Hypernetted Chain Approximation Research Articles

Martini 3D-OZ: A Theoretical Investigation of Solvation Shell Structures and Solvation Free Energies of Martini Coarse-Grained Proteins.

We investigate the properties of aqueous solutions using integral equation theories and molecular dynamics (MD) simulations within the framework of the MARTINI coarse-grained force field. The integral equation theory used in the present work is based on the Ornstein-Zernike equation coupled with the hypernetted chain (HNC) and Kovalenko-Hirata (KH) closures. Overall, the solvation shell structures and solvation thermodynamics in the HNC approximation are shown to be in better agreement with those from the MD simulation than the KH results. Especially, through the analysis of spatial distribution functions of water around a protein, we have demonstrated that the HNC approximation can provide the highly anisotropic structure of the solvation shell of the protein. On the other hand, the KH approximation works well for simple particle solutes, but the results for highly hydrated proteins deviate quite significantly from the MD results. We further explore in detail the reason underlying the deviation caused by the KH approximation. Lastly, a potential application of the integral equation theory with the MARTINI model is outlined.

Spatial distribution of reduced density of hard spheres near a hard-sphere dimer: Results from three-dimensional Ornstein–Zernike equations coupled with several different closures and from grand canonical Monte Carlo simulation

We calculated the spatial distribution function, which is the one-body reduced density distribution of solvent particles around a nonspherical solute, using the three-dimensional Ornstein–Zernike equations coupled with closures. The solvent was a hard-sphere fluid, and the contact dimer of the solvent particles was the nonspherical solute. Two traditional closures, the Percus–Yevick and hypernetted-chain approximations, and three closures with bridge functions (BFs) were examined. These spatial distribution functions were compared with the results obtained by grand canonical Monte Carlo (GCMC) simulations. Three closures with BFs, such as the modified hypernetted-chain (MHNC) closure using a bridge function proposed by Kinoshita, gave precise spatial distribution functions. These results were much more accurate than those obtained by the traditional closures. The deviations from the spatial distribution function obtained by the GCMC simulation appeared only near the concave surface of the solute dimer. However, the deviations were minor compared with those between the simulation and the predictions of the two traditional closures. By contrast, the three-body correlation functions for the closures with BFs were not more accurate than those obtained by the traditional closures.

Collision frequency measurement from optical reflectivity of laser indirect-driven CH/Al/diamond

A collision frequency measurement from the optical reflectivity of laser indirect-driven CH/Al/diamond on the SG-10kJ laser facility is presented. The optical reflectivity and the Al/diamond interface velocity were measured simultaneously by the velocity interferometer. The aluminum rear surface density was deduced from the interface velocity by analyzing the wave interaction. The deduced sample state was compared with the simulation and quite good agreement was found. The electron collision frequency was deduced by fitting the sample state to the optical reflectivity, and it is found that the experimental collision frequency agrees with a semi-empirical result within the error bar, but is larger than the simulated result based on the average-atom model with the hypernetted chain approximation.

Applicability of a central force water model to study adsorption in disordered hydrophobic matrices—replica Ornstein–Zernike theory

A simple central force water model to study adsorption in random Lennard-Jones-like matrices has been studied using the replica Ornstein–Zernike integral equation theory in hypernetted-chain (HNC) approximation and in HNC+bridge approximation. The structure of water in obstacle matrices of different sizes was studied, showing that the model appropriately accounts for the hydrophobic hydration. By calculating the chemical potential of water in the model adsorbent, we have constructed the adsorption isotherm. Except for the cases of highly dispersed matrices, water gets excluded from the crowded hydrophobic environment, as expected experimentally.

A study on the extension of correlation functions obtained from molecular dynamics simulations by the Ornstein–Zernike theory for modeled molten salts

We extend the correlation functions obtained by molecular dynamics (MD) simulation for a molten salt modeled as a superposition of the Lennard-Jones (LJ) and Coulomb potentials using the hybrid closure method, which employs the Ornstein–Zernike (OZ) theory coupled with a closure relation. An appropriate distance for switching the short-range MD part and the long-range OZ part is determined by monitoring the isothermal compressibility, excess internal energy, and pressure. The Kobryn–Gusarov–Kovalenko (KGK) closure relation is mainly employed for the hybrid closure method (MD–KGK hybrid closure). The hybrid closure with either the hypernetted chain (HNC) or Kovalenko–Hirata (KH) closure was also tested to confirm that the performance was almost equivalent to one another among the MD–HNC, MD–KH, and MD–KGK methods. The bridge function for the model molten salt is extracted using the MD–KGK hybrid closure method. At a high-density state, the bridge function shows a steep increase in the repulsive core region, as is often observed for simple fluids, whereas when the density is relatively low, the bridge function for the cation–anion pair shows a downward-sloping behavior. Furthermore, the accuracies of excess internal energy, pressure, and isothermal compressibility were also examined for the HNC, KH, and KGK approximations. For molten salt systems, these approximations exhibited a similar behavior to those for monatomic LJ fluids, especially in the high-density state. The analysis of the integrand for excess internal energy and pressure is also discussed.

Morse potential parameters of dissipative particle dynamics force fields for non-polarizable water models

The Morse potential parameters compatible with dissipative particle dynamics (DPD) for water are determined by the Ornstein-Zernike (OZ) equation with the hyper-netted chain (HNC) closure. The present study verifies that the HNC approximation is accurate for a DPD system described by the soft repulsive and Morse potentials. Firstly, we demonstrate that the modified form of the Morse potential is necessary for the numerical solution of the OZ/HNC equations. Subsequently, the Morse potential parameters for mimicking SPCE, TIP3P, TIP4P, and TIP5P water models within the DPD framework are determined by mapping the radial distribution function (RDF) in the HNC approximation to those of all-atom simulations. The first-peak positions and heights of the center-of-mass RDFs for all-atom water models are successfully reproduced by DPD simulations using the determined parameters. The OZ/HNC results for the RDFs and pressures, calculated for the obtained water models, are in satisfactory agreement with the DPD results. Moreover, we investigate how the addition of the Morse potential term to the solute-solvent interaction manifests itself in the solution properties. Major effects of the Morse potential are found in the solute-solvent RDF and the potential of mean force (PMF) between a pair of solutes. We discuss the molecular mechanism underlying the effects induced by the Morse potential in detail.

The Influence of the Effects of the Bound Electrons on the Ion Structural and Thermodynamic Properties

In this article, a strongly coupled one-component plasma (OCP), where ions interact with a background of weakly coupled electrons, is considered. The thermodynamic properties of this plasma were studied on the basis of three types of interaction potentials. The first potential with ionic core obtained from <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$ab initio$ </tex-math></inline-formula> simulations takes into account the effects of the bound electrons, the second is the well-known Yukawa potential, and the third is the effective potential derived by the method of dielectric response function taking into account electron screening and quantum-mechanical diffraction effect of electrons. The radial distribution functions (RDFs) were calculated for the first potential using molecular dynamics (MD) simulations and were additionally compared with the RDFs obtained for all three potentials by solving the integral Ornstein–Zernike equation in the hypernetted chain (HNC) approximation and show that the use of the effective potential is not applicable at warm dense matter (WDM) conditions, when the ion core is significant.

Spiers Memorial Lecture: Towards understanding of iontronic systems: electroosmotic flow of monovalent and divalent electrolyte through charged cylindrical nanopores.

In many practical applications, ions are the primary charge carrier and must move through either semipermeable membranes or through pores, which mimic ion channels in biological systems. In analogy to electronic devices, the "iontronic" ones use electric fields to induce the charge motion. However, unlike the electrons that move through a conductor, motion of ions is usually associated with simultaneous solvent flow. A study of electroosmotic flow through narrow pores is an outstanding challenge that lies at the interface of non-equilibrium statistical mechanics and fluid dynamics. In this paper, we will review recent works that use dissipative particle dynamics simulations to tackle this difficult problem. We will also present a classical density functional theory (DFT) based on the hypernetted-chain approximation (HNC), which allows us to calculate the velocity of electroosmotic flows inside nanopores containing 1 : 1 or 2 : 1 electrolyte solution. The theoretical results will be compared with simulations. In simulations, the electrostatic interactions are treated using the recently introduced pseudo-1D Ewald summation method. The zeta potentials calculated from the location of the shear plane of a pure solvent are found to agree reasonably well with the Smoluchowski equation. However, the quantitative structure of the fluid velocity profiles deviates significantly from the predictions of the Smoluchowski equation in the case of charged pores with 2 : 1 electrolyte. For low to moderate surface charge densities, the DFT allows us to accurately calculate the electrostatic potential profiles and the zeta potentials inside the nanopores. For pores with 1 : 1 electrolyte, the agreement between theory and simulation is particularly good for large ions, for which steric effects dominate over the ionic electrostatic correlations. The electroosmotic flow is found to depend very strongly on the ionic radii. In the case of pores containing 2 : 1 electrolyte, we observe a reentrant transition in which the electroosmotic flow first reverses and then returns to normal as the surface change density of the pore is increased.

Integral equation theories for fluid with very short-range screened Coulomb plus power series interactions

Monte Carlo NVT simulations have been performed to obtain the thermodynamic and structural properties of fluids with a potential consisting in a hard-core and screened Coulomb plus power series (HC-SPS) interactions for three sets of the potential parameters. These ‘exact’ data are used to check the accuracy of two integral equation theories and a local excess chemical potential formula for very short-ranged potentials. The theories considered are the reference hypernetted-chain (RHNC) approximation and the local self-consistent OZ approximation (LSCOZA). The main conclusions of our study are described below. (i) The RHNC always underestimates the first peak of the radial distribution, which directly leads to an underestimation of the pressure and an overestimation of excess internal energy. In general, as the potential range becomes shorter ranged and the temperature decreases, the performance of the RHNC worsens and even becomes qualitatively incorrect. (ii) Local pressure self-consistency is of crucial importance to ensure that the hard sphere bridge function with an effective hard sphere diameter yields very accurate results for the pressure and excess internal energy even for very short-range potentials and at temperatures very close to the critical temperature. (iii) The accuracy of the excess chemical potential, obtained by thermodynamic integration of the excess internal energy, is also remarkable even for very short-ranged potentials and superior to that of a local excess chemical potential formula previously proposed [S. Zhou, Theor. Chem. Acc. 117, 555(2007)].

THERMODYNAMIC PROPERTIES OF DENSE HYDROGEN PLASMAS WITH PARTIALLY DEGENERATE SEMICLASSICAL IONS

In this article, we investigated the thermodynamic properties of a one-component dense non-ideal hydrogen plasma with semiclassical non-ideal ions against the background of degenerate quantum electrons. The effective potential of the ion-ion interaction that takes into account the quantum mechanical effects of diffraction was used as a model for the interaction. Ion-ion radial distribution functions were calculated based on the solution of the Ornstein-Zernike integral equation in the hypernetted-chain (HNC) approximation. In the calculations, the effective interaction potential was used as a micropotential since only the screening due to the uniform background (i.e., electrons) was taken into account, but not the screening due to the interaction between the ions themselves. In addition, the effective potential takes into account only the quantum effect of ion diffraction. Thermodynamic properties, such as the correlation energy and the non-ideal component of the equation of state, were calculated based on the obtained radial distribution functions and the interparticle interaction potential. Thermodynamic properties were presented as a percentage between the results based on the effective potential and the Yukawa potential depending on the density and coupling parameter values. The difference between the results is greater in denser plasma and at higher values of the coupling parameters.

Dust-Acoustic Wave Dispersion in Thermal Dusty Plasmas at Weak and Moderate Couplings

The dispersion of dust-acoustic waves (DAWs) in weakly and moderately coupled thermal dusty plasmas is studied in the framework of the linear density-response formalism with the static local-field correction for interdust interactions. The plasma medium composition and the charge of dust particles are simultaneously determined within a recently developed chemical model (Physical Review E, vol. 101, 063203, 2020) based on minimizing the Helmholtz free energy of the system under investigation. Stemming from the generalized Poisson-Boltzmann equation, the renormalization procedure is consistently applied to derive an interdust screened potential that takes into account the finiteness of dust grains. Within the framework of the Ornstein-Zernike relationship in the hypernetted chain approximation, the static structure factor of the dust component is evaluated to manifest the appearance of local extrema on its wavenumber dependence, thereby indicating the short-range order formation in the arrangement of dust particles with respect to one another. It is shown that the DAW dispersion law is completely governed by the static structure factor and, therefore, exhibits a nonmonotonic dependence on the wavenumber as well. In the long-wavelength limit, the acoustic-like behavior of the DAW dispersion is strictly confirmed and the corresponding phase speed, reduced in units of the dust thermal velocity, is ultimately expressed via the static structure factor at zero wavenumber.

Ionic self-diffusion coefficient and shear viscosity of high-Z materials in the hot dense regime

High-Z materials exhibit a broad range of variation of the charge state in the hot dense regime, and so ionic structures become complex with increasing density and temperature owing to ionization. Taking high-Z uranium as example, we study its electronic and ionic structures in the hot dense regime by combining an average-atom model with the hypernetted chain approximation. The electronic structure is described by solving the Dirac equation, taking account of relativistic effects, including broadening of the energy levels, and the effect of other ions via correlation functions. On the basis of the electronic distribution around a nucleus, the ion pair potential is constructed using the modified Gordon–Kim model in the frame of temperature-dependent density functional theory. Because of the presence of ion–ion strong coupling, the bridge function is included in the hypernetted chain approximation, which is used to calculate the correlation functions. To take account of the influence on transport properties of the strong correlation of electrons with highly charged ions, we perform both classical and Langevin molecular dynamics simulations to determine ion self-diffusion coefficients and the shear viscosity, using the Green–Kubo relation and an ion–ion pair potential with good convergence. We show that the influence of electron–ion collisions on transport properties becomes more important as the free electron density increases owing to thermal ionization.

Influence of different charge-state ion distribution on elastic X-ray scattering in warm dense matter

The study of warm dense matter is very important for the evolution of celestial bodies and inertial confinement fusion, which often contains a mixture of multiple elements and different charge-state ions. The ionic structure and distribution of different charge-states directly affect the diagnosis and physical properties of warm dense matter. At the same time, the influence of high-temperature dense plasma on the ionic structure should be considered when we study the physical properties from the first-principle calculation of electron structure. In the present work, the radial distribution functions of multiple charge-state ions (gold, carbon-hydrogen mixture, and aluminum) are developed in the hypernetted-chain approximation, and elastic x-ray scattering of different charge-state ions are calculated in the warm dense matter regime. Firstly, the electron structure of different charge-state ions is self-consistently computed in the ionic sphere, in which the ion-sphere radii are determined by the plasma density and their charges. And then the ionic fraction is obtained by solving the modified Saha equation, with the interactions among different charge-state ions taken into account, and ion-ion pair potentials are obtained by Yukawa model. Finally, the ion features of x-ray elastic scattering for Al are calculated on the basis of electronic distribution around the nuclei and ionic radial distribution function. By comparing the results of different charge-sate ions with the result of mean charge-sate ion, it is shown that different statistical methods can affect the physical properties which are dependent on the electronic and ionic structure.

Radial Distribution Function for a Hard Sphere Liquid: A Modified Percus-Yevick and Hypernetted-Chain Closure Relations

Establishment of the radial distribution function by solving the Ornstein-Zernike equation is still an important problem, even more than a hundred years after the original paper publication. New strategies and approximations are common in the literature. A crucial step in this process consists in defining a closure relation which retrieves correlation functions in agreement with experiments or molecular simulations. In this paper, the functional Taylor expansion, as proposed by J. K. Percus, is applied to introduce two new closure relations: one that modifies the Percus‑Yevick closure relation and another one modifying the Hypernetted-Chain approximation. These new approximations will be applied to a hard sphere system. An improvement for the radial distribution function is observed in both cases. For some densities a greater accuracy, by a factor of five times compared to the original approximations, was obtained.

Assessment of the Wolf method using the Stillinger-Lovett sum rules: From strong electrolytes to weakly charged colloidal dispersions.

The Ewald method has been the cornerstone in molecular simulations for modeling electrostatic interactions of charge-stabilized many-body systems. In the late 1990s, Wolf and collaborators developed an alternative route to describe the long-range nature of electrostatic interactions; from a computational perspective, this method provides a more efficient and straightforward way to implement long-range electrostatic interactions than the Ewald method. Despite these advantages, the validity of the Wolf potential to account for the electrostatic contribution in charged fluids remains controversial. To alleviate this situation, in this contribution, we implement the Wolf summation method to both electrolyte solutions and charged colloids with moderate size and charge asymmetries in order to assess the accuracy and validity of the method. To this end, we verify that the proper selection of parameters within the Wolf method leads to results that are in good agreement with those obtained through the standard Ewald method and the theory of integral equations of simple liquids within the so-called hypernetted chain approximation. Furthermore, we show that the results obtained with the original Wolf method do satisfy the moment conditions described by the Stillinger-Lovett sum rules, which are directly related to the local electroneutrality condition and the electrostatic screening in the Debye-Hückel regime. Hence, the fact that the solution provided by the Wolf method satisfies the first and second moments of Stillinger-Lovett proves, for the first time, the reliability of the method to correctly incorporate the electrostatic contribution in charge-stabilized fluids. This makes the Wolf method a powerful alternative compared to more demanding computational approaches.

On the overlap between configurations in glassy liquids.

The overlap, or similarity, between liquid configurations is at the core of the mean-field description of the glass transition and remains a useful concept when studying three-dimensional glass-forming liquids. In liquids, however, the overlap involves a tolerance, typically of a fraction a/σ of the inter-particle distance, associated with how precisely similar two configurations must be for belonging to the same physically relevant "state." Here, we systematically investigate the dependence of the overlap fluctuations and of the resulting phase diagram when the tolerance is varied over a large range. We show that while the location of the dynamical and thermodynamic glass transitions (if present) is independent of a/σ, that of the critical point associated with a transition between a low- and a high-overlap phase in the presence of an applied source nontrivially depends on the value of a/σ. We rationalize our findings by using liquid-state theory and the hypernetted-chain approximation for correlation functions. In addition, we confirm the theoretical trends by studying a three-dimensional glass-former by computer simulations. We show, in particular, that a range of a/σ below what is commonly considered maximizes the temperature of the critical point, pushing it up in a liquid region where viscosity is low and computer investigations are easier due to a significantly faster equilibration.

Density functional theory of fluid-solid phase transition in a two-dimensional system of superparamagnetic colloids in tilted magnetic field

We have used density functional theory (DFT) of freezing to investigate the fluid-solid phase transition in a system of super-paramagnetic colloidal particles confined to a two dimensional plane and exposed to an external magnetic field which is tilted from the direction parallel to the plane. Magnetic field induces magnetic moment in the particles which are directed along the field. Because of the tilted induced magnetic moments, particles interact via anisotropic repulsive dipole-dipole interaction potential. We solve the Ornstein-Zernike(OZ)equation within the hypernetted chain (HNC) approximation to obtain the pair-correlation functions (PCF), which are used as structural input in DFT. All the pair functions were expanded in a suitable orthogonal basis and the expansion coefficients were determined by using the method of Fries and Patey [32] which leads to an unphysical negative contact value of radial distribution functions. We have employed a method similar to that of Hansen and Hayter [35], to resolve this issue. The accuracy of the resulting PCFs are tested by performing NVT Monte Carlo simulation. We found that for the tilt angle of the external field in the range 900 to 750, the fluid freezes into triangular solid, beyond which DFT fails to stabilize the solids considered in this work.

Indirect Solution of Ornstein-Zernike Equation Using the Hopfield Neural Network Method

Microscopic information, such as the pair distribution and direct correlation functions, can be obtained from experimental data. From these correlation functions, thermodynamical quantities and the potential interaction function can be recovered. Derivations of Ornstein-Zernike equation and Hopfield Neural Network method are given first, as a theoretical background to follow the present work. From these two frameworks, structural information, such as the radial distribution (g(r)) and direct correlation (C(r)) functions, were retrieved from neutron scattering experimental data. The problem was solved considering simple initial conditions, which does not require any previous information about the system, making it clear the robustness of the Hopfield Neural Network method. The pair interaction potential was estimated in the Percus-Yevick (PY) and hypernetted chain (HNC) approximations and a poor agreement, compared with the Lennard-Jones 6-12 potential, was observed for both cases, suggesting the necessity of a more accurate closure relation to describe the system. In this sense, the Hopfield Neural Network together with experimental information provides an alternative approach to solve the Ornstein-Zernike equations, avoiding the limitations imposed by the closure relation.

Simple Parameter-Free Bridge Functionals for Molecular Density Functional Theory. Application to Hydrophobic Solvation.

Computer simulations have been fundamental in understanding the fine details of hydrophobic solvation and hydrophobic interactions. Alternative approaches based on liquid-state theories have been proposed, but are not yet at the same degree of completeness and accuracy. In this vein, a classical, molecular density functional theory approach to hydrophobic solvation is introduced. The lowest, second-order approximation of the theory, equivalent to the hypernetted chain approximation in integral equations, fails in describing correctly cavitation free-energies. It is corrected here by two simple, angular-independent, so-called bridge functionals; they are parameter-free in the sense that all variables can be fixed unambiguously from the water bulk properties, including pressure, isothermal compressibility, and liquid-gas surface tension. A hard-sphere bridge functional, based on the known functional of a reference hard fluid system, turns out to face strong limitations for water. A simpler weighted density approximation is shown to properly reproduce the solvation free energy of hydrophobes of various sizes, from microscopic ones to the nanoscale, and predicting the solvation free energy of a data set of more than 600 model hydrophobic molecules having a variety of shapes and sizes with an accuracy of a quarter of kBT compared to Monte Carlo simulations values. It constitutes an excellent starting point for a general functional describing accurately both hydrophobic and hydrophilic solvation, and making it possible to study nonidealized hydrophobic interactions.

Predicting Hydration Free Energies of the FreeSolv Database of Drug-like Molecules with Molecular Density Functional Theory.

We assess the performance of molecular density functional theory (MDFT) to predict hydration free energies of the small drug-like molecules benchmark, FreeSolv. The MDFT in the hypernetted chain approximation (HNC) coupled with a pressure correction predicts experimental hydration free energies of the FreeSolv database within 1 kcal/mol with an average computation time of 2 cpu·min per molecule. This is the same accuracy as for simulation-based free energy calculations that typically require hundreds of cpu·h or tens of gpu·h per molecule.