Hypernetted-chain Method Research Articles

A perturbation theory for equation of state of hydrogen in warm and hot dense regimes

In the warm dense regime of hydrogen plasma where the ions and the electrons are strongly coupled, few theoretical models for equation of state (EOS) exist which are accurate enough and simple to implement. In the recent past, we have developed a method for calculating the Helmholtz free energy of a mixture of ions and electrons by combining the orbital-free quantum hypernetted chain (QHNC) method with perturbation theory treating the electron-ion interaction as a perturbation over a reference system of one component plasma (OCP) and uniform electron gas (UEG). However, it was assumed that electrons were at zero Kelvin and a crude formula for the free energy of the UEG was used. This limited the applicability of the method to a small density-temperature regime. In the present work, a finite temperature extension to the orbital-free QHNC theory is derived and is employed in the perturbation theory we developed. Also, an accurate formula given by Ichimaru is used for the free energy of the UEG. EOS, ionic, and electronic structures of fully ionized hydrogen plasma are obtained using the improved method for a wide range of densities and temperatures in the warm and hot dense plasma regimes. The results of the present method are in excellent agreement with those of simulation data for pressures above 1Mbar. It is observed that, above 5 eV, the reference system, i.e., the OCP + UEG, gives the most dominant contribution to EOS and the contribution of perturbation terms is limited to 5%.

The influence of the poly(ethylene glycol) on the mean activity coefficients of NaCl aqueous solutions. The application of the MSA and HNC method

A primitive model electrolyte in mixture with uncharged hard spheres was used to model the concentration dependence of the mean activity coefficient of sodium chloride in aqueous solutions of poly(ethylene glycol) (PEG) of different degree of polymerization and concentration. Ornstein-Zernike integral equation was numerically solved using the hypernetted chain (HNC) closure and the results were compared with the results of an analytical solution valid within the mean spherical approximation (MSA) closure. Good agreement between the HNC and MSA activity coefficients was obtained. However, at higher NaCl concentrations the convergence of the numerical solution was not satisfied for large model PEG molecules. The presence of the neutral component in the solution increases the mean activity coefficients in the whole concentration range due to the confinement. For a given NaCl and PEG concentration, decreasing the dielectric constant of the solution diminishes the activity coefficient due to stringer interaction between the ions. Mean activity coefficients of NaCl in two aqueous solutions differing in the concentrations of PEG (molecular weight of 12,000gmol−1) were determined also experimentally and compared with MSA results. The agreement was semi-qualitative, mainly due to difficulties in estimating good model parameters for the PEG molecules and the potential decrease of the dielectric constant of the solvent.

Average atom model based on Quantum Hyper-Netted Chain method

The study shows how to define, without any ad hoc assumption, the average ion charge ZI in the electron-ion model for plasmas and liquid metals: this definition comes out of the condition that a plasma consisting of electrons and nuclei can be described as an electron–ion mixture. Based on this definition of the average ion charge, the Quantum Hyper-Netted Chain (QHNC) method takes account of the thermal ionization and the resonant-state contribution to the bound electrons forming an ion.On the other hand, Blenski and Cichocki (2007) have derived a formula to determine the uniform electron density in a plasma as an electron–ion mixture by using the variational method with the help of the local density approximation. Without use of any approximation, we derived the formula determining the electron density in an extended form on the basis of the density functional theory. This formula is shown to be valid also for the QHNC method.

Comparing different coarse-grained potentials for star polymers

We compare different coarse-grained single-blob models for star polymers. We find that phenomenological models inspired by the Daoud-Cotton theory reproduce quite poorly the thermodynamics of these systems, even if the potential is assumed to be density dependent, as done in the analysis of experimental results. Using the numerically determined coarse-grained potential, we also determine the minimum value f(c) of the functionality of the star polymer for which a fluid-solid transition occurs. By applying the Hansen-Verlet criterion we find 35 < f(c) ≲ 40. This result is confirmed by an analysis that uses the modified (reference) hypernetted chain method and is qualitatively consistent with previous work.

Periodic box Fermi hypernetted chain calculations of neutron star crustal matter

Neutron star crustal matter, whose properties are relevant in many models aimed at explaining observed astrophysical phenomena, has so far always been studied using a mean field approach. In order to check the results obtained in this way, a sensible next step is to make use of a realistic nuclear potential. The present paper extends the periodic-box Fermi HyperNetted Chain method to include longitudinal-isospin dependence of the correlations, making feasible a study of asymmetric crustal matter. Results are presented for the symmetry energy, the low-density neutron star equation of state and the single particle neutron and proton energies.

Charged-boson fluid at zero-temperature in the fractional dimensional space

In the fractional dimensional space, the ground state properties of the charged-boson fluid are studied in a theory which goes beyond the random phase approximation by including correlations through a static local-field-factor. The structure factor and static local field-factor are obtained in the self-consistent method. Qualitative agreement with the Monte Carlo results on static screening is achieved by imposing self-consistency on the compressibility of the fluid in addition to self-consistency on the structure factor. Using the structure factors obtained in these methods, several properties of the charged boson system such as the energy spectrum of elementary excitations, the pair-correlation function, the ground state energy, the pressure, the chemical potential and the compressibility are obtained in several dimensions including both integer and fractional values. Results obtained in the later method in two and three dimensional systems are close to the Monte Carlo and hypernetted-chain methods. The Wigner crystallisation in the lower dimensional system is found at higher density of bosons. However, it vanishes in any lower dimensional system whose dimension lies below 2.

Charge Inversion and Ion−Ion Correlation Effects at the Mercury/Aqueous MgSO4 Interface: Toward the Solution of a Long-Standing Issue

Charge inversion is the phenomenon in which an electric double layer contains more counterions than needed to compensate the surface charge. For colloidal particles this has the consequence that the apparent surface charge, as inferred from electrophoresis or interaction studies, has a sign opposite to that of the actual surface charge, obtainable by titration. This phenomenon has been known for over a century. According to the traditional interpretation, the inversion is caused by (chemical) specific adsorption of ions. However, beginning in the early 1980s it has been predicted by a large number of workers that charge inversion should occur as a consequence of many-body ion−ion correlations. For surfaces of sufficiently high surface charge density in the presence of divalent or multivalent counterions, charge inversion is expected to be ubiquitous even in the absence of specific adsorption. Testing this prediction has proved difficult because chemical specific adsorption is a very common phenomenon and can outweigh the effects of ion correlations. So far, no experimental systems have been thoroughly investigated where strong specific adsorption could be unambiguously ruled out under conditions where charge inversion due to ion−ion correlations is predicted. Here, we solve this problem by studying the mercury/aqueous MgSO4 interface. This system has the advantage that highly accurate double layer data are available for a variety of conditions, including some where chemical specific adsorption is known to be absent (or at least very weak). From precise data for this system [Harrison, J. A.; Randles, J. E. B.; Schiffrin, D. J. J. Electroanal. Chem. 1970, 25, 197] one can establish the ionic components of charge and surface charge density. To extract quantitative theoretical predictions about the consequences of ion−ion correlations, we use the highly accurate anisotropic hypernetted chain (AHNC) method, where ion−ion correlations in the double layer are taken into account in a fully self-consistent fashion. It is found that for moderate to large negative surface charge densities and not too high concentration, the variation in the ionic components of charge with the surface charge density can to a large extent be quantitatively explained by enrichment of ions close to the surface due to ion−ion correlations. That chemical specific adsorption of Mg2+ is negligible is supported by considering the properties of the double layer close to the electrocapillary maximum. In view of the large body of evidence indicating that the counterions tend to specifically adsorb on the mercury surface for positive polarization but not for negative, the agreement between theory and experiment for negative surface charge constitutes strong evidence for ion−ion correlations as the origin of charge inversion.

Ion correlation forces between uncharged dielectric walls

The interaction pressure between two uncharged planar walls immersed in various electrolyte solutions containing mono- and/or divalent ions is investigated. The solution is treated as a primitive model electrolyte, and the wall surfaces constitute dielectric discontinuities. Ionic image charge and ion-wall dispersion interactions are included. The interaction parameters are appropriate for hydrocarbon (polystyrene)/water interfaces, and the electrolyte concentrations considered lie between 0.250M and 1.00M. The anisotropic hypernetted chain method is used to self-consistently calculate the ion density profiles and the ion-ion correlation functions in the inhomogeneous electrolyte. Thereby, the effects of image charge interactions and dispersion interactions on the pressure and the electrolyte structure are included in a fully consistent manner. The explicit consideration of correlations between the ions in the presence of image charges ensures that the screening of the zero-frequency van der Waals interaction is taken into account. Of special interest are the effects of asymmetries between anions and cations with respect to valency and/or dispersion interaction with the walls. Such asymmetries create an electric double layer in the electrolyte outside each electroneutral surface. This causes the wall-wall interaction for large surface separations to be similar to the interaction between charged surfaces. For intermediate separations, around 1-2 nm, a substantial repulsive peak appears in the ionic pressure. In some cases the repulsion is larger than the van der Waals attraction between the walls, which implies that there is a repulsive barrier in the total pressure despite that the surfaces are uncharged. The strongest repulsion is found for 2:1 electrolytes where the monovalent anions interact strongly with the walls via dispersion forces. In general, ion-wall dispersion forces acting on ions of lower valency have a much greater effect than equally strong dispersion forces acting on ions of higher valency. This is mainly due to the more strongly repulsive image charge forces on ions of higher valency that counteract the attractive dispersion forces. Effects of confinement on the ion-ion correlations also contribute to this difference. For all electrolytes the interaction pressure from the ions is attractive for small surface separations. The main cause is a depletion of ions between the walls from the self-image repulsion and confinement effects. For totally symmetric electrolytes the attractive pressure extends to large separations in most cases.

Pair-distribution functions of two-temperature two-mass systems: Comparison of molecular dynamics, classical-map hypernetted chain, quantum Monte Carlo, and Kohn-Sham calculations for dense hydrogen

Two-temperature, two-mass quasiequilibrium plasmas may occur in electron-ion plasmas, nuclear-matter, as well as in electron-hole condensed-matter systems. Dense two-temperature hydrogen plasmas straddle the difficult partially degenerate regime of electron densities and temperatures which are important in astrophysics, in inertial-confinement fusion research, and other areas of warm dense-matter physics. Results from quantum Monte Carlo (QMC) are used to benchmark the procedures used in classical molecular-dynamics simulations and hypernetted chain (HNC) and classical-map HNC (CHNC) methods to derive electron-electron and electron-proton pair-distribution functions. Where QMC is not available, we used Kohn-Sham results as the reference calculation. Then, nonequilibrium molecular dynamics for two-temperature, two-mass plasmas are used to obtain pair distribution functions without specifying the interspecies cross temperature. Using these results, the correct HNC and CHNC procedures for the evaluation of pair-distribution functions in two-temperature two-mass two-component charged fluids are established and results for a mass ratio of 1:5, typical of electron-hole fluids, are presented.

Ion‐Ion Correlation Effect on the Neutrino‐Nucleus Scattering in Supernova Cores

We calculate the ion-ion correlation effect on the neutrino-nucleus scattering in supernova cores, which is an important opacity source for the neutrinos and plays a vital role in the supernova explosion. In order to calculate the ion-ion correlation effect we use the results of the improved hypernetted-chain method calculations of the classical one-component plasma. As in the preceding studies on this effect, we find a dramatic decrease of the effective neutrino-nucleus scattering cross section for relatively low energy neutrinos with E < 20MeV. As a matter of fact, our calculation shows a much more dramatic reduction of the effective neutrino-nucleus scattering cross section for the low energy neutrinos with E < 10MeV than the results of Horowitz. Therefore, the ion-ion correlation effect will be more important than has hitherto been recognized. We present an accurate analytic fitting formula that summarizes our numerical results. This fitting formula will facilitate the application of the present results to the supernova explosion simulations.

Electron correlation in two-dimensional systems: CHNC approach to finite-temperature and spin-polarization effects

Applying the classical-map hypernetted-chain method (CHNC) developed recently by Dharma-wardana and Perrot, we have studied the temperature and spin-polarization effects on electron correlation in the uniform quantum two-dimensional gas (2DEG) over a wide range of temperature T and spin-polarization ζ. The quantum fluid at the temperature T is mapped to a classical fluid at the temperature T cf given by T cf 2= T 2+ T q 2, where the quantum temperature T q is determined by comparing the calculated correlation energy to that of Monte Carlo results for the fully spin-polarized quantum system at zero temperature. By the iterative solution of the modified HNC equation and the Ornstein–Zernike equation, we have obtained the pair distribution function (PDF) and correlation energy for the two-component classical 2DEG with a classical fluid temperature T cf. The anti-parallel bridge function B 12( r) appearing in the modified HNC equation is determined by using the Monte Carlo correlation energy at T=0 or STLS (Singwi–Tosi–Land-Sjölander) result at T>0 and the numerical solution to the Percus–Yevick (PY) equation for the system of hard disks. By calculating the Pauli potential, the bridge function, PDFs, structure factors and correlation energy, we have shown that in some cases, the properties of the uniform quantum 2DEG depend remarkably on the temperature and spin-polarization.

Improved variational calculations of nucleon matter

Variational calculations of nucleon matter, either symmetric nuclear or pure neutron matter, use Fermi hypernetted chain and single-operator chain summation methods to sum approximately the contributions of clusters of $&gt;~3$ nucleons to the energy expectation value. The cluster contributions summed by these methods are discussed in detail, and it is shown that for realistic interactions the 3-body cluster contribution is larger than the sum of $&gt;~4$-body contributions. We present a new method, based on representing cluster wave functions by multidimensional vectors in spin-isospin space, as is common in quantum Monte Carlo calculations of light nuclei, to calculate exactly the 3-body cluster contribution including 3-body forces and all but spin-orbit correlations. The variational energies obtained with the Argonne $v18$ 2- and Urbana IX 3-nucleon interactions, using the exact 2- and 3-body cluster contributions and the approximate $&gt;~4$-body contributions summed with chain summation techniques are lower, closer to the empirical values for symmetric nuclear matter than in previous calculations using the operator chain summation approximation for the large 3-body cluster. In pure neutron matter the operator chain summation approximation is found to be fairly accurate for the 3-body cluster; the present results are only slightly higher than the previous ones. We also report on the results for the Argonne $v14$ 2-nucleon interaction without any 3-body interaction. This case has been studied with Brueckner's method including 2, 3, and parts of 4-hole line terms by Day and Wiringa [Phys. Rev. C 32, 1057 (1985)]. Our results are significantly lower than theirs.

Fermi hypernetted-chain study of half-filled Landau levels with broken rotational symmetry

We investigate broken rotational symmetry (BRS) states at half-filling of the valence Landau level (LL). We generalize Rezayi and Read's (RR) trial wave function, a special case of Jain's composite fermion (CF) wave functions, to include anisotropic coupling of the flux quanta to electrons, thus generating a nematic order in the underlying CF liquid. Using the Fermi hypernetted-chain method, which readily gives results in the thermodynamic limit, we determine the properties of these states in detail. By using the anisotropic pair distribution and static structure functions we determine the correlation energy and find that, as expected, RR's state is stable in the lowest LL, whereas BRS states may occur at half-filling of higher LL's, with a possible connection to the recently discovered quantum Hall liquid crystals.

2D electron gas at arbitrary spin polarizations and coupling strengths: exchange-correlation energies, distribution functions, and spin-polarized phases.

We use the classical mapping of quantum fluids [Phys. Rev. Lett. 84, 959 (2000)] for a study of the uniform 2D electron gas. The "quantum temperature" T(q) of the classical 2D Coulomb fluid having the same correlation energy E(c) as the quantum fluid is determined as a function of the density parameter r(s). Using this T(q), spin-dependent pair-distribution functions are determined, and an analytic fit to the E(c) is given. The T(q) for 2D and 3D electron fluids is found to be approximately the same universal function of the classical coupling constant Gamma. The coupling Gamma increases more rapidly with density in 2D; hence, we present a scheme for including bridge terms in the 2D hypernetted-chain method. The Helmholtz free energies of the spin-polarized and unpolarized phases are calculated, and the existence of a ferromagnetic phase near r(s) approximately 30 is found to be a marginal possibility.

Application of Fermi-hypernetted-chain theory to composite-fermion quantum Hall states

The Fermi-hypernetted-chain ~FHNC! theory and the effective hypernetted-chain method are applied to study the composite-fermion ~CF! states of the fractional quantum Hall effect. Using this theory we compute, in the thermodynamic limit, the correlation energy, radial distribution function, and static structure factor for all unprojected CF wave functions. The unprojected excitation gaps for n51/3,1/5 were obtained by adopting in the FHNC a scheme previously used to compute nuclear matter excitation spectra. The results obtained so far are consistent with Monte Carlo simulations and small-number exact diagonalizations. @S0163-1829~97!04444-5#

Structure of the metal-liquid interface: self-consistent combination of the first-principles metal calculation and an integral equation method

This letter contributes to development of microscopic theories for the metal/liquid interface. We report results of the first attempt to combine the first-principles calculation for the metal surface and the reference hypernetted-chain method for the liquid in a fully self-consistent manner. The dipolar liquid/Pt(111) interface is considered. The electron density, liquid structure, and electrostatic potential across the interface are briefly discussed.

Ground-state properties of two-dimensional quantum fluid helium: Variational hypernetted chain methods

We calculate the ground-state energies and the radial distribution functions of two-dimensional quantum fluid4He, mass 3 boson3He and3He within a hypernetted chain and a Fermi hypernetted chain summation method. To get more accurate results, we include three-body correlation functions in the trial wavefunctions and consider the contributions arising from elementary diagrams via scaling approximations. The two-dimensional4He system has a binding energy of 0.84 K at an equilibrium density 0.042 A−2, which is closer to the result of a Green's function Monte Carlo calculation than previous ones, while mass 3 boson3He has 0.286 K at 0.025 A−2. The3He system is not self-bound and therefore not a liquid state due to its statistics and light mass of the3He atom. We calculate the ground-state energy of two-dimensional liquid4He through a Lennard-Jones potential and that of Aziz to compare the results of both. Aziz's potential gives lower and more accurate energies than the Lennard-Jones potential, as in three dimensions.

Local-field corrections of three-dimensional electron liquids

The static local-field corrections satisfying the compressibility, f-, and third moment sum rules exactly have been obtained for three-dimensional electron liquids. The authors results also satisfy the non-negativity condition of the pair correlation function at origin, because they reproduce the static structure factors or the pair correlations obtained by the Fermi hypernetted chain method. An appropriate dynamic local-field correction has been introduced to satisfy the third moment sum rule exactly. The authors check the moment sum rules numerically for q=1.5qF, for example.

Bose quantum fluids at finite temperatures: A variational density-matrix approach.

The variational density-matrix approach introduced by Campbell, K\urten, Ristig, and Senger [Phys. Rev. B 30, 3728 (1984)], to study finite-temperature Bose fluids is reformulated to avoid dealing directly with the entropy of the trial density matrix. We obtain exact expressions for the Euler-Lagrange equations for a finite-temperature trial statistical density matrix, which is the simplest finite-temperature generalization of a Jastrow trial ground-state wave function. A multicomponent hypernetted-chain method is developed to solve approximately these Euler-Lagrange equations at low temperatures. In leading order the results are shown to be equivalent to those obtained by Campbell et al. The current formalism provides a scheme to improve results of the previous approach.

Electrostatics of membrane adhesion

We consider electrical double layer interaction under the conditions typically encountered during membrane fusion. Within the physiological concentration range of monovalent electrolytes the interaction is repulsive and the Poisson-Boltzmann calculation may be used to evaluate the force. When divalent counterions are added, strong ion-ion correlations make the Poisson-Boltzmann approximation inapplicable. We use the anisotropic hypernetted chain method to show that in the presence of small amounts of divalent counterions in adsorption equilibrium with the surfaces, the double layer interaction turns into attraction. This attractive electrostatic force may be the balancing contribution controlling membrane adhesion.