The Onsager Model

In their classic review on Continuous Distributions of the Solvent , Tomasi and Persico (1994) identify four groups of approaches to dealing with the solvent. First, there are methods based on the elaboration of physical functions this includes approaches based on the virial equation of state and methods based on perturbation theory with particularly simple reference systems. For many years 

In the second group come molecular dynamics and Monte Carlo simulations, especially those where the solvent is modelled without being explicitly included. Their fourth class is the related supermolecule class, where we actually include solvent molecules in the simulation, and treat the entire array of molecules according to the rules of quantum mechanics or whatever. 

The phenomenon is referred to as dielectric polarization. The induced dipole moment per volume is called the polarization P, and the charge reorganization always acts so as to reduce the field inside the dielectric. The phenomenon is treated in all elementary books on electromagnetism. 

For very many materials, the polarization can be related to the applied field by 

Suppose now, that the material is a gas comprising N atoms each of polarizability o in a volume V. When an electric field E is applied, each atom acquires an induced dipole moment aE and so the polarization is 

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This can be solved analytically only for a few simplified systems. The Onsager model uses one of the known analytic solutions. 

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. 

Fig. 7. The field-dependence of the charge-generation efficiency of a 2.0- lm thick (0) a l.l-).tm thick ( ), and 1.8-).tm thick (A) fuUerene/PMPS film obtained with positive charging and 340-nm irradiation (A). The soHd lines are calculated from the Onsager model. The best-fit curve is obtained with Tq = 2.7 nm and = 0.85. Also plotted is the charge-generation efficiency of a fuUerene/PVK film (+) obtained with positive charging and 340-nm irradiation (B). The soHd lines are calculated from the Onsager model. The best-fit curve is obtained with = 1.9 nm and = 0.9 (13).

The initial election—hole separation distance, and the quantum yield, ( )q, are derived by fitting with the Onsager model. When the initial quantum yield,... 

Experimental Values of Charge-Generation Efficiencies. In this section the charge-generation efficiencies of many polymeric photoconductors are compared (Table 3). When the experimental data has been fitted to the Onsager model, the initial electron—hole separation distance,... 

We ran an SCRF single point energy calculation for gauche dichloroethane conformers in cyclohexane (e=2.0), using the Onsager model at the Hartree-Fock and MP2 levels of theory (flfl=3.65) and using the IPCM model at the B3LYP level. The 6-31+G(d) basis set was used for all jobs. We also ran gas phase calculations for both conformations at the same model chemistries, and an IPCM calculation for the trans conformation (SCRF=Dipole calculations are not necessary for the trans conformation since it has no dipole moment). 

A molecular volume calculation to estimate Oq for the Onsager model. 

Remember that the trans form of dichloroethane has no dipole moment, so it is not necessary to compute its energy in solution with the Onsager model. 

As the plot of AE indicates, the energy difference between the two forms decreases in more polar solvents, and becomes nearly zero in acetonitrile. The left plot illustrates the fact that the IPCM model (at the B3LYP/6-31+G(d) level of theory) does a much better job of reproducing the observed solvent effect than the two Onsager SCRF models. In contrast, the Onsager model at the MP2 level treats the solvated systems more accurately than it does the gas phase system, leading to a poorer value for the solvent effect. ... 

Molecules do not consist of rigid arrays of point charges, and on application of an external electrostatic field the electrons and protons will rearrange themselves until the interaction energy is a minimum. In classical electrostatics, where we deal with macroscopic samples, the phenomenon is referred to as the induced polarization. I dealt with this in Chapter 15, when we discussed the Onsager model of solvation. The nuclei and the electrons will tend to move in opposite directions when a field is applied, and so the electric dipole moment will change. Again, in classical electrostatics we study the induced dipole moment per unit volume. 

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. 

Ab initio Simulating MeCN (the Onsager Model), semiempirical The COSMO Model, MOPAPC 93 keywords NSPA = 60, EPS = 35.9, TS, PRECISE. 

Both of these substitution pathways in MeCN solution have been simulated using the Onsager model (Tables IV and V). Whereas pathway b is favored in the gas phase, inclusion of solvent effects in the calculations causes pathway a to be energetically favored. Substitution of Cl via pathway a is now 1.6 kcal/mol more favorable. In addition, TS(X)/TS(Pyr) calculations (Scheme 15) for the OMe (40) and OSiMes (41) cations have been performed. TS(X) of both 40 and 41 remain significantly disfavored (+66.9 kcal/mol and +46.6 kcakmol, respectively), thus indicating that pathway b should be preferred in MeCN.Tliese calculations are in complete agreement with experimental observations. 

Using the Onsager model, the function Av-l(t) can be calculated for all time domains of dielectric relaxation of solvents measured experimentally for commonly used liquids (see, for example, [39]). Such simulations, for example, give for alcohols, at least, three different time components of spectral shift during relaxation, which are due to appropriate time domains of solvents relaxation. 

Of course, there are some uncertainties in this procedure, as the Onsager model describes the structures of solution and a solute only approximately. It can be noted that there is a good opportunity to calculate dipole moments, exactly, their ratio, in the simpler way using the relative shifts of absorption, and fluorescence spectra. As follows from (16) and (17), dividing them by proper parts we may obtain the following relation ... 

As the temperature is decreased, the chains become increasingly rigid zc then approaches 1 if we assume that there is only one fully ordered crystalline structure and Zconf for the liquid becomes smaller than 1. This means that, at this level of approximation, the disordered state becomes less favorable than the crystalline ground state. A first-order disorder-order phase transition is expected to occur under these conditions. Flory interpreted this phase transition as the spontaneous crystallization of bulk semiflexible polymers [12], However, since the intermolecular anisotropic repulsion essential in the Onsager model is not considered in the calculation, only the short-range intramolecular interaction is responsible for this phase transition. 

At this stage, it should be recalled that although the S/I ratio has been derived in the Onsager model for a geminate pair, there is extensive Monte Carlo simulation work to extend the same value for multiple-ion-pair cases and, therefore, to the entire track as well (see Sect. 9.3). Hence, the comparison of the experimental value of S/l with the theoretical value of Onsager is meaningful. For example, Hummel and Allen (1967) measure S/I = 6.0 x 10-5 cm/V in n-hexane at 298 K, whereas the theoretical value is 5.81 x 10-5 cm/V. 

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... 

Equation 32 expresses the influence of the dielectric constant of the medium in the case of infinite dilution of solutions. The first term is attributed to the ionic charge and is of the Born form. Born (11) obtained a corresponding term in A (see Equation 24) in his derivation of the heat of hydration, and Scatchard (12) introduced it in the theory of activities. The second and third terms represent the influence of the dipolar part. Their form is essentially affected by the use of the Onsager model (5). 

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. 

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,. 

Similarly, the reaction field, R (88-90), associated with a group of solvent molecules with cholesteric phase order is much larger when operating on a triplet of BN R increases with increasing a. Hie limitations of the Onsager model to the very anisotropic environment experienced by 2BN preclude a reasonable quantitative discussion. The solute cavity Is not spherical BN may be described better for the purposes of elucidating its interactions with neighboring solvent molecules as a quadrupole... 

Nevertheless, the use of spherical-cavity SCRF models finally led to a dead end, because no meaningful spherical cavities can be defined for non-spherical molecules. Even the attempt to overcome this problem by an extension of the Onsager model to ellipsoidal cavities did not really solve the problem, because this introduces additional fit parameters, while only a small portion of real molecules can still be considered as approximately ellipsoidal. Thus, the need for molecular-shaped SCRF models became more and more obvious. 

Correlation and solvation effects on heterocyclic equilibria in aqueous solution were analyzed with the use of SM2/AM1 and Onsager models. It was found that the Onsager model was inferior to SM2/AM1,because it underestimates the solvation of the syn -form. Local bond moments, as shown by SM2/AM1, had significant effects on the bulk electric polarizaton, even when they largely cancel in the net dipole moment. Moreover, the equilibrium shifts calculated with SM2 /AMI, due to the effects of methyl substitution on the isoxazole ring, were consistent with the available experimental data [79]. 

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