Long-range Part Research Articles

Long-range correlations of polarization and number densities in dilute electrolytes

In dilute electrolytes, we calculate the pair correlation functions among the solvent polarization p, the solvent density n1, the cation density n2, and the anion density n3. We set up a simple Ginzburg-Landau free energy for these variables, so our results are valid at distances longer than the molecular size σ. In particular, we reproduce the Høye-Stell result for the polarization correlation ⟨pα(r)pβ(0)⟩ (α, β = x, y, z) [J. S. Høye and G. Stell, J. Chem. Phys. 68, 4145 (1978)], which is proportional to the second derivative ∂2(e-κr/r)/∂xα∂xβ for r ≫ σ with κ being the Debye wave number. We also show that size asymmetry between the cations and the anions gives rise to similar long-range correlations in ⟨pα(r)δn1(0)⟩ and ⟨δni(r)δn1(0)⟩ (i = 1, 2, 3). Moreover, we calculate the polarization time-correlation function. As a unique feature in dynamics, the longitudinal polarization fluctuations (∝∇ · p) consist of rapidly decaying and slowly decaying components, where the latter relax with the charge density ρ. As a result, the long-range part of the equal-time polarization correlation changes into a different long-ranged and long-lived form after the shorter polarization relaxation.

Bound states of an ultracold atom interacting with a set of stationary impurities

In this manuscript we analyse properties of bound states of an atom interacting with a set of static impurities. We begin with the simplest system of a single atom interacting with two static impurities. We consider two types of atom-impurity interaction: (i) zero-range potential represented by regularized delta, (ii) more realistic polarization potential, representing long-range part of the atom-ion interaction. For the former we obtain analytical results for energies of bound states. For the latter we perform numerical calculations based on the application of finite element method. Then, we move to the case of a single atom interacting with one-dimensional (1D) infinite chain of static ions. Such a setup resembles Kronig-Penney model of a 1D crystalline solid, where energy spectrum exhibits band structure behaviour. For this system, we derive analytical results for the band structure of bound states assuming regularized delta interaction, and perform numerical calculations, considering polarization potential to model atom-impurity interaction. Both approaches agree quite well when separation between impurities is much larger than characteristic range of the interaction potential.

Asymptotic behavior of correlation functions of one-dimensional polar-molecules on optical lattices

We combine a slave-spin approach with a mean-field theory to develop an approximate theoretical scheme to study the density, spin, and, pairing correlation functions of fermionic polar molecules. We model the polar molecules subjected to a one-dimensional periodic optical lattice potential using a generalized t–J model, where the long-range part of the interaction is included through the exchange interaction parameter. For this model, we derive a set of self-consistent equations for the correlation functions, and evaluate them numerically for the long-distance behavior. We find that the pairing correlations are related to spin correlations through the density and the slave-spin correlations. Further, our calculations indicates that the long-range character of the interaction can be probed through these correlation functions. In the absence of exact solutions for the one-dimensional t–J model, our approximate theoretical treatment can be treated as a useful tool to study one dimensional long-range correlated fermions.

Interactions in the 8-orbital model for twisted bilayer graphene

We calculate the interactions between the Wannier functions of the 8-orbital model for twisted bilayer graphene (TBG). In this model, two orbitals per valley centered at the AA regions, the AA-p orbitals, account for the most part of the spectral weight of the flats bands. Exchange and assisted-hopping terms between these orbitals are found to be small. Therefore, the low energy properties of TBG will be determined by the density-density interactions. These interactions decay with the distance much faster than in the two orbital model, following a 1/r law in the absence of gates. The magnitude of the largest interaction in the model, the onsite term between the flat band orbitals, is controlled by the size of the AA regions and is estimated to be ~ 40 meV. To screen this interaction, the metallic gates have to be placed at a distance smaller than 5 nm. For larger distances only the long-range part of the interaction is substantially screened. The model reproduces the band deformation induced by doping found in other approaches within the Hartree approximation. Such deformation reveals the presence of other orbitals in the flat bands and is sensitive to the inclusion of the interactions involving them.

The Axial Hartree–Fock + BCS Code SkyAx

The nuclear mean-field model based on Skyrme forces can predict a variety of properties of nuclear ground states. We present the Code SkyAx solving the Hartree–Fock equations in two spatial dimensions assuming axial symmetry. Pairing can be included in the BCS approximation. The code is implemented with a view on computational speed. Program summaryProgram title: SkyAxCPC Library link to program files:http://dx.doi.org/10.17632/fd453hc4jb.1Licensing provisions: GPLv3Programming language: Fortran 90 and parallel version with OpenMP.External routines/libraries: BLAS, LAPACK.Nature of problem: The Hartree–Fock equations can be used to determine static properties of nuclei all over the nuclear chart, e.g., nuclear masses, charge radii and deformations. This code implements the widely used Skyrme forces as interaction model and offers in addition to include the pairing interaction through the BCS theory. Due to its two-dimensional nature, the code allows only for axial deformations, which is suitable for most nuclei in the chart of nuclides.Solution method: The nucleonic wave functions are represented on a two-dimensional mesh assuming axial symmetry. The Coulomb potential is calculated for an isolated charge distribution by splitting the problem into a short-range and a long-range part. All spatial derivatives are evaluated using the finite Fourier transform method. The code solves the static Hartree–Fock equations with a damped gradient iteration method. It also allows for constraint iterations, where the monopole, quadrupole, octupole and hexadecapole moments can be fixed.Additional comments including restrictions and unusual features: The current implementation is restricted to even–even nuclei. Furthermore the Hartree–Fock + BCS model is valid only for well-bound nuclei. For nuclei near the neutron and proton drip lines, a full Hartree–Fock–Bogolyubov treatment would be more suitable.The code allows for multipole constraints up to l=4. Furthermore the code can be used to calculate fission paths or landscapes of deformations (potential energy surfaces) in one single run.

Electron States of Group-V Donors in Germanium: Variational Calculation Taking into Account the Short-Range Potential

The wave functions of low-lying 1s (A1), 2s, 2p0, 2p±, and 3p0 states of P, As, and Sb shallow donor centers in germanium are calculated in the scope of the envelope-function approximation taking into account the short-range impurity potential. The latter is constructed individually for each impurity allowing for the spatial permittivity dispersion and the difference between the ion cores of germanium and the impurity center. The envelope-function equation is solved using the Ritz variational method; herewith, the selected test functions of orbitally nondegenerate s states are characterized by two spatial scales. The first scale, on the order of the donor effective Bohr radius, corresponds to the long-range part of the potential, and the second scale, which is smaller by an order of magnitude, simulates the electron response to the short-range part of the donor potential. The electron density in the donor ground state is shifted to the nucleus, which is due to allowance made for the attracting “central cell” potential. The envelope functions of p states are in turn constructed so that they are orthogonal to envelopes in the ground states for each impurity centers and are different for various donors in contrast with previous works.

Perturbative chiral nucleon–nucleon potential for the 3P0 partial wave

We study perturbativeness of chiral nuclear forces in the 3P0 channel. In previous works, the focus has been on the one-pion exchange, and the applicable window of perturbative pion exchanges has been shown to span from the threshold to center-of-mass momentum k ≃ 180 MeV. We will examine, instead, whether the cancellation of short- and long-range parts can sufficiently soften the 3P0 chiral force to make it more amenable to perturbation theory. The result is encouraging, as the combined 3P0 force is shown to be perturbative up to k ≃ 280 MeV, covering many nuclear-structure calculations.

A parabolic quasi-Sturmian approach to quantum scattering by a Coulomb-like potential

A computational method in parabolic coordinates is proposed to treat the scattering of a charged particle from both spherically and axially symmetric Coulomb-like potentials. Specifically, the long-range part of the Hamiltonian is represented in parabolic quasi-Sturmian basis functions, while the short-range part is approximated by a Sturmian $$L^2$$ -basis-set truncated expansion. We establish an integral representation of the Coulomb Green’s function in parabolic coordinates from which we derive a convenient closed form for its matrix elements in the chosen $$L^2$$ basis set. From the Green’s function, we build quasi-Sturmian functions that are also given in closed form. Taking advantage of their adequate built-in Coulomb asymptotic behavior, scattering amplitudes are extracted as simple analytical sums that can be easily computed. The scheme, based on the proposed quasi-Sturmian approach, proves to be numerically efficient and robust as illustrated with converged results for three different scattering potentials, one of spherical and two of axial symmetry.

Probing the pinning strength of magnetic vortex cores with sub-nanometer resolution

Understanding interactions of magnetic textures with defects is crucial for applications such as racetrack memories or microwave generators. Such interactions appear on the few nanometer scale, where imaging has not yet been achieved with controlled external forces. Here, we establish a method determining such interactions via spin-polarized scanning tunneling microscopy in three-dimensional magnetic fields. We track a magnetic vortex core, pushed by the forces of the in-plane fields, and discover that the core (~ 104 Fe-atoms) gets successively pinned close to single atomic-scale defects. Reproducing the core path along several defects via parameter fit, we deduce the pinning potential as a mexican hat with short-range repulsive and long-range attractive part. The approach to deduce defect induced pinning potentials on the sub-nanometer scale is transferable to other non-collinear spin textures, eventually enabling an atomic scale design of defect configurations for guiding and reliable read-out in race-track type devices.

Crystal potentials and dispersion forces in additive approximation

On the basis of the additive approximation model, a formula for determining the van der Waals constant was obtained and the possibility of correctly describing the long-range part of the interaction potential of solidified inert gases crystals was indicated. Using the modified Lennard-Jones potential and experimental data on the dissociation energy and the equilibrium radius, the dispersion constants of systems of inert gas pairs and an alkali metal–inert gas are determined. The results of the calculations are in general agreement with the experiment.

Long-time self-diffusion in quasi-two-dimensional colloidal fluids of paramagnetic particles.

The effect of hydrodynamic interactions (HI) on the long-time self-diffusion in quasi-two-dimensional fluids of paramagnetic colloidal particles is investigated using a combination of experiments and Brownian dynamics (BD) simulations. In the BD simulations, the direct interactions (DI) between the particles consist of a short-ranged repulsive part and a long-ranged part that is proportional to 1/r^{3}, with r the interparticle distance. By studying the equation of state, the simulations allow for the identification of the regime where the properties of the fluid are fully controlled by the long-ranged interactions, and the thermodynamic state solely depends on the dimensionless interaction strength Γ. In this regime, the radial distribution functions from the simulations are in quantitative agreement with those from the experiments for different fluid area fractions. This agreement confirms that the DI in the experiments and simulations are identical, which thus allows us to isolate the role of HI, as these are not taken into account in the BD simulations. Experiment and simulation fall onto a master curve with respect to the Γ dependence of D_{L}^{★}=D_{L}/(D_{0}Γ^{1/2}), with D_{0} the self-diffusion coefficient at infinite dilution and D_{L} the long-time self-diffusion coefficient. Our results thus show that, although HI affect the short-time self-diffusion, for a quasi-two-dimensional system with 1/r^{3} long-ranged DI, the reduced quantity D_{L}^{★} is effectively not affected by HI. Interestingly, this is in agreement with prior work on quasi-two-dimensional colloidal hard spheres.

Automatic Generation of Flexible-Monomer Intermolecular Potential Energy Surfaces.

A method is developed for automatic generation of nonreactive intermolecular two-body potential energy surfaces (PESs) including intramonomer degrees of freedom. This method, called flex-autoPES, is an extension of the autoPES method developed earlier, which assumes rigid monomers. In both cases, the whole PES development proceeds without any human intervention. The functional form used is a sum of products of site-site functions (both atomic and off-atomic sites can be used). The leading terms with sites involving different monomers are of physically motivated form. The long-range part of a PES is computed from monomer properties without using any dimer information. The close-range part is fitted to dimer interaction energies computed using electronic structure methods. Virtually any method can be used in such calculations, but the use of symmetry-adapted perturbation theory provides a seamless connection to the long-range part of the PES. The performance of the flex-autoPES code was tested by developing a full-dimensional PES for the water dimer and PESs including only some soft intramonomer degrees of freedom for the ethylene glycol dimer and for the ethylene glycol-water dimer. In the case of the water dimer, the root-mean-square error (RMSE) of the PES from the data points with negative total energies is 0.03 kcal/mol, and we expect this PES to be more accurate than any previously published PES of this type. For the ethylene glycol dimer and the ethylene glycol-water dimers, the analogous RMSEs are 0.25 and 0.1 kcal/mol, respectively.

Vibrational Quenching of Weakly Bound Cold Molecular Ions Immersed in Their Parent Gas

Hybrid ion–atom systems provide an excellent platform for studies of state-resolved quantum chemistry at low temperatures, where quantum effects may be prevalent. Here we study theoretically the process of vibrational relaxation of an initially weakly bound molecular ion due to collisions with the background gas atoms. We show that this inelastic process is governed by the universal long-range part of the interaction potential, which allows for using simplified model potentials applicable to multiple atomic species. The product distribution after the collision can be estimated by making use of the distorted wave Born approximation. We find that the inelastic collisions lead predominantly to small changes in the binding energy of the molecular ion.

Potential Energy Surface for the CH4-H2 van der Waals Interaction.

Methane is an ubiquitous molecule, present as a minor component in many environments, including the Earth and planet atmospheres. Its van der Waals interaction with the main gases is an important ingredient for the understanding of radiative properties for those atmospheres. We present here the first precise determination of the interaction between CH4 and H2. We compute the interaction in an ab initio coupled cluster formalism, with extended atomic bases. We compare a pure CCSD(T) approach to an explicitly correlated CCSD(T)-F12a formalism. The full geometry of scattering two rigid molecules is used, resulting in a potential energy surface depending on 5 degrees of freedom. The long-range part of the ab initio computation is compared with an analytic multipolar expansion. The potential is fit onto a suitable formalism for subsequent scattering dynamics. The potential well is computed at -92.92 cm-1, an intermediate value if compared with other methane-diatomic molecule interactions.

Molecule-surface interaction from van der Waals-corrected semilocal density functionals: The example of thiophene on transition-metal surfaces

Semi-local density functional approximations are widely used. None of them can capture the long-range van der Waals (vdW) attraction between separated subsystems, but they differ remarkably in the extent to which they capture intermediate-range vdW effects responsible for equilibrium bonds between neighboring small closed-shell subsystems. The local density approximation (LDA) often overestimates this effect, while the Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation (GGA) underestimates it. The strongly-constrained and appropriately normed (SCAN) meta-GGA often estimates it well. All of these semi-local functionals require an additive non-local correction such as the revised Vydrov-Van Voorhis 2010 (rVV10) to capture the long-range part. This work reports adsorption energies and the corresponding geometry of the aromatic thiophene (C$_4$H$_4$S) bound to transition metal surfaces. The adsorption process requires a genuine interplay of covalent and weak binding and requires a simultaneously accurate description of surface and adsorption energies with the correct prediction of the adsorption site. All these quantities must come from well balanced short and long-range correlation effects for a universally applicable method for weak interactions with chemical accuracy.

Multiband k·p Model for Tetragonal Crystals: Application to Hybrid Halide Perovskite Nanocrystals.

We investigate the theoretical band structure of organic-inorganic perovskites APbX3 with tetragonal crystal structure. Using D4h point group symmetry properties, we derive a general 16-band Hamiltonian describing the electronic band diagram in the vicinity of the wave-vector point corresponding to the direct band gap. For bulk crystals, a very good agreement between our predictions and experimental physical parameters, as band gap energies and effective carrier masses, is obtained. Extending this description to three-dimensional confined hybrid halide perovskite, we calculate the size dependence of the excitonic radiative lifetime and fine structure. We describe the exciton fine structure of cube-shaped nanocrystals by an interplay of crystal-field and electron-hole exchange interaction (short- and long-range parts) enhanced by confinement. Using very recent experimental results on FAPbBr3 nanocrystals, we extract the bulk short-range exchange interaction in this material and predict its value in other hybrid compounds. Finally, we also predict the bright-bright and bright-dark splittings as a function of nanocrystal size.

Электронные состояния доноров V группы в германии: вариационный расчет с учетом короткодействующего потенциала

In the framework of the envelope function approximation, the wave functions of low-lying 1s(A1), 2s, 2p0, 2p±, 3p0 states of shallow donor centers P, As, Sb in germanium are calculated considering the short-range part of the impurity potential. The latter is constructed individually for each impurity, taking into account the spatial dispersion of the dielectric function and the difference between the ionic cores of germanium and the impurity center. The envelope function equation was solved using the Ritz variational method, and selected trial wave functions of the orbitally non-degenerate s-states are characterized by two spatial scales: the first one is of the order of the donor effective Bohr radius and corresponds to the long-range part of the potential, and the second one, which is an order of magnitude less, simulates the electron response to the short-range part of the donor potential. The electron density in the donor ground state is shifted to the nucleus due to the attractive “central cell” correction. The envelope functions of p-states, in turn, are constructed in such a way they are orthogonal to the ground state envelope functions for each impurity center, and, unlike previous works, are different for various donors.

Near-field infrared microscopy of graphene on metal substrate

Graphene plasmons, collective oscillation modes of electrons in graphene, have recently attracted intense attention in both the fundamental researches and the applications because of their strong field confinement, low loss and excellent tunability. The dispersion of graphene plasmons can be significantly modified in the system of graphene on metal substrate, in which the screening of the long-range part of the electron-electron interactions by nearby metal can lead to many novel quantum effects, such as acoustic plasmons, quantum nonlocal effects and renormalization of band structure. Scattering-type scanning near-field optical microscopy (s-SNOM) which consists of a laser coupled to the tip of an atomic force microscopy (AFM), is an effective technique to directly probe plasmons in two-dimensional materials including graphene, and the graphene plasmons can be observed visually by real-space imaging. But so far the detailed s-SNOM studies of graphene/metal system have not been reported. One potential challenge is that the near-field response of highly conductive metal substrate may partially or entirely obscure that of graphene, making it difficult to further explore graphene by using s-SNOM. Here in this paper, we report the direct observation of near-field optical response of graphene in a graphene/metal system excited by a mid-infrared quantum cascade laser. From a close examination of the data of graphene/Cu compared with that of h-BN/Cu, we are able to identify experimental features due to the near-field response of graphene. Surprisingly, two completely different behaviors are observed in the s-SNOM data for different graphene samples on Cu substrates with similar surface step geometries. These results suggest that the near-field response of graphene/metal system is not completely dominated by the metal substrate, and that two completely different near-field response behaviors of graphene may be attributed to their intrinsic properties affected by metal substrates themselves rather than surface step geometries of metal substrate. In addition, following this approach it is possible to distinguish the near-field optical responses of graphene from that of graphene/metal system. Our work reveals the clear signatures of the near-field optical response of graphene on metal substrate, which provides the foundation for probing plasmons in these systems by using the s-SNOM and understanding many novel quantum phenomena therein.

Effect of Projectile Charge State on Electron Emission Spectra from He Atom

The single ionization of He atom by the impact of highly charged dressed projectile nucleus is investigated by applying the three-body formalism of three-Coulomb wave (3CW) model and the first Born approximation (FBA). In the 3CW formalism, the interaction between the dressed projectile and the active electron is represented by a model potential containing both the long-range and short-range part of the Coulomb potential. The target is treated as a one-electron atom, where the interaction of the active electron with the rest of the target is represented by a similar type of model potential. The correlated motion of each pair of particles, interacting through long-range Coulomb potential, is taken into account in the final state. In addition, an independent particle model and a single-zeta Roothaan-Hartree-Fock 1s wavefunction, describing helium in its ground state, is used to carry out the calculations, where the effective charge of the target nuclei is obtained from the initial orbital energy. The energy distributions of the double-differential cross sections (DDCS) at low and intermediate energy electron emission from helium atom have been investigated. Our theoretical results are compared with both the available experimental measurements and theoretical calculations using other methods. There is good overall agreement of the present calculations with experimental data except for low-energy electron emission (less than 10 eV). For target ionization by a dressed projectile, we find that the cross sections depend on the projectile charge. For distant collisions involving high energy ions, the primary effect of bound projectile electrons is to screen the projectile nuclear charge, thereby reducing the strength of the interaction potential. Finally, the energy distribution of DDCS ratio at different angles has also been studied.

Microscopic three-cluster study of light exotic nuclei

I develop a microscopic three-cluster model for exotic light nuclei. I use the hyperspherical formalism, associated with the Generator Coordinate Method. This model is well adapted to halo nuclei, since the long-range part of the radial wave functions is accurately reproduced. The core wave functions are described in the shell model, including excited states. This technique provides large bases, expressed in terms of projected Slater determinants. Matrix elements involve seven-dimension integrals, and therefore require long calculation times. I apply the model to 11Li, 14Be, 15B, and 17N described by two neutrons surrounding a 9Li, 12Be, 13B and 15N core, respectively. The 17Ne (as 15O+p+p) and 15Ne (as 13O+p+p) mirror nuclei are briefly discussed. I present the spectra and some spectroscopic properties, such as r.m.s. radii or E2 transition probabilities. I also analyze the importance of core excitations.