Multipole Moments Research Articles

Transition moments beyond the electric-dipole approximation: Visualization and basis set requirements.

In the simulation of x-ray absorption spectroscopy, the validity of the electric-dipole approximation comes into question. Three different schemes exist to go beyond this approximation: the first scheme is based on the full semi-classical light-matter interaction, whereas the latter two schemes, referred to as the generalized length and velocity representation, are based on truncated multipole expansions. Even though these schemes have been successfully implemented in several quantum chemistry codes, their basis set requirements remained largely unknown. Here, we assess basis set requirements of these three schemes. We have considered 1s1/2 and 7s1/2 → 7p1/2 transitions in the radium atom, representative of core and valence excitations, respectively, and carried out calculations with dyall.aeXz (X = 2, 3, 4) basis sets at the four-component relativistic TD-HF level of theory. Our basis set study was greatly facilitated by the generation and visualization of radial distributions of transition moment densities, allowing for a straightforward comparison with equivalent finite-difference calculations. Pertaining to the truncated interaction, we find that the length representation electric multipole is the easiest to converge, requiring the dyall.ae2z basis for low-order multipoles and the dyall.ae4z basis at higher orders. The magnetic multipole moments follow a similar trend although they are more difficult to converge. The velocity representation electric multipoles are the most difficult to converge: at high orders, the dyall.ae3z and dyall.ae4z basis sets introduce artificial peaks and oscillations, which increase the overall error. These artifacts are associated with linear dependence issues in the small component space of larger basis sets. The full interaction operator, however, does not suffer from these problems, and we therefore recommend its use in the simulation of x-ray spectroscopy.

Non-spherical sources of Schwarzschild space-time

While it is known that any spherical fluid distribution may only source the spherically symmetric Schwarzschild space-time, the inverse is not true. Thus, in this manuscript, we find exact axially symmetric and static fluid (interior) solutions to Einstein equations, which match smoothly on the boundary surface to the Schwarzschild (exterior) space-time, even though the fluid distribution is not endowed with spherical symmetry. The solutions are obtained by using the general approach outlined in Hernández-Pastora et al. (Class Quantum Gravity 33:235005, 2016), and satisfy the usual requirements imposed to any physically admissible interior solution. A discussion about the physical and geometric properties of the source is presented. The relativistic multipole moments (RMM) are explicitly calculated in terms of the physical variables, allowing to prove that spherical sources can only match to the Schwarzschild space-time. The complexity of the source is evaluated through the complexity factors. It is shown that there is only one independent complexity factor, as in the spherically symmetric case.

DIFFRACTION OF X-RAYS IN CRYSTALS: A TENSOR APPROACH

The use of X-ray synchrotron radiation makes it possible to observe the polarization, spectral, and angular dependences for diffraction reflections. Their theoretical study calls for application of a tensor approach to describe the interaction of X-rays with atoms of matter. Various representations of the tensor atomic scattering amplitude, results of experimental observations of the anisotropy of resonant X-ray scattering, and the relationship of the electric and magnetic multipole moments on atoms with the properties of forbidden resonant reflections are considered.

Electromagnetic scattering by arbitrary-shaped magnetic particles and multipole decomposition: Analytical and numerical approaches

We develop a theoretical approach that makes it possible to analyze the role of multipole contributions in the scattering spectra of arbitrarily shaped magnetic particles with the relative permittivity and permeability given in a general tensor-valued form. The method for calculating the exact multipole moments with the inclusion of induced electric polarization and magnetization is suggested and explicitly applied for dipole and quadrupole terms. The final expressions for the dipole and quadrupole moments are obtained in such a form that they can be easily implemented in numerical solvers. Successful verification of the analytical expressions for multipole moments and the multipole decomposition of the scattered field and scattering cross section is provided by comparing the results obtained for differently shaped magnetic particles using the analytical Mie theory, numerical discrete dipole approximation, and comsol multiphysics software. As a particular example, the manifestation of the Faraday effect for fields scattered by a ferrite sphere is discussed in the framework of the multipole decomposition method with the derived magnetization terms. Accounting for the magnetic properties of particles in the multipole analysis of the electromagnetic scattering significantly expands the method's capabilities for studying and modeling metamaterial properties in the spectral ranges where natural materials can have a relative permeability different from unity.

Breakdown of dipole Born approximation and the role of Rydberg’s predissociation for the electron-induced ion-pair dissociation to oxygen in the presence of background gases

We study the electron-induced ion-pair dissociation to gas-phase oxygen molecules using a state-of-the-art velocity-map ion-imaging technique. The analysis is entirely based on the conical time-gated wedge-shaped velocity slice images of O-/O2 nascent anionic fragments, and the resulting observations are in favor of Van Brunt et al.'s report [R. J. Van Brunt and L. J. Kieffer, J. Chem. Phys. 60, 3057 (1974)]. A new image reconstruction method, Jacobian over parallel slicing, is introduced to overcome the drawback of ion exaggeration in determining the kinetic energy distribution from the time-gated parallel slicing technique, which offers an alternative approach to the wedge slicing method. Most importantly, the role of the quintet-heavy Rydberg state has been drawn out to the complex ion-pair formalism. The extracted kinetic energy and angular distributions from the wedge slice images reveal a high momentum transfer during the ion-pair dissociation process, which could be the finest rationale to observe the breakdown of dipole Born approximation driven by multipole moment associated with the incident electron beam. Three distinct dissociative momentum bands have been precisely identified for O- dissociation. However, radiationless Rydberg's predissociation continuum (≥15%) has become an inherent character of electron-induced ion-pair dissociation, which could be dealt with using the beyond Born-Oppenheimer treatment. The incoherent sum of Σ and Π symmetric-associated ion-pair final states has been precisely identified by modeling the angular distribution of O-/O2 for each of the kinetic energy bands. A negligibly small amount of forward-backward asymmetry is observed in the angular distribution of O-/O2, which might be explained by the dissociative state-specific quantum coherence mechanism as reported [Krishnakumar et al., Nat. Phys. 14, 149 (2018); Kumar et al., arXiv:2206.15024 (2022)] by Prabhudesai et al.

Adiabatic extraction of nonlinear optical properties from real-time time-dependent electronic-structure theory.

Real-time simulations of laser-driven electron dynamics contain information about molecular optical properties through all orders in response theory. These properties can be extracted by assuming convergence of the power series expansion of induced electric and magnetic multipole moments. However, the accuracy relative to analytical results from response theory quickly deteriorates for higher-order responses due to the presence of high-frequency oscillations in the induced multipole moment in the time domain. This problem has been ascribed to missing higher-order corrections. We here demonstrate that the deviations are caused by nonadiabatic effects arising from the finite-time ramping from zero to full strength of the external laser field. Three different approaches, two using a ramped wave and one using a pulsed wave, for extracting electrical properties from real-time time-dependent electronic-structure simulations are investigated. The standard linear ramp is compared to a quadratic ramp, which is found to yield highly accurate results for polarizabilities, and first and second hyperpolarizabilities, at roughly half the computational cost. Results for the third hyperpolarizability are presented along with a simple, computable measure of reliability.

Multipolar quantum electrodynamics of localized charge-current distributions: Spectral theory and renormalization

We formulate a nonrelativistic quantum field theory to model interactions between quantized electromagnetic fields and localized charge-current distributions. The electronic degrees of freedom are encoded in microscopic polarization and magnetization field operators whose moments are identified with the multipole moments of the charge-current distribution. The multipolar Hamiltonian is obtained from the minimal coupling Hamiltonian through a unitary transformation, often referred to as the Power-Zienau-Woolley transformation; we renormalize this Hamiltonian using perturbation theory, the result of which is used to compute the leading-order radiative corrections to the electronic energy levels due to interactions between the electrons and quantum vacuum fluctuations in the electromagnetic field. Our renormalized energy shift constitutes a generalization of the Lamb shift in atomic hydrogen, valid for general localized assemblies of atoms and molecules, possibly with net charge but absent a free current. By expanding the fields in a series of multipole moments, our results can be used to study contributions to this energy shift coming from specific multipole moments of arbitrary order.

Cluster Toroidal Multipoles Formed by Electric-Quadrupole and Magnetic-Octupole Trimers: A Possible Scenario for Hidden Orders in Ca5Ir3O12

Cluster multipole orderings composed of atomic high-rank multipole moments are theoretically investigated with a 5$d$-electron compound Ca$_5$Ir$_3$O$_{12}$ in mind. Ca$_5$Ir$_3$O$_{12}$ exhibits two hidden orders: One is an intermediate-temperature phase with time-reversal symmetry and the other is a low-temperature phase without time-reversal symmetry. By performing the symmetry and augmented multipole analyses for a $d$-orbital model under the hexagonal point group $D_{\rm 3h}$, we find that the 120$^{\circ}$-type ordering of the electric quadrupole corresponds to cluster electric toroidal dipole ordering with the electric ferroaxial moment, which can become the microscopic origin of the intermediate-temperature phase in Ca$_5$Ir$_3$O$_{12}$. Furthermore, based on ${}^{193}$Ir synchrotron-radiation-based M\"{o}ssbauer spectroscopy, we propose that the low-temperature phase in Ca$_5$Ir$_3$O$_{12}$ is regarded as a coexisting state with cluster electric toroidal dipole and cluster magnetic toroidal quadrupole, the latter of which is formed by the 120$^{\circ}$-type ordering of the magnetic octupole and accompanies a small uniform magnetization as a secondary effect. Our results provide a clue to two hidden phases in Ca$_5$Ir$_3$O$_{12}$.

Spin currents with unusual spin orientations in noncollinear Weyl antiferromagnetic Mn3Sn

There are intensive efforts to search for mechanisms that lead to spin-orbit torque with unusual spin orientation, particularly out-of-plane spin orientation which can efficiently switch perpendicular magnetizations. Such a phenomenon has been observed in materials with low structural symmetry, ferromagnetic materials, and antiferromagnets with noncollinear spin structures. Here, we demonstrate the observation of, in addition to out-of-plane spin orientation, spin orientation along the charge current direction in ${\mathrm{Mn}}_{3}\mathrm{Sn}$, a noncollinear antiferromagnet and Weyl semimetal. The mechanism arises from noncollinear spin structure with spin-orbit coupling and it can be viewed as spin rotation around the octupole moment, the lowest order of cluster multipole moment pertaining to the ${\mathrm{Mn}}_{3}\mathrm{Sn}$ crystal group.

Higher multipoles of highly symmetric charge distributions over Platonic solids

We analyze higher multipole moments (beyond dipole and quadrupole) in the expansion of the electrostatic potential produced by discrete as well as volume, surface, and linear continuous distributions of charges over five Platonic solids. In the case of tetrahedron, the lowest non-zero multipole moment is octupole. As the number of polyhedron vertices grows, the first non-zero elements arise at higher multipoles. Namely, for octahedron and cube, the lowest non-zero multipole moment is hexadecapole (24-pole). For icosahedron and dodecahedron, non-zero elements appear first in the 26-pole moment tensors.

Is a black hole shadow a reliable test of the no-hair theorem?

Capturing the image of the shadow cast by the event horizon of an illuminated black hole is, at the most basic level, an experiment of extreme light deflection in a strongly curved spacetime. As such, the properties of an imaged shadow can be used to probe the general relativistic Kerr nature of astrophysical black holes. As an example of this prospect, it is commonly asserted that a shadow can test the validity of the theory's famous ``no hair theorem'' for the black hole's mass and spin multipole moments. In this paper, we assess this statement by calculating the shadow's equatorial radius in spacetimes with an arbitrary multipolar structure and within a slow rotation approximation. We find that when moments higher than the quadrupole are taken into account, the shadow acquires a high degree of degeneracy as a function of the deviation from the Kerr multipole moments. The results of our analysis suggest that dark objects with strongly non-Kerr multipolar structure could nevertheless produce a Kerr-like shadow with its characteristic quasicircular shape.

Hexagonal photonic crystal with nontrivial quadrupole topology

Topological theory of multipole moments in condensed matter physics have recently been generalized to classical systems such as photonic and sonic crystals and several types of quadrupole topological photonic crystals (QTPCs) have already been investigated. However, preceding reports relevant to this issue dealt mainly with square lattice. In this work, we reveal that quadrupole topological phases can also be appeared in hexagonal photonic lattices with the C3v-crystalline symmetry. For this purpose, we extend the topological theory of bulk and Wannier-sector polarizations to the affine coordinate system for analyzing the topological characteristics of non-square lattice systems. The calculation results show that the hexagonal photonic crystals with C3-crystalline symmetry have the quadrupole moments quantized to 1/3, provided that all photonic bands below the gap in consideration are non-degenerate. The hexagonal QTPCs support topological edge and corner states, which are much stronger confined compared with the edge and corner modes originating from nontrivial bulk polarizations. The presented results have fundamental importance and pave the way towards the broader applications of higher-order photonic topological insulators.

Revisiting theN(N + 1)/2‐sites‐type Gaussian charge model for permutationally invariant polynomial fitting of global molecular tensor surfaces

AbstractThe exact expressions for the dipole, quadrupole, and octupoles of a collection ofNpoint charges involve summations of corresponding tensors over theNsites weighted by their charge magnitudes. When the point charges are atoms (in a molecule) theN‐site formula is an approximation, and one must integrate over the electron density to recover the exact multipoles. In the present work we revisit theN(N + 1)/2‐site point charge density model of Hall (Chem. Phys. Lett.6, 501, 1973) for the purpose of fitting ab initio derived multipole moment hypersurfaces using permutationally invariant polynomials (PIP). We examine new approaches in PIP‐fitting procedures for the dipole, quadrupole, octupole moments, and polarizability tensor surfaces (DMS, QMS, OMS and PTS, respectively) for a non‐polar CCl4and a polar CHCl3and show that compared to the primitiveN‐site model theN(N + 1)/2‐site model appreciably improves the relative RMSE of the DMS and does much more substantially so, by an order of magnitude, for the corresponding ones of QMS and OMS. Training datasets are obtained by sampling potential energies up to 18 000 cm−1above the global minima, generated by molecular dynamics simulations at the DFT B3LYP/aug‐cc‐pVDZ level of theory.

Biorthogonal harmonics for the decomposition of gravitational radiation. I. Angular modes, completeness, and the introduction of adjoint-spheroidal harmonics

The estimation of radiative modes is a central problem in gravitational wave theory, with essential applications in signal modeling and data analysis. This problem is complicated by most astrophysically relevant systems not having modes that are analytically tractable. A ubiquitous workaround is to use not modes, but multipole moments defined by spin-weighted spherical harmonics. However, spherical multipole moments are only related to the modes of systems without angular momentum. As a result, they can obscure the underlying physics of astrophysically relevant systems, such as binary black hole merger and ringdown. In such cases, spacetime angular momentum means that radiative modes are not spherical, but spheroidal in nature. Here, we work through various problems related to spheroidal harmonics. We show for the first time that spheroidal harmonics are not only capable of representing arbitrary gravitational wave signals, but that they also possess a kind of orthogonality not used before in general relativity theory. Along the way we present a new class of spin-weighted harmonic functions dubbed ``adjoint-spheroidal'' harmonics. These new functions may be used for the general estimation of spheroidal multipole moments via complete biorthogonal decomposition (in the angular domain). By construction, adjoint-spheroidal harmonics suppress mode-mixing effects known to plague spherical harmonic decomposition; as a result, they better approximate a system's true radiative modes. We discuss potential applications of these results. Lastly, we briefly comment on the challenges posed by the analogous problem with Teukolsky's radial functions.

The gravitational field outside a spatially compact stationary source in a generic fourth-order theory of gravity

By applying the symmetric and trace-free formalism in terms of the irreducible Cartesian tensors, the metric for the external gravitational field of a spatially compact stationary source is provided in F(X, Y, Z) gravity, a generic fourth-order theory of gravity, where X := R is Ricci scalar, Y := RμνRμν is Ricci square, and Z := RμνρσRμνρσ is Riemann square. A new type of gauge condition is proposed so that the linearized gravitational field equations of F(X, Y, Z) gravity are greatly simplified, and then, the stationary metric in the region exterior to the source is derived. In the process of applying the result, integrations are performed only over the domain occupied by the source. The multipole expansion of the metric potential in F(X, Y, Z) gravity for a spatially compact stationary source is also presented. In the expansion, the corrections of F(X, Y, Z) gravity to General Relativity are Yukawa-like ones, dependent on two characteristic lengths. Two additional sets of mass-type source multipole moments appear in the corrections and the salient feature characterizing them is that the integrations in their expressions are always modulated by a common radial factor related to the source distribution.

Multipole Analysis of the Extinction Cross Section and Circular Dichroism of Chiral Metamolecules with Optical Theorem

AbstractPlasmonic chirality of metamolecules attracts increasing attention because of the tunable circular dichroism (CD) and near‐field enhancement for chiral molecule detection. Although various mechanisms have been proposed to explain the plasmonic CD, the quantitative analysis of particles with sub‐wavelength dimensions remains to be refined. Unlike biomolecules and small particles for which only absorption effects are considered, both multipolar scattering and absorption of sub‐wavelength metamolecules need to be analyzed. Here, the extinction and CD of chiral metamolecules are experimentally measured, and the contribution of excited multipole moments is theoretically investigated via combining multipole scattering and optical theorem. It is found that for the low‐order resonance mode, the absorption dominates the CD signal. While for high‐order resonance modes, both scattering and absorption significantly influence the CD, and the high‐order electric quadrupole moment that is commonly ignored plays a crucial role. Moreover, the intrinsic mechanism of the chiral response is also intuitively explained with linear polarization analysis. The mechanism proposed here is not only applicable for biomolecules and tiny particles with low‐order resonance modes but also works well for sub‐wavelength particles with high‐order modes, where conventional mixed electric and magnetic polarizability mechanisms usually do not work very well.

Construction of a Gaussian Process Regression Model of Formamide for Use in Molecular Simulations

FFLUX, a novel force field based on quantum chemical topology, can perform molecular dynamics simulations with flexible multipole moments that change with geometry. This is enabled by Gaussian process regression machine learning models, which accurately predict atomic energies and multipole moments up to the hexadecapole. We have constructed a model of the formamide monomer at the B3LYP/aug-cc-pVTZ level of theory capable of sub-kJ mol-1 accuracy, with the maximum prediction error for the molecule being 0.8 kJ mol-1. This model was used in FFLUX simulations along with Lennard-Jones parameters to successfully optimize the geometry of formamide dimers with errors smaller than 0.1 Å compared to those obtained with D3-corrected B3LYP/aug-cc-pVTZ. Comparisons were also made to a force field constructed with static multipole moments and Lennard-Jones parameters. FFLUX recovers the expected energy ranking of dimers compared to the literature, and changes in C═O and C-N bond lengths associated with hydrogen bonding were found to be consistent with density functional theory.

Gaussian Process Regression Models for Predicting Atomic Energies and Multipole Moments.

Developing a force field is a difficult task because its design is typically pulled in opposite directions by speed and accuracy. FFLUX breaks this trend by utilizing Gaussian process regression (GPR) to predict, at ab initio accuracy, atomic energies and multipole moments as obtained from the quantum theory of atoms in molecules (QTAIM). This work demonstrates that the in-house FFLUX training pipeline can generate successful GPR models for six representative molecules: peptide-capped glycine and alanine, glucose, paracetamol, aspirin, and ibuprofen. The molecules were sufficiently distorted to represent configurations from an AMBER-GAFF2 molecular dynamics run. All internal degrees of freedom were covered corresponding to 93 dimensions in the case of the largest molecule ibuprofen (33 atoms). Benefiting from active learning, the GPR models contain only about 2000 training points and return largely sub-kcal mol-1 prediction errors for the validation sets. A proof of concept has been reached for transferring the model produced through active learning on one atomic property to that of the remaining atomic properties. The prediction of electrostatic interaction can be assessed at the intermolecular level, and the vast majority of interactions have a root-mean-square error of less than 0.1 kJ mol-1 with a maximum value of ∼1 kJ mol-1 for a glycine and paracetamol dimer.

Light Guidance Aided by the Toroidal Dipole and the Magnetic Quadrupole in Silicon Slotted-Disk Chains.

Far-field scattering of high-index nanoparticles can be hugely reduced via interference of multipolar moments giving rise to the so-called anapole states. It has been suggested that this reduced scattering can contribute to efficient transmission along periodic chains of such nanoparticles. In this work, we analyze via numerical simulation and experiments the transmission of light along chains of regular and slotted silicon disks in the frequency region over the light cone. We do not observe transmission at wavelengths corresponding to the excitation of the first electric anapole for regular disks. However, large transmission along straight and curved chains is observed for slotted disks due to the simultaneous excitation of the toroidal dipole and magnetic quadrupole modes in the disks. Photonic band calculations unveil that such large transmission can be ascribed to leaky resonances, though bound states in the continuum do not appear in the structures under analysis. Experiments at telecom wavelengths using silicon disk chains confirm the numerical results for straight and bent chains. Our results provide new insights into the role of radiationless states in light guidance along nanoparticle chains and offer new avenues to utilize Mie resonances of simple nanophotonic structures for on-chip dielectric photonics.

All-dielectric metasurface-based multimode sensing with symmetry-protected and accidental bound states in the continuum

Symmetry-protected and accidental bound states in the continuum (BICs) are generated by various triggering mechanisms, and it is challenging to use them together for sensing. This study designed an all-dielectric metasurface that produced BICs in the terahertz band. Moreover, we investigated eight mode distributions by energy band analysis, of which four modes had symmetry-protected BICs at the high-symmetry Γ point, and three had accidental BICs at the non-Γ point, all with quality factors up to 109. Using multipole analysis and parameter adjustment, we investigated the contribution of multipole moments to the far field, the electromagnetically induced transparency caused by the intermodal coupling, and the merging of accidental BICs. Finally, we used six BICs for multimode sensing, where the figure of merit (FOM) of the symmetry-protected BICs reached 108, and the FOM of the accidental BICs reached 105, providing a new perspective on terahertz linear sensor devices. These findings will open a new avenue for the study of terahertz multimode sensors with ultra-high FOM.