Polarizable dielectric medium

A discussion in an earlier section pointed out how reorientations of the two groups involved in a H-bond can shift the relative pKs. That is, a change in angular character of the H-bond can cause the equilibrium position of the bridging proton to shift from one group to its partner. Since these shifts 

The ability to model reactions within dielectric media continues to improve. Some of the more recent developments permit electron correlation to be included explicitly, more general cavity shapes, and even geometry optimizations within the context of the medium [118,119], and can be applied to biological systems such as nucleic acid bases [120]. 

A segment of the molecular biology community has taken this general observation as the basis for recent proposals that H-bonds of this type can make major contributions to enzymatic catalysis. This catalytic enhancement would occur by stabilization of particular states such as an enzyme-intermediate complex or a transition state [128-131]. The proponents of these ideas have dubbed these H-bonds by various acronyms, including Low Barrier H-bond (LBHB), Very Short H-bond (VSHB), and Short Strong H-bond (SSHB). The central idea behind the catalytic enhancement [131] starts with a weak, or normal, H-bond between the substrate and enzyme when they 

There is some experimental evidence that H-bonds with low barriers to proton transfer do occur in enzymatic systems. A H-bond with a low barrier has been detected between a Tyr residue of the enzyme A -S-ketosteroid isomerase and an analog of the intermediate [132], for example. The proton appears to move freely between the Asp and His residues in chymotrypsin 

The central question, however, is not whether H-bonds with low barriers exist they surely do in certain circumstances. What is relevant here is the energetic value of the transition from a so-called normal H-bond to one with a low barrier. Whereas previous experimental studies have been able to successfully identify H-bonds with a low barrier they have had much less luck in answering this important question. Estimates have been indirect at best. While some studies have suggested enhancements of as large as 7 kcal/mol [132], others have capped this effect at 1 or 2 kcal/mol [137,138], or as much as 4 or 5 kcal/mol [135,139]. (In fact, a quantum chemical study [140] later added theoretical verification to some of the energetic conclusions of Schwartz and Dmeckhammer [139].) 

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In more recent work, Johnston and co-workers (17,18,20,27,32) showed quantitatively that the local fluid density about the solute is greater than the bulk density. In these papers, results were presented for CQ2, C2H4, CF3H, and CF3C1. Local densities were recovered by comparison of the observed spectral shift (or position) to that expected for a homogeneous polarizable dielectric medium. Clustering manifests itself in deviation from the expected linear McRae continuum model (17,18,20,27,32,56,57). These data were subsequently interpreted using an expression derived from Kirkwood-Buff solution theory (20). Detailed theoretical... 

The solvent is described as an isotropic polarizable dielectric medium at equilibrium at a given pressure and temperature. 

A general description of quantum-chemical models based on a representation of the solvent as an infinite polarizable dielectric medium (dielectric continuum) is given below. All of these models are derived from the direct solution of the Poisson equations [21] defining the total electrostatic potential as... 

In the Polarizable Continuum Model for solvation, the molecular solute is hosted in a cavity of a polarizable dielectric medium representing the solvent The cavity is accurately modeled on the shape of the molecular solute (Miertus et al. 1981), and the dielectric medium is characterized by the dielectric permittivity e of the bulk solvent The physics of the model is very simple. The solute charge distribution polarizes the dielectric medium, which in turn acts back on the solute, in a process of mutual polarization that continues until self-consistence is reached. The polarization of the solvent is represented by an apparent charge distribution (ASC) spread on the cavity surface. In computational practice the ASC is discretized to a set of NTS point charges and the solute-solvent interaction is expressed as in terms of the interaction between these and the charge distribution of the molecular solute. 

The QM/MM method, and the polarizable continuum method as well, are usually considered as prototypical examples of the so-called multi-scale approaches. They combine two different description levels for the chemical system in both cases, a quantum part interacts with a classical part. Indeed, the QM/MM method can easily be extended to multi-scale schemes that include more than two description levels. Examples of three level schemes are the QM/MM/Continuum [47] and QM/QM7 MM approaches [48, 49]. In the later case, the system is divided into two QM parts, which may be described with the same or different methods, and a classical MM part. Dielectric continuum models for liquid interfaces are already available [43,50, 51] and a QM/MM/Continuum partition could be imagined in this case too, for instance to describe a solute-solvent cluster interacting with a polarizable dielectric medium. Here, however, we will focus on the QM/QM /MM partition. There is not a general scheme for this kind of approach and different algorithms can be employed to describe the interaction between subsystems. The main issue is the calculation of the interaction between two quantum subsystems that are described at QM (possibly different) theoretical levels. 

Solvation effects have been incorporated into the calculations of anionic proton transfer potentials in a number of ways. The simplest is the microsolvation model where a few solvent molecules are included to form a supermolecular system that is directly characterized by quantum mechanical calculations. This has the advantage of high accuracy, but is limited to small systems. Moreover, one must assume that a limited number of solvent molecules can adequately model a tme solution. A more realistic approach is to explicitly describe the inner solvation shell with quantum calculations and then treat the outer solvation sphere and bulk solvent as a continuum (infinite polarizable dielectric medium). In this way, the specific interactions can be treated by high-level calculations, but the effect of the bulk solvent and its dielectric is not neglected. An ej tension of this approach is to characterize the reaction partners by quantum mechanics and then treat the solvent with a molecular mechanics approach (hybrid quantum mechanics/molecular mechanics QM/MM). The low-cost of the molecular mechanics treatment allows the solvent to be involved in molecular dynamics simulations and consequently free energies can be calculated. In more recent work, solvent also has been treated with a frozen or constrained density functional theory approach. ... 

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 ... 

The OWB model describes the solute as a classical polarizable point dipole located in a spherical or ellipsoidal cavity in an isotropic and homogeneous dielectric medium representing the solvent. In the presence of a macroscopic Maxwell field E, the solute experiences an internal (or local) field E given by a superposition of a cavity field Ec and a reaction field ER. In terms of Fourier components E -n, Ec,n, ER,n of the fields we have... 

The OWB equations obtained in this semiclassical scheme analyse the effective polarizabilities in term of solvent effects on the polarizabilities of the isolated molecules. Three main effects arise due to (a) a contribution from the static reaction field which results in a solute polarizability, different from that of the isolated molecules, (b) a coupling between the induced dipole moments and the dielectric medium, represented by the reaction field factors FR n, (c) the boundary of the cavity which modifies the cavity field with respect the macroscopic field in the medium (the Maxwell field) and this effect is represented by the cavity field factors /c,n. 

The dispersion interaction between an atom and a metal surface was first calculated by Lennard-Jones in 1932, who considered the metal as a perfect conductor for static and time-dependent fields, using a point dipole for the molecule [44], Although these results overestimate the dispersion energy, the correct l/d3 dependence was recovered (d is the metal-molecule distance). Later studies [45 17] extended the work of Lennard-Jones to dielectrics with a frequency-dependent dielectric constant [48] (real metals may be approximated in this way) and took into account electromagnetic retardation effects. Limiting ourselves to small molecule-metal distances, the dispersion interaction of a molecule characterized by a frequency-dependent isotropic polarizability a embedded in a dielectric medium with permittivity esol (note that no cavity is built around the molecule) reads ... 

The mutual polarization process between the solute and the polarizable medium is obtained by solving a system of two coupled equations, i.e., the QM Schrodinger equation for the solute in presence of the polarized dielectric, and the electrostatic Poisson equation for the dielectric medium in presence of the charge distribution (electrons and nuclei) of the solute. The solute occupies a molecular shaped cavity within the dielectric continuum, whose polarization is represented by an apparent surface charge (ASC) density spread on the cavity surface. The solute-solvent interaction is then represented by a QM operator, the solvent reaction potential operator, Va, corresponding to the electrostatic interaction of the solute electrons and nuclei with the ASC density of the solvent. 

Molecular distortion polarizability is a measure of the ease with which atomic nuclei within molecules tend to be displaced from their zero-field positions by the applied electric field. (3) Orientation polarizability is a measure of the ease with which dipolar molecules tend to align against the applied electric field. The electron polarizability of an individual molecule is related to the -> permittivity (relative) of a dielectric medium by the -> Clausius-Mossotti relation. 

Significant progress has been achieved in the theoretical calculation of these interactions. The most advanced theoretical approach to the problem relies upon the use of the Poisson-Debye equation for polarizable solutes of known structure embedded in a dielectric medium (Klapper et al., 1986 Sharp and Honig, 1990 Bashford and Karplus, 1990 Bajorath et al., 1991, Aqvist et al., 1991 Tidor and Karplus,1991 Sharp et al., 1992 Gilson, 1993 Loewenthal et al., 1993 Yang et al., 1993 Scott et al., 1994 Anni et al., 1994, Hechtetal., 1995 Honig and Nicholls, 1995 ... 

Rigorous formulations of the problems associated with solvation necessitate approximations. From the computational point of view, we are forced to consider interactions between a solute and a large number of solvent molecules which requires approximate models [75]. The microscopic representation of solvent constitutes a discrete model consisting of the solute surrounded by individual solvent molecules, generally only those in close proximity. The continuous model considers all the molecules surrounding the solvent but not in a discrete representation. The solvent is represented by a polarizable dielectric continuous medium characterized by macroscopic properties. These approximations, and the use of potentials, which must be estimated with empirical or approximate computational techniques, allows for calculations of the interaction energy [75],... 

The Polarizable Continuum Model (PCM)[18] describes the solvent as a structureless continuum, characterized by its dielectric permittivity e, in which a molecular-shaped empty cavity hosts the solute fully described by its QM charge distribution. The dielectric medium polarized by the solute charge distribution acts as source of a reaction field which in turn polarizes back the solute. The effects of the mutual polarization is evaluated by solving, in a self-consistent way, an electrostatic Poisson equation, with the proper boundary conditions at the cavity surface, coupled to a QM Schrodinger equation for the solute. 

In addition to being polarizable, it is also paramount that the dielectric medium be of low conductivity i.e., have a large bandgap - an offset with Si of > 1 eV), to minimize carrier injection into its bands. For example, although the dielectric... 

The solvent also acts as a dielectric medium, which determines the field diji/dx and the energy of Interaction between charges. Now, the dielectric constant e depends on the inherent properties of the molecules (mainly their permanent dipole moment and polarizability) and on the structure of the solvent as a whole. Water is unique in this sense. It is highly associated in the liquid phase and so has a dielectric constant of 78 (at 25 C), which is much higher than that expected from the properties of the individual molecules. When it is adsorbed on the surface of an electrode, inside the compact double layer, the structure of bulk water is destroyed and the molecules are essentially immobilized... 

In this equation, po is the permanent dipole moment of the molecule, a is the linear polarizability, 3 is the first hyperpolarizability, and 7 is the second hyperpolarizability. a, and 7 are tensors of rank 2, 3, and 4 respectively. Symmetry requires that all terms of even order in the electric field of the Equation 10.1 vanish when the molecule possesses an inversion center. This means that only noncentrosymmetric molecules will have second-order NLO properties. In a dielectric medium consisting of polarizable molecules, the local electric field at a given molecule differs from the externally applied field due to the sum of the dipole fields of the other molecules. Different models have been developed to express the local field as a function of the externally applied field but they will not be presented here. In disordered media,... 

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