Continuum methods

Electrostatic interactions play an important role in many aspects of biology, such as protein structural stability, enzyme function, gene expression, ion transport, and protein-protein interactions. Consequently, the number of publications studying these interactions continues to grow every year. Continuum methods are by far the most common approach to studying electrostatic interactions. In continuum methods, the solute is usually represented as a 

The relative basic advantages and disadvantages of each method have been discussed previously. For a comprehensive review of methodology before the annual period of this review the interested reader is directed to reviews by Allewell and Oberoi84 and Warshel and Papazyan.85 

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A yet more realistic cavity shape is that obtained from the van der Waals radii of the atoms of the solute. This is the approach taken in the polarisable continuum method (PCM) [Miertus et al. 1981], which has been implemented in a variety of ab initio and semi-empirical quantu/rt mechanical programs. Due to the non-analytical nature of the cavity shapes in the PCM approach, it is necessary to calculate numerically. The cavity surface is divided... 

The most popular of the SCRF methods is the polarized continuum method (PCM) developed by Tomasi and coworkers. This technique uses a numerical integration over the solute charge density. There are several variations, each of which uses a nonspherical cavity. The generally good results and ability to describe the arbitrary solute make this a widely used method. Flowever, it is sensitive to the choice of a basis set. Some software implementations of this method may fail for more complex molecules. 

The conductor-like screening model (COSMO) is a continuum method designed to be fast and robust. This method uses a simpler, more approximate equation for the electrostatic interaction between the solvent and solute. Line the SMx methods, it is based on a solvent accessible surface. Because of this, COSMO calculations require less CPU time than PCM calculations and are less likely to fail to converge. COSMO can be used with a variety of semiempirical, ah initio, and DFT methods. There is also some loss of accuracy as a result of this approximation. 

OPW (orthogonalized plane wave) a band-structure computation method P89 (Perdew 1986) a gradient corrected DFT method parallel computer a computer with more than one CPU Pariser-Parr-Pople (PPP) a simple semiempirical method PCM (polarized continuum method) method for including solvation effects in ah initio calculations... 

SCF (self-consistent field) procedure for solving the Hartree-Fock equations SCI-PCM (self-consistent isosurface-polarized continuum method) an ah initio solvation method... 

Given the diversity of different SCRF models, and the fact that solvation energies in water may range from a few kcal/mol for say ethane to perhaps 100 kcal/mol for an ion, it is difficult to evaluate just how accurately continuum methods may in principle be able to represent solvation. It seems clear, however, that molecular shaped cavities must be employed, the electiostatic polarization needs a description either in terms of atomic charges or quite high-order multipoles, and cavity and dispersion terms must be included. Properly parameterized, such models appear to be able to give absolute values with an accuracy of a few kcal/mol." Molecular properties are in many cases also sensitive to the environment, but a detailed discussion of this is outside the scope of this book. ... 

In literature, some researchers regarded that the continuum mechanic ceases to be valid to describe the lubrication behavior when clearance decreases down to such a limit. Reasons cited for the inadequacy of continuum methods applied to the lubrication confined between two solid walls in relative motion are that the problem is so complex that any theoretical approach is doomed to failure, and that the film is so thin, being inherently of molecular scale, that modeling the material as a continuum ceases to be valid. Due to the molecular orientation, the lubricant has an underlying microstructure. They turned to molecular dynamic simulation for help, from which macroscopic flow equations are drawn. This is also validated through molecular dynamic simulation by Hu et al. [6,7] and Mark et al. [8]. To date, experimental research had "got a little too far forward on its skis however, theoretical approaches have not had such rosy prospects as the experimental ones have. Theoretical modeling of the lubrication features associated with TFL is then urgently necessary. 

Zemach, C., a continuum method for modeling surface tension, J. Comput. 

The last thirty years have seen a flowering of simulation techniques based on explicit treatments of solvent molecules (some references are given above). Such methods provide new insight into the reasons why continuum methods work or don t work. However they have not and never will replace continuum models. In fact, continuum models are sometimes so strikingly successful that hubris may be the most serious danger facing their practitioners. One of the goals of this present chapter will be to diffuse (but not entirely deflate ) any possible overconfidence. 

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). 

These problems were partially solved through the inclusion of multipole expansions in ellipsoidal cavities [23] or through the use of the polarizable continuum method... 

Tomasi, J., Bonaccorsi, R., Cammi, R. and Olivares del Valle, F. J. Theoretical chemistry in solution. Some results and perspectives of the continuum methods and in particular of the polarizable continuum model,. J.Mol.Struct., 234 (1991), 401-424... 

Davidson, M. M., Hillier, I. El., Hall, R. J. and Burton, N. A. Effect of solvent on the Claisen rearrangement of allyl vinyl ether using ab initio continuum methods, J.Am.Chem.Soc., 116 (1994), 9294-9297... 

AGhydration For Metal Ions, In Kcal/Mole, Computed By Two Hybrid Discrete Molecular/Continuum Methods... 

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