Hypernetted Chain Equation Research Articles

A unified derivation of Percus–Yevick and hyper-netted chain integral equations in liquid state theory

A general one-parameter integral equation for liquids is derived that incorporates intrinsically both the famous Percus–Yevick and hyper-netted chain equations as special cases. To examine its accuracy and reliability, the equation is solved self-consistently to find the structure and thermodynamics of Lennard–Jones fluid successfully.

Bound electron screening effect on ion-ion potential of warm and hot dense matter.

The effects of bound electron screening in warm and hot dense matter are investigated analytically and a theoretical description of screened short-range repulsion is given meanwhile. An empirical ion-ion potential including the classic charge screening and chemical bond attraction at various temperatures and densities is proposed. By solving hypernetted chain equations and comparing the obtained radial distribution function (RDF) with ab initio simulations, the proposed ion-ion potential is found to be promising over a wide range of temperatures and densities for warm dense aluminum and iron. The elastic scattering amplitude and the x-ray absorption near the edge structure of warm dense aluminum calculated from the obtained RDF are in good agreement with experiment results.

Linked cluster expansion of the many-body path integral

We develop an approach of calculating the many-body path integral based on the linked cluster expansion method. First, we derive a linked cluster expansion and we give the diagrammatic rules for calculating the free-energy and the pair distribution function $g(r)$ as a systematic power series expansion in the particle density. We also generalize the hypernetted-chain (HNC) equation for $g(r)$, known from its application to classical statistical mechanics, to a set of quantum HNC equations (QHNC) for the quantum case. The calculated $g(r)$ for distinguishable particles interacting with a Lennard-Jones potential in various attempted schemes of approximation of the diagrammatic series compares very well with the results of path integral Monte Carlo simulation even for densities as high as the equilibrium density of the strongly correlated liquid $^4$He. Our method is applicable to a wide range of problems of current general interest and may be extended to the case of identical particles and, in particular, to the case of the many-fermion problem.

Closure for the Ornstein-Zernike equation with pressure and free energy consistency.

The Ornstein-Zernike (OZ) integral equation theory is a powerful approach to simple liquids due to its low computational cost and the fact that, when combined with an appropriate closure equation, the theory is thermodynamically complete. However, approximate closures proposed to date exhibit pressure or free energy inconsistencies that produce inaccurate or ambiguous results, limiting the usefulness of the Ornstein-Zernike approach. To address this problem, we combine methods to enforce both pressure and free energy consistency to create a new closure approximation and test it for a single-component Lennard-Jones fluid. The closure is a simple power series in the direct and total correlation functions for which we have derived analytical formulas for the excess Helmholtz free energy and chemical potential. These expressions contain a partial molar volumelike term, similar to excess chemical potential correction terms recently developed. Using our bridge approximation, we have calculated the pressure, Helmholtz free energy, and chemical potential for the Lennard-Jones fluid using the Kirkwood charging, thermodynamic integration techniques, and analytic expressions. These results are compared with those from the hypernetted chain equation and the Verlet-modified closure against Monte Carlo and equations-of-state data for reduced densities of ρ^{*}<1 and temperatures of T^{*}=1.5, 2.74, and 5. Our closure shows consistency among all thermodynamic paths, except for one expression of the Gibbs-Duhem relation, whereas the hypernetted chain equation and the Verlet-modified closure exhibit consistency between only a few relations. Accuracy of the closure is comparable to the Verlet-modified closure and a significant improvement to results obtained from the hypernetted chain equation.

Correction of Kovalenko-Hirata closure in Ornstein-Zernike integral equation theory for Lennard-Jones fluids

We have studied the accuracy of Ornstein-Zernike (OZ) integral equation theory in terms of the solvation free energy (SFE) for a two-component Lennard-Jones system at the infinite dilution limit. We have employed the hypernetted chain (HNC), Kovalenko-Hirata (KH), Kobryn-Gusarov-Kovalenko (KGK), and Percus-Yevick (PY) closure equations. Further, we have extended the Verlet-modified (VM) closure to the two-component system to examine the accuracy of this method in terms of the SFE. Molecular dynamics simulations were employed to compare the results with the above mentioned OZ theories. The HNC and KH approximations significantly overestimate the SFE, whereas the PY approximation tends to underestimate it. The overestimation of the SFE by the KGK approximation becomes significant when the solvent density is relatively high. In contrast, the VM approximation is found to be rather accurate at all studied conditions. An analysis of the integrand for the SFE reveals that, to improve the SFE, the first rising (FR) region in the radial distribution function (RDF) must be corrected. We have also tested the sigma-enlarging-bridge (SEB) function that we proposed previously [T. Miyata and Y. Ebato, J. Mol. Liq. 217 (2016) 75] for the correction of the FR region of the RDF. In this study, we applied the SEB function to both HNC-type and KH-type closures and found that these combinations (SEB-HNC and SEB-KH) improve the SFE significantly.

Primitive electrolyte solutions around a charged colloid: density functional approach

ABSTRACTWe have developed the density functional theory, which is based both on the modified fundamental-measure theory for the hard spheres and on the weighted-density approximation for the electrostatic residual contribution, for studying the structure of the size-symmetric and asymmetric electrolyte solutions around a charged hard colloid. The calculation shows that the present theory provides an excellent description of the primitive electrolyte solutions and indicates the suitability of the approximations used in the formalism. The preset theory provides better the structural results than the density functional theories based on the excess free energy functional expansion with respect to the bulk ionic density and the hypernetted-chain equation based on the mean spherical approximation over a wide range of colloid sizes, ion concentrations and valence of a colloid. We have confirmed that the present theory is able to predict the phenomena of charge reversal and overcharging occurring near a colloidal interface, in agreement with previous primitive model Monte Carlo simulations and theoretical studies reported in the literature.

Coexistence of low and high overlap phases in a supercooled liquid: An integral equation investigation.

The pair structure, free energy, and configurational overlap order parameter Q of an annealed system of two weakly coupled replicas of a supercooled "soft sphere" fluid are determined by solving the hypernetted-chain (HNC) and self-consistent Rogers-Young (RY) integral equations over a wide range of thermodynamic conditions ρ (number-density), T (temperature), and inter-replicas couplings ε12. Analysis of the resulting effective (or Landau) potential W(ρ,T; Q) and of its derivative with respect to Q confirms the existence of a "precursor transition" between weak and strong overlap phases below a critical temperature Tc well above the temperature To of the "ideal glass" transition observed in the limit ε12→0. The precursor transition is signalled by a loss of convexity of the potential W(Q) and by a concomitant discontinuity of the order parameter Q just below Tc, which crosses over to a mean-field-like van der Waals loop at lower temperatures. The HNC and RY equations lead to the same phase transition scenario, with quantitative differences in the predicted temperatures Tc and To.

The Hypernetted Chain Equations for Periodic Systems

Starting from the general inhomogeneous Fermi hypernetted chain equations, the equations for periodic systems are derived by simple Fourier transform. It is shown how the symmetry reduces the size of the involved quantities. First results for a one-dimensional (1D) model system are presented. The results allow a reliable estimation of the numerical demand even for realistic 3D systems, such as solids. It is shown that treatment of this systems is feasible with moderate computational resources.

Use of Two-Body Correlated Basis Functions with van der Waals Interaction to Study the Shape-Independent Approximation for a Large Number of Trapped Interacting Bosons

We study the ground state and the low-lying excitations of a trapped Bose gas in an isotropic harmonic potential for very small ($\sim 3$) to very large ($\sim 10^7$) particle numbers. We use the correlated two-body basis functions and the shape-dependent van der Waals interaction in our many-body calculations. We present an exhaustive study of the effect of inter-atomic correlations and the accuracy of the mean-field equations considering a wide range of particle numbers. We calculate the ground state energy and the one-body density for different values of the van der Waals parameter $C_{6}$. We compare our results with those of the modified Gross-Pitaevskii results, the correlated Hartree hypernetted-chain equations (which also utilize the two-body correlated basis functions), as well as of the Diffusion Monte Carlo for hard sphere interactions. We observe the effect of the attractive tail of the van der Waals potential in the calculations of the one-body density over the truly repulsive zero-range potential as used in the Gross-Pitaevskii equation and discuss the finite-size effects. We also present the low-lying collective excitations which are well described by a hydrodynamic model in the large particle limit.

Multi-average ion model for hot dense plasmas derived from finite temperature density-functional theory

We present a formulation for plasmas in local thermal equilibrium (LTE) as a system of consisting of (ZA + 1) kinds of average ions, i.e., one for each charge state 0; 1; 2; …; ZA and each with a uniform electronic charge, thus generating a multi-average ion model or Multi-AIM model. Here the bound electron density of average atoms and their pair correlation functions is obtained self-consistently by means of minimizing the free energy of the whole system established by the finite temperature density functional theory (FTDFT). The pair correlation function is obtained by solving the hypernetted chain equation (HNC). We show the charge state distribution of Al and Fe plasmas for several temperatures and densities determined by the Multi-AIM.

Electric double layer for a size-asymmetric electrolyte around a spherical colloid

We have studied the structure of a size-asymmetric electrolyte on charged colloids by a density functional perturbation theory. The hard-sphere contribution has been approximated as the direct pair correlation function with the coupling parameter, whereas the electronic contribution has been approximated as the mean-spherical approximation in the bulk phase. The calculated results for the ionic density distributions and mean electrostatic potentials are in very good agreement with the computer simulations over a wide range of colloid sizes and electrolyte concentrations. The present theory provides better structural results than the hypernetted-chain equation based on the mean spherical approximation. We have confirmed that the overcharging appears when the counterions are larger than the coions. The overcharging disappears everywhere when the electrostatic repulsion becomes strong enough, while the charge reversal is observed when the coions are larger than the counterions, and the reversal effect appears for a size-asymmetric electrolyte at high surface charge densities. The charge reversal occurs even for the point of zero charge, mainly due to the depletion force between two ions. The present theory is able to provide interesting insights about the charge reversal and overcharging phenomena occurring at the interface.

From square-well to Janus: Improved algorithm for integral equation theory and comparison with thermodynamic perturbation theory within the Kern-Frenkel model

Building upon past work on the phase diagram of Janus fluids [F. Sciortino, A. Giacometti, and G. Pastore, Phys. Rev. Lett. 103, 237801 (2009)], we perform a detailed study of integral equation theory of the Kern-Frenkel potential with coverage that is tuned from the isotropic square-well fluid to the Janus limit. An improved algorithm for the reference hypernetted-chain (RHNC) equation for this problem is implemented that significantly extends the range of applicability of RHNC. Results for both structure and thermodynamics are presented and compared with numerical simulations. Unlike previous attempts, this algorithm is shown to be stable down to the Janus limit, thus paving the way for analyzing the frustration mechanism characteristic of the gas-liquid transition in the Janus system. The results are also compared with Barker-Henderson thermodynamic perturbation theory on the same model. We then discuss the pros and cons of both approaches within a unified treatment. On balance, RHNC integral equation theory, even with an isotropic hard-sphere reference system, is found to be a good compromise between accuracy of the results, computational effort, and uniform quality to tackle self-assembly processes in patchy colloids of complex nature. Further improvement in RHNC however clearly requires an anisotropic reference bridge function.

Theoretical study of interactions of BSA protein in a NaCl aqueous solution

Bovine Serum Albumine (BSA) aqueous solutions in the presence of NaCl are investigated for different protein concentrations and low to intermediate ionic strengths. Protein interactions are modeled via a charge-screened colloidal model, in which the range of the potential is determined by the Debye-Hückel constant. We use Monte Carlo computer simulations to calculate the structure factor, and assume an oblate ellipsoidal form factor for BSA. The theoretical scattered intensities are found in good agreement with the experimental small angle X-ray scattering intensities available in the literature. The performance of well-known integral equation closures to the Ornstein-Zernike equation, namely the mean spherical approximation, the Percus-Yevick, and the hypernetted chain equations, is also assessed with respect to computer simulation.

Simple electrolyte solutions: Comparison of DRISM and molecular dynamics results for alkali halide solutions

Using the dielectrically consistent reference interaction site model (DRISM) of molecular solvation, we have calculated structural and thermodynamic information of alkali-halide salts in aqueous solution, as a function of salt concentration. The impact of varying the closure relation used with DRISM is investigated using the partial series expansion of order-n (PSE-n) family of closures, which includes the commonly used hypernetted-chain equation (HNC) and Kovalenko-Hirata closures. Results are compared to explicit molecular dynamics (MD) simulations, using the same force fields, and to experiment. The mean activity coefficients of ions predicted by DRISM agree well with experimental values at concentrations below 0.5 m, especially when using the HNC closure. As individual ion activities (and the corresponding solvation free energies) are not known from experiment, only DRISM and MD results are directly compared and found to have reasonably good agreement. The activity of water directly estimated from DRISM is nearly consistent with values derived from the DRISM ion activities and the Gibbs-Duhem equation, but the changes in the computed pressure as a function of salt concentration dominate these comparisons. Good agreement with experiment is obtained if these pressure changes are ignored. Radial distribution functions of NaCl solution at three concentrations were compared between DRISM and MD simulations. DRISM shows comparable water distribution around the cation, but water structures around the anion deviate from the MD results; this may also be related to the high pressure of the system. Despite some problems, DRISM-PSE-n is an effective tool for investigating thermodynamic properties of simple electrolytes.

Testing ion structure models with x-ray Thomson scattering

We investigate the influence of various ionic structure models on the interpretation of the X-ray Thomson scattering signal. For the calculation of the ion structure, classical hypernetted chain equations are used applying different effective inter-particle potentials. It is shown that the different models lead to significant discrepancies in the theoretically predicted weight of the Rayleigh peak, in particular for small k-values where correlation effects are important. Here, we propose conditions which might allow for an experimental verification of the theories under consideration of experimental constraints of k-vector blurring.

Extraction of effective ion pair interactions in warm dense beryllium and helium plasmas within integral equation theory

Under hypernetted chain (HNC) approximation, effective ion pair interaction potentials for the warm dense matter are extracted by using available radial distribution functions (RDFs). The effective ion pair potentials extracted from first-principles simulation results are found containing the short-ranged attraction (SRA) component for both warm dense helium and beryllium plasmas. The SRA potentials can be well represented by Gaussian functions in both cases and then the extracted effective ion potentials are well fitted. As an application, the well fitted potentials are used to describe ion-ion interactions in classical molecular dynamics simulations. The yield RDFs are in excellent agreement with those computed by HNC equations and first-principles simulations, respectively.

Enhancement of depletion forces by electrostatic depletant repulsion

A large variety of engaging phenomena stems from the occurrence of short-ranged attractive depletion forces. Yet, so far, most experimental studies have been interpreted on the basis of the simple Asakura-Oosawa model, where the depletion agent can be regarded as ideal. Here, conversely, we focus on a system where strong electrostatic coupling is present in the suspension. Specifically, from measurements of equilibrium sedimentation profiles, we obtain an equation of states for a colloidal system where depletion forces are tuned by the addition of a surfactant. At fixed colloid volume fraction, colloidal aggregation takes place when the surfactant concentration reaches a critical value which rises for increasing ionic strength. Screening repulsive electrostatic interactions inhibits the depletion mechanism and weakens the effective colloid-colloid attraction. The metastable coexistence curve displays the universal scaling behavior predicted for short-ranged potentials. The experimental data are compared with the theoretical predictions of a simple model which includes only electrostatic interactions. The effective depletion force on the colloids is evaluated by using the hypernetted-chain equation of liquid state theory at different salt concentrations. This model provides a convincing interpretation of the observed enhancement of the depletion mechanism by Coulomb repulsion.

An accurate equation of state for the one-component plasma in the low coupling regime

An accurate equation of state of the one-component plasma is obtained in the low coupling regime 0 ⩽ Γ ⩽ 1. The accuracy results from a smooth combination of the well-known hypernetted chain integral equation, Monte Carlo simulations and asymptotic analytical expressions of the excess internal energy u. In particular, special attention has been brought to describe and take advantage of finite size effects on Monte Carlo results to get the thermodynamic limit of u. This combined approach reproduces very accurately the different plasma correlation regimes encountered in this range of values of Γ. This paper extends to low Γ's, an earlier Monte Carlo simulation study devoted to strongly coupled systems for 1 ⩽ Γ ⩽ 190 (J-M Caillol 1999 J. Chem. Phys. 111 6538). Analytical fits of u(Γ) in the range 0 ⩽ Γ ⩽ 1 are provided with a precision that we claim to be not smaller than p = 10−5. The hypernetted chain equation and exact asymptotic expressions are shown to give reliable results for u(Γ) only in narrow Γ intervals, i.e. 0 ⩽ Γ ⩽ sssim0.5 and 0 ⩽ Γ ⩽ sssim0.3 respectively.

Dynamic many-body theory: Pair fluctuations in bulkH4e

Dynamical excitations in bulk liquid $^{4}\text{H}\text{e}$ are investigated by using a manifestly microscopic theory of excitations that includes multiple-phonon scattering. The wave function of the dynamic system is represented in terms of one- and two-body excitation amplitudes. Equations of motion for the linear response of boson liquids to a scalar external field are then derived from a stationarity principle. For a consistent treatment of long- and short-wavelength properties of the excitation amplitudes we derive and solve three sets of generic ``hypernetted chain'' equations determining the basic ingredients of the theory. From those ingredients, we calculate a dynamic structure function for $^{4}\text{H}\text{e}$ at saturation density. It is shown that the complete solution of the hypernetted chain equations leads, partly by the cancellation of errors, to an insignificantly improved theoretical prediction for the dynamic structure function compared with approximations introduced by Jackson, Feenberg, and Campbell. The implications of this result and the need for including higher-order multiparticle fluctuations are discussed.

Reference interaction site model investigation of homonuclear hard dumbbells under simple fluid theory closures: Comparison with Monte Carlo simulations

We revisit the thermodynamic and structural properties of fluids of homonuclear hard dumbbells in the framework provided by the reference interaction site model (RISM) theory of molecular fluids. Besides the previously investigated Percus-Yevick (PY) approximation, we test the accuracy of other closures to the RISM equations, imported from the theory of simple fluids; specifically, we study the hypernetted chain (HNC), the modified HNC (MHNC) and, less extensively, the Verlet approximations. We implement our approach for models characterized by several different elongations, up to the case of tangent diatomics, and investigate the whole fluid density range. The theoretical predictions are assessed against Monte Carlo simulations, either available from literature or newly generated by us. The HNC and PY equations of state, calculated via different routes, share on the whole the same level of accuracy. The MHNC is applied by enforcing an internal thermodynamic consistency constraint, leading to good predictions for the equation of state as the elongation of the dumbbell increases. As for the radial distribution function, the MHNC appears superior to other theories, especially for tangent diatomics in the high density limit; the PY approximation is better than the HNC and Verlet closures in the high density or elongation regime. Our structural analysis is supplemented by an accurate inversion procedure to reconstruct from Monte Carlo data and RISM the "exact" direct correlation function. In agreement with such calculations and consistent with the forecast of rigorous diagrammatic analysis, all theories predict the occurrence in the direct correlation function of a first cusp inside the dumbbell core and (with the obvious exception of the PY) of a second cusp outside; the cusps' heights are also qualitatively well reproduced by the theories, except at high densities.