility, (-w w,0,0), intrinsic permutation symmetry in the last two indices holds, xltJ zx X xx- The most general Kerr susceptibility of an isotropic medium therefore has only two independent components, x9 zz and x9 xx Likewise, the EFISHG susceptibility (-2w w, w,0), important for the evaluation of second-order molecular polarizabilities in solution (see pp. 158 and 162), has only two independent components, x zz and x9]txz, because of intrinsic permutation symmetry in the second and third indices. 

The different symmetry properties considered above (p. 131) for macroscopic susceptibilities apply equally for molecular polarizabilities. The linear polarizability a - w w) is a symmetric second-rank tensor like Therefore, only six of its nine components are independent. It can always be transformed to a main axes system where it has only three independent components, and If the molecule possesses one or more symmetry axes, these coincide with the main axes of the polarizability ellipsoid. Like /J is a third-rank tensor with 27 components. All coefficients of third-rank tensors vanish in centrosymmetric media effects of the molecular polarizability of second order may therefore not be observed in them. Solutions and gases are statistically isotropic and therefore not useful technically. However, local fluctuations in solutions may be used analytically to probe elements of /3 (see p. 163 for hyper-Rayleigh scattering). The number of independent and significant components of /3 is considerably reduced by spatial symmetry. The non-zero components for a few important point groups are shown in (42)-(44). 

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). 

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