Continuum dielectric

1 Dielectric Continuum. - In the one approach one assumes that the interactions between the solvent and the solute are mainly of an electrostatic nature and, therefore, that there are no bonds between the solute and the solvent molecules (including hydrogen bonds which could be important, e.g., when the solvent is water). Then the solvent is modelled as a dielectric (polarizable) continuum with a cavity occupied by the solute. The continuum responds to the charge distribution of the solute by becoming polarized which in turn may lead to a redistribution of the charge distribution of the solute, and so on. Therefore, the electron density of a dissolved molecule may be different from that of the molecule in the gas phase, which ultimately may even lead to changes in its 

This approach has been used by Andzelm et al.83 who studied various molecules in the gas phase and dissolved in a continuum that was supposed to model water. Here, we shall just mention a few of their results. 

As one example they studied relative basicities for different molecules. These are defined as the negative enthalpy of the protonation process A + H+ — AH+. They studied the set of molecules NH3 x(CH3) with 0 x 3. Relative to the values for NH3 the basicities drop from 10-20 kcal mol-1 in the gas phase to 1-2 kcal mol-1 in the solvent according to experiment. This change is well reproduced by the calculations. 

Secondly, they studied two isomers of acetic acid CH3COOH and found that upon solvation the structure changed slightly. But the changes in the bond lengths never exceeded 0.02 A. Furthermore, the dipole moment changed from about 1.7 D to 2.6 D for the sy/z-form and from about 4.25 D to about 6.15 D for the anti-form upon solvation. 

Finally, they showed that for glycine a different isomer is the stabler one in the solution than in the gas phase. 

---

Within the framework of the same dielectric continuum model for the solvent, the Gibbs free energy of solvation of an ion of radius and charge may be estimated by calculating the electrostatic work done when hypothetically charging a sphere at constant radius from q = 0 q = This yields the Bom equation [13]... 

Another way of calculating the electrostatic component of solvation uses the Poisson-Boltzmann equations [22, 23]. This formalism, which is also frequently applied to biological macromolecules, treats the solvent as a high-dielectric continuum, whereas the solute is considered as an array of point charges in a constant, low-dielectric medium. Changes of the potential within a medium with the dielectric constant e can be related to the charge density p according to the Poisson equation (Eq. (41)). 

Consider an alchemical transformation of a particle in water, where the particle s charge is changed from 0 to i) (e.g., neon sodium q = ). Let the transformation be performed first with the particle in a spherical water droplet of radius R (formed of explicit water molecules), and let the droplet then be transferred into bulk continuum water. From dielectric continuum theory, the transfer free energy is just the Born free energy to transfer a spherical ion of charge q and radius R into a continuum with the dielectric constant e of water ... 

The continuum model, in which solvent is regarded as a continuum dielectric, has been used to study solvent effects for a long time [2,3]. Because the electrostatic interaction in a polar system dominates over other forces such as van der Waals interactions, solvation energies can be approximated by a reaction field due to polarization of the dielectric continuum as solvent. Other contributions such as dispersion interactions, which must be explicitly considered for nonpolar solvent systems, have usually been treated with empirical quantity such as macroscopic surface tension of solvent. 

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. 

A reasonable alternative to the PDLD method can be obtained by approaches that represent the solvent as a dielectric continuum and evaluate the electric field in the system by discretized continuum approaches (see Ref. 15). Note, however, that the early macroscopic studies (including the... 

Chipman M (2002) Computation of pKa from Dielectric Continuum Theory. J Phys Chem A 106 7413-7422. 

PPII helix-forming propensities have been measured by Kelly et al. (2001) and A. L. Rucker, M. N. Campbell, and T. P. Creamer (unpublished results). In the simulations the peptide backbone was constrained to be in the PPII conformation, defined as (0,VO = ( — 75 25°, +145 25°), using constraint potentials described previously (Yun and Hermans, 1991 Creamer and Rose, 1994). The AMBER/ OPLS potential (Jorgensen and Tirado-Rives, 1988 Jorgensen and Severance, 1990) was employed at a temperature of 298° K, with solvent treated as a dielectric continuum of s = 78. After an initial equilibration period of 1 x 104 cycles, simulations were run for 2 x 106 cycles. Each cycle consisted of a number of attempted rotations about dihedrals equal to the total number of rotatable bonds in the peptide. Conformations were saved for analysis every 100 cycles. Solvent-accessible surface areas were calculated using the method of Richmond (1984) and a probe of 1.40 A radius. 

Equation (15) permits a straightforward analysis of dielectric continuum models of hydration that have become popular in recent decades. The dielectric model, also called the Bom approximation, for the hydration free energy of a spherical ion of radius R with a charge q at its center is... 

We can exploit the new results for packing contributions to reconsider the outer shell contribution in Eq. (33). For ionic solutes, the outer shell term would represent the Born contribution because it describes a hard ion stripped of any inner shell ligands. A Born model based on a picture of a dielectric continuum solvent is reasonable (see Section III,B, and Fig. 9, color insert). With that motivation, we first separate the outer shell term into an initial packing contribution and an approximate electrostatic contribution as... 

The electrostatic contributions, identified as /t x born and modeled on the basis of a dielectric continuum, are typically a substantial part of the... 

Truong, T. N. and E. V. Stefanovich. 1995. Hydration Effects on Reaction Profiles An ab initio Dielectric Continuum Study on the SN2CL + CH3C1 Reaction. J. Phys. Chem. 99, 14700. 

In this section, we discuss applications of the FEP formalism to two systems and examine the validity of the second-order perturbation approximation in these cases. Although both systems are very simple, they are prototypes for many other systems encountered in chemical and biological applications. Furthermore, the results obtained in these examples provide a connection between molecular-level simulations and approximate theories, especially those based on a dielectric continuum representation of the solvent. 

This result implies that AA should be a quadratic function of the ionic charge. This is exactly what is predicted by the Bom model, in which the ion is a spherical particle of radius a and the solvent is represented as a dielectric continuum characterized by a dielectric constant e [1]... 

Archontis, G. Simonson, T., Proton binding to proteins a free energy component analysis using a dielectric continuum model, Biophys. J. 2005, 88, 3888-3904... 

---