Onsager continuum

Jepsen and Friedman found, however, that for microscopic impurities, (2.34a) and (2.34b)—in contrast to (2.34c)—no longer appeared to be satisfied beyond the lowest order iny in the low-density approximation they were considering, which left open the asymptotic form the microscopic results would have. Equation 2.33 reveals that only if the Onsager approximation (2.30d) were satisfied in the molecular solvent would (2.33) and (2.34) be the same. The reason for this will become clear in our discussion of the y->0 limit below, where we show that only in the Onsager continuum limit, in which (2.30d) becomes exact, is the dielectric response to each solvent dipole that of a vacuum in a macroscopic sphere surrounding the solvent dipole. Thus only in the Onsager continuum limit are the assumptions satisfied under which one can identify each solvent particle as a macrosphere within which 6= 1, and so assure the identity of the full set of ratios in (2.33) to (2.34). 

To get more insight into the properties of the Onsager continuum, we consider first (2.30c), recalling that to recover Onsager s result (2.30d) one has to put... 

Next we investigate the physical interpretation of the Onsager continuum. We start by considering the discussion of the field set up by test dipoles in a dielectric medium given in Section II.B. We compute the polarization of the medium for r[Pg.211]

Investigation in the literature in this field shows that special efforts were done in the past to investigate the relation between IR intensities in the gas (A ) and liquid (A ° ) phases in the case of pure liquids [175-177] and systems in solution [169,178-181]. Almost all the classical models for solvent effects on IR intensities, such as the ones due to Buckingham [169, 170], Mecke [182], Polo and Wilson [176], Mirone [181], and Warner and Wolfsberg [183], are based on the Onsager continuum model. 

The Onsager model describes the system as a molecule with a multipole moment inside of a spherical cavity surrounded by a continuum dielectric. In some programs, only a dipole moment is used so the calculation fails for molecules with a zero dipole moment. Results with the Onsager model and HF calculations are usually qualitatively correct. The accuracy increases significantly with the use of MP2 or hybrid DFT functionals. This is not the most accurate method available, but it is stable and fast. This makes the Onsager model a viable alternative when PCM calculations fail. 

A variety of methodologies have been implemented for the reaction field. The basic equation for the dielectric continuum model is the Poisson-Laplace equation, by which the electrostatic field in a cavity with an arbitrary shape and size is calculated, although some methods do not satisfy the equation. Because the solute s electronic strucmre and the reaction field depend on each other, a nonlinear equation (modified Schrddinger equation) has to be solved in an iterative manner. In practice this is achieved by modifying the electronic Hamiltonian or Fock operator, which is defined through the shape and size of the cavity and the description of the solute s electronic distribution. If one takes a dipole moment approximation for the solute s electronic distribution and a spherical cavity (Onsager s reaction field), the interaction can be derived rather easily and an analytical expression of theFock operator is obtained. However, such an expression is not feasible for an arbitrary electronic distribution in an arbitrary cavity fitted to the molecular shape. In this case the Fock operator is very complicated and has to be prepared by a numerical procedure. 

Continuum models of solvation treat the solute microscopically, and the surrounding solvent macroscopically, according to the above principles. The simplest treatment is the Onsager (1936) model, where aspirin in solution would be modelled according to Figure 15.4. The solute is embedded in a spherical cavity, whose radius can be estimated by calculating the molecular volume. A dipole in the solute molecule induces polarization in the solvent continuum, which in turn interacts with the solute dipole, leading to stabilization. 

Equation (2.39) leads to the prediction that AA should be proportional to p2. For a bulk solvent, this can be considered as a molecular equivalent of the well-known Onsager formula derived for the continuum dielectric model [12],... 

Continuum models have a long and honorable tradition in solvation modeling they ultimately have their roots in the classical formulas of Mossotti (1850), Clausius (1879), Lorentz (1880), and Lorenz (1881), based on the polarization fields in condensed media [32, 57], Chemical thermodynamics is based on free energies [58], and the modem theory of free energies in solution is traceable to Bom s derivation (1920) of the electrostatic free energy of insertion of a monatomic ion in a continuum dielectric [59], and Kirkwood and Onsager s... 

From a computational view point, chemical reactions in solution present a yet not solved challenge. On one hand, some of the solvent effects can be approximated as if the solute molecule would be in a continuum with a given dielectric characterization of the liquid, and this view point has been pioneered by Bom [1], later by Kirkwood [2] and Onsager [3] and even later by many computational quantum chemists [4-9], On the other hand, the continuum model fails totally when one is interested in the specific... 

In the quantum mechanical continuum model, the solute is embedded in a cavity while the solvent, treated as a continuous medium having the same dielectric constant as the bulk liquid, is incorporated in the solute Hamiltonian as a perturbation. In this reaction field approach, which has its origin in Onsager s work, the bulk medium is polarized by the solute molecules and subsequently back-polarizes the solute, etc. The continuum approach has been criticized for its neglect of the molecular structure of the solvent. Also, the higher-order moments of the charge distribution, which in general are not included in the calculations, may have important effects on the results. Another important limitation of the early implementations of this method was the lack of a realistic representation of the cavity form and size in relation to the shape of the solute. 

Various treatments of these effects have been developed over a period of years. The conductance equations of Fuoss and Onsager l, based on a model of a sphere moving through a continuum, are widely used to interpret conductance data. Similar treatments n 3, as well as more rigorous statistical mechanical approaches 38>, will not be discussed here. For a comparison of these treatments see Ref. 11-38) and 39>. The Fuoss-Onsager equations are derived in Ref.36), and subsequently modified slightly by Fuoss, Onsager and Skinner in Ref. °). The forms in which these equations are commonly expressed are... 

Figure 5. Cascade construction with a continuum of copies of the system with Onsager matrix Lij located in [y, y + dy], along with corresponding heat baths for the conversion of a heat flow Q y) into a power contribution dW(y).

Historically, one of the central research areas in physical chemistry has been the study of transport phenomena in electrolyte solutions. A triumph of nonequilibrium statistical mechanics has been the Debye—Hiickel—Onsager—Falkenhagen theory, where ions are treated as Brownian particles in a continuum dielectric solvent interacting through Cou-lombic forces. Because the ions are under continuous motion, the frictional force on a given ion is proportional to its velocity. The proportionality constant is the friction coefficient and has been intensely studied, both experimentally and theoretically, for almost 100... 

The Onsager cavity description of solvation treats the solvent as a dielectric continuum. The dielectric dynamics of the solvent is typically characterized by the frequency-dependent complex dielectric constant s(co). The measurement of (co) for a neat solvent is conventionally called a dielectric dispersion measurement. 

An Evaluation of the Debye-Onsager Model. The simplest treatment for solvation dynamics is the Debye-Onsager model which we reviewed in Section II.A. It assumes that the solvent (i) is well modeled as a uniform dielectric continuum and (ii) has a single relaxation time (i.e., the solvent is a Debye solvent ) td (Eq. (18)). The model predicts that C(t) should be a single... 

A number of theoretical models for solvation dynamics that go beyond the simple Debye Onsager model have recently been developed. The simplest is an extension of Onsager model to include solvents with a non-Debye like (dielectric continuum and the probe can be represented by a spherical cavity. Newer theories allow for nonspherical probes [46], a nonuniform dielectric medium [45], a structured solvent represented by the mean spherical approximation [38-43], and other approaches (see below). Some of these are discussed in this section. Attempts are made where possible to emphasize the comparison between theory and experiment. 

These results allow a test of the Onsager cavity model for a uniform dielectric continuum solvent with a dielectric response that is well modeled by Eq. (24). Our group recently tested this model for methanol. In this case, both high frequency (co) data (see Barthel et al. [Ill]) and short time resolution C(t) data [32] exist. 

Bagchi and co-workers [47-50] have explored the role of translational diffusion in the dynamics of solvation by employing a Smoluchowski-Vlasov equation (see also Calef and Wolyness [37] and Nichols and Calef [42]). A significant contribution to polarization relaxation is observed in certain cases. It is found that the Onsager inverted snowball model is correct only when the rotational diffusion mechanism of solvation dominates the polarization relaxation. The Onsager model significantly breaks down when there is an important translational contribution to the polarization relaxation [47-50]. In fact, translational effects can rapidly accelerate solvation near the probe. In certain cases, the predicted behavior can actually approach the uniform continuum result that rs = t,. 

According to the Onsager model1886, the dipole of the neutral spherical cavity solute embedded in a continuum dielectric induces a dipole within the dielectric. The induced dipole, in mm, changes the dipole of the solute and then both dipoles are iterated to self-consistency. The energy of the interaction s() v is represented by equation 49... 

Born s idea was taken up by Kirkwood and Onsager [24,25], who extended the dielectric continuum solvation approach by taking into account electrostatic multipole moments, Mf, i.e., dipole, quadrupole, octupole, and higher moments. Kirkwood derived the general formula ... 

As a first approximation, solvent effects can be described by models where the solvent is represented by a dielectric continuum, e.g., the Onsager reaction-field model. 

Since the development of the Onsager model, there have been a number of elaborations on the model [4,5]. For example, the spherical cavity has been replaced by molecularly-shaped cavities. The state of the art within the field of solvent effects described by continuum solvent models is now implemented in, e.g., the Gaussian program package. 

In practice, empirical or semi-empirical interaction potentials are used. These potential energy functions are often parameterized as pairwise additive atom-atom interactions, i.e., Upj(ri,r2,..., r/v) = JT. u ri j), where the sum runs over all atom-atom distances. An all-atom model usually requires a substantial amount of computation. This may be reduced by estimating the electronic energy via a continuum solvation model like the Onsager reaction-field model, discussed in Section 9.1. 

Still within continuum solvation models, Wang et al. [5] have used an ab initio SCRF Onsager model to compute vibrational frequencies at different levels of the ab initio QM molecular theory, the G-COSMO model has been used by Stefanovich and Truong to calculate vibrational frequencies at the DFT level [6], and the multipole SCRF model, developed by the group of Rivail, has been extended to the calculation of frequency shifts at the HF, MP2 and DFT levels, including nonequilibrium effects [7],... 

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 ... 

In order to formulate a theory for the evaluation of vibrational intensities within the framework of continuum solvation models, it is necessary to consider that formally the radiation electric field (static, Eloc and optical E[jc) acting on the molecule in the cavity differ from the corresponding Maxwell fields in the medium, E and Em. However, the response of the molecule to the external perturbation depends on the field locally acting on it. This problem, usually referred to as the local field effect, is normally solved by resorting to the Onsager-Lorentz theory of dielectric polarization [21,44], In such an approach the macroscopic quantities are related to the microscopic electric response of... 

Within the dielectric continuum model, the electrostatic interactions between a probe and the surrounding molecules are described in terms of the interaction between the charges contained in the molecular cavity, and the electrostatic potential these changes experience, as a result of the polarization of the environment (the so-called reaction field). A simple expression is obtained for the case of an electric dipole, /a0, homogeneously distributed within a spherical cavity of radius a embedded in an anisotropic medium [10-12], by generalizing the Onsager model [13]. For the dipole parallel (perpendicular) to the director, the reaction field is parallel (perpendicular) to the dipole, and can be calculated as [10] ... 

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