Onsager cavity model

Several authors have discussed how C(r) can be calculated from the Onsager cavity model. Briefly, we need to consider the time-dependent reaction field, which was related above (Eqs. (12) and (15)) to C(t). For simplicity we consider the case of a probe with ps = 0 and 0. If the probe... 

The physical meaning of the relationship described in the previous subsection becomes apparent when we consider the popular special case of the Onsager cavity model that arises if we assume that the solvent s dielectric properties are well described by a Debye form. 

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. 

The energy curves in Figure 22 are closely related to the Marcus-Hush theory for electron transfer. The formalism we employ emphasizes a dipole model for the solute solvent interaction, i.e., an Onsager cavity model. However, a Born charge model based on ion solvation as something in between [135] would be essentially equivalent because we do not attempt to calculate Bop and Bor but rather determine them empirically. 

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). 

If the term in the derivative of the field factor were negligible the expression on the left of this equation would be defined completely in terms of macroscopic measurable quantities. The specifics of the chosen cavity model enter the field factor derivative where Lorentz-Lorenz and Onsager factors may be mixed. The most commonly used procedure is to employ Onsager for the static field and Lorentz factors for the optical fields. For Fj (i = 0,1),... 

In addition, the molecules properties are changed due to the interaction with the surrounding medium. Several computational schemes have been proposed to address these effects. Tliey are essentially based on the extension of the Onsager reaction field cavity model and give effective hyperpolarizabilities, i.e. molecular hyperpolarizabilities induced by the external fields that include the modifications due to the surrounding molecules as well as local (cavity) field effects [40 2]. These condensed-phase effects have, however, not yet been included in the SFG hyperpolarizability calculations, which are therefore strictly gas-phase calculations. 

The chemical shifts of polar molecules are frequently found to be solvent dependent. Becconsall and Hampson have studied the solvent effects on the shifts of methyl iodide and acetonitrile. The results obtained from dilution studies in various solvents may be explained as arising from a reaction field around the solute molecules. The spherical cavity model due to Onsager was used to describe this effect, and this model was completely consistent with the experimental data when a modified value for the dielectric constant, s, of the particular solvent was used. 

Figure 2 Solute cavity model used by Kirkwood, Block, Walker, and Onsager.

For electrically neutral solutes, only the dipolar interactions contribute to the solvation energy. In the Onsager s spherical cavity model, the Fock operator Fmn is then modified by adding the response of a dielectric medium, resulting in... 

The concept of reaction field, originally formulated by Onsager [194], has been proved to be fruitful in the quantum chemical treatment of polar subsystems (solutes) embedded in polarizable environment (solvent) [195]. Simple cavity models, where the solvent is represented by a continuous dielectric medium and the solute is sitting in a cavity inside this dielectric, has numerous application in the framework of semiempirical [196-200] and ab initio [201-205] methods. The utility of this concept in the modelisation of biochemical processes was pointed out by Tapia and his coworkers [206]. 

The earlier attempt to approach the electrostatic contribution to the free energy of solvation is due to Kirkwood (1934). This model is based on a multipole expansion of the charge distribution of the solute at the center of a spherical cavity surroimded by a continuum represented by the dielectric permittivity of the solvent. When this expansion is limited to rank 1 which corresponds to a pure dipole fi, one finds the Onsager model (Onsager 1936) in which the electrostatic contribution to the free energy of solvation by a solvent of dielectric constant e of a molecule having a dipole moment in a cavity of radius a takes the expression ... 

The local field factors depend on the dielectric constant of the solvent and on the shape of the solute cavity. The experimental values reported in the literature for y " in solution are obtained from the corresponding macroscopic quantities by exploiting approximate expressions of / " (Onsager and/or Lorentz formulas) based on spherical cavity models. 

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. 

The simplest SCRF model is the Onsager reaction field model. In this method, the solute occupies a fixed spherical cavity of radius Oq within the solvent field. A dipole in the molecule will induce a dipole in the medium, and the electric field applied by the solvent dipole will in turn interact with the molecular dipole, leading to net stabilization. 

Optimize the two equilibrium structures in solution, using the Onsager SCRF method and the RHF/6-31G(d) model chemistry. You ll of course need to determine the appropriate cavity radius first. 

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. 

The spherical cavity, dipole only, SCRF model is known as the OnMger model.The Kirkwood model s refers to a general multipole expansion, if the cavity is ellipsoidal the Kirkwood—Westheimer model arise." A fixed dipole moment of yr in the Onsager model gives rise to an energy stabilization. 

FL, and the difference in dipole moments determined from the plot is 2.36 D if the Onsager radius is 0.33 nm [53]. The Onsager cavity radius was obtained from molecular models where the molar volumes were calculated by CAChe WS 5.0 computer program. The simplest method to estimate the cavity radius is to assume a = (3y/47r) 3, where V is the volume of the solute. 

In the Onsager s SCRF model, the solute is placed in a cavity immersed in a continuous medium with a dielectric constant e. The molecular dipole of the solute induces a dipole in the solvent, which in turn interacts with the molecular dipole, leading to a net stabilization effect. 

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. 

On the basis of an Onsager cavity (23) model of dielectrics applied to a polar solute with an intrinsic dipole movement /xr° in its rth electronic state, Mazurenko gives an equation for the orientational free energy of the solute molecule in a pure polar solvent environment, which can be identified as equivalent to u/jlpe chem, thus 2... 

Since our treatment of the ionic atmosphere around a dipolar molecule makes use of the Onsager model, it becomes necessary to adopt a similar model for the ion. Consequently we are going to assume that the ion is also represented by a spherical cavity in the surrounding dielectric with a point charge at its center. Then the constants by the ordinary boundary conditions become... 

The physical significance of these variables is apparent when they are evaluated in the Onsager cavity description of solvation, which treats the solute as a sphere (which we will assume here is unpolarizable) of radius a. The solvent is modeled as a uniform dielectric medium with a static dielectric constant s and an optical dielectric constant op. The following relationships apply in the Onsager cavity description... 

A simple model for C(t). In this subsection we explore the relationship of C(r) to dynamic properties of the solvent, in terms of the Onsager cavity description, following the work in the literature on this subject [12-14, 53-57]. Theories that go beyond the Onsager model are described in Sections II.E and II.D. 

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. 

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