Three-body Distribution Functions Research Articles

Measuring many-body distribution functions in fluids using test-particle insertion.

We derive a hierarchy of equations, which allow a general n-body distribution function to be measured by test-particle insertion of between 1 and n particles. We apply it to measure the pair and three-body distribution functions in a simple fluid using snapshots from Monte Carlo simulations in the grand canonical ensemble. The resulting distribution functions obtained from insertion methods are compared with the conventional distance-histogram method: the insertion approach is shown to overcome the drawbacks of the histogram method, offering enhanced structural resolution and a more straightforward normalization. At high particle densities, the insertion method starts breaking down, which can be delayed by utilizing the underlying hierarchical structure of the insertion method. Our method will be especially useful in characterizing the structure of inhomogeneous fluids and investigating closure approximations in liquid state theory.

Nuclear matter calculations with the phenomenological three-nucleon interaction

Employing the concept of three-body radial distribution function and using the two-body correlation functions, calculated based on the lowest order constrained variational method, we investigated the effect of the three-body force (TBF) on the nuclear matter properties, for Argonne and Urbana v14 potentials. As such, the results for nuclear matter density, incompressibility, energy per nucleon, and symmetry energy are presented at the saturation point. The inclusion of a phenomenological TBF resulted in closer values of the saturation density, incompressibility, and symmetry energy to the empirical ones for the symmetric nuclear matter. This is especially the case for the Urbana v14 potential. In addition, an empirically-verified parabolic approximation of the interaction energy was utilized to perform an approximate study of the nuclear matter with neutron excess. Hence, at densities higher than about 0.3 fm−3 and for proton-to-neutron density ratios close to the symmetric nuclear matter, the inclusion of TBF resulted in an extra attraction for the Argonne as compared to the Urbana v14 potential.

Descriptors representing two- and three-body atomic distributions and their effects on the accuracy of machine-learned inter-atomic potentials.

When determining machine-learning models for inter-atomic potentials, the potential energy surface is often described as a non-linear function of descriptors representing two- and three-body atomic distribution functions. It is not obvious how the choice of the descriptors affects the efficiency of the training and the accuracy of the final machine-learned model. In this work, we formulate an efficient method to calculate descriptors that can separately represent two- and three-body atomic distribution functions, and we examine the effects of including only two- or three-body descriptors, as well as including both, in the regression model. Our study indicates that non-linear mixing of two- and three-body descriptors is essential for an efficient training and a high accuracy of the final machine-learned model. The efficiency can be further improved by weighting the two-body descriptors more strongly. We furthermore examine a sparsification of the three-body descriptors. The three-body descriptors usually provide redundant representations of the atomistic structure, and the number of descriptors can be significantly reduced without loss of accuracy by applying an automatic sparsification using a principal component analysis. Visualization of the reduced descriptors using three-body distribution functions in real-space indicates that the sparsification automatically removes the components that are less significant for describing the distribution function.

A mixed radial, angular, three-body distribution function as a tool for local structure characterization: Application to single-component structures.

A mixed radial, angular three-body distribution function g3(rBC, θABC) is introduced, which allows the local atomic order to be more easily characterized in a single graph than with conventional correlation functions. It can be defined to be proportional to the probability of finding an atom C at a distance rBC from atom B while making an angle θABC with atoms A and B, under the condition that atom A is the nearest neighbor of B. As such, our correlation function contains, for example, the likelihood of angles formed between the nearest and the next-nearest-neighbor bonds. To demonstrate its use and usefulness, a visual library for many one-component crystals is produced first and then employed to characterize the local order in a diverse body of elemental condensed-matter systems. Case studies include the analysis of a grain boundary, several liquids (argon, copper, and antimony), and polyamorphism in crystalline and amorphous silicon including that obtained in a tribological interface.

Kernel-Based Machine Learning for Efficient Simulations of Molecular Liquids.

Current machine learning (ML) models aimed at learning force fields are plagued by their high computational cost at every integration time step. We describe a number of practical and computationally efficient strategies to parametrize traditional force fields for molecular liquids from ML: the particle decomposition ansatz to two- and three-body force fields, the use of kernel-based ML models that incorporate physical symmetries, the incorporation of switching functions close to the cutoff, and the use of covariant meshing to boost the training set size. Results are presented for model molecular liquids: pairwise Lennard-Jones, three-body Stillinger–Weber, and bottom-up coarse-graining of water. Here, covariant meshing proves to be an efficient strategy to learn canonically averaged instantaneous forces. We show that molecular dynamics simulations with tabulated two- and three-body ML potentials are computationally efficient and recover two- and three-body distribution functions. Many-body representations, decomposition, and kernel regression schemes are all implemented in the open-source software package VOTCA.

Solvation properties and behaviour of lutetium(III) in aqueous solution—A quantum mechanical charge field (QMCF) study

This work presents the first ab initio molecular dynamics study of trivalent lutetium in aqueous solution. The hybrid quantum and molecular mechanics simulation has been carried out on Hartree-Fock level and the results were compared to extended X-ray absorption fine structure and X-ray diffraction data. In addition to the structural characterisation via radial and angular distribution functions, the influence of the ion on the surrounding solvent was further investigated by local-density-corrected three-body distribution functions and frequency calculations. The obtained results for the mean Lu-O bond distance and force constant were in very good agreement with the literature. Furthermore, deeper insight into the dynamics and geometry of the solvation shell and the number of involved solvent molecules was obtained.

Structure and dynamics of the Th4+-ion in aqueous solution – An ab initio QMCF-MD study

An ab initio quantum mechanical charge-field molecular dynamics simulation of the tetravalent thorium ion in aqueous environment is presented. Including the first and second hydration shell in the quantum mechanical treatment to enhance accuracy, this study yields very good results for a wide range of characteristics, as compared to experimental data. 20ps of simulation time were used to investigate structural and dynamical properties of the hydrate by numerous methods, including radial and angular distribution functions, three-body distribution functions, ligand exchange analysis and vibrational spectra. The hydrate proved stable during the simulation and did not show hydrolysis. The first solvation shell was found to be a very flexible one, with a number of intrashell ligand rearrangements occurring along the simulation. The data of the simulation also indicated the existence of a third hydration layer. Vibrational analysis yielded an average ion–oxygen frequency of 420cm−1, which is in excellent agreement with experimental data.

Structural and Dynamical Aspects of the Unsymmetric Hydration of Sb(III): An ab initio Quantum Mechanical Charge Field Molecular Dynamics Simulation

An ab initio quantum mechanical charge field molecular dynamics (QMCF MD) simulation was performed to investigate the behavior of the Sb(3+) ion in aqueous solution. The simulation reveals a significant influence of the residual valence shell electron density on the solvation structure and dynamics of Sb(3+). A strong hemidirectional behavior of the ligand binding pattern is observed for the first hydration shell extending up to the second hydration layer. The apparent domain partitioned structural behavior was probed by solvent reorientational kinetics and three-body distribution functions. The three-dimensional hydration space was conveniently segmented such that domains having different properties were properly resolved. The approach afforded a fair isolation of localized solvent structural and dynamical motifs that Sb(3+) seems to induce to a remarkable degree. Most intriguing is the apparent impact of the lone pair electrons on the second hydration shell, which offers insight into the mechanistic aspects of hydrogen bonding networks in water. Such electronic effects observed in the hydration of Sb(3+) can only be studied by applying a suitable quantum mechanical treatment including first and second hydration shell as provided by the QMCF ansatz.

Local density corrected three-body distribution functions for probing local structure reorganization in liquids

Three-body distribution functions are calculated for metal ions in an aqueous medium in order to investigate and characterise solvent structure reorganization. Based on the existing formulation of three body correlation function, a local density correction is introduced to enable a comparison of different sub-regions within a solvate as well as different systems, thus taking into account the varying density arising from the influence of the solute.

Numerical Analysis of Triangle-Lattice Alignments Formed by Magnetic Dipole Interactions among Feeble Magnetic Particles under High Magnetic Fields

We performed a two-dimensional numerical simulation of triangle-lattice alignments formed by feeble magnetic particles with a diameter of 10 µm to 1 mm, where Brownian motion of particles is negligible. To quantitatively evaluate the distance between particles in triangle-lattice structures in which some spacing resulted from the interactions among induced magnetic dipoles, we used a three-body distribution function. It is confirmed that the feeble magnetic particles form a more orderly triangle-lattice structure as the number of particles increases and also particles become smaller, and that the distance between the particles in the triangle-lattice structures is more homogeneous as particles become smaller.

Evidence for symmetric cationic sites in zirconium-bearing oxide glasses

The local environment of Zr in a borosilicate glass has been investigated by combining molecular dynamics (MD) simulations with Zr $K$-edge x-ray absorption spectroscopy (XAS) measurements. Two- and three-body distribution functions in agreement with the experimental data were obtained. The oxygen first neighbors around Zr are found to be narrowly distributed, revealing almost perfectly regular $\mathrm{Zr}{\mathrm{O}}_{6}$ octahedra. Significant three-body contributions (Zr-O-O) to the XAS signal are observed, probing the existence of highly symmetric cationic sites in a glassy network. The ${g}_{\mathrm{Zr}\mathrm{O}}(r)$ first peak derived from the experimental data is narrower than the one given by the MD model, thus providing elements for further improvements of the MD potentials.

Experiment and numerical simulation of interactions among magnetic dipoles induced in feeble magnetic substances under high magnetic fields

Using a two-dimensional closed vessel filled with an aqueous solution, we present triangle-lattice alignments with some spacing formed by interactions among magnetic dipoles induced in feeble magnetic substances under high magnetic fields. We conducted an experiment and a numerical simulation using Au particles with a 1 mm diameter dispersed in a MnCl 2 aqueous solution and compared their configurations quantitatively using a three-body distribution function. We confirmed that the numerical simulation demonstrates the formation of triangle-lattice alignments in a two-dimensional plane as obtained in the experiment and also verified that the interaction among induced magnetic dipoles is a significant force that governs the structure formed by feeble magnetic substances under high magnetic fields. On the basis of the results obtained in the experiment and the numerical simulation, it is clear that the distance among particles and the interaction force depend on the number of particle at a steady state and they are closely correlative with each other. The distance among particles can be estimated from the correlative relation.

Triplet correlations in two-dimensional colloidal model liquids

Three-body distribution functions in classical fluids have been theoretically investigatedmany times, but have never been measured directly. Here we present experimentalthree-point correlation functions that are computed from particle configurations measuredby means of video microscopy in two types of quasi-two-dimensional colloidal model fluids:a system of charged colloidal particles and a system of paramagnetic colloids. In the firstsystem the particles interact via a Yukawa potential, in the second via a potentialΓ/r3. Varying the particles’ density in the charged system or the interaction strengthΓ in the magnetic system, one can systematically explore how triplet correlations behave ifthe coupling between the particles changes. We find for both systems very similar results:on increasing the coupling between the particles one observes the gradual formation of acrystal-like local order due to triplet correlations, even though the system is still deepinside the fluid phase. These are mainly packing effects, as is evident from the closeresemblance between the results for the two systems having completely differentpair-interaction potentials. To demonstrate that triplet correlations are significant not onlylocally but also when integrated over the whole volume, we consider the Born–Greenequation and show that in a strongly interacting system this equation can be satisfied onlywith the full triplet correlation function but not with three-body distribution functionsobtained from superposing pair correlations (Kirkwood superposition approximation).

Three-particle correlations in simple liquids.

We use videomicroscopy to follow the phase-space trajectory of a two-dimensional colloidal model liquid and calculate three-point correlation functions from the measured particle configurations. Approaching the fluid-solid transition by increasing the strength of the pair-interaction potential, one observes the gradual formation of a crystal-like local order due to triplet correlations, while being still deep inside the fluid phase. Furthermore, we show that in a strongly interacting system the Born-Green equation can be satisfied only with the full triplet correlation function but not with three-body distribution functions obtained from superposing pair correlations (Kirkwood superposition approximation).

Hydrogen and higher shell contributions in Zn2+, Ni2+, and Co2+ aqueous solutions: an X-ray absorption fine structure and molecular dynamics study.

A detailed investigation of the hydration structure of Zn2+, Ni2+, and Co2+ in water solutions has been carried out combining X-ray absorption fine structure (EXAFS) spectroscopy and Molecular Dynamics (MD) simulations. The first quantitative analysis of EXAFS from hydrogen atoms in 3d transition metal ions in aqueous solutions has been carried out and the ion-hydrogen interactions have been found to provide a detectable contribution to the EXAFS spectra. An accurate determination of the structural parameters associated with the first hydration shell has been performed and compared with previous experimental results. No evidence of significant contributions from the second hydration shell to the EXAFS signal has been found for these solutions, while the inclusion of the hydrogen signal has been found to be important in performing a quantitative analysis of the experimental data. The high-frequency contribution present in the EXAFS spectra has been found to be due to multiple scattering (MS) effects inside the ion-oxygen first coordination shell. MD has been used to generate three-body distribution functions from which a reliable analysis of the MS contributions to the EXAFS spectra of these systems has been carried out.

Fermion hypernetted chain equations for the three-body distribution

The fermion hypernetted chain equations for the three-body distribution function are derived in this paper. These equations are relatively complicated and some care is required for their derivation. The formulas are presented explicitly and an outline for their practical implementation discussed.

Direct determination of two-body potentials from measured pair structures

Abstract We developed and tested a numerical scheme for solving a systematically discretized version of the Born-Green-Yvon equation. We collected histograms for the two- and three-body distribution functions from Monte Carlo simulations of the Lennard-Jones liquid. The discretized pair-potential were obtained by solving a set of linear equations where the number of unknowns was given by the number of bins of the pair-correlation function. Using this scheme we were able to recover the Lennard-Jones pair-potential of the simulations with good accuracy. We found that the technique is not very sensitive for the statistics of the correlation functions but requires small bin size. We applied the method for three-particle correlation functions calculated from Reverse Monte Carlo (RMC) simulations to test the performance of different possible RMC algorithms.

The 12C(p, p′3α) Breakup Reaction Induced by 14, 18 and 26 MeV Protons

Double-differential cross sections (DDXs) of emitted protons and αs particles were measured for proton-induced reactions on 12C at 14, 18 and 26MeV in order to investigate the 12C(p, p′3α) breakup reaction. The experimental DDXs were analyzed on the assumption that the (p, p′3α) reaction proceeds through two major processes, a three-body simultaneous breakup (3BSB) process (i.e., p+12C→p+α+8Be*2094Mev) and sequential decay processes from unstable nuclei produced by (p, p′) and (p, α) reactions (i.e., 12C, 9B, 8Be and 5Li). Partial cross sections for the various reaction channels were extracted from the measured DDXs by a least squares method with two fitting functions: a three-body phase distribution function for the 3BSB process and a Breit-Wigner function for the transition to a discrete level in the first step of the sequential decay process. The result showed that the 12C(p, p′3α) reaction took place predominantly via nearly isotropic particle emission in the 3BSB process, and emission of protons and α particles with low energies was explained well by the sequential decay via 12C* and 9Bg.s Experimental (p, p) and (p, p′) scattering cross sections at 26 MeV were reproduced well by the coupled-channels calculation with the soft-rotator model. The experimental DDXs were compared with a Monte Carlo calculation based on the SCINFUL/DDX code and the latest intermediate energy nuclear data evaluation library LA150.

Three-body distribution functions in hard sphere fluids. Comparison of excluded-volume-anisotropy model predictions and Monte Carlo simulation

Hard sphere three-body distribution functions predicted by the recently developed Excluded-Volume-Anisotropy (EVA) model are compared with Monte Carlo computer simulation measurements. Two types of simulations, both based on the Widom insertion method, are performed as a function of solvent density (0.1⩽ρσ3⩽0.8), solute structure (linear, triangular, and bent 3-bead chain), and solute–solvent sphere diameter ratio (0⩽σ/σS⩽3). Comparisons of these results with those of previous studies illustrate the accuracy of the EVA model in predicting multi-body distribution functions near contact separations (and inside of contact), where the Kirkwood-Superposition-Approximation is least accurate.

Many-body components in the integrated far-infrared absorption coefficient of diatomic molecules in spherical solvents

In this paper we present a quantitative study of the static cancellation effects between two- and three-body components of the total integrated absorption coefficient in the far-infrared spectra of several systems (diluted solutions of diatomic molecules in spherical solvents) with important electric multipolar induced contributions to absorption. This static cancellation decreases with an increasing order of the multipolar induced mechanism. Even more, for hexadecapole-induced dipole contributions, cancellation transforms into enhancement for all the systems considered (CO–Ar, N2–Ar, and N2–Xe at different densities and temperatures). These results are obtained first by computer simulations and second by means of the knowledge of the static structure of the fluid, that is two- and three-body static distribution functions. From both procedures results are similar.