Discretized continuum models

Claverie P, J P Daudey, J Lmglet, B Pullman, D Piazzola and M J Huron 1978. Studies of Solvent Effects. I. Discrete, Continuum and Discrete-Continuum Models and Their Comparison for Some Simple Cases NH, CH3OH and substituted NH4. Journal of Physical Chemistry 82 405-418. 

Claverie, P. Daudey, J.P. Langlet, J. Pullman, B. Piazzola, D. Huron, M.J., Studies of solvent effects. 1. Discrete, continuum, and discrete-continuum models and their comparison for some simple cases NH4, CH3OH, and substituted NH4, J. Phys. Chem. 1978, 82, 405-418... 

The molecular mechanism of the Hoffmann elimination involving (iV-Cl)-N-methyl-ethanolamine has been theoretically characterized by using DFT at the B3LYP/ 6-31++G computing level.49 The role of water as a solvent has been analysed by using both discrete and hybrid discrete-continuum models. The rearrangement proceeds by a water-assisted asynchronous concerted mechanism. 

Equation (3.21) shows that the potential of the mean force is an effective potential energy surface created by the solute-solvent interaction. The PMF may be calculated by an explicit treatment of the entire solute-solvent system by molecular dynamics or Monte Carlo methods, or it may be calculated by an implicit treatment of the solvent, such as by a continuum model, which is the subject of this book. A third possibility (discussed at length in Section 3.3.3) is that some solvent molecules are explicit or discrete and others are implicit and represented as a continuous medium. Such a mixed discrete-continuum model may be considered as a special case of a continuum model in which the solute and explicit solvent molecules form a supermolecule or cluster that is embedded in a continuum. In this contribution we will emphasize continuum models (including cluster-continuum models). 

Pratt and co-workers have proposed a quasichemical theory [118-122] in which the solvent is partitioned into inner-shell and outer-shell domains with the outer shell treated by a continuum electrostatic method. The cluster-continuum model, mixed discrete-continuum models, and the quasichemical theory are essentially three different names for the same approach to the problem [123], The quasichemical theory, the cluster-continuum model, other mixed discrete-continuum approaches, and the use of geometry-dependent atomic surface tensions provide different ways to account for the fact that the solvent does not retain its bulk properties right up to the solute-solvent boundary. Experience has shown that deviations from bulk behavior are mainly localized in the first solvation shell. Although these first-solvation-shell effects are sometimes classified into cavitation energy, dispersion, hydrophobic effects, hydrogen bonding, repulsion, and so forth, they clearly must also include the fact that the local dielectric constant (to the extent that such a quantity may even be defined) of the solvent is different near the solute than in the bulk (or near a different kind of solute or near a different part of the same solute). Furthermore... 

The halogenation reaction of ethylene has been modeled by many researchers [170, 172-176], For chlorination in apolar solvents (or in the gas phase), the formation of two radical species requires the use of flexible CASSCF and MRCI electronic structure methods, and such calculations have been reported by Kurosaki [172], In aqueous solution, Kurosaki has used a mixed discrete-continuum model to show that the reaction proceeds through an ionic mechanism [174], The bromination reaction has also received attention [169,170], However, only very recently was a reliable theoretical study of the ionic transition state using PCM/MP2 liquid-phase optimization reported by Cammi et al. [176], These authors calculated that the free energy of activation for the ionic bromination of the ethylene in aqueous solution is 8.2 kcalmol-1, in good agreement with the experimental value of 10 kcalmol-1. 

Calculations at the B3LYP/6-31G level were used to show how hydrogen bond formation influences the chemical reactivity of ketones.70 The effect of the chloroform on the activation energies was modelled by means of discrete-continuum models. Explicit hydrogen bond formation to chloroform lowers the gas-phase activation barrier. A DFT analysis of the global electrophilicity of the reagents provided a sound explanation of the catalytic effects of chloroform (see Table 6 and Chart 3). The electrophilicity of acetone... 

Traditional continuum, discrete, and mixed discrete-continuum models developed during the past few years have been reviewed in a number of papers (Bonaccorsi et al., 1982 Claverie, 1982 Tapia, 1982). In spile of their obvious shortcomings, there is still hope that the models proposed so far may be suitable for describing, at least partially, some of the features of the solvation phenomena. We shall present an application of theoretical models of solvation to the estimation of the influence of the environment on the shift of the tautomeric equilibrium A B. First, some comments on the models are given. 

Solvent effects (i.e. THF, s = 7.4257) were introduced by a discrete-continuum model two THF molecules were explicitly included in the calculations as potential ligands (see above), and the effect of the bulk solvent was considered with a continuum method, the PCM approach [39], by means of single point calculations at all optimized gas phase geometries. In this method, the radii of the spheres employed to create the cavity for the solute were defined with the UFF model,. which is the default in Gaussian09. 

In the PCM the solvent is described as a homogeneous dielectric which is polarized by the solute. The latter is placed within a cavity in the solvent medium (built as the envelope of spheres centered on the solute atoms) and the proper electrostatic problem at the cavity surface is solved using a boundary element approach [78]. In the PCM framework, the solvent loses its molecularity and, especially in hydrogen-bonding solvents, the explicit inclusion of solute-solvent interactions is very important for getting accurate results. In these cases, as discussed in detail in the next section, mixed discrete/continuum models, where a limited number of solvent molecules are included in the computational model, usually provide accurate results. 

Actually, our mixed discrete-continuum model is not limited to the study of UV-vis spectra, but it has been already successfully employed to model solvent effects on several different spectral properties, such as electron paramagnetic resonance (EPR) hyperfine coupling constants, nuclear magnetic resonance (NMR) chemical shifts, and so on [45, 121]. 

Because we will focus especially on molecular liquids, it is of primary importance to define an accurate model for the treatment of solvent effects. In this respect, we have adopted a discrete/continuum model, which is well suited for solute-solvent systems and is nicely consistent with the time-dependent approach described in the following... 

Jeon J, Quaranta V, Cummings P (2010) An off-lattice hybrid discrete-continuum model of tumor growth and invasion. Biophys J 98(1) 37-47... 

As a matter of fact, the four water molecules forming specific hydrogen bonds with the carbonyl and nitrogen moieties of uracil (see Fig. 17.3) introduce an additional contribution to the total solvent shift. Therefore, a quantitative reproduction of the total solvent shift is achieved only using an integrated discrete/continuum model, which can be viewed as an application of Eq. 17.19 in which the QM level of theory is used for the treatment of both the solute and the first solvation shell, while the MF layer is used for the bulk solvent effects. 

The mixed solvent models, where the first solvation sphere is accounted for by including a number of solvent molecules, implicitly include the solute-solvent cavity/ dispersion terms, although the corresponding tenns between the solvent molecules and the continuum are usually neglected. Once discrete solvent molecules are included, however, the problem of configuration sampling arises. Nevertheless, in many cases the first solvation shell is by far the most important, and mixed models may yield substantially better results than pure continuum models, at the price of an increase in computational cost. 

The continuum model with the Hamiltonian equal to the sum of Eq. (3.10) and Eq. (3.12), describing the interaction of electrons close to the Fermi surface with the optical phonons, is called the Takayama-Lin-Liu-Maki (TLM) model [5, 6], The Hamiltonian of the continuum model retains the important symmetries of the discrete Hamiltonian Eq. (3.2). In particular, the spectrum of the single-particle states of the TLM model is a symmetric function of energy. 

The theoretical methods can be divided into two fundamental groups. The so-called continuum models are characterized by assuming that the medium is a structureless and polarizable dielectricum described only by macroscopic physical constants. On the other hand there are the so-called discrete models. The main advantage of... 

Another aspect that has been theoretically studied109,124,129 is experimental evidence that Diels-Alder reactions are quite sensitive to solvent effects in aqueous media. Several models have been developed to account for the solvent in quantum chemical calculations. They may be divided into two large classes discrete models, where solvent molecules are explicitly considered and continuum models, where the solvent is represented by its macroscopic magnitudes. Within the first group noteworthy is the Monte Carlo study... 

The methods used for modeling pure granular flow are essentially borrowed from that of a molecular gas. Similarly, there are two main types of models the continuous (Eulerian) models (Dufty, 2000) and discrete particle (Lagrangian) models (Herrmann and Luding, 1998 Luding, 1998 Walton, 2004). The continuum models are developed for large-scale simulations, where the controlling equations resemble the Navier-Stokes equations for an ordinary gas flow. The discrete particle models (DPMs) are typically used in small-scale simulations or... 

Bokkers, G. A., Van Sint Annaland, M., and Kuipers, J. A. M., Comparison of continuum models using kinetic theory of granular flow with discrete particle models and experiments extent of particle mixing induced by bubbles. Proceedings of Fluidization XI, May 9-14, 2004, 187-194, Naples, Italy (2004). 

Even better agreement is observed between calorimetric and elastic Debye temperatures. The Debye temperature is based on a continuum model for long wavelengths, and hence the discrete nature of the atoms is neglected. The wave velocity is constant and the Debye temperature can be expressed through the average speed of sound in longitudinal and transverse directions (parallel and normal to the wave vector). Calorimetric and elastic Debye temperatures are compared in Table 8.3 for some selected elements and compounds. 

The two avenues above recalled, namely ab-initio computations on clusters and Molecular Dynamics on one hand and continuum model on the other, are somewhat bridged by those techniques where the solvent is included in the hamiltonian at the electrostatic level with a discrete representation [13,17], It is important to stress that quantum-mechanical computations imply a temperature of zero K, whereas Molecular Dynamics computations do include temperature. As it is well known, this inclusion is of paramount importance and allows also the consideration of entropic effects and thus free-energy, essential parameters in any reaction. 

It can be seen from Table 26.1 that various methods used to model the effect of a solvent can be broadly classified into three types (1) those which treat the solvent as continuous medium, (2) those which describe the individual solvent molecules (discrete/explicit solvation), and (3) combinations of (1) and (2) treatments. The following section provides a brief introduction to continuum models. 

This volume of Modem Aspects covers a wide spread of topics presented in an authoritative, informative and instructive manner by some internationally renowned specialists. Professors Politzer and Dr. Murray provide a comprehensive description of the various theoretical treatments of solute-solvent interactions, including ion-solvent interactions. Both continuum and discrete molecular models for the solvent molecules are discussed, including Monte Carlo and molecular dynamics simulations. The advantages and drawbacks of the resulting models and computational approaches are discussed and the impressive progress made in predicting the properties of molecular and ionic solutions is surveyed. 

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