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Multipole expansion

About: Multipole expansion is a research topic. Over the lifetime, 9675 publications have been published within this topic receiving 214783 citations.


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TL;DR: In this paper, a new classical empirical potential is proposed for water, which uses a polarizable atomic multipole description of electrostatic interactions, and a modified version of Thole's interaction model is used to damp induction at short range.
Abstract: A new classical empirical potential is proposed for water. The model uses a polarizable atomic multipole description of electrostatic interactions. Multipoles through the quadrupole are assigned to each atomic center based on a distributed multipole analysis (DMA) derived from large basis set molecular orbital calculations on the water monomer. Polarization is treated via self-consistent induced atomic dipoles. A modified version of Thole's interaction model is used to damp induction at short range. Repulsion−dispersion (vdW) effects are computed from a buffered 14−7 potential. In a departure from most current water potentials, we find that significant vdW parameters are necessary on hydrogen as well as oxygen. The new potential is fully flexible and has been tested versus a variety of experimental data and quantum calculations for small clusters, liquid water, and ice. Overall, excellent agreement with experimental and high level ab initio results is obtained for numerous properties, including cluster st...

1,315 citations

Journal ArticleDOI
TL;DR: In this article, a unified notation for the multipole formalisms for gravitational radiation is presented, which includes scalar, vector, and tensor spherical harmonics used in the general relativity literature, including Regge-Wheeler harmonics, the symmetric, trace-free ("STF") tensors of Sachs and Pirani, the Newman-Penrose spin-weighted harmonics and the Mathews-Zerilli Clebsch-Gordan-coupled harmonics.
Abstract: This paper brings together, into a single unified notation, the multipole formalisms for gravitational radiation which various people have constructed. It also extends the results of previous workers. More specifically: Part One of this paper reviews the various scalar, vector, and tensor spherical harmonics used in the general relativity literature—including the Regge-Wheeler harmonics, the symmetric, trace-free ("STF") tensors of Sachs and Pirani, the Newman-Penrose spin-weighted harmonics, and the Mathews-Zerilli Clebsch-Gordan-coupled harmonics—which include "pure-orbital" harmonics and "pure-spin" harmonics. The relationships between the various harmonics are presented. Part One then turns attention to gravitational radiation. The concept of "local wave zone" is introduced to facilitate a clean separation of "wave generation" from "wave propagation." The generic radiation field in the local wave zone is decomposed into multipole components. The energy, linear momentum, and angular momentum in the waves are expressed as infinite sums of multipole contributions. Attention is then restricted to sources that admit a nonsingular, spacetime-covering de Donder coordinate system. (This excludes black holes.) In such a coordinate system the multipole moments of the radiation field are expressed as volume integrals over the source. For slow-motion systems, these source integrals are re-expressed as infinite power series in L / λ≡(size of source ) / (reduced wavelength of waves ). The slow-motion source integrals are then specialized to systems with weak internal gravity to yield (i) the standard Newtonian formulas for the multipole moments, (ii) the post-Newtonian formulas of Epstein and Wagoner, and (iii) post-post-Newtonian formulas. Part Two of this paper derives a multipole-moment wave-generation formalism for slow-motion systems with arbitrarily strong internal gravity, including systems that cannot be covered by de Donder coordinates. In this formalism one calculates, by any means, the source's instantaneous, near-zone, external gravitational field as a solution of the time-independent Einstein field equations. One then reads off of this near-zone field the source's instantaneous multipole moments; and one plugs those time-evolving moments into the standard radiation formulae given in Part One of this paper. As building blocks for this formalism, Part Two also does the following things: (1) In the linearized theory of gravity, for the vacuum exterior of an isolated system, it derives the general solution of the field equations (a result due to Sachs, Bergmann, and Pirani). (2) In full nonlinear general relativity, for the vacuum near-zone exterior of an isolated system, it derives the structure of the general solution of the Einstein field equations. That structure is expressed as a sum of products of multipole contributions. It also matches this near-zone field onto an outgoing-wave radiation field. (3) In full nonlinear general relativity, for the vacuum exterior of a stationary isolated system, (a) it presents a definition of multipole moments which meshes naturally with gravitational-wave theory; (b) it introduces the concept of "asymptotically Cartesian and mass centered" (ACMC) coordinate systems; and (c) it shows how to deduce the multipole moments of a source from the form of its metric in an ACMC coordinate system. As an example, the lowest few (l ≤ 3) multipole moments of the Kerr metric are computed.

1,253 citations

Journal ArticleDOI
TL;DR: In this paper, a general formulae for the energy of a Bloch wave with reduced wave vector k is obtained by the application of the dynamical theory of lattice interferences to electron waves.

1,083 citations

Journal ArticleDOI
TL;DR: In this paper, the authors extend the theory of dipole moments in crystalline insulators to higher multipole moments, and describe the topological invariants that protect these moments.
Abstract: We extend the theory of dipole moments in crystalline insulators to higher multipole moments. In this paper, we expand in great detail the theory presented in Ref. 1, and extend it to cover associated topological pumping phenomena, and a novel class of 3D insulator with chiral hinge states. In quantum-mechanical crystalline insulators, higher multipole bulk moments manifest themselves by the presence of boundary-localized moments of lower dimension, in exact correspondence with the electromagnetic theory of classical continuous dielectrics. In the presence of certain symmetries, these moments are quantized, and their boundary signatures are fractionalized. These multipole moments then correspond to new SPT phases. The topological structure of these phases is described by "nested" Wilson loops, which reflect the bulk-boundary correspondence in a way that makes evident a hierarchical classification of the multipole moments. Just as a varying dipole generates charge pumping, a varying quadrupole generates dipole pumping, and a varying octupole generates quadrupole pumping. For non-trivial adiabatic cycles, the transport of these moments is quantized. An analysis of these interconnected phenomena leads to the conclusion that a new kind of Chern-type insulator exists, which has chiral, hinge-localized modes in 3D. We provide the minimal models for the quantized multipole moments, the non-trivial pumping processes and the hinge Chern insulator, and describe the topological invariants that protect them.

1,045 citations

Journal ArticleDOI
TL;DR: In this paper, a spin-weighted analysis of the cosmic microwave background (CMB) is presented, where linear polarization is a second-rank symmetric and traceless tensor, which can be decomposed on a sphere into spin $\ifmmode\pm\else\textpm\fi{}2$ spherical harmonics.
Abstract: Using the formalism of spin-weighted functions we present an all-sky analysis of polarization in the cosmic microwave background (CMB). Linear polarization is a second-rank symmetric and traceless tensor, which can be decomposed on a sphere into spin $\ifmmode\pm\else\textpm\fi{}2$ spherical harmonics. These are the analogues of the spherical harmonics used in the temperature maps and obey the same completeness and orthogonality relations. We show that there exist two linear combinations of spin $\ifmmode\pm\else\textpm\fi{}2$ multipole moments which have opposite parities and can be used to fully characterize the statistical properties of polarization in the CMB. Magnetic-type parity combination does not receive contributions from scalar modes and does not cross correlate with either temperature or electric-type parity combination, so there are four different power spectra that fully characterize statistical properties of CMB. We present their explicit expressions for scalar and tensor modes in the form of line of sight integral solution and numerically evaluate them for a representative set of models. These general solutions differ from the expressions obtained previously in the small scale limit both for scalar and tensor modes. A method to generate and analyze all-sky maps of temperature and polarization is given and the optimal estimators for various power spectra and their corresponding variances are discussed.

1,042 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
2023309
2022587
2021310
2020298
2019263
2018296