Onsager reaction field theory

The Kamlet-Taft u polarity/polarizability scale is based on a linear solvation energy relationship between the n it transition energy of the solute and the solvent polarity ( 1). The Onsager reaction field theory (11) is applicable to this type of relationship for nonpolar solvents, and successful correlations have previously been demonstrated using conventional liquid solvents ( 7 ). The Onsager theory attempts to describe the interactions between a polar solute molecule and the polarizable solvent in the cybotatic region. The theory predicts that the stabilization of the solute should be proportional to the polarizability of the solvent, which can be estimated from the index of refraction. Since carbon dioxide is a nonpolar fluid it would be expected that a linear relationship... 

Figure 6. Qustering of CHFj about (dimethylamino)benzonitrile at 50 C based on solvatochromic shifts in fluorescence (A data from ref. 26, — Onsager reaction field theory).

The simplest continuum model is based on die classical Onsager reaction field theory assuming the spherical or ellipsoidal form of cavities for die solute molecules in dielectric... 

The cluster-continuum moder iUuslrated in scheme 1 was used to simulate the most common electrolytes of LIBs. The supra-molecular cluster applied to the inner ring (scheme 1) incorporates the mutual effect between salt and solvent molecules by including the first solvation shell of the salt. Due to the minor effect that the salt anion XT has on the reductive behavior of solvent molecules coordinated with Li, XT and the surrounding solvent molecules are not discussed in the current chapter. The bulk solvent effect in the second ring of scheme 1 is treated by polarized continuum models, such as PCM, conductor-like PCM (CPCM), isodensity PCM (IPCM), and self-consistent isodensity PCM (SCl-PCM), which were developed on the basis of the Onsager reaction field theory and are recognized to provide reliable results for systems without specific interactions such as hydrogen bond. 

The Onsager s reaction field theory [3] has been incorporated into MO calculations by Tapia and Goscinski [6], The model has been applied to different problems using either semiempirical [51] or ab-initio MO theory [52], or correlated ab-initio techniques [52],... 

According to the McRae-Bayliss model of solvatochromism [69, 70] which is directly evolved from Onsager s reaction field theory [80], the electronic transition from ground [g) to excited state (e) of a solvatochromic solute is given by Eq. (6-1) [318] ... 

The expression most commonly used in fluorescence spectroscopy is, however, the somewhat simphfied Eq. (6-5b), first developed by Lippert [47, 488] and Mataga [14, 489]. It is based on Onsager s reaction-field theory, which assumes that the fluorophore is a point dipole residing in the center of a spherical cavity with radius a in a homogeneous and isotropic dielectric with relative permittivity e,. The so-called Lippert-Mataga equation is as follows ... 

The first theoretical treatment of infrared solvent shifts was given in 1937 by Kirkwood [166] and by Bauer and Magat [167], Eq. (6-8) - known as the Kirkwood-Bauer-Magat (KBM) relationship - has been derived on the basis of Onsager s reaction field theory [80] using the simple model of a diatomic oscillator within a spherical cavity in an isotropic medium of macroscopic relative permittivity r. 

Born—Kirkwood—Onsager Reaction Field The theory underlying the implementation of the BKO model at the semiempirical level is no different from that presented in Equations [22] and [23], although the approximations inherent to various levels of semiempirical theory make certain technicalities of the... 

The degree of clustering may be determined by a direct comparison of the experimental value of the solvatochromic parameter, for example. Ex, with the value which is calculated for a honx)geneous polarizable dielectric. Diis calculation has been done using the McRae-Bayliss nKxlel(221 which is based on the continuum reaction field theory of Onsager. 

Onsager-McRae reaction field theory, — V2 data for... 

In view of the approximations inherent in the derivation of the reaction field theory, it is not surprising that some instances are known in which a non-linear relationship exists between the solvent shift and dielectric constant in polar solvents. As pointed out by Buckingham, the reaction field model is only valid for a solute that reacts in no way with the solvent or with other solute molecules but simply presents a continuum of certain dielectric properties. Protons are normally on the surface of the molecule and are therefore exposed to direct contact with the surrounding molecules, so that the Onsager model is a poor approximation of the actual reaction field acting on a molecule. 

Various models have been developed for explaining the solvent-induced changes in the Xe shielding. For example, it has been proposed, based on the reaction field theory of Onsager, that the medium shift is proportional to the function f(n) = [ n - 1)/ 2n +l)] (this is called the van der Waals continuum model), where n is the refractive index of the solvent. Part of the experimental data indeed follows this... 

Onsager Theory for C(t) for Non-Debye Solvents. Generally solvents have more complex dielectric responses than described by the Debye equation (Eq. (18)). To obtain the time dependence of the reaction field R from Eqs. (12, (15), (16) and (7) an appropriate model for dielectric behavior of a specific liquid should be employed. One of the most common dielectric relaxation is given by the Debye-type form, which is applicable to normal alcohols. 

Solvent continuum models are now routinely used in quantum mechanical (QM) studies to calculate solvation effects on molecular properties and reactivity. In these models, the solvent is represented by a dielectric continuum that in the presence of electronic and nuclear charges of the solute polarizes, creating an electrostatic potential, the so-called reaction field . The concept goes back to classical electrostatic schemes by Martin [1], Bell [2] and Onsager [3] who made fundamental contributions to the theory of solutions. Scholte [4] and Kirkwood [5] introduced the use of multipole moment distributions. The first implementation in QM calculations was reported in a pioneer work by Rivail and Rinaldi [6,7], Other fundamental investigations were carried out by Tapia and Goscinski [8], Hilton-McCreery et al. [9] and Miertus et al. [10], Many improvements have been made since then (for a review,... 

In case (I), the theory that has been used to describe the EFISH analysis in earlier sections, the isolated molecule calculation provides the pz value and the internal field factors adjust the fields to allow for the polarization on the cavity surface. The elfect of reaction fields due to the additional polarization on the cavity walls induced by the permanent and induced dipoles on the central molecule is implicitly included in the low frequency Onsager field factor through the dielectric constant values. The choice of the high frequency dielectric constant ( o) in this formulation is rather ill defined and no account is taken of changes in the static or dynamic polarizabilities of the molecule as a result of the surrounding fields. 

The spectral position of absorption and fluorescence are influenced by the dielectric properties of the medium in which observations are made. Figure 5 shows that the vapour phase 0-0 bands in absorption and fluorescence of a molecule are identical, whereas in solution with solvent of static dielectric constant e, refractive index n, the bands are no longer coincident. The differences can be rationalized as follows. From Onsager theory, a solute molecule of dipole moment ju in a spherical cavity of radius a polarizes the dielectric of the solvent, producing a reaction field. This is given for the ground-state of the solute molecule (of dipole moment iiq), by (22). Upon excitation, and invoking the Franck-Condon principle, the electronic excitation is much more rapid than the dielectric relaxation time of... 

In the classical reaction-field model, the solute molecule is considered embedded in a cavity inside a homogeneous dielectric medium. From the Onsager theory (Onsager, 1936) the electronic reaction field at the center of the solute molecule is given by... 

Among the few determinations of of molecular crystals, the CPHF/ INDO smdy of Yamada et al. [25] is unique because, on the one hand, it concerns an open-shell molecule, the p-nitrophenyl-nitronyl-nitroxide radical (p-NPNN) and, on the other hand, it combines in a hybrid way the oriented gas model and the supermolecule approach. Another smdy is due to Luo et al. [26], who calculated the third-order nonlinear susceptibility of amorphous thinmultilayered films of fullerenes by combining the self-consistent reaction field (SCRF) theory with cavity field factors. The amorphous namre of the system justifies the choice of the SCRF method, the removal of the sums in Eq. (3), and the use of the average second hyperpolarizability. They emphasized the differences between the Lorentz Lorenz local field factors and the more general Onsager Bbttcher ones. For Ceo the results differ by 25% but are in similar... 

While this result confirmed the feasibility of the general approach, it did not precipitate wider exploration of dielectric medium effects. Recently, however, Wiberg et al. have incorporated the Onsager self-consistent reaction-field model into ab initio MO theory in an implementation which provides analytical gradients and second derivatives. The model considers just the dipole of the solute molecules and a spherical cavity whose radius is chosen for a given solute molecule from the molecular volume estimated at the 0.001 eB electron-density contour (B is the Bohr radius), plus an empirical constant 0.5 A to account for the nearest approach of solvent molecules [164]. Cieplak and Wiberg have used this model to probe solvent effects on the transition states for nucleophilic additions to substituted acetaldehydes [165]. For each... 

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