The simulation of systems with long-ranged electrostatic interactions using conventional treatments such as Ewald sums or fast multipole methods is very involved and computationally demanding. We have shown that that our LMF theory can lead to significantly more efficient simulations by permitting us to accurately simulate nonuniform charged systems using solely the minimum periodic image of the simulation cell. In contrast, using the minimum image method along with simple truncations of the Coulomb interaction, say by reaction field methods or shifted force truncation, is known to produce significant errors in many physically relevant cases.

The key idea is the introduction an effective external field, the LMF, that accounts for the averaged effects of properly chosen long ranged components of the Coulomb interactions ignored in simple truncation schemes. LMF theory provides a general conceptual framework for treating systems with long ranged interactions that allows us to circumvent most of the limitations of the conventional methods.

In one of the first applications of LMF theory to Coulomb systems we showed that it provides a physically suggestive and very accurate treatment of the general problem of counterion-induced effective attraction between like charged objects in nonuniform ionic fluids. We considered in particular the force on two equally charged hard walls with neutralizing counterions in between as a function of the wall spacing and the coupling strength (temperature or ion valence). The theory predicts that at strong coupling and intermediate separations of the walls there is an effective attraction between the walls, in agreement with detailed computer simulations of the full ionic system by Moria and Netz. While certain limiting cases were understood previously, the LMF theory provides the first unified perspective that can accurately describe all regimes.

We also showed that the mimic system can give very accurate description of correlation functions in the size asymmetric primitive model at strong coupling both at high densities and in the dilute ion-pairing regime. The latter has proved particularly difficult to describe using conventional methods, and yet an accurate description of ion pairing is required to understand basic features such as Coulomb critical phenomena. Different aspects of this work were published in Proceedings of the National Academy of Sciences and in Physical Review Letters in 2006.

In a second major project we have generalized the LMF approach to apply to the Simple Point Charge (SPC/E and related models) water models. We first replace each charge by the short-ranged Coulomb core component and refer to the resulting short-ranged model as short water. Spherical truncation of site-site point-charge water models alone is not a new idea, and as shown by previous workers, it is often possible to obtain atomic and charge density profiles for bulk water and water confined between hydrophobic walls that appear accurate to the eye by using short water. Using our LMF motivated short water truncation we get exceptionally accurate results for the radial distribution functions of bulk short water when compared to full simulations using Ewald sums. Even dipole angle correlations in bulk water are well described by the short water model. However, thermodynamic properties for bulk short water are less accurate and short water gives very poor results for the electrostatic potential of water confined between hydrophobic walls.

In new work we have also shown that when electric fields are applied, short water predicts quite incorrect atomic density profiles and even yields a negative dielectric constant! Clearly naive truncations of water interactions alone cannot be trusted in many relevant cases. Results such as these have convinced most workers that a careful treatment of electrostatics in water models by Ewald sum or related methods is essential.

However, all these difficulties associated with short-ranged truncations of water in confined situations result from the neglect of the net cumulative effects of the long- ranged forces in nonuniform environments due to charge density outside of the range of the Coulomb core cutoff. LMF theory very naturally corrects for exactly these issues by using a mean field average over the slowly varying long ranged part of the Coulomb interactions. This same idea allows us to correct the thermodynamic corrections for bulk short water by taking account of the average corrections due to the asymptotic long ranged part of the interaction potential analogous to corrections we previously introduced for ionic systems based on the Stillinger-Lovett moment conditions. We have shown that when the LMF is taken into account all the above problems disappear. In particular LMF theory for the first time gives quantitatively accurate results for the electrostatic potential of short water near hydrophobic walls as well as for the dielectric constant of water using a model with only short-ranged intermolecular interactions. Moreover LMF theory quantitatively corrects the very poor result of short water in an electric field. Some of these results are described in a new article recently published in PNAS in 2008.

One of the most important general finding we have made connects the basic LMF theory to ideas of classical electrostatics. For reasons that we have fully appreciated only in the past two years, it turns out that the basic LMF mapping becomes even more powerful and also much simpler to carry out when it is used only for the basic Coulomb 1/r interaction (with no splitting, e.g., of Lennard-Jones interactions as well). We can then take advantage of Coulomb symmetries between charges of different signs and magnitudes interacting with the same basic 1/r potential and determine the effective field using only the total charge density and not separate number densities for each species or interaction site as before.

We have shown that the LMF equation for the renormalized part of the effective field then reduces exactly to a modified Poissons equation defined with a Gaussian smoothed charge density. This is a beautiful and physically suggestive result that helps us understand why LMF theory can give such accurate results for electrostatic properties of water and strongly coupled ionic systems in particular and why certain apparently crude approximations to the charge density can still give good results. Classical electrostatics often posits some sort of smoothing of the molecular charge density. LMF theory shows precisely how such smoothing can be carried out based on sound principles of statistical mechanics and using realistic molecular models containing other strong intermolecular forces. It thus connects the idealized textbook treatments of electrostatics to state of the art molecular models and simulations and provides important new insights in both areas. A general description of the LMF theory and its application to electrostatics has appeared in an article in Journal of Physics, Condensed Matter in 2008.

A very different system on which our work is still in progress is a polyelectrolyte chain immersed in a low molecular weight ionic solution. At weak coupling, where the counterions are relatively far from the polyelectrolyte, the polymer configuration is extended because of repulsive interactions from the negative charges of the backbone. However, at strong coupling the counterions can cluster or condense near the backbone, and the resulting effectively neutral polymer can fold in the usual way. Capturing this phenomena in a computer simulation using only short ranged intermolecular forces provides a major test of both the conceptual foundation of LMF theory and of its practical use in a more realistic polymer system.

Our model incorporates several new features which we believe make it a more faithful representation of real polyelectrolyte systems than the models considered so far. In particular, to obtain a well-defined limit as the system size approaches infinity, we consider an equilibrium between the ions in the simulation box, including the polymer counterions, and those coming from a bulk reservoir. We also account for the fact that the ion size is typically much smaller than the size of the polymer segments. This results in a very large number of ions --- of the order of 10000 ions in the volume which is relevant to a model polymer chain comprised of 100 polymer segments --- which means that a large part of the simulation time is actually spent on the low molecular weight species. Since we are ultimately interested in the polymer conformational changes, the presence of such numbers of ions makes the simulations computationally very demanding. We have carried out extensive simulations on this and related polymer systems using computer resources available at the University of Maryland and also on the NSF TeraGrid system.

An initial analysis of the free energy pathways for folding of the uncharged polymer has been published in Physical Review Letters and we are preparing a manuscript detailing the changes that charges induce to the folding pathways and dynamics. Click here for more information.

This material is based upon work supported by the National Science Foundation under Grants No. CHE-0111104 and CHE-0517818. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.