Continuum reaction field model

Most coordination complexes are not well modeled as spheres of uniform charge (e.g., metallo-protein active sites are an extreme example). Therefore, more sophisticated dielectric continuum reaction field models are generally employed to take into account the shape of the complex and even account for the dielectric properties of the ligands themselves. Furthermore, the charge distribution and/or polarity of the complex can be used to model discretely the reaction field energies at various points on the exterior surface of the solute. There are many variations of such solvent models, some of which are available in standard quantum chemistry programs. 

Any of the methods used in classical Monte Carlo and molecular dynamics simulations may be borrowed in the combined QM/MM approach. However, the use of a finite system in condensed phase simulations is always a severe approximation, even when appropriate periodic or stochastic boundary conditions are employed. A further complication is the use of potential function truncation schemes, particular in ionic aqueous solutions where the long-range Coulombic interactions are significant beyond the cutoff distance.Thus, it is alluring to embed a continuum reaction field model in the quantum mechanical calculations in addition to the explicit solute—solvent interaaions to include the dielectric effect beyond the cutoff distance. - uch an onion shell arrangement has been used in spherical systems, whereas Lee and Warshel introduced an innovative local reaction field method for evaluation of long-... 

Solvent effects on chemical equilibria and reactions have been an important issue in physical organic chemistry. Several empirical relationships have been proposed to characterize systematically the various types of properties in protic and aprotic solvents. One of the simplest models is the continuum reaction field characterized by the dielectric constant, e, of the solvent, which is still widely used. Taft and coworkers [30] presented more sophisticated solvent parameters that can take solute-solvent hydrogen bonding and polarity into account. Although this parameter has been successfully applied to rationalize experimentally observed solvent effects, it seems still far from satisfactory to interpret solvent effects on the basis of microscopic infomation of the solute-solvent interaction and solvation free energy. 

As discussed in Section 2, one key assumption of reaction field models is that the polarization field of the solvent is fully equilibrated with the solute. Such a situation is most likely to occur when the solute is a long-lived, stable molecular structure, e g., the electronic ground state for some local minimum on a Bom-Oppenheimer potential energy surface. As a result, continuum solvation models... 

As a first approximation, solvent effects can be described by models where the solvent is represented by a dielectric continuum, e.g., the Onsager reaction-field model. 

In practice, empirical or semi-empirical interaction potentials are used. These potential energy functions are often parameterized as pairwise additive atom-atom interactions, i.e., Upj(ri,r2,..., r/v) = JT. u ri j), where the sum runs over all atom-atom distances. An all-atom model usually requires a substantial amount of computation. This may be reduced by estimating the electronic energy via a continuum solvation model like the Onsager reaction-field model, discussed in Section 9.1. 

There are various methods for treating solvation, ranging from a detailed description at the molecular level to reaction field models where the solvent is modelled as a continuum method.125... 

Probably the simplest quantum-mechanical operators that include interaction with a continuum are the Born and Onsager reaction field models. In the case of neutral solutes, the model could be express as... 

In the reaction field model (Onsager, 1936), a solute molecule is considered as a polarizable point dipole located in a spherical or ellipsoidal cavity in the solvent. The solvent itself is considered as an isotropic and homogeneous dielectric continuum. The local field E at the location of the solute molecule is represented by (78) as a superposition of a cavity field E and a reaction field (Boettcher, 1973). 

Various solvent effect theories concerning HFS constants in ESR spectra using various reaction held approaches have been developed by Reddoch et al. [385] and Abe et al. [392]. According to Reddoch et al., none of the continuum reaction held models is entirely satisfactory. Therefore, a dipole-dipole model using a held due only to a dipole moment of one solvent molecule instead of various reaction fields was proposed, and applied to di-t-butyl aminyloxide [385]. However, Abe et al. found that the HFS constants are proportional to the reaction held of Block and Walker [393] when protic solvents are excluded [392]. This relationship has been successfully applied to di-t-butyl and diaryl aminyloxides, to the 4-(methoxycarbonyl)-l-methylpyridinyl radical cf. Fig. 6-10), and to the 4-acetyl-1-methylpyridinyl radical (see below) [392]. For another theoretical approach to the calculation of gr-values and HFS constants for di-t-butyl aminyloxide, see reference [501]. 

To examine this, we make use of the self-consistent reaction field model [29-31] that treats the surrounding polarizable matter as a dielectric continuum. This is, of course, a very simple approach, but it is not possible to actually include enough of the protein and its surroundings to model this effect directly. 

In view of the approximations inherent in the derivation of the reaction field theory, it is not surprising that some instances are known in which a non-linear relationship exists between the solvent shift and dielectric constant in polar solvents. As pointed out by Buckingham, the reaction field model is only valid for a solute that reacts in no way with the solvent or with other solute molecules but simply presents a continuum of certain dielectric properties. Protons are normally on the surface of the molecule and are therefore exposed to direct contact with the surrounding molecules, so that the Onsager model is a poor approximation of the actual reaction field acting on a molecule. 

A major area of theoretical interest has been on solvent effects, and several techniques have been applied to the calculation of NLO properties. " The most common (and simplest) method is the reaction field model, where the solute molecule is in a cavity of solvent, which is treated as a uniform dielectric medium. Cavity approaches are problematic. How do you pick the cavity size How do you pick the cavity shape How do you model stronger, specific interactions (such as hydrogen bonding) The work of Willetts and Rice " illustrated the inability of reaction field models to adequately treat solvent effects even though they tried both spherical and ellipsoidal cavities. Mikkelsen et al. attempted to provide specific interactions with their solvent model by explicitly including solvent molecules inside the cavity. These and related issues need to be addressed further if computational chemists are to develop truly useful procedures capable of including solvent effects in NLO calculations. Recent work by Cammi, Tomasi, and co-workers " has attempted to address these issues within the polarized continuum model (PCM) and have included studies of frequency-dependent hyperpolarizabilities. 

Lange, A. W., and Herbert, J. M. (2010). Polarizable continuum reaction-field solvation models affording smooth potential energy surfaces, J. Phys. Chem. Lett. 1, pp. 556-561. 

The absolute solvation Gibbs free energy of a proton can also be calculated using high-level gas phase calculations with a supermolecule-continuum approach, involving a self-consistent reaction field model. 

A many-body perturbation theory (MBPT) approach has been combined with the polarizable continuum model (PCM) of the electrostatic solvation. The first approximation called by authors the perturbation theory at energy level (PTE) consists of the solution of the PCM problem at the Hartree-Fock level to find the solvent reaction potential and the wavefunction for the calculation of the MBPT correction to the energy. In the second approximation, called the perturbation theory at the density matrix level only (PTD), the calculation of the reaction potential and electrostatic free energy is based on the MBPT corrected wavefunction for the isolated molecule. At the next approximation (perturbation theory at the energy and density matrix level, PTED), both the energy and the wave function are solvent reaction field and MBPT corrected. The self-consistent reaction field model has been also applied within the complete active space self-consistent field (CAS SCF) theory and the eomplete aetive space second-order perturbation theory. ... 

Bartkowiak and Misiaszek used the eoupled perturbed Hartree-Fock method and the sum-over-modes formalism to ealeulate the eleetronie and vibrational /i-tensors for 4-nitroaniline, 4-nitro-4 -aminostilbene, 4-amino-4 -nitrobiphenyl and 4-amino-4 -nitrophenylacetylene, all typieal push-pull eonjugated molecules of the kind that have been associated with seeond order optieal non-linearities derived from their large P values. Their ealeulations refer to the gas phase and to chloroform and aqueous solutions, the solvent effects being included through a continuum self-consistent reaction field model. They demonstrate that the solvent effects are much greater for the vibrational hyperpolarizability than for the electronic contribution. 

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