Kirkwood-Onsager model

The exciplex or CIP is treated as a dipole of radius q and dipole moment p. The last term in Eq. (15) describes the energy of this dipole, based on the Kirkwood-Onsager model (assuming formation of a spherical complex) [14]. Thus, an exciplex or CIP is stabilized by Coulombic interactions and by solvation. The solvation energy is expected to be favored by increasing solvent polarity and a large dipole... 

The Born—Kirkwood—Onsager model, however, is particularly simple to implement its advantages include the ability to use correlated wavefunc-tions220,222 and to calculate analytic first and second derivatives.- T2i9,22o such, the BKO or Onsager model (the latter considers only the dipole-—not the charge—and hence is appropriate only for neutral solutes) is available in many standard programs,2i . 228,229 and it is widely employed. 

Applications of the Born—Kirkwood-Onsager model at the ab initio level include investigations of solvation effects on sulfamic acid and its zwitterion,23i an examination of the infrared spectra of formamide and formamidic acid,222 and a number of studies focusing on heterocyclic tautomeric equilibria.222,232,233 a more detailed comparison of some of the heterocyclic results is given later. The gas phase dipole moment depends on basis set, and systematic studies of this dependence are available. Furthermore, the effects of basis set choice and level of correlation analysis have been explored in solvation studies as well,222,233 but studies to permit identification of particular trends in their impact on the solvation portion of the calculation are as yet insufficient. 

Reaction Fields from Higher Order Multipolar Expansions Generalizations of the Born—Kirkwood—Onsager model have appeared which extend the multipole series to arbitrarily high order.20,62,144,234-236 ybis approach yields... 

The aqueous solvation free energies of the four tautomers available to the 5-(2H)-isoxazolone system have also been studied using a variety of continuum models (Table 7). Hillier and co-workers - " have provided data at the ab initio level using the Born-Kirkwood-Onsager model, the classical multipolar expansion model (up to I = 7), and an ab initio polarized continuum model. We examined the same BKO model with a different cavity radius and the AMl-SMl and AMl-SMla o- models, and Wang and Ford have performed calculations with the AMl-PCM model. 

Nearly symmetrical molecules deserve special mention. Benzene and piperazine are uncharged and have no dipole moment, so the Born-Kirkwood-Onsager model predicts AGs — 0- However, AM1-SM2 predicts —0.5 and —7.8 kcal/mol, respectively, in good agreement with the experimental -0.9 and -7.4 kcal/mol. In benzene the result comes as the sum of a hydrophobic AGcds = 1.4 kcal/mol and a hydrophilic AGe p = —2.0 kcal/mol whereas in piperazine both terms are hydrophilic (AG ds 4.1 kcal/mol, AGg p = —3.7 kcal/ mol), and they reinforce each other. Similar reinforcement occurs in many other compounds [e.g., p-bromophenol (AGs -4.4 kcal/mol, AGg p = —2.7 kcal/mol)], in which case AM1-SM2 predicts AGs 7.0 kcal/mol versus an experimental value of —7.1 kcal/mol. 

M. V. Basilevsky and G. E. Chudinov, Dynamics of charge transfer chemical reactions in a polar medium within the scope of the Born-Kirkwood-Onsager model, Chem. Phys. 157, 327-344 (1991). 

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. 

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 ... 

As emphasized, the Born—Kirkwood—Onsager (BKO) approach includes only the solute s monopole and dipole interaction with the continuum. That is, the full classical multipolar expansion of the total solute charge distribution is truncated at the dipole term. This simplification of the electronic distribution fails most visibly for neutral molecules whose dipole moments vanish as a result of symmetry. A distributed monopole or distributed dipole model is more... 

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... 

Kirkwood-Westheimer model arise. A fixed dipole moment of fj, in the Onsager model gives rise to an energy stabilization. 

More recently, Harris and Alder,24 keeping the general principles of Kirkwood s theory, have tried to calculate the polarization effects more rigorously. Unfortunately their final equation does not coincide as it should, with Onsager s equation when it is assumed that there are no short-range interactions cos y) — 0). This is because some of Kirkwood s equations are only valid when the assumptions of the author are justified, and cannot be used as was done by Harris and Alder, when a deformation polarization is superimposed on the orientation polarization. For instance, in presence of deformation effects boundary conditions cannot be introduced in the same manner as in Kirkwood s model (cf. Frdhlich 21). 

The traditional treatment regarding medium effects is the dielectric continuum model in the Kirkwood-Onsager theory [76]. This simple model assumes that the solvation energy arises from the electrostatic interaction between the solutes and... 

The quantum Onsager model, which has also been termed the Self-Consistent Reaction Field (SCRF) method, is the simplest of the continuum models used in solvation studies. In this model, which dates from the work of Kirkwood[44] and Onsager[45] in the 1930s, the solvent is represented by a continuous rmifonn dielectric with a static dielectric constant, e, surrounding a solute in a spherical cavity[46] - [48]. 

We begin by making contact with the work of Booth [7] to which reference has already been made. Booth makes use of earlier models derived independently by Kirkwood, Onsager and Frohlich [24,25,26] and the most convenient place to introduce the work is by starting from Eq. (11.18) and proceeding as follows ... 

For such a comparison one has to consider the step of the density at the phase transition. All the other relations which connect molecular parameters such as the molecular dipole moment /i, the polarizability a and the angle between the molecular long axis and fj, with each other have a general problem the calculation of the internal field and its anisotropy. Therefore, all the equations given in Vol. 1, Chap. VII.2 are necessary and useful but one has to take always into account the limitations of the models. Nevertheless, the Onsager theory [27] (basis for the Maier-Meier model [28]) and Kirkwood-Frbhlich model [29] have been... 

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 ... 

SASA), a concept introduced by Lee and Richards [9], and the electrostatic free energy contribution on the basis of the Poisson-Boltzmann (PB) equation of macroscopic electrostatics, an idea that goes back to Born [10], Debye and Htickel [11], Kirkwood [12], and Onsager [13]. The combination of these two approximations forms the SASA/PB implicit solvent model. In the next section we analyze the microscopic significance of the nonpolar and electrostatic free energy contributions and describe the SASA/PB implicit solvent model. 

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