Two-body Potential Research Articles

Multiconfigurational time-dependent Hartree method for fermions: Implementation, exactness, and few-fermion tunneling to open space

We report on an implementation of the multiconfigurational time-dependent Hartree method (MCTDH) for spin-polarized fermions (MCTDHF). Our approach is based on a mapping for opera- tors in Fock space that allows a compact and efficient application of the Hamiltonian and solution of the MCTDHF equations of motion. Our implementation extends, builds on and exploits the recursive implementation of MCTDH for bosons (R-MCTDHB) package. Together with R-MCTDHB, the present implementation of MCTDHF forms the MCTDH-X package. We benchmark the accuracy of the algorithm with the harmonic interaction model and a time-dependent generalization thereof. These models consider parabolically trapped particles that interact through a harmonic interaction potential. We demonstrate, that MCTDHF is capable of solving the time-dependent many-fermion Schr\"odinger equation to an in principle arbitrary degree of precision and can hence yield numerically exact results even in the case of Hamiltonians with time-dependent one-body and two-body potentials. As an application we study the problem of two initially parabolically confined and charged fermions tunneling through a barrier to open space. We demonstrate the validity of a model proposed previously for the many-body tunneling to open space of bosonic particles with contact interactions [Proc. Natl. Acad. Sci. USA 109, 13521-13525 (2012)]. The many-fermion tunneling can be built up from sequentially happening single-fermion tunneling processes. The characteristic momenta of these processes are determined by the chemical potentials of trapped subsystems of smaller particle numbers: the escaped fermions convert the different chemical potentials into kinetic energy. Using the two-body correlation function, we present a detailed picture of the sequentiality of the process and are able to tell tunneling from over-the-barrier escape.

Variational calculations with explicit energy functionals for fermion systems at zero temperature

The variational method with explicit energy functionals (EEFs) is applied to infinite neutron matter. Starting from the Argonne v8’ two-body potential and the repulsive part of the Urbana IX three-body potential, the energy per neutron is expressed explicitly with the spin-dependent central, tensor, and spin-orbit distribution functions. This EEF is constructed as an extension of the previously proposed one for the Argonne v6’ potential, including central and tensor forces. The Euler-Lagrange equations derived from this EEF are solved numerically. The obtained fully minimized energy with this EEF is in good agreement with that obtained from the auxiliary field diffusion Monte Carlo calculation.

Dynamics of sound waves in an interacting Bose gas

We consider a non-relativistic quantum gas of N bosonic atoms confined to a box of volume Λ in physical space. The atoms interact with each other through a pair potential whose strength is inversely proportional to the density, ρ=NΛ, of the gas. We study the time evolution of coherent excitations above the ground state of the gas in a regime of large volume Λ and small ratio Λρ. The initial state of the gas is assumed to be close to a product state of one-particle wave functions that are approximately constant throughout the box. The initial one-particle wave function of an excitation is assumed to have a compact support independent of Λ. We derive an effective non-linear equation for the time evolution of the one-particle wave function of an excitation and establish an explicit error bound tracking the accuracy of the effective non-linear dynamics in terms of the ratio Λρ. We conclude with a discussion of the dispersion law of low-energy excitations, recovering Bogolyubov's well-known formula for the speed of sound in the gas, and a dynamical instability for attractive two-body potentials.

Efimov Physics with $${1/2}$$ 1 / 2 Spin-Isospin Fermions

The structure of few-fermion systems having $1/2$ spin-isospin symmetry is studied using potential models. The strength and range of the two-body potentials are fixed to describe low energy observables in the angular momentum $L=0$ state and spin $S=0,1$ channels of the two-body system. Successively the strength of the potentials are varied in order to explore energy regions in which the two-body scattering lengths are close to the unitary limit. This study is motivated by the fact that in the nuclear system the singlet and triplet scattering lengths are both large with respect to the range of the interaction. Accordingly we expect evidence of universal behavior in the three- and four-nucleon systems that can be observed from the study of correlations between observables. In particular we concentrate in the behavior of the first excited state of the three-nucleon system as the system moves away from the unitary limit. We also analyze the dependence on the range of the three-body force of some low-energy observables in the three- and four-nucleon systems.

Knowledge-based three-body potential for transcription factor binding site prediction.

A structure-based statistical potential is developed for transcription factor binding site (TFBS) prediction. Besides the direct contact between amino acids from TFs and DNA bases, the authors also considered the influence of the neighbouring base. This three-body potential showed better discriminate powers than the two-body potential. They validate the performance of the potential in TFBS identification, binding energy prediction and binding mutation prediction.

A New Treatment Below the Three-Body Break up Threshold in the NNπSystem

The two-body threshold behavior at NN′ and π D are investigated by using the multi-channel Lippmann-Schwinger equations with an energy dependent two-body quasi potential, which are analytically continued from the three-body Faddeev equations at the three-body break up threshold. Our calculated NN′ and π D scattering lengths show better agreement with the experimental data for NN and π D systems than those from the original NNπ three-body Faddeev equations.

Structural change of Na<sub>2</sub>O-doped SiO<sub>2</sub> glasses by melting

The structural change of Na2O·5SiO2 and Na2O·3SiO2 glasses by melting were analyzed. From their droplet specimens that were levitated by N2 gas flow and melted by CO2 laser irradiation, the X-ray diffraction spectra were successfully obtained, using 61.41 keV high energy X-ray from synchrotron radiation. The obtained spectra were corrected, normalized, and then Fourier-transformed to radial distribution functions (RDFs). It was found that the Si–O, Na–O, O–O, and Si–Si peaks in the RDFs got broadened and shifted to longer distance compared with the solid glasses, while no significant difference was found between the RDFs of the melts at 900 and 1200°C. In the classical molecular dynamics simulations using Born–Mayer–Huggins type two-body potential with optimized parameters, the trends of RDF change by melting described above were well reproduced. The simulated structural models indicated that the distortion of SiO4 tetrahedra, increase of Na–O distance, and cleavage of Si–O and Na–O bonds are occurring in the melts.

The External Field Dependence of the BCS Critical Temperature

We consider the Bardeen-Cooper-Schrieffer free energy functional for particles interacting via a two-body potential on a microscopic scale and in the presence of weak external fields varying on a macroscopic scale. We study the influence of the external fields on the critical temperature. We show that in the limit where the ratio between the microscopic and macroscopic scale tends to zero, the next to leading order of the critical temperature is determined by the lowest eigenvalue of the linearization of the Ginzburg-Landau equation.

Systematic investigations of deep sub-barrier fusion reactions using an adiabatic approach

To describe fusion hindrance observed in fusion reactions at extremely low incident energies, I propose a novel extension of the standard CC model by introducing a damping factor that describes a smooth transition from sudden to adiabatic processes. I demonstrate the performance of this model by systematically investigating various deep sub-barrier fusion reactions. I extend the standard CC model by introducing a damping factor into the coupling matrix elements in the standard CC model. I adopt the Yukawa-plus-exponential (YPE) model as a basic heavy ion-ion potential, which is advantageous for a unified description of the one- and two-body potentials. For the purpose of these systematic investigations, I approximate the one-body potential with a third-order polynomial function based on the YPE model. Calculated fusion cross sections for the medium-heavy mass systems of $^{64}$Ni + $^{64}$Ni, $^{58}$Ni + $^{58}$Ni, and $^{58}$Ni + $^{54}$Fe, the medium-light mass systems of $^{40}$Ca + $^{40}$Ca, $^{48}$Ca + $^{48}$Ca, and $^{24}$Mg + $^{30}$Si, and the mass-asymmetric systems of $^{48}$Ca + $^{96}$Zr and $^{16}$O + $^{208}$Pb are consistent with the experimental data. The astrophysical S factor and logarithmic derivative representations of these are also in good agreement with the experimental data. Since the results calculated with the damping factor are in excellent agreement with the experimental data in all systems, I conclude that the smooth transition from the sudden to adiabatic processes occurs and that a coordinate-dependent coupling strength is responsible for the fusion hindrance. In all systems, the potential energies at the touching point $V_{\rm Touch}$ strongly correlate with the incident threshold energies for which the fusion hindrance starts to emerge, except for the medium-light mass systems.

Theoretical prediction of the structural, elastic, mechanical and phonon properties of bismuth telluride under pressure

Bismuth telluride (Bi2Te3) is one of the most intricate materials with its semiconducting, insulating and pressure-induced superconducting properties. Although different theoretical works have been carried out to understand the confusing properties of Bi2Te3, information about the high pressure structural, elastic, mechanical and phonon properties of this significant material is still rare. Unlike earlier theoretical approaches, two-body interatomic potentials in the Morse potential form have been employed for the first time to predict the density, phase transition pressure, elastic constants, bulk, shear and Young moduli and elastic wave velocities of Bi2Te3under pressures up to 12 GPa. [Formula: see text] phase transition pressure of Bi2Te3was found to be 10 GPa. The results of above elastic quantities agree well with experiments and are better than some of the published theoretical data. In addition, the effect of pressure on the phonon dispersion and density of states (DOS) were also evaluated with the same potential and their results are satisfactory, especially for the low-frequency acoustic portions of phonons.

Hyperspherical three-body model calculation for the bound 3,1S-states of Coulombic systems

In this paper, hyperspherical three-body model formalism has been applied for the calculation of energies of the low-lying bound [Formula: see text]-states of neutral helium and helium like Coulombic three-body systems having nuclear charge (z) in the range [Formula: see text]. Energies of [Formula: see text]-states are also calculated for those having nuclear charge in the range [Formula: see text]. The calculation of the coupling potential matrix elements of the two-body potentials has been simplified by the use of Raynal–Revai Coefficients (RRC). The three-body wave function in the Schrödinger equation when expanded in terms of hyperspherical harmonics (HH), leads to an infinite set of coupled differential equation (CDE) which for practical purposes is truncated to a finite set and the truncated set of CDE’s are solved by renormalized Numerov method (RNM) to get the energy (E). The calculated energy is compared with the ones of the literature.

Conformation of a flexible polymer in explicit solvent: Accurate solvation potentials for Lennard-Jones chains.

The conformation of a polymer chain in solution is coupled to the local structure of the surrounding solvent and can undergo large changes in response to variations in solvent density and temperature. The many-body effects of solvent on the structure of an n-mer polymer chain can be formally mapped to an exact n-body solvation potential. Here, we use a pair decomposition of this n-body potential to construct a set of two-body potentials for a Lennard-Jones (LJ) polymer chain in explicit LJ solvent. The solvation potentials are built from numerically exact results for 5-mer chains in solvent combined with an approximate asymptotic expression for the solvation potential between sites that are distant along the chain backbone. These potentials map the many-body chain-in-solvent problem to a few-body single-chain problem and can be used to study a chain of arbitrary length, thereby dramatically reducing the computational complexity of the polymer chain-in-solvent problem. We have constructed solvation potentials at a large number of state points across the LJ solvent phase diagram including the vapor, liquid, and super-critical regions. We use these solvation potentials in single-chain Monte Carlo (MC) simulations with n ≤ 800 to determine the size, intramolecular structure, and scaling behavior of chains in solvent. To assess our results, we have carried out full chain-in-solvent MC simulations (with n ≤ 100) and find that our solvation potential approach is quantitatively accurate for a wide range of solvent conditions for these chain lengths.

Dipole response inPb208within a self-consistent multiphonon approach

Background: The electric dipole strength detected around the particle threshold and commonly associated to the pygmy dipole resonance offers a unique information on neutron skin and symmetry energy, and is of astrophysical interest. The nature of such a resonance is still under debate. Purpose: We intend to describe the giant and pygmy resonances in 208 Pb by enhancing their fragmentation with respect to the random-phase approximation. Method: We adopt the equation of motion phonon method to perform a fully self-consistent calculation in a space spanned by one-phonon and two-phonon basis states using an optimized chiral two-body potential. A phenomenological density dependent term, derived from a contact three-body force, is added in order to get single-particle spectra more realistic than the ones obtained by using the chiral potential only. The calculation takes into full account the Pauli principle and is free of spurious center of mass admixtures. Results: We obtain a fair description of the giant resonance and obtain a dense low-lying spectrum in qualitative agreement with the experimental data. The transition densities as well as the phonon and particle-hole composi- tion of the most strongly excited states support the pygmy nature of the low-lying resonance. Finally, we obtain realistic values for the dipole polarizability and the neutron skin radius. Conclusions: The results emphasize the role of the two-phonon states in enhancing the fragmentation of the strength in the giant resonance region and at low energy, consistently with experiments. For a more detailed agreement with the data, the calculation suggests the inclusion of the three-phonon states as well as a fine tuning of the single-particle spectrum to be obtained by a refinement of the nuclear potential.

Many interacting fermions in a one-dimensional harmonic trap: a quantum-chemical treatment

We employ ab initio methods of quantum chemistry to investigate spin-1/2 fermions interacting via a two-body contact potential in a one-dimensional harmonic trap. The convergence of the total energy with the size of the one-particle basis set is analytically investigated for the two-body problem and the same form of the convergence formula is numerically confirmed to be valid for the many-body case. Benchmark calculations for two to six fermions with the full configuration interaction method equivalent to the exact diagonalization approach, and the coupled cluster (CC) method including single, double, triple, and quadruple excitations are presented. The convergence of the correlation energy with the level of excitations included in the CC model is analyzed. The range of the interaction strength for which single-reference CC methods work is examined. Next, the CC method restricted to single, double, and noniterative triple excitations, CCSD(T), is employed to study a two-component Fermi gas composed of 6–80 atoms in a one-dimensional harmonic trap. The density profiles of trapped atomic clouds are also reported. Finally, a comparison with experimental results for few-fermion systems is presented. Upcoming possible applications and extensions of the presented approach are discussed.

DFTB Parameters for the Periodic Table, Part 2: Energies and Energy Gradients from Hydrogen to Calcium.

In the first part of this series, we presented a parametrization strategy to obtain high-quality electronic band structures on the basis of density-functional-based tight-binding (DFTB) calculations and published a parameter set called QUASINANO2013.1. Here, we extend our parametrization effort to include the remaining terms that are needed to compute the total energy and its gradient, commonly referred to as repulsive potential. Instead of parametrizing these terms as a two-body potential, we calculate them explicitly from the DFTB analogues of the Kohn-Sham total energy expression. This strategy requires only two further numerical parameters per element. Thus, the atomic configuration and four real numbers per element are sufficient to define the DFTB model at this level of parametrization. The QUASINANO2015 parameter set allows the calculation of energy, structure, and electronic structure of all systems composed of elements ranging from H to Ca. Extensive benchmarks show that the overall accuracy of QUASINANO2015 is comparable to that of well-established methods, including PM7 and hand-tuned DFTB parameter sets, while coverage of a much larger range of chemical systems is available.

Mass Spectrum of Baryons in a Relativistic Hypercentral Constituent Quark Model

We have investigated the light and strange baryons as a relativistic three-body bound system. Inspired by lattice QCD calculations, we treated the baryons as a spin-independent three-quark system. For this purpose we introduced a spin independent relativistic description for the SU (6) invariant part of the spectrum by presenting the analytical solution of the three-particle Klein-Gordon equation. We carried out calculations within the constituent quark model based on a hypercentral approach, which takes into account three-body force effects and standard two-body potential contributions. Herewith the average energy values of the nonstrange and strange resonances are reproduced. Then we used the generalized Gursey-Radicati mass formula as a SU (6) breaking interaction to describe the splittings within the SU (6)-multiplets and the hyperfine structure of the baryon. The considered SU (6)-invariant potential is the well-known ＂Coulomb-plus-linear＂ potential. The resulting description of the baryon spectrum is fairly good and comparable with the experimental spectrum.

Force and heat current formulas for many-body potentials in molecular dynamics simulations with applications to thermal conductivity calculations

We derive expressions of interatomic force and heat current for many-body potentials such as the Tersoff, the Brenner, and the Stillinger-Weber potential used extensively in molecular dynamics simulations of covalently bonded materials. Although these potentials have a many-body nature, a pairwise force expression that follows Newton's third law can be found without referring to any partition of the potential. Based on this force formula, a stress applicable for periodic systems can be unambiguously defined. The force formula can then be used to derive the heat current formulas using a natural potential partitioning. Our heat current formulation is found to be equivalent to most of the seemingly different heat current formulas used in the literature, but to deviate from the stress-based formula derived from two-body potential. We validate our formulation numerically on various systems descried by the Tersoff potential, namely three-dimensional silicon and diamond, two-dimensional graphene, and quasi-one-dimensional carbon nanotube. The effects of cell size and time used in the simulation are examined.

Exact Diagonalization Study of Bose-Condensed Gas with Finite-Range Gaussian Interaction

We investigate a system of $N$ spinless bosons confined in quasi-two-dimensional harmonic trap with repulsive two-body finite-range Gaussian interaction potential of large $s$-wave scattering length. Exact diagonalization of the Hamiltonian matrix is carried out to obtain the $N$-body ground state as well as low-lying excited states, using Davidson algorithm in beyond lowest-Landau-level approximation. We examine the finite-range effects of the interaction potential on the many-body ground state energy as also the degree of condensation of the Bose-condensed gas. The results obtained indicate that the finite-range Gaussian interaction potential enhances the degree of condensation compared to the zero-range interaction potential. We further analyze the effect of finite-range interaction potential on the breathing mode collective excitation. Our theoretical results may be relevant for experiments currently conducted on quasi-two-dimensional Bose gas with more realistic interaction potential.

Numerical study of long-time dynamics and ergodic-nonergodic transitions in dense simple fluids.

Since the mid-1980s, mode-coupling theory (MCT) has been the de facto theoretic description of dense fluids and the transition from the fluid state to the glassy state. MCT, however, is limited by the approximations used in its construction and lacks an unambiguous mechanism to institute corrections. We use recent results from a new theoretical framework--developed from first principles via a self-consistent perturbation expansion in terms of an effective two-body potential--to numerically explore the kinetics of systems of classical particles, specifically hard spheres governed by Smoluchowski dynamics. We present here a full solution for such a system to the kinetic equation governing the density-density time correlation function and show that the function exhibits the characteristic two-step decay of supercooled fluids and an ergodic-nonergodic transition to a dynamically arrested state. Unlike many previous numerical studies--and in stark contrast to experiment--we have access to the full time and wave-number range of the correlation function with great precision and are able to track the solution unprecedentedly close to the transition, covering nearly 15 decades in scaled time. Using asymptotic approximation techniques analogous to those developed for MCT, we fit the solution to predicted forms and extract critical parameters. We find complete qualitative agreement with known glassy behavior (e.g. power-law divergence of the α-relaxation time scale in the ergodic phase and square-root growth of the glass form factors in the nonergodic phase), as well as some limited quantitative agreement [e.g. the transition at packing fraction η*=0.60149761(10)], consistent with previous static solutions under this theory and with comparable colloidal suspension experiments. However, most importantly, we establish that this new theory is able to reproduce the salient features seen in other theories, experiments, and simulations but has the advantages of being derived from first principles and possessing a clear mechanism for making systematic corrections.

Effective potential for many-body interactions in some properties of the HFD-like solids

We have determined the room-temperature equation of state for the HFD-like solids (such as Ar, Kr, Xe, CH4, CO2, N2, and O2) using the two-body HFD-like potentials from molecular dynamics simulation at high pressures. A simple and accurate empirical many-body expression has also been used with the two-body potential for the HFD-like solids without requiring an expensive three-body calculation. The configurational energy and the exact equation of state using the new model have been obtained in good agreement with the experiment. Our predicted results of the elastic constants also indicated that our new effective (many-body corrected) potential improves the two-body values of solids Ar, Kr, and CO2 to get better agreement with the experiment.