Spherical cavity model

The solvent reaction field calculations involve several different aspects. We would like concentrate on the points required to make these models successful as well as on the facts that limit their accuracy. One of them is the shape of the molecular cavity, which can be modelled spherically or according to the real shape of the solute molecule. First, we discuss the papers in which spherical cavity models were applied. The studies utilizing the solute-shaped cavity models are collected the second group. Finally, the approaches employing explicit treatment of the first-solvation shell molecules combined with the continuum models are discussed. 

The spherical cavity model requires parameters for the solvent and the cavity to calculate the reorganization energy. The dimensions of the molecule have been obtained from models created in MacSparten. The cavity radius was calculated as r = (y/d dj )/I, where d-, are the three principle diameters of the molecule (including van der Waals end radius for each atom) and are 7.8, 11.26 and 11.26, respectively, with r = 5.0 A. The Re atom is placed off center by... 

More recently, there has been renewed interest in confined atomic systems in several areas of research for reviews and references see Jaskolski [7], Sako and Diercksen [8,9] and Dolmatov et al. [10], as well as papers here in the present volume. In addition to the spherical box model, the hydrogen atom has also been studied under various types of confinement (see, for example, Ley-Koo and Rubinstein [4], Froman et al. [5], Connerade et al. [11] and Saha et al. [12]), and off-centre investigations of the spherical cavity model have been performed as well [13]. 

Two other approaches are used less often. The original SCRF spherical cavity model... 

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. 

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 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. 

The self-consistent reaction held (SCRF) method is an adaptation of the Poisson method for ah initio calculations. There are quite a number of variations on this method. One point of difference is the shape of the solvent cavity. Various models use spherical cavities, spheres for each atom, or an isosurface... 

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. 

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 simplest reaction field model is a spherical cavity, where only the net charge and dipole moment of the molecule are taken into account, and cavity/dispersion effects are neglected. For a net charge in a cavity of radius a, the difference in energy between vacuum and a medium with a dielectric constant of e is given by the Bom model. ... 

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. 

Quiben JM, Thome JR (2007b) Flow pattern based two-phase pressure drop model for horizontal tubes. Part II. New phenomenological model. Int. J. Heat and Fluid Flow. 28(5) 1060-1072 Rayleigh JWS (1917) On the pressure developed in a liquid during the collapse of a spherical cavity. Phil Mag 34 94-98... 

Fig. 5. Probabilities pn of observing n water-oxygen atoms in spherical cavity volumes v. Results from Monte Carlo simulations of SPC water are shown as symbols. The parabolas are predictions using the flat default model in Eq. (11). The center-to-center exclusion distance d (in nanometers) is noted next to the curves. The solute exclusion volume is defined by the distance d of closest approach of water-oxygen atoms to the center of the sphere. (Hummer et al., 1998a)... 

A fascinating insight into the impact that modelling can make in polymer science is provided in an article by Miiller-Plathe and co-workers [136]. They summarise work in two areas of experimental study, the first involves positron annihilation studies as a technique for the measurement of free volume in polymers, and the second is the use of MD as a tool for aiding the interpretation of NMR data. In the first example they show how the previous assumptions about spherical cavities representing free volume must be questioned. Indeed, they show that the assumptions of a spherical cavity lead to a systematic underestimate of the volume for a given lifetime, and that it is unable to account for the distribution of lifetimes observed for a given volume of cavity. The NMR example is a wonderful illustration of the impact of a simple model with the correct physics. 

Fig. 2.2 Self-Consistent Reaction Field (SCRF) model for the inclusion of solvent effects in semi-empirical calculations. The solvent is represented as an isotropic, polarizable continuum of macroscopic dielectric e. The solute occupies a spherical cavity of radius ru, and has a dipole moment of p,o. The molecular dipole induces an opposing dipole in the solvent medium, the magnitude of which is dependent on e.

Two tautomeric equilibria have been considered for substituted imidazoles, that between 2-imidazolone 3 and its 2-hydroxyimidazole tautomer 4 [268] and also that between the 1H and 3H tautomers of 4-nitroimidazole, 6 and 5, respectively [269, 270], Karelson et al. used the D02 model with a spherical cavity of 2.5 A radius and found 2-imidazolone to be better solvated than its tautomer by 7.7 kcal/mol at the AMI level. [The asterisk in D02 indicates that the reaction field... 

Karelson et al. [268] used the AMI D02 method with a spherical cavity of 2.5 A, radius to study tautomeric equilibria in the 3-hydroxyisoxazole system (the keto tautomer 13 is referred to as an isoxazolone). AMI predicts 13 to be 0.06 kcal/mol lower in energy than 14 in the gas phase. However, the AMI dipole moments are 3.32 and 4.21 D for 13 and 14, respectively. Hydroxy tautomer 14 is better solvated within the D02 model, and is predicted to be 2.6 kcal/mol lower in energy than 13 in a continuum dielectric with e = 78.4. Karelson et al. note, however, that the relative increase in dipole moment upon solvation is larger for 13 than for 14 (aqueous AMI dipole moments of 5.05 and 5.39 D, respectively). This indicates that the relative magnitude of gas-phase dipole moments will not always be indicative of which tautomer will be better solvated within a DO solvation approach — the polarizability of the solutes must also be considered. In any case, the D02 model is consistent with the experimental observation [266] of only the hydroxy tautomer in aqueous solution. 

We begin with a comparison of the various DO models to each other. Based on a parametric procedure that takes account of the molecular volume encompassed by the 0.001 a.u. electron density envelope, Wong et al. [297] suggested that an appropriate spherical cavity radius is 3.8 A. Szafran et al. [157]... 

Young et al. [195] have provided a calculation in which they compared expanding the multipole series up to /= 6 in a spherical cavity of 3.8 A. These results may be compared directly to those of Wong et al. [297] at the identical level of theon asis set in order to assess the effect of including higher moments. In each case, the differential solvation free energy increases by about 40%. This illustrates nicely the relationship between cavity radius and model... 

One point of particular interest is that it is not clear from the electrostatics-only models whether non-electrostatic phenomena affect the aqueous tautomeric equilibria. For instance, the DO results of Wong et al. [297] would suggest there are differentiating non-electrostatic phenomena, while the results of Young et al. [195] for a multipole expansion in a spherical cavity suggest that there are not. Since the SMI, SM2, and SM3 GB/ST models use Mulliken charges rather than... 

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