Effective density terms in proper integral equations

Integral Equation Theory of Biomolecules and Electrolytes

The so-called three-dimensional version (3D-RISM) can be used to describe the interactions of solvent components (here we treat water and ions) with a chemical or biomolecular solute of arbitrary size and shape. Here we give an overview of the current status of such models, describing some aspects of “pure” electrolytes (water plus simple ions) and of ionophores, proteins and nucleic acids in the presence of water and salts. Here we focus primarily on interactions with water and dissolved salts; as a practical matter, the discussion is mostly limited to monovalent ions, since studies of divalent ions present many difficult problems that have not yet been addressed. This is not a comprehensive review, but covers a few recent examples that illustrate current issues.

Integral equations in the study of polar and ionic interaction site fluids

In this review article we consider some of the current integral equation approaches and application to model polar liquid mixtures. We consider the use of multidimensional integral equations and in particular progress on the theory and applications of three dimensional integral equations. The IEs we consider may be derived from equilibrium statistical mechanical expressions incorporating a classical Hamiltonian description of the system. We give example including salt solutions, inhomogeneous solutions and systems including proteins and nucleic acids.

Protein solvation from theory and simulation: Exact treatment of Coulomb interactions in three-dimensional theories

Solvation forces dominate protein structure and dynamics. Integral equation theories allow a rapid and accurate evaluation of the effect of solvent around a complex solute, without the sampling issues associated with simulations of explicit solvent molecules. Advances in integral equation theories make it possible to calculate the angle dependent average solvent structure around an irregular molecular solution. We consider two methodological problems here: the treatment of long-ranged forces without the use of artificial periodicity or truncations and the effect of closures. We derive a method for calculating the long-ranged Coulomb interaction contributions to three-dimensional distribution functions involving only a rewriting of the system of integral equations and introducing no new formal approximations. We show the comparison of the exact forms with those implied by the supercell method. The supercell method is shown to be a good approximation for neutral solutes whereas the new method does not exhibit the known problems of the supercell method for charged solutes. Our method appears more numerically stable with respect to thermodynamic starting state. We also compare closures including the Kovalenko-Hirata closure, the hypernetted-chain (HNC) and an approximate three-dimensional bridge function combined with the HNC closure. Comparisons to molecular dynamics results are made for water as well as for the protein solute bovine pancreatic trypsin inhibitor. The proposed equations have less severe approximations and often provide results which compare favorably to molecular dynamics simulation where other methods fail.

An Integral Equation Study of the Hydrophobic Interaction between Graphene Plates

The hydrophobic association of two parallel graphene sheets is studied using the 3D-RISM HNC integral equations with several theoretical methods for the solvent distribution functions. The potential of mean force is calculated to study the effects of the aqueous solvent models and methods on the plates as a function of distance. The results of several integral equations (IE) are compared to MD simulations for the same model. The 3D-IEs are able to qualitatively reproduce the nature of the solvent effects on the potential of mean force but not quantitatively. The local minima in the potential of mean force occur at distances allowing well defined layers of solvent between the plates but are not coincident with those found in simulation of the same potential regardless of the theoretical methods tested here. The dewetting or drying transition between the plates is generally incorrectly dependent on steric effects with these methods even for very hydrophobic systems without solute-solvent attractions, in contradiction with simulation.

Solutions of the Optimized Closure Integral Equation Theory: Heteronuclear Polyatomic Fluids

Recently, we developed a thermodynamically optimized integral equation method which has been successfully tested on both simple and homonuclear diatomic Lennard-Jones fluids [J. Chem. Phys. 2007, 126, 124107]. The systematic evaluation of correlation functions required by the optimization of the chemical potential has shown a clear need for more efficient algorithms to solve these integral equations. In the present paper we introduce a high-performance algorithm which is found to be faster and more efficient than the direct Picard iteration. Here we have utilized this to solve the aforementioned optimized theory for molecules more complex than those considered previously. We analyzed representative models for heteronuclear diatomic and triatomic polar molecular fluids. We include results for several modified SPC-like models for water, obtaining site-site correlation functions in good agreement with simulation data.

A site-renormalized molecular fluid theory

The orientation-dependent pair distribution function for molecular fluids on site-site potentials is expanded in a topological analog of the diagrammatically proper site-site theory of liquids [D. Chandler et al., Mol. Phys. 46, 1335 (1982)]. The resulting functions are then used to diagrammatically renormalize the molecular fluid theory. A result is that the diagrammatically proper interaction site model theory is shown to be a linearized, minimal angular basis set approximation to this site-renormalized molecular theory. This framework is used to propose a new, exact, and proper closure to the diagrammatically proper interaction site model theory. The resulting equation system contains a bridge function expansion in the proper site-site theory. In addition, the construction of the theory is such that the molecular pair distribution function, in full dimensionality, is intrinsic to the theory. Furthermore, the theory is equivalent to the molecular Ornstein-Zernike treatment of site-site molecules in the basis set expansion of Blum and Torruella [J. Chem. Phys. 56, 303 (1971)]. A significant formal result of the theory is the demonstration that certain classes of diagrams which would otherwise be considered improper in the interaction site model formalism are included in the angular expansion of molecular interactions. Numerical results for several apolar homonuclear models and an apolar heteronuclear model are shown to quantitatively improve upon those of reference interaction site model and our recent proper variant with respect to simulation. Significant numerical results are that the various thermodynamic quantities obey the exact symmetries and sum rules within numerical error for the different sites in the heteronuclear case, even for the low order approximation used in this work, and the theory is independent of the so-called auxiliary site problem common to previous site-site theories.

Optimized theory for simple and molecular fluids

An optimized closure approximation for both simple and molecular fluids is presented. A smooth interpolation between Perkus-Yevick and hypernetted chain closures is optimized by minimizing the free energy self-consistently with respect to the interpolation parameter(s). The molecular version is derived from a refinement of the method for simple fluids. In doing so, a method is proposed which appropriately couples an optimized closure with the variant of the diagrammatically proper integral equation recently introduced by this laboratory [K. M. Dyer et al., J. Chem. Phys. 123, 204512 (2005)]. The simplicity of the expressions involved in this proposed theory has allowed the authors to obtain an analytic expression for the approximate excess chemical potential. This is shown to be an efficient tool to estimate, from first principles, the numerical value of the interpolation parameters defining the aforementioned closure. As a preliminary test, representative models for simple fluids and homonuclear diatomic Lennard-Jones fluids were analyzed, obtaining site-site correlation functions in excellent agreement with simulation data.

An angle dependent site-renormalized theory for the conformations of n-butane in a simple fluid

The angular dependent site-renormalized integral equation theory is developed to compute the dihedral conformation distribution and intermolecular pair distributions of n-butane at infinite dilution in a Lennard-Jones solvent. The equations take advantage of the topological diagrammatic expansion of the full angular dependent molecular system by resumming the series in conjunction with the intramolecular degree of freedom. To first order in an angular basis set, the numerical results of these site-renormalized equations are a systematic quantitative improvement over previous methods. In particular, the thermodynamics and conformational distribution of the solute are essentially indistinguishable from simulation.