Dielectric continuum methods

Prior to addressing the results of simulations on the issues exposed in the last section, we will now develop in this section a simple model perspective [5c,21,22,43]. Its purpose is both to shed light on the interpretation in terms of solvation of those results and to emphasize the interconnections (and differences) that may exist. The development given below is suitable for charge transfer reaction systems, which have pronounced solute-solvent electrostatic coupling it is not appropiate for, e.g., neutral reactions in which the solvent influence is mainly of a collisional character. (Although we do not pursue it here, the various frequencies that arise in the model can be easily evaluated by dielectric continuum methods [21,431). 

Mujika, J.I., Mercero, J.M., and Lopez, X., A theoretical evaluation of the pKa for twisted amides using density functional theory and dielectric continuum methods, J. Phys. Chem. A, 107, 6099-6107, 2003. 

Theoretical calculations by Siegbahn and Crabtree [4] found the barrier for the reaction via the [PP (R)(H)] intermediate to be a little lower in energy compared with a one-step mechanism, while a study by Hill and Puddephatt favors type interactions [5]. The most recent theoretical study was conducted by Hush and co-workers using density functional theory (B3LYP functional) calculations with double-f to polarized double- basis sets [6]. They also studied solvation effects by a dielectric continuum method. 

The most rigorous dielectric continuum methods employ numerical solutions to the Poisson-Boltzmann equation [55]. As these methods are computationally quite expensive, in particular in connection with calculations of derivatives, much work has been concentrated on the development of computationally less expensive approximate continuum models of sufficient accuracy. One of the most widely used of these is the Generalized Born Solvent Accessible Surface Area (GB/SA) model developed by Still and coworkers [56,57]. The model is implemented in the MacroModel program [17,28] and parameterized for water and chloroform. It may be used in conjunction with the force fields available in MacroModel, e.g., AMBER, MM2, MM3, MMFF, OPTS. It should be noted that the original parameterization of the GB/SA model is based on the OPLS force field. 

Many of the available computations on radicals are strictly applicable only to the gas phase they do not account for any medium effects on the molecules being studied. However, in many cases, medium effects cannot be ignored. The solvated electron, for instance, is all medium effect. The principal frameworks for incorporating the molecular environment into quantum chemistry either place the molecule of interest within a small cluster of substrate molecules and compute the entire cluster quantum mechanically, or describe the central molecule quantum mechanically but add to the Hamiltonian a potential that provides a semiclassical description of the effects of the environment. The 1975 study by Newton (28) of the hydrated and ammoniated electron is the classic example of merging these two frameworks Hartree-Fock wavefunctions were used to describe the solvated electron together with all the electrons of the first solvent shell, while more distant solvent molecules were represented by a dielectric continuum. The intervening quarter century has seen considerable refinement in both quantum chemical techniques and dielectric continuum methods relative to Newton s seminal work, but many of his basic conclusions... 

Solvation effects have been included using a variety of simple models [16— 23]. These models have been based on exposed surface area, dielectric continuum methods, and screened or modified Coulomb interactions. The validity... 

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... 

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. 

As seen in Fig. 1.1, and as I will explain in detail in this book, the dielectric continuum solvation model COSMO and the subsequent COSMO-based thermodynamics COSMO-RS are two clearly separable and very different steps. However, I have found that many researchers in this field refer to both methods as COSMO, which is both inaccurate and confusing. To avoid this confusion, I find it necessary to emphasize the importance of using the correct notations—COSMO and COSMO-RS-for these methods in all discussions and written literature on these subjects. 

Born s idea of the dielectric continuum solvation approximation became very popular, and many researchers worked on its further development. Hence a brief overview of the most important development steps will be given, but it is impossible to mention all the different modifications and all workers who have been contributing to this field. Readers who seek a broader overview are referred to some reviews on continuum solvation methods, e.g., by Cramer and Truhlar [22] or by Tomasi and Persico [23]. The goal of the history given here is to enable... 

There are two other main directions for the calculation of the electrostatic interaction between the solute and a surrounding dielectric continuum for molecular-shaped cavities. Both require intensive numerical calculations and are thus slower than GB methods. The first direction is the direct numerical solution of the Poisson equation for the volume polarization P(r) at a position r of the dielectric medium ... 

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