Local Molecular Field Theory Research Articles

Modeling nuclear quantum effects on long-range electrostatics in nonuniform fluids

Nuclear quantum effects play critical roles in a variety of molecular processes, especially in systems that contain hydrogen and other light nuclei, such as water. For water under ambient conditions, nuclear quantum effects are often interpreted as local effects resulting from a smearing of the hydrogen atom distribution. However, the orientational structure of water at interfaces determines long-range effects, such as electrostatics, through the O–H bond ordering that is impacted by nuclear quantum effects. In this work, I examine nuclear quantum effects on long-range electrostatics of water confined between hydrophobic walls using path integral simulations. To do so, I combine concepts from local molecular field theory with path integral methods at varying levels of approximation to develop efficient and physically intuitive approaches for describing long-range electrostatics in nonuniform quantum systems. Using these approaches, I show that quantum water requires larger electrostatic forces to achieve interfacial screening than the corresponding classical system. This work highlights the subtleties of electrostatics in nonuniform classical and quantum molecular systems, and the methods presented here are expected to be of use to efficiently model nuclear quantum effects in large systems.

Quantifying the Relevance of Long-Range Forces for Crystal Nucleation in Water.

Understanding nucleation from aqueous solutions is of fundamental importance in a multitude of fields, ranging from materials science to biophysics. The complex solvent-mediated interactions in aqueous solutions hamper the development of a simple physical picture, elucidating the roles of different interactions in nucleation processes. In this work, we make use of three complementary techniques to disentangle the role played by short- and long-range interactions in solvent-mediated nucleation. Specifically, the first approach we utilize is the local molecular field (LMF) theory to renormalize long-range Coulomb electrostatics. Second, we use well-tempered metadynamics to speed up rare events governed by short-range interactions. Third, the deep learning-based State Predictive Information Bottleneck approach is employed in analyzing the reaction coordinate of the nucleation processes obtained from the LMF treatment coupled with well-tempered metadynamics. We find that the two-step nucleation mechanism can largely be captured by the short-range interactions, while the long-range interactions further contribute to the stability of the primary crystal state under ambient conditions. Furthermore, by analyzing the reaction coordinate obtained from the combined LMF-metadynamics treatment, we discern the fluctuations on different time scales, highlighting the need for long-range interactions when accounting for metastability.

Local Molecular Field Theory for Coulomb Interactions in Aqueous Solutions

Coulomb interactions play a crucial role in a wide array of processes in aqueous solutions but present conceptual and computational challenges to both theory and simulations. We review recent developments in an approach addressing these challenges─local molecular field (LMF) theory. LMF theory exploits an exact and physically suggestive separation of intermolecular Coulomb interactions into strong short-range and uniformly slowly varying long-range components. This allows us to accurately determine the averaged effects of the long-range components on the short-range structure using effective single particle fields and analytical corrections, greatly reducing the need for complex lattice summation techniques used in most standard approaches. The simplest use of these ideas in aqueous solutions leads to the short solvent (SS) model, where both solvent-solvent and solute-solvent Coulomb interactions have only short-range components. Here we use the SS model to give a simple description of pairing of nucleobases and biologically relevant ions in water.

Influence of Long-Range Forces on the Transition States and Dynamics of NaCl Ion-Pair Dissociation in Water.

We study NaCl ion-pair dissociation in a dilute aqueous solution using computer simulations both for the full system with long-range Coulomb interactions and for a well-chosen reference system with short-range intermolecular interactions. Analyzing results using concepts from Local Molecular Field (LMF) theory and the recently proposed AI-based analysis tool "State predictive information bottleneck" (SPIB), we show that the system with short-range interactions can accurately reproduce the transition rate for the dissociation process, the dynamics for moving between the underlying metastable states, and the transition state ensemble. Contributions from long-range interactions can be largely neglected for these processes because long-range forces from the direct interionic Coulomb interactions are almost completely canceled (>90%) by those from solvent interactions over the length scale where the transition takes place. Thus, for this important monovalent ion-pair system, short-range forces alone are able to capture detailed consequences of the collective solvent motion, allowing the use of physically suggestive and computationally efficient short-range models for the dissociation event. We believe that the framework here should be applicable to disentangling mechanisms for more complex processes such as multivalent ion disassociation, where previous work has suggested that long-range contributions may be more important.

Ion-dependent protein–surface interactions from intrinsic solvent response

The phyllosilicate mineral muscovite mica is widely used as a surface template for the patterning of macromolecules, yet a molecular understanding of its surface chemistry under varying solution conditions, required to predict and control the self-assembly of adsorbed species, is lacking. We utilize all-atom molecular dynamics simulations in conjunction with an electrostatic analysis based in local molecular field theory that affords a clean separation of long-range and short-range electrostatics. Using water polarization response as a measure of the electric fields that arise from patterned, surface-bound ions that direct the adsorption of charged macromolecules, we apply a Landau theory of forces induced by asymmetrically polarized surfaces to compute protein-surface interactions for two muscovite-binding proteins (DHR10-mica6 and C98RhuA). Comparison of the pressure between surface and protein in high-concentration KCl and NaCl aqueous solutions reveals ion-specific differences in far-field protein-surface interactions, neatly capturing the ability of ions to modulate the surface charge of muscovite that in turn selectively attracts one binding face of each protein over all others.

Dielectric response with short-ranged electrostatics

The dielectric nature of polar liquids underpins much of their ability to act as useful solvents, but its description is complicated by the long-ranged nature of dipolar interactions. This is particularly pronounced under the periodic boundary conditions commonly used in molecular simulations. In this article, the dielectric properties of a water model whose intermolecular electrostatic interactions are entirely short-ranged are investigated. This is done within the framework of local molecular-field theory (LMFT), which provides a well-controlled mean-field treatment of long-ranged electrostatics. This short-ranged model gives a remarkably good performance on a number of counts, and its apparent shortcomings are readily accounted for. These results not only lend support to LMFT as an approach for understanding solvation behavior, but also are relevant to those developing interaction potentials based on local descriptions of liquid structure.

Local Molecular Field Theory for Nonequilibrium Systems.

We provide a framework for extending equilibrium local molecular field (LMF) theory to a statistical ensemble evolving under a time-dependent applied field. In this context, the self-consistency of the original LMF equation is achieved dynamically, which provides an efficient method for computing the equilibrium LMF potential, in addition to providing the nonequilibrium generalization. As a concrete example, we investigate water confined between hydrophobic or charged walls, systems that are very sensitive to the treatment of long-ranged electrostatics. We then analyze confined water in the presence of a time-dependent applied electric field, generated by a sinusoidal or abrupt variation of the magnitudes of uniform charge densities on each wall. Very accurate results are found from the time-dependent LMF formalism even for strong static fields and for time-dependent systems that are driven far from equilibrium where linear response methods fail.

Short solvent model for ion correlations and hydrophobic association

Coulomb interactions play a major role in determining the thermodynamics, structure, and dynamics of condensed-phase systems, but often present significant challenges. Computer simulations usually use periodic boundary conditions to minimize corrections from finite cell boundaries but the long range of the Coulomb interactions generates significant contributions from distant periodic images of the simulation cell, usually calculated by Ewald sum techniques. This can add significant overhead to computer simulations and hampers the development of intuitive local pictures and simple analytic theory. In this paper, we present a general framework based on local molecular field theory to accurately determine the contributions from long-ranged Coulomb interactions to the potential of mean force between ionic or apolar hydrophobic solutes in dilute aqueous solutions described by standard classical point charge water models. The simplest approximation leads to a short solvent (SS) model, with truncated solvent-solvent and solute-solvent Coulomb interactions and long-ranged but screened Coulomb interactions only between charged solutes. The SS model accurately describes the interplay between strong short-ranged solute core interactions, local hydrogen-bond configurations, and long-ranged dielectric screening of distant charges, competing effects that are difficult to capture in standard implicit solvent models.

Self-consistent field theory of density correlations in classical fluids

More than half of a century has passed since the free energy of classical fluids defined by second Legendre transform was derived as a functional of density-density correlation function. It is now becoming an increasingly significant issue to develop the correlation functional theory that encompasses the liquid state theory, especially for glassy systems where out of equilibrium correlation fields are to be investigated. Here we have formulated a field theoretic perturbation theory that incorporates two-body fields (both of density-density correlation field and its dual field playing the role of two-body interaction potential) into a density functional integral representation of the Helmholtz free energy. Quadratic density fluctuations are only considered in the saddle-point approximation of two-body fields as well as density field. We have obtained a set of self-consistent field equations with respect to these fields, which simply reads a modified mean-field equation of density field where the bare interaction potential in the thermal energy unit is replaced by minus the direct correlation function given in the mean spherical approximation. Such replacement of the interaction potential in the mean-field equation belongs to the same category as the local molecular field theory proposed by Weeks and co-workers. Notably, it has been shown that even the mean-field part of the free energy functional given by the self-consistent field theory includes information on short-range correlations between fluid particles, similarly to the formulation of the local molecular field theory. The advantage of our field theoretic approach is not only that the modified mean-field equation can be improved systematically, but also that fluctuations of two-body fields in nonuniform fluids may be considered, which would be relevant especially for glass-forming liquids.

A third and fourth order perturbation theory for dipolar hard spheres

In this work, we develop and analyze a third order and a fourth order perturbation theory for non-polarizable dipolar hard spheres. The electrostatic potential is split into a short-ranged and a long-ranged part. A perturbation expansion is applied for the short-ranged potential where the contribution of higher order terms is approximated through a [2,1]-Padé approximation for the third order theory and by a [2,2]-Padé approximation for the fourth order theory. Rodgers and Weeks [J. Chem. Phys. 131, 244108 (2010)] developed the Local Molecular Field (LMF) theory for the long-ranged contribution of the electrostatic energy. The LMF theory requires the relative permittivity of the considered fluid. We apply a perturbation theory for the relative permittivity and show that the LMF is then in very good agreement with the results for the long-ranged energy from molecular simulations. The short-ranged contribution to the Helmholtz energy from the third order and from the fourth order perturbation theory is, for densities ρ* ≤ 1 and for dipole moments of μ*2 ≤ 1, in very good agreement with the data from molecular simulations. For larger dipole densities (up to μ*2 ≤ 4 for ρ* ≲ 1), we observe significantly improved results for the fourth order perturbation theory compared to the third order perturbation theory.

Role of Solute Attractive Forces in the Atomic-Scale Theory of Hydrophobic Effects.

The role that van der Waals (vdW) attractive forces play in the hydration and association of atomic hydrophobic solutes such as argon (Ar) in water is reanalyzed using the local molecular field (LMF) theory of those interactions. In this problem, solute vdW attractive forces can reduce or mask hydrophobic interactions as measured by contact peak heights of the ArAr correlation function compared to reference results for purely repulsive core solutes. Nevertheless, both systems exhibit a characteristic hydrophobic inverse temperature behavior in which hydrophobic association becomes stronger with increasing temperature through a moderate temperature range. The new theoretical approximation obtained here is remarkably simple and faithful to the statistical mechanical LMF assessment of the necessary force balance. Our results extend and significantly revise approximations made in a recent application of the LMF approach to this problem and, unexpectedly, support a theory of nearly 40 years ago.

Alchemical free energy calculations and umbrella sampling with local molecular field theory

Understanding the thermodynamic driving forces underlying any chemical process requires a description of the underlying free energy surface. However, computation of free energies is difficult, often requiring advanced sampling techniques. Moreover, these computations can be further complicated by the evaluation of any long-ranged interactions in the system of interest, such as Coulomb interactions in charged and polar media. Local molecular field theory is a promising approach to avoid many of the conceptual and computational difficulties associated with long-ranged interactions. We present frameworks for performing alchemical free energy calculations and non-Boltzmann sampling with local molecular field theory. We demonstrate that local molecular field theory can be used to perform these free energy calculations with accuracy comparable to traditional methodologies while eliminating the need for explicit treatment of long-ranged interactions in simulations.

Deconstructing Free Energies in the Law of Matching Water Affinities.

The law of matching water affinities (LMWA) is explored in classical molecular dynamics simulations of several alkali halide ion pairs, spanning the size range from small kosmotropes to large chaotropes. The ion-ion potentials of mean force (PMFs) are computed using three methods: the local molecular field theory (LMFT), the weighted histogram analysis method (WHAM), and integration of the average force. All three methods produce the same total PMF for a given ion pair. In addition, LMFT-based partitioning into van der Waals and local and far-field electrostatic free energies and assessment of the enthalpic, entropic, and ion-water components yield insights into the origins of the observed free energy profiles in water. The results highlight the importance of local electrostatic interactions in determining the shape of the PMFs, while longer-ranged interactions enhance the overall ion-ion attraction, as expected in a dielectric continuum model. The association equilibrium constants are estimated from the smooth WHAM curves and compared to available experimental conductance data. By examining the variations in the average hydration numbers of ions with ion-ion distance, a correlation of the water structure in the hydration shells with the free energy features is found.

Free Energies of Ion Binding in the Bacterial CLC-ec1 Chloride Transporter with Implications for the Transport Mechanism and Selectivity.

The chloride channel/transporter family of proteins facilitates anion transport across biological membranes. There is extensive physiological and bioinformatic evidence that the channels and transporters are closely related. Each monomer of a homodimeric CLC transport protein contains a narrow selectivity filter. Investigating the ion binding properties inside the filter is crucial for understanding key mechanistic states during ion transit. Here computer simulations are used to explore the free energies of Cl(-) ions in the binding sites of the wild-type CLC-ec1 transporter and its mutant E148A. Specifically, a local molecular field theory approach for free energy calculations is exploited to compute the absolute free energies in water and in the protein binding sites. The calculations indicate a close synergy between anion binding and protonation of the external glutamate gate. Electrostatic differences between the bacterial CLC-ec1 and eukaryotic CmCLC transporters revealed by these and other simulations help to rationalize the observed differing structures in the pore region. In addition, quantum chemical calculations on the F(-), Cl(-), and Br(-) ions in the central binding site are used to examine ion selectivity. The calculations show a significant extent of charge transfer from the ion to the nearby residues. The computed free energies, in conjunction with experimental measurements, place constraints on proposed mechanisms for the transport cycle.

Molecular Theory and the Effects of Solute Attractive Forces on Hydrophobic Interactions.

The role of solute attractive forces on hydrophobic interactions is studied by coordinated development of theory and simulation results for Ar atoms in water. We present a concise derivation of the local molecular field (LMF) theory for the effects of solute attractive forces on hydrophobic interactions, a derivation that clarifies the close relation of LMF theory to the EXP approximation applied to this problem long ago. The simulation results show that change from purely repulsive atomic solute interactions to include realistic attractive interactions diminishes the strength of hydrophobic bonds. For the Ar-Ar rdfs considered pointwise, the numerical results for the effects of solute attractive forces on hydrophobic interactions are opposite in sign and larger in magnitude than predicted by LMF theory. That comparison is discussed from the point of view of quasichemical theory, and it is suggested that the first reason for this difference is the incomplete evaluation within LMF theory of the hydration energy of the Ar pair. With a recent suggestion for the system-size extrapolation of the required correlation function integrals, the Ar-Ar rdfs permit evaluation of osmotic second virial coefficients B2. Those B2's also show that incorporation of attractive interactions leads to more positive (repulsive) values. With attractive interactions in play, B2 can change from positive to negative values with increasing temperatures. This is consistent with the puzzling suggestions of decades ago that B2 ≈ 0 for intermediate cases of temperature or solute size. In all cases here, B2 becomes more attractive with increasing temperature.

Relationship between local molecular field theory and density functional theory for non-uniform liquids

The local molecular field theory (LMF) developed by Weeks and co-workers has proved successful for treating the structure and thermodynamics of a variety of non-uniform liquids. By reformulating LMF in terms of one-body direct correlation functions we recast the theory in the framework of classical density functional theory (DFT). We show that the general LMF equation for the effective reference potential φ(R)(r) follows directly from the standard mean-field DFT treatment of attractive interatomic forces. Using an accurate (fundamental measures) DFT for the non-uniform hard-sphere reference fluid we determine φ(R)(r) for a hard-core Yukawa liquid adsorbed at a planar hard wall. In the approach to bulk liquid-gas coexistence we find the effective potentials exhibit rich structure that can include damped oscillations at large distances from the wall as well as the repulsive hump near the wall required to generate the low density "gas" layer characteristic of complete drying. We argue that it would be difficult to obtain the same level of detail from other (non-DFT based) implementations of LMF. LMF emphasizes the importance of making an intelligent division of the interatomic pair potential of the full system into a reference part and a remainder that can be treated in mean-field approximation. We investigate different divisions for an exactly solvable one-dimensional model where the pair potential has a hard-core plus a linear attractive tail. Results for the structure factor and the equation of state of the uniform fluid show that including a significant portion of the attraction in the reference system can be much more accurate than treating the full attractive tail in mean-field approximation. We discuss further aspects of the relationship between LMF and DFT.

Deconstructing Classical Water Models at Interfaces and in Bulk

Using concepts from perturbation and local molecular field theories of liquids we divide the potential of the SPC/E water model into short and long ranged parts. The short ranged parts define a minimal reference network model that captures very well the structure of the local hydrogen bond network in bulk water while ignoring effects of the remaining long ranged interactions. This deconstruction can provide insight into the different roles that the local hydrogen bond network, dispersion forces, and long ranged dipolar interactions play in determining a variety of properties of SPC/E and related classical models of water. Here we focus on the anomalous behavior of the internal pressure and the temperature dependence of the density of bulk water. We further utilize these short ranged models along with local molecular field theory to quantify the influence of these interactions on the structure of hydrophobic interfaces and the crossover from small to large scale hydration behavior. The implications of our findings for theories of hydrophobicity and possible refinements of classical water models are also discussed.

Hydration Free Energies by Energetic Partitioning of the Potential Distribution Theorem

In the quasichemical theory of molecular solutions, the hydration free energy is spatially partitioned in a three-step thermodynamic process: cavity formation, solute insertion into the cavity, and relaxation of the cavity constraint. In the alternative local molecular field theory that focuses on the relationship of fluid structure and forces, the interaction energies are directly partitioned into local and far-field components; a restructured local potential incorporates information from the far-field interactions at the mean-field level. Here the quasichemical and local molecular field theories are related via energetic partitioning of the potential distribution theorem free energy. The resulting theory leads to a free energy division in which the local contribution requires direct evaluation, but the far-field component can be accurately estimated at the Gaussian level. A numerical approach for computing hydration free energies is developed that employs interaction energy distributions from several sampling states. Classical model problems of nonpolar, polar, and ionic hydration are presented to illustrate the theory. Extensions of the theory for estimating free energies at the quantum level are also discussed.

On the efficient and accurate short-ranged simulations of uniform polar molecular liquids

We show that spherical truncations of the 1/r interactions in models for water and acetonitrile yield very accurate results in bulk simulations for all site–site pair correlation functions as well as dipole–dipole correlation functions. This good performance in bulk simulations contrasts with the generally poor results found with the use of such truncations in nonuniform molecular systems. We argue that Local Molecular Field (LMF) theory provides a general theoretical framework that gives the necessary corrections to simple truncations in most nonuniform environments and explains the accuracy of spherical truncations in uniform environments by showing that these corrections are very small. LMF theory is derived from the exact Yvon–Born–Green (YBG) hierarchy by making physically-motivated and well-founded approximations. New and technically interesting derivations of both the YBG hierarchy and LMF theory for a variety of site–site molecular models are presented in appendices. The main paper focuses on understanding the accuracy of these spherical truncations in uniform systems both phenomenologically and quantitatively using LMF theory.

Accurate thermodynamics for short-ranged truncations of Coulomb interactions in site-site molecular models

Coulomb interactions are present in a wide variety of all-atom force fields. Spherical truncations of these interactions permit fast simulations but are problematic due to their incorrect thermodynamics. Herein we demonstrate that simple analytical corrections for the thermodynamics of uniform truncated systems are possible. In particular, results for the simple point charge/extended (SPC/E) water model treated with spherically truncated Coulomb interactions suggested by local molecular field theory [J. M. Rodgers and J. D. Weeks, Proc. Natl. Acad. Sci. U.S.A. 105, 19136 (2008)] are presented. We extend the results developed by Chandler [J. Chem. Phys. 65, 2925 (1976)] so that we may treat the thermodynamics of mixtures of flexible charged and uncharged molecules simulated with spherical truncations. We show that the energy and pressure of spherically truncated bulk SPC/E water are easily corrected using exact second-moment-like conditions on long-ranged structure. Furthermore, applying the pressure correction as an external pressure removes the density errors observed by other research groups in NPT simulations of spherically truncated bulk species.