Onsager’s reaction field

Onsager s reaction field model in its original fonn offers a description of major aspects of equilibrium solvation effects on reaction rates in solution that includes the basic physical ideas, but the inlierent simplifications seriously limit its practical use for quantitative predictions. It smce has been extended along several lines, some of which are briefly sunnnarized in the next section. 

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 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],... 

Sules respectively, and a is the radius of the cavity in which the solute nolecule resides. The latter is defined by Onsager s reaction field Assuming the solvent as continuous dielectric. 

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. 

The electric fields of electric multipoles are extensively discussed in Sections 2 and 3. Here, it should only be noted that in many a case, with the aim of performing rapid numerical evaluations, one can replace the molecular field Fj by Onsager s reaction field which, for dipolar molecules, amounts to ... 

Onsager s original reaction field method imposes some serious lunitations the description of the solute as a point dipole located at the centre of a cavity, the spherical fonn of the cavity and the assumption that cavity size and solute dipole moment are independent of the solvent dielectric constant. 

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. 

R is the radius of the cavity, p and a are the dipole moment and polarizability of the solute, and s the dielectric constant of the solvent. Equation (35) does address the polarization of the solute molecule by the reaction field, although not carrying this to self-consistency. (It is interesting that Onsager s paper, the sixth-most-cited in the history of the Journal of the American Chemical Society, was rejected by the Physikalische Zeitshrift, to which it had initially been submitted.)89... 

The key differences between the PCM and the Onsager s model are that the PCM makes use of molecular-shaped cavities (instead of spherical cavities) and that in the PCM the solvent-solute interaction is not simply reduced to the dipole term. In addition, the PCM is a quantum mechanical approach, i.e. the solute is described by means of its electronic wavefunction. Similarly to classical approaches, the basis of the PCM approach to the local field relies on the assumption that the effective field experienced by the molecule in the cavity can be seen as the sum of a reaction field term and a cavity field term. The reaction field is connected to the response (polarization) of the dielectric to the solute charge distribution, whereas the cavity field depends on the polarization of the dielectric induced by the applied field once the cavity has been created. In the PCM, cavity field effects are accounted for by introducing the concept of effective molecular response properties, which directly describe the response of the molecular solutes to the Maxwell field in the liquid, both static E and dynamic E, [8,47,48] (see also the contribution by Cammi and Mennucci). 

Local ordering effects have long been recognised experimentally in measurements of dipole moments of polar solutes in non-polar solvents, where the value obtained on the basis of the simple model differs from the value obtained for the pure solute in the gas phase, even when the results are extrapolated to infinite dilution. This so-called solvent effect is due to the Onsager reaction field. If there is no strong local ordering, Onsager s formula (2.52) is valid and the apparent solution moment is related to the isolated molecule or gas moment by... 

The expansion of the electrostatic potential into spherical harmonics is at the basis of the first quantum-continuum solvation methods (Rinaldi and Rivail, 1973 Tapia and Goschinski, 1975 Hylton McCreery et al., 1976). The starting points are the seminal Kirkwood s and Onsager s papers (Kirkwood 1934 Onsager 1936) the first one introducing the concept of cavity in the dielectric, and of the multipole expansion of the electrostatic potential in that spherical cavity, the second one the definition of the solvent reaction field and of its effect on a point dipole in a spherical cavity. The choice of this specific geometrical shape is not accidental, since multipole expansions work at their best for spherical cavities (and, with a little additional effort, for other regular shapes, such as ellipsoids or cylinders). 

The strength of the N H- -O bridge of ort/zo-amino-furanaldehyde, ealeu-lated as 26.02 kJ/mol at B3LYP/6-31+G level (with respect to the Z con-former) was found about 5 kJ/mol lower than that of the N H- -S hydrogen bond of ort/zo-amino-furanthioaldehyde (30.92 kJ/mol) [238]. Such values become 14.06 and 16.86 kJ/mol, respectively, when calculated in DMSO solution (a = 46.7) following the Onsager self-consistent reaction field method. 

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. 

---