Multipole Expansion Research Articles

Physics-Inspired Equivariant Descriptors of Nonbonded Interactions.

One essential ingredient in many machine learning (ML) based methods for atomistic modeling of materials and molecules is the use of locality. While allowing better system-size scaling, this systematically neglects long-range (LR) effects such as electrostatic or dispersion interactions. We present an extension of the long distance equivariant (LODE) framework that can handle diverse LR interactions in a consistent way and seamlessly integrates with preexisting methods by building new sets of atom centered features. We provide a direct physical interpretation of these using the multipole expansion, which allows for simpler and more efficient implementations. The framework is applied to simple toy systems as proof of concept and a heterogeneous set of molecular dimers to push the method to its limits. By generalizing LODE to arbitrary asymptotic behaviors, we provide a coherent approach to treat arbitrary two- and many-body nonbonded interactions in the data-driven modeling of matter.

Tightly focused linearly and radially polarized beam effect on the LSPR peak with varying particle size

Interactions of gold and silver spherical nanoparticles (NPs) of a wide range of radius size with tightly focused beams are investigated for a wide range of wavelengths. The scattering of tightly focused beams with a single NP of varying size is examined using a generalized Mie theory and average intensity enhancement in the near-field due to localized surface plasmons is examined for different tightly focused beams. A multipole expansion approach is used to observe the contribution of modes in shaping the scattered light intensity. Influence of particle size and tightly focused beam on intensity enhancement factor is investigated and a shift in localized surface plasmon resonance (LSPR) peak towards higher wavelength is observed with increasing particle size. Maximum intensity enhancement factor due to scattering of different incident beams by a metallic NP is theoretically examined and reported. Gold and silver NP having radius size ranging from 30–60 nm and 1–15 nm respectively showed a maximum intensity enhancement factor in different media. A significant enhancement for the tightly focused radially polarized beam is observed for gold and silver NPs of radius size between 50 nm and 100 nm. Variation towards higher wavelength in LSPR peak due to involvement of higher order of multipoles, and the surrounding medium are also examined in detail.

Slow Rotation of Coaxial Slip Colloidal Spheres about Their Axis

The flow field around a straight chain of multiple slip spherical particles rotating steadily in an incompressible Newtonian fluid about their line of centers is analyzed at low Reynolds numbers. The particles may vary in radius, slip coefficient, and angular velocity, and they are permitted to be unevenly spaced. Through the use of a boundary collocation method, the Stokes equation governing the fluid flow is solved semi-analytically. The interaction effects among the particles are found to be noteworthy under appropriate conditions. For the rotation of two spheres, our collocation results for their hydrodynamic torques are in good agreement with the analytical asymptotic solution in the literature obtained by using a method of twin multipole expansions. For the rotation of three spheres, the particle interaction effect indicates that the existence of the third particle can influence the torques exerted on the other two particles noticeably. The interaction effect is stronger on the smaller or less slippery particles than on the larger or more slippery ones. Torque results for the rotation of chains of many particles visibly show the shielding effect among the particles.

LVC/MM: A Hybrid Linear Vibronic Coupling/Molecular Mechanics Model with Distributed Multipole-Based Electrostatic Embedding for Highly Efficient Surface Hopping Dynamics in Solution.

We present a theoretical framework for a hybrid linear vibronic coupling model electrostatically embedded into a molecular mechanics environment, termed the linear vibronic coupling/molecular mechanics (LVC/MM) method, for the surface hopping including arbitrary coupling (SHARC) molecular dynamics package. Electrostatic embedding is realized through the computation of interactions between environment point charges and distributed multipole expansions (DMEs, up to quadrupoles) that represent each electronic state and transition densities in the diabatic basis. The DME parameters are obtained through a restrained electrostatic potential (RESP) fit, which we extended to yield higher-order multipoles. We also implemented in SHARC a scheme for achieving roto-translational invariance of LVC models as well as a general quantum mechanics/molecular mechanics (QM/MM) interface, an OpenMM interface, and restraining potentials for simulating liquid droplets. Using thioformaldehyde in water as a test case, we demonstrate that LVC/MM can accurately reproduce the solvation structure and energetics of rigid solutes, with errors on the order of 1-2 kcal/mol compared to a BP86/MM reference. The implementation in SHARC is shown to be very efficient, enabling the simulation of trajectories on the nanosecond time scale in a matter of days.

Multipole Expansion of the Fundamental Solution of a Fractional Degree of the Laplace Operator

Multipole Expansion of the Fundamental Solution of a Fractional Degree of the Laplace Operator

An Investigation of Harmonic Oscillation of an Ion Particle in a Non-uniform Electrostatic Field

In this study, we investigate the possibility to trap an ion particle in a non-uniform electrostatic field generated by two ionic rings, with the aim to obtain and verify general results regarding our proposition. To develop the equation of motion, we first derive the electric field generated by each ring, based on the multipole expansion technique under the azimuthally symmetrical charge distribution speculation and superposition principle. Then, we establish the equation of motion regarding the Lagrangian formalism. From the results, we observe that an ion particle can exhibit harmonic motion, if the stability condition is satisfied by an appropriate parametric setting. The results are confirmed by displacement responses and phase trajectory plots, which validate our belief in the prospect to trap an ion in an electrostatic field. Furthermore, the velocity amplitude is significantly influenced by both the charge concentration of each ring and its size.

Convergence of the Laplace and the alternative multipole expansion approximation series for the Coulomb potential

Multipole expansion is a powerful technique used in many-body physics to solve dynamical problems involving correlated interactions between constituent particles. The Laplace multipole expansion series of the Coulomb potential is well established in literature. We compare its convergence with our recently developed perturbative and analytical alternative multipole expansion series of the Coulomb potential. In our working, we confirm that the Laplace and the analytical alternative multipole expansion series are equivalent as expected. In terms of performance, the perturbative alternative multipole expansion series underapproximate the expected results to some extent while the Laplace and the analytical alternative multipole expansion series yield results which are relatively accurate but oscillatory in nature even with a higher number of angular momentum terms employed. As a practical example, we have evaluated the Slater double integrals for two-electron systems using the multipole expansion techniques and a mean field approximation. The estimated results show that only spherical interactions are dominant while the higher-order interactions are negligible. To highlight the discrepancy in the application of each of the formulations of the multipole expansion series for the electron-electron interaction potential, an estimation of the non-relativistic groundstate energies of some helium-like systems, evaluated using the spherical approximation of the multipole potential, is provided. Our findings are likely to be useful in the treatment of the Coulomb potential in electronic structure calculations as well as in celestial mechanics.

HCS background magnetic field study HYTARO+

Context. HYTARO+ is an analytical model developed to evaluate the influence of the solar magnetic dipole in the topology of the local structure of the heliospheric current sheet (HCS). Aims. Statistical methods such as MVA, which are typically used in the study of the HCS come with two limitations. On one hand, they do not provide any information about the physical properties of the current sheet and, on the other, they do not offer the possibility of defining the local structure of the HCS from the physical conditions that prevail in the two sectors that separate the sheet. HYTARO model describes the interaction between the magnetic field that defines the structure of the current sheet and the plasma confined within it. HYTARO+ goes deeper into the study of the background magnetic field that is included in the HYTARO model with the aim of identifying its source. Methods. Multipole expansion is used to provide a general approach to any physical phenomenon of some system, thereby allowing for a potential function representation. In our case, we considered the dipolar and quadrupolar Sun magnetic field contribution in the HCS. Results. In the development of the HYTARO+ model, we estimated the components of the dipole and quadrupole magnetic fields at 1 AU. In the present study, we establish that the Z component of the dipole field and the X component of the quadrupole field are the more relevant terms of the background field present in the data that correspond to the discontinuities. A summary of the contribution of the multipolar components along the 23 solar cycle is incorporated.

Magnetic field recovery technique based on distance weighting multipole expansion method

A space-borne gravitational wave detector requires the test mass (TM) to be in an ultra-low disturbance state. However, magnetic field fluctuations will disturb the TM and produce acceleration noise. To assess the influence of the magnetic field on the TM, it is necessary to monitor and reconstruct the magnetic field near the TM in real time. In this paper, a distance weighting multipole expansion (DWME) method was proposed, and its magnetic field reconstruction accuracy was analyzed. The results demonstrated that the proposed DWME method significantly improved the reconstruction precision compared to traditional methods. It reduced the average reconstruction error of the sensitive axial magnetic field from 1.2% to 0.8% and the maximum error from 16% to 8%. In the in-orbit situation, the DWME method also outperforms traditional methods.

An advanced multipole model of the (130) Elektra quadruple system

Context. The Ch-type asteroid (130) Elektra is orbited by three moons, making it the first quadruple system in the main asteroid belt. Aims. We aim to characterise the irregular shape of Elektra and construct a complete orbital model of its unique moon system. Methods. We applied the All-Data Asteroid Modelling (ADAM) algorithm to 60 light curves of Elektra, including our new measurements, 46 adaptive-optics (AO) images obtained by the VLT/SPHERE and Keck/Nirc2 instruments, and two stellar occultation profiles. For the orbital model, we used an advanced N-body integrator, which includes a multipole expansion of the central body (with terms up to the order ℓ = 6), mutual perturbations, internal tides, and the external tide of the Sun acting on the orbits. We fitted the astrometry measured with respect to the central body and also relatively, with respect to the moons themselves. Results. We obtained a revised shape model of Elektra with the volume-equivalent diameter (201 ± 2) km. Of two possible pole solutions, (λ,β) = (189; −88) deg is preferred, because the other one leads to an incorrect orbital evolution of the moons. We also identified the true orbital period of the third moon S/2014 (130) 2 as P2 = (1.642112 ± 0.000400) days, which is in between the other periods, P1 ≃ 1.212days, P3 = 5.300 days, of S/2014 (130) 1 and S/2003 (130) 1, respectively. The resulting mass of Elektra, (6.606-0.013+0.007) ×1018 kg, is precisely constrained by all three orbits. Its bulk density is then (1.536 ± 0.038) g cm−3. The expansion with the assumption of homogeneous interior leads to the oblateness J2 = −C20 ≃ 0.16. However, the best-fit precession rates indicate a slightly higher value, ≃0.18. The number of nodal precession cycles over the observation time span 2014–2019 is 14, 7, and 0.5 for the inner, middle, and outer orbits. Conclusions. Future astrometric or interferometric observations of Elektra’s moons should constrain these precession rates even more precisely, allowing the identification of possible inhomogeneities in primitive asteroids.

Fast Barnes–Hut-based algorithm in 2D vortex method of computational hydrodynamics

In the present paper, the efficient modification of the Barnes–Hut algorithm is proposed based on ideas of multipole and local expansions. Such an approach allows for forming a hierarchy of fast method schemes with different accuracy and computational complexity depending on the number of expansion terms. All the necessary formulae for computational are derived. The parallel implementation is developed using OpenMP and CUDA technologies, which allows performing computations on multicore processors and graphics accelerators.

Time-Efficient Multiplex Orbital Angular Momentum Beam Synthesis With Arbitrary-Shaped Array Using Multipole Moment Expansion

Despite the significant interest in pattern synthesis over the decades, the beam synthesis of orbital angular momentum (OAM) has received relatively little attention. In this paper, we present a time-efficient method for OAM beam synthesis based on multipole moment expansion. The approach introduces the multipole moment term as a transforming level, which separates the excitation coefficients and radiation pattern. We evaluate the performance of our proposed method through several examples and demonstrate its high capability to synthesize multiplex OAM beams with arbitrary-shaped arrays, ensuring reasonable accuracy. The theoretical analysis shows a significant improvement in computational efficiency and a great reduction in computational complexity compared to conventional synthesis methods. Finally, we design, fabricate, and measure a double-ring concentric circular patch antenna array using our proposed method, and the measured results confirm the validity of our approach. The work provides a valuable contribution to the field of OAM beam synthesis and offers a promising avenue for future research.

The multipolar elastic fields of ellipsoidal and polytopal plastic inclusions

The article offers a general formulation for the multipolar field expansion of an inclusion. The multipolar moments of arbitrary order for the ellipsoidal inclusion are derived, showing known features of their elastic fields in both the far and near field. Using integer point enumeration theory, the same formulation is extended to all polytopal inclusions, which serve to model faceted inclusions found in, e.g. igneous and metamorphic rocks, particle reinforced composite materials, or as intermetallic precipitates in alloys. A general algorithm for the calculation of the multipolar moments of polytopes of any number of vertices is derived. A number of examples for tetrahedral, cuboidal and dodecahedral inclusions are given. It is shown that the parity of the axial moment function of a polytopal inclusion mirrors the symmetry of the inclusion. This represents the main properties of their elastic fields, including the long- and near-field decay. If the inclusion is symmetric with respect to a given axes, the fields will decay with 1 / r 2 in the far field and 1 / r 4 in the near field. If symmetry along that direction is lost, as is expected in most real inclusions, the decay rate progresses with 1 / r 2 but the near field with 1 / r 3 .

Multipole expansion for the electron-nucleus scattering at high energies in the unified electroweak theory

The paper presents the multipole expansion for the electron-nucleus scattering cross section at high energies within the framework of the unified electroweak theory. The electroweak currents of the nucleus are expanded into the simple components with definite angular momenta, called the multipole form factors. The multipole expansion of the cross section is a consequence of the above expansion. Besides the familiar electromagnetic form factors, there are also the vector and axial form factors, respectively, related to weak interactions. To determine multipole form factors, general formulas for the calculation of reduced matrix elements are established using the fractional parentage coefficient method and the multiparticle shell model. Calculation of them enables us to obtain more detailed information about the nuclear structure and elucidate the role played by the weak interaction in the high-energy reaction mechanisms.

An Iterative Implementation of the Signal Space Separation Method for Magnetoencephalography Systems with Low Channel Counts.

The signal space separation (SSS) method is routinely employed in the analysis of multichannel magnetic field recordings (such as magnetoencephalography (MEG) data). In the SSS method, signal vectors are posed as a multipole expansion of the magnetic field, allowing contributions from sources internal and external to a sensor array to be separated via computation of the pseudo-inverse of a matrix of the basis vectors. Although powerful, the standard implementation of the SSS method on MEG systems based on optically pumped magnetometers (OPMs) is unstable due to the approximate parity of the required number of dimensions of the SSS basis and the number of channels in the data. Here we exploit the hierarchical nature of the multipole expansion to perform a stable, iterative implementation of the SSS method. We describe the method and investigate its performance via a simulation study on a 192-channel OPM-MEG helmet. We assess performance for different levels of truncation of the SSS basis and a varying number of iterations. Results show that the iterative method provides stable performance, with a clear separation of internal and external sources.

Analysis of inhomogeneous magnetization using a spinner magnetometer

The assumption for measuring a discrete sample by spinner magnetometer in paleomagnetism is that the sample under study is a magnetic dipole so that the magnetization vector can be calculated by successive measurements of components along three orthogonal axes. This assumption may not hold for samples with inhomogeneous magnetization, irregular shape, or both, where non-dipole components are no longer ignorable. This study draws attention to their effect on the measurement of remanent magnetization based on an offset dipole model using multipole expansion of magnetic potential. Results from this model are compared with the experimental data measured by a new type of spinner magnetometer featured by high spatial resolution and the capability of measuring the fundamental wave component as well as the harmonic wave components. By using the relative amplitude of the harmonic waves to the fundamental wave, the contribution of non-dipole components can be quantified, leading to assessing how dipolar a sample is. The new analyses and instrumentation will be useful as a new diagnostic tool to obtain more reliable data from the samples collected from tectonically active regions with complicated geological histories.

Relegation-free closed-form perturbation theory and the domain of secular motions in the restricted three-body problem

We propose a closed-form (i.e., without expansion in the orbital eccentricities) scheme for computations in perturbation theory in the restricted three-body problem (R3BP) when the massless particle is in an orbit exterior to the one of the primary perturber. Starting with a multipole expansion of the barycentric (Jacobi-reduced) Hamiltonian, we carry out a sequence of normalizations in Delaunay variables by Lie series, leading to a secular Hamiltonian model without use of relegation. To this end, we introduce a book-keeping analogous to the one proposed in Cavallari and Efthymiopoulos (Celest Mech Dyn Astron 134(2):1–36, 2022) for test particle orbits interior to the one of the primary perturber, but here adapted, instead, to the case of exterior orbits. We give numerical examples of the performance of the method in both the planar circular and the spatial elliptic restricted three-body problem, for parameters pertinent to the Sun-Jupiter system. In particular, we demonstrate the method’s accuracy in terms of reproducibility of the orbital elements’ variations far from mean-motion resonances. As a basic outcome of the method, we show how, using as criterion the size of the series’ remainder, we reach to obtain an accurate semi-analytical estimate of the boundary (in the space of orbital elements) where the secular Hamiltonian model arrived at after eliminating the particle’s fast degree of freedom provides a valid approximation of the true dynamics.

Multipole expansion of gravitational waves: memory effects and Bondi aspects

In our previous work, we proposed an algorithm to transform the metric of an isolated matter source in the multipolar post-Minkowskian approximation in harmonic (de Donder) gauge to the Newman-Unti gauge. We then applied this algorithm at linear order and for specific quadratic interactions known as quadratic tail terms. In the present work, we extend this analysis to quadratic interactions associated with the coupling of two mass quadrupole moments, including both instantaneous and hereditary terms. Our main result is the derivation of the metric in Newman-Unti and Bondi gauges with complete quadrupole-quadrupole interactions. We rederive the displacement memory effect and provide expressions for all Bondi aspects and dressed Bondi aspects relevant to the study of leading and subleading memory effects. Then we obtain the Newman-Penrose charges, the BMS charges as well as the second and third order celestial charges defined from the known second order and novel third order dressed Bondi aspects for mass monopole-quadrupole and quadrupole-quadrupole interactions.

Estimating the Net Magnetic Moment of Geological Samples From Planar Field Maps Using Multipoles

AbstractRecent advances in magnetic microscopy have enabled studies of geological samples whose weak and spatially nonuniform magnetizations were previously inaccessible to standard magnetometry techniques. A quantity of central importance is the net magnetic moment, which reflects the mean direction and the intensity of the magnetization states of numerous ferromagnetic crystals within a certain volume. The planar arrangement of typical magnetic microscopy measurements, which originates from measuring the field immediately above the polished surface of a sample to maximize sensitivity and spatial resolution, makes estimating net moments considerably more challenging than with spherically distributed data. In particular, spatially extended and nonuniform magnetization distributions often cannot be adequately approximated by a single magnetic dipole. To address this limitation, we developed a multipole fitting technique that can accurately estimate net moment using spherical harmonic multipole expansions computed from planar data. Given that the optimal location for the origin of such expansions is unknown beforehand and generally unconstrained, regularization of this inverse problem is critical for obtaining accurate moment estimates from noisy experimental magnetic data. We characterized the performance of the technique using synthetic sources under different conditions (noiseless data, data corrupted with simulated white noise, and data corrupted with measured instrument noise). We then validated and demonstrated the technique using superconducting quantum interference device microscopy measurements of impact melt spherules from Lonar crater, India and dusty olivine chondrules from the CO chondrite meteorite Dominion Range 08006.

Basis set customization for modeling noncovalent interactions

AbstractIn this article, a simple approach is proposed to reduce the basis set requirement for computing accurate interaction energies for bimolecular noncovalent complexes. A distributed multipole expansion is used to estimate the importance of basis functions in a given atom. Moderate truncation of basis sets results in reduced computational cost without compromising the accuracy. The approach was found fruitful for wavefunction‐based methods and density functional theory approximation using the Dunning, Karlsruhe, and Jensen basis sets.