Multipole Representation Research Articles

Unified Description of Electronic Orderings and Cross Correlations by Complete Multipole Representation

Unified Description of Electronic Orderings and Cross Correlations by Complete Multipole Representation

A Multipole Representation for the Gravitational Field of Asteroid (433) Eros

A Multipole Representation for the Gravitational Field of Asteroid (433) Eros

Understanding Electronic Ordering and Cross Correlations with Multipole Representation

This study reviews the recent advancements in research of multipole representations for understanding electronic orderings and cross-correlations in materials and presents future research directions.

Release of windowed multipole library for ENDF/B-VIII.0 in support of OpenMC criticality and depletion analysis

Major evaluations have been updated in the latest release of the Evaluated Nuclear Data library (ENDF/B-VIII.0) and is the currently recommended pointwise library for criticality and depletion analysis in the open-source Monte Carlo code — OpenMC. However, the currently released OpenMC library does not yet include an accompanying windowed multipole library (WMP) that enables on-the-fly Doppler broadening. This paper revisits the windowed multipole representation and fixes a major drawback that has prevented its use for depletion analysis. The new WMP library presented in this paper explicitly treats key absorption reactions as separate channels such that proper reaction rates are provided during depletion analysis. Based on this improvement, the new WMP library is processed for ENDF/B-VIII.0 and then verified against the pointwise data through direct comparison, criticality benchmarks and fuel assembly depletion. The numerical results show good agreement and efficiency with the pointwise library.

Multipole Representation of the Gravitational Field of the Asteroid (16) Psyche

Multipole Representation of the Gravitational Field of the Asteroid (16) Psyche

Resonance Scattering Treatment with the Windowed Multipole Formalism

A new method for directly sampling the resonance upscattering effect is presented. Alternatives have relied on inefficient rejection sampling techniques or large tabular storage of relative velocities. None of these approaches, which require pointwise energy data, are particularly well suited to the windowed multipole cross-section representation. The new method, called multipole analytic resonance scattering, overcomes these limitations by inverse transform sampling from the target relative velocity distribution where the cross section is expressed in the multipole formalism. The closed-form relative speed distribution contains a novel special function we deem the incomplete Faddeeva function, and we present the first results on its efficient numerical evaluation.

Definite complete invariant parametrization of R -matrix theory

We establish a definite parametrization of $R$-matrix theory, which is complete and invariant. Compared to the traditional parametrization of Wigner and Eisenbud, our parametrization has the major advantage of having no arbitrary boundary condition ${B}_{c}$, and of being constituted of scattering matrix poles ${\mathcal{E}}_{\ensuremath{\lambda}}$, which are physical quantities and hence invariant with the choice of arbitrary channel radii ${a}_{c}$. Moreover, being the poles of the scattering matrix, the definite levels ${\mathcal{E}}_{\ensuremath{\lambda}}$ correspond exactly to the nuclear resonances. Our definite parametrization is also global and complete, meaning a finite number of definite parameters---the same number as the Wigner and Eisenbud ones, minus the boundary conditions---can fully describe the scattering matrix on the whole complex plane [it is thus not a local description restricted to an energy window as the previous Windowed Multipole Representation of Ducru et al., Phys. Rev. C 103, 064610 (2021)]. These benefits come at the cost of requiring all parameters to now be complex numbers without an explicit set of constraints, which significantly complicates their direct nuclear data evaluation. We show that our parametrization also gives rise to shadow poles, though we prove they can be ignored and still completely reconstruct the scattering matrix with all its poles, and thus describe nuclear cross sections exactly. This means our parametrization only requires as many scattering matrix poles ${\mathcal{E}}_{\ensuremath{\lambda}}$ as there are Wigner-Eisenbud resonance levels ${E}_{\ensuremath{\lambda}}$, thereby establishing a one-to-one correspondence between the traditional Wigner-Eisenbud and our definite parametrization of $R$-matrix theory. Remarkably, we show these same cross sections can also be obtained using the shadow poles instead of the principal poles. We observe evidence of these phenomena in the spin-parity group ${J}^{\ensuremath{\pi}}=1/{2}^{(\ensuremath{-})}$ of xenon isotope $^{134}\mathrm{Xe}$.

Multipolar Force Fields for Amide-I Spectroscopy from Conformational Dynamics of the Alanine Trimer.

The dynamics and spectroscopy of N-methyl-acetamide (NMA) and trialanine in solution are characterized from molecular dynamics simulations using different energy functions, including a conventional point charge (PC)-based force field, one based on a multipolar (MTP) representation of the electrostatics, and a semiempirical DFT method. For the 1D infrared spectra, the frequency splitting between the two amide-I groups is 10 cm-1 from the PC, 13 cm-1 from the MTP, and 47 cm-1 from self-consistent charge density functional tight-binding (SCC-DFTB) simulations, compared with 25 cm-1 from experiment. The frequency trajectory required for the frequency fluctuation correlation function (FFCF) is determined from individual normal mode (INM) and full normal mode (FNM) analyses of the amide-I vibrations. The spectroscopy, time-zero magnitude of the FFCF C(t = 0), and the static component Δ02 from simulations using MTP and analysis based on FNM are all consistent with experiments for (Ala)3. Contrary to this, for the analysis excluding mode-mode coupling (INM), the FFCF decays to zero too rapidly and for simulations with a PC-based force field, the Δ02 is too small by a factor of two compared with experiments. Simulations with SCC-DFTB agree better with experiment for these observables than those from PC-based simulations. The conformational ensemble sampled from simulations using PCs is consistent with the literature (including PII, β, αR, and αL), whereas that covered by the MTP-based simulations is dominated by PII with some contributions from β and αR. This agrees with and confirms recently reported Bayesian-refined populations based on 1D infrared experiments. FNM analysis together with a MTP representation provides a meaningful model to correctly describe the dynamics of hydrated trialanine.

The Multipole Structure and Symmetry Classification of Even-Type Deviators Decomposed from the Material Tensor.

The number of distinct components of a high-order material/physical tensor might be remarkably reduced if it has certain symmetry types due to the crystal structure of materials. An nth-order tensor could be decomposed into a direct sum of deviators where the order is not higher than n, then the symmetry classification of even-type deviators is the basis of the symmetry problem for arbitrary even-order physical tensors. Clearly, an nth-order deviator can be expressed as the traceless symmetric part of tensor product of n unit vectors multiplied by a positive scalar from Maxwell’s multipole representation. The set of these unit vectors shows the multipole structure of the deviator. Based on two steps of exclusion, the symmetry classifications of all even-type deviators are obtained by analyzing the geometric symmetry of the unit vector sets, and the general results are provided. Moreover, corresponding to each symmetry type of the even-type deviators up to sixth-order, the specific multipole structure of the unit vector set is given. This could help to identify the symmetry types of an unknown physical tensor and possible back-calculation of the involved physical coefficients.

A geodesic multipolar parameterization-based representation for 3D face recognition

Here, we propose a new 3D representation designed in the objective of face recognition independently of the expressions. It is defined on localities presenting limited deformations following expression variations. The proposed representation is based on the multipolar parameterization that we recently introduced in Jribi et al. (2019) which is relative invariant under three dimensional Euclidean transformations and robust to the original mesh. A choice of the number of reference points of each multipolar parameterization is made according to the shape of the two types of region of the face namely those of nose and the eyes. The main curvature field is estimated on the parameterizations of each region. The parameters of dimensional reduction algorithms applied to the overall description are adjusted so that the recognition rates remain efficient. The experiments are carried out on three challenging 3D face databases: the FRGC v2.0, the BU-3DFE and Bosphorus. Very high rates are obtained for both identification and verification scenarios. These results are very competitive with state of the art methods.

Multipole representation for anisotropic Coulomb interactions

Multipole representation is proposed for the anisotropic Coulomb interactions in solids. Any local interactions can be expressed as the product of two multipole operators, and the interaction parameters are systematically classified based on the point group symmetry. The form of the multipole interactions are restricted not only by the symmetry and Hermiticity but also by the spatial structure of the interaction, which is closely related to the presence or absence of the odd-rank multipoles. As an exemplary demonstration, the screened Coulomb interaction for SrVO$_3$ is considered, where only a few parameters are necessary for its description. By comparing it with the unscreened version, the totally symmetric $A_1$ representation is found to be strongly suppressed, but the $A_1$ component still gives a dominant contribution for the anisotropic part of the interaction. The anisotropic interactions are also applied to the localized two $f$-electron wave functions, which give the same-order contribution as the one-body level splitting estimated by the band structure calculation.

Windowed multipole representation of R -matrix cross sections

Nuclear cross sections are basic inputs to any nuclear computation. Campaigns of experiments are fitted with the parametric R-matrix model of quantum nuclear interactions, and the resulting cross sections are documented - both point-wise and as resonance parameters (with uncertainties) - in standard evaluated nuclear data libraries (ENDF, JEFF, BROND, JENDL, CENDL, TENDL): these constitute our common knowledge of fundamental nuclear physics. In the past decade, a collaborative effort has been deployed to establish a new nuclear cross section library format - the Windowed Multipole Library - with the goal of considerably reducing the cost of cross section calculations in nuclear transport simulations. This article lays the theoretical foundations underpinning these efforts. From general R-matrix scattering theory, we derive the windowed multipole representation of nuclear cross sections. Though physically and mathematically equivalent, the windowed multipole representation is particularly well suited for subsequent temperature treatment of angle-integrated cross sections: we show that accurate Doppler broadening can be performed analytically up to the first reaction threshold; and we derive cross sections temperature derivatives to any order. Furthermore, we here establish a way of converting the R-matrix resonance parameters uncertainty (covariance matrices) into windowed multipole parameters uncertainty. We show that generating stochastic nuclear cross sections by sampling from the resulting windowed multipole covariance matrix can reproduce the cross section uncertainty in the original nuclear data file. Through this foundational article, we hope to make the Windowed Multipole Representation accessible, reproducible, and usable for the nuclear physics community, as well as provide the theoretical basis for future research on expanding its capabilities.

Optical band engineering via vertical stacking of honeycomb plasmonic lattices

Inspired by recent advances in atomic homo and heterostructures, we consider the vertical stacking of plasmonic lattices as a new degree of freedom to create a coupled system showing a modified optical response concerning the monolayer. The precise design of the stacking and the geometrical parameters of two honeycomb plasmonic lattices tailors the interaction among their metallic nanoparticles. Based on the similarity of the lattice symmetry, analogies can be drawn with stacked atomic crystals, such as graphene. We use the multipolar spectral representation to study the plasmonic vertical stack's optical response in the near-field regime, emphasizing symmetry properties. The strong coupling of certain optical bands shows up as anticrossings in the dispersion diagram, resulting in the polarization exchange of the interacting bands. By leveraging these effects, we engineer the near-field intensity distribution. Additionally, lifting band degeneracy at specific points of the Brillouin zone is obtained with the consequent opening of minigaps. These effects are understood by quantifying the multipolar coupling among nanospheres belonging to the same and different sublattices, as well as the interlayer and intralayer nanoparticle interactions. Differences with the atomic case are also analyzed and explained in terms of the stack's interaction matrix. Finally, we predict the absorption spectrum projected on the two orthogonal linear polarizations.

Frequency-dependent transmission boundary condition in the acoustic time-domain nodal discontinuous Galerkin model

Accurate modeling of boundary conditions is of critical importance for acoustic simulations. Recently, the time-domain nodal discontinuous Galerkin (TD-DG) method has emerged as a potential wave-based method for acoustic modeling. Although the acoustic reflection behavior of various time-domain impedance boundaries has been studied extensively, the modeling of the sound transmission across a locally-reacting layer of impedance discontinuity is far less developed. This paper presents a formulation of broadband time-domain transmission boundary conditions for locally-reacting surfaces in the framework of the TD-DG method. The formulation simulates the acoustic wave behavior at each of the boundary nodes using the plane-wave theory. Through the multi-pole model representation of the transmission coefficient, various types of transmission layers can be simulated. One-dimensional numerical examples demonstrate the capability of the proposed formulation to accurately simulate the reflection and transmission characteristics of the limp wall and the porous layer, where quantitative error behavior against analytical results is presented. Furthermore, to demonstrate the applicability, two scenarios of two-dimensional acoustic environment are considered. One is the sound transmission between two rooms partitioned by a limp panel and the other is the sound propagation through a transmissive noise barrier. Comparison of the predicted results from the proposed method against the results from the frequency-domain finite element simulations further verifies the formulation.

Biofilms at interfaces: microbial distribution in floating films.

Cellular motility is a key function guiding microbial adhesion to interfaces, which is the first step in the formation of biofilms. The close association of biofilms and bioremediation has prompted extensive research aimed at comprehending the physics of microbial locomotion near interfaces. We study the dynamics and statistics of microorganisms in a 'floating biofilm', i.e., a confinement with an air-liquid interface on one side and a liquid-liquid interface on the other. We use a very general mathematical model, based on a multipole representation and probabilistic simulations, to ascertain the spatial distribution of microorganisms in films of different viscosities. Our results reveal that microorganisms can be distributed symmetrically or asymmetrically across the height of the film, depending on their morphology and the ratio of the film's viscosity to that of the fluid substrate. Long-flagellated, elongated bacteria exhibit stable swimming parallel to the liquid-liquid interface when the bacterial film is less viscous than the underlying fluid. Bacteria with shorter flagella on the other hand, swim away from the liquid-liquid interface and accumulate at the free surface. We also analyze microorganism dynamics in a flowing film and show how a microorganism's ability to resist 'flow-induced-erosion' from interfaces is affected by its elongation and mode of propulsion. Our study generalizes past efforts on understanding microorganism dynamics under confinement by interfaces and provides key insights on biofilm initiation at liquid-liquid interfaces.

On Alfvén wave propagation along a circle on dipolar coordinates

The multipolar representation of the magnetic field has, for the lowest-order term, a magnetic dipole that dominates the far field. Thus the far-field representation of the magnetic field of the Earth, Sun and other celestial bodies is a dipole. Since these bodies consist of or are surrounded by plasma, which can support Alfvén waves, their propagation along dipole magnetic field lines is considered using a new coordinate system: dipolar coordinates. The present paper introduces multipolar coordinates, which are an example of conformal coordinates; conformal coordinates are orthogonal with equal scale factors, and can be extended from the plane to space, for instance as cylindrical or spherical dipolar coordinates. The application considered is to Alfvén waves propagating along a circle, that is a magnetic field line of a dipole, with transverse velocity and magnetic field perturbations; the various forms of the wave equation are linear second-order differential equations, with variable coefficients, specified by a background magnetic field, which is force free. The absence of a background magnetic force leads to a mean state of hydrostatic equilibrium, specified by the balance of gravity against the pressure gradient, for a perfect gas or incompressible liquid. The wave equation is simplified to a Gaussian hypergeometric type in the case of zero frequency, otherwise, for non-zero frequency, an extended Gaussian hypergeometric equation is obtained. The solution of the latter specifies the magnetic field perturbation spectrum, and also, via a polarisation relation, the velocity perturbation spectrum; both are plotted, over half a circle, for three values of the dimensionless frequency.

An SE(3) invariant description for 3D face recognition

Here, we intend to introduce a novel 3D face description which is invariant under the Special Euclidean group SE(3) and independent to the original surface parameterization. It is well known that it is too difficult to define a relative invariant parameterization of a general curved surface. In the present work, we introduce the multi-polar geodesic representation of ℝ3 surfaces. It allows to reach an isotropic canonical parameterization relatively to SE(3) due to the fact that the face surfaces can be easily assumed to be a graph of a function from ℝ2 to ℝ. The principal curvature fields according to a three-polar parameterization are considered. A statistical study is made to choose a good configuration of reference points. The performances of the novel description are tested on the standard database FRGC v2.0. Many recognition scenarios are established. The obtained results are very competitive with the state of the art.

Representation of the QM Subsystem for Long-Range Electrostatic Interaction in Non-Periodic Ab Initio QM/MM Calculations.

In QM/MM calculations, it is essential to handle electrostatic interactions between the QM and MM subsystems accurately and efficiently. To achieve maximal efficiency, it is convenient to adopt a hybrid scheme, where the QM electron density is used explicitly in the evaluation of short-range QM/MM electrostatic interactions, while a multipolar representation for the QM electron density is employed to account for the long-range QM/MM electrostatic interactions. In order to avoid energy discontinuity at the cutoffs, which separate the short- and long-range QM/MM electrostatic interactions, a switching function should be utilized to ensure a smooth potential energy surface. In this study, we benchmarked the accuracy of such hybrid embedding schemes for QM/MM electrostatic interactions using different multipolar representations, switching functions and cutoff distances. For test systems (neutral and anionic oxyluciferin in MM (aqueous and enzyme) environments), the best accuracy was acquired with a combination of QM electrostatic potential (ESP) charges and dipoles and two switching functions (long-range electrostatic corrections (LREC) and Switch) in the treatment of long-range QM/MM electrostatics. It allowed us to apply a 10Å distance cutoff and still obtain QM/MM electrostatics/polarization energies within 0.1 kcal/mol and time-dependent density functional theory (TDDFT)/MM vertical excitation energies within 10−3 eV from theoretical reference values.

Fast algorithms for evaluating the stress field of dislocation lines in anisotropic elastic media

In dislocation dynamics (DD) simulations, the most computationally intensive step is the evaluation of the elastic interaction forces among dislocation ensembles. Because the pair-wise interaction between dislocations is long-range, this force calculation step can be significantly accelerated by the fast multipole method (FMM). We implemented and compared four different methods in isotropic and anisotropic elastic media: one based on the Taylor series expansion (Taylor FMM), one based on the spherical harmonics expansion (Spherical FMM), one kernel-independent method based on the Chebyshev interpolation (Chebyshev FMM), and a new kernel-independent method that we call the Lagrange FMM. The Taylor FMM is an existing method, used in ParaDiS, one of the most popular DD simulation softwares. The Spherical FMM employs a more compact multipole representation than the Taylor FMM does and is thus more efficient. However, both the Taylor FMM and the Spherical FMM are difficult to derive in anisotropic elastic media because the interaction force is complex and has no closed analytical formula. The Chebyshev FMM requires only being able to evaluate the interaction between dislocations and thus can be applied easily in anisotropic elastic media. But it has a relatively large memory footprint, which limits its usage. The Lagrange FMM was designed to be a memory-efficient black-box method. Various numerical experiments are presented to demonstrate the convergence and the scalability of the four methods.

Spontaneous ordering at the free surface of molecular clusters

The present paper seeks to establish a theoretical framework to account for the spontaneous molecular ordering at free surfaces of molecular aggregates. It is shown that within a multipole representation of the molecular interactions, ordering is the effect of the interplay of two fundamental causes: the translational symmetry breaking of the bulk phase at the interface and the interactions of multipoles of different parity. Quantitative analysis of the order parameter demonstrates the building up of ordering at different length scales with range depending on the strength of the interactions.