Multipole Representation Research Articles

The exact multicenter multipolar part of a molecular charge distribution and its simplified representations

We study the problem of representing the molecular charge distribution in a convenient way for practical applications and we propose, instead of a single representation, a flexible procedure for building approximations with an arbitrary level of accuracy as concerns the long-range part of the electrostatic potential. We first discuss the splitting of the total electrostatic potential into a multipolar part (long-range) and a penetration part (shortrange) in connection with the usual one-center multipole expansion: at large enough distances, this expansion precisely converges towards the multipolar part. However, this representation is not practically efficient as soon as the molecule departs from a spherical shape, and we therefore consider a so-called multicenter multipole representation. In the MO-LCAO framework, the use of a basis of Gaussian atomic orbitals {χα} generates such a representation in a natural way: indeed, each elementary distribution χ*αχβ then reduces to a one-center distribution with a finite number of multipoles, hence an exact multipolar representation (corresponding to the approximate MO-LCGO wave function) involving a finite (albeit large, in general) number of centers, each one bearing a finite (small) number of multipoles. From this representation used as a reference, we then generate simplified multicenter multipole representations through a systematic procedure of reduction of the number of centers (having chosen a reduced set of new centers, we represent the multipoles of each suppressed old center through suitable new multipoles located on a few new centers closest to the old one under consideration). The number and locations of the new centers are arbitrary, hence the great flexibility of the method. We checked the accuracy of this procedure for molecules of different polarity (water, benzene, adenine), and we noticeably found that a representation involved as centers, the atoms plus one point per chemical bond exhibited quite a satisfactory accuracy. In the conclusion, we briefly discuss the possibility of extending the method to other kinds of basis functions (e.g., Slater orbitals), in connection with the problem of representing the charge distribution associated with the exact molecular wave function, we also discuss the choice of the number centers according to the desired accuracy and we briefly describe a way of applying the method to the problem of representing a large molecule in terms of molecular fragments.

Dynamics of buried pipes in back-filled trench

Dynamic response of pipelines lying in a back-filled trench in a semi-infinite medium is investigated. The pipelines are modelled as circular cylindrical shells of small thickness. The problems associated are the ones of plane strain, in which it is assumed that the waves are propagating perpendicular to the pipe-axes. Since no exact solution is obtainable for the problems considered here, a numerical technique that combines the finite element method with multipolar representation of the scattered field is used. Numerical results are presented for the normal displacements, as well as the hoop stress in the pipewall either in the presence or absence of another pipe lying nearby.

Determination of Gross Plasma Equilibrium from Magnetic Multipoles

A new approximate technique to determine the gross plasma equilibrium parameters, major radius, minor radius, elongation and triangularity for an up-down symmetric plasma is developed. It is based on a multipole representation of the externally applied poloidal magnetic field, relating specific terms to the equilibrium parameters. The technique shows reasonable agreement with free boundary MHD equilibrium results. The method is useful in dynamic simulation and control studies.

Dynamic stress intensity factors for buried planar and nonplanar cracks

The scattering of time-harmonic plane longitudinal, shear and Rayleigh waves by a crack in two dimensions embedded in a semi-infinite homogeneous isotropic elastic half-space has been studied in this paper. Two problems have been considered: a straight crack and a Y-shaped crack. A hybrid numerical technique combining a multipolar representation of the scattered field in the half-space with the finite element method has been used to obtain the stress-intensity factors for the crack tips. Numerical results for the various crack geometries and the incident waves are discussed in this paper.

Surface displacements due to elastic wave scattering by buried planar and non-planar cracks

The scattering of time-harmonic plane longitudinal, shear, and Rayleigh waves by a crack in two dimensions embedded in a semi-infinite homogeneous isotropic elastic half-space has been studied in this paper. Two problems have been considered: a straight crack and a Y-shaped crack. A hybrid numerical technique combining a multipolar representation of the scattered field in the half-space with the finite element method has been used to obtain the far-field displacements as well as the stress-intensity factors for the crack tips. Results for vertical displacement on the free surface of the half-space are presented in this paper.

Simplified approach to probe-corrected spherical near-field scanning

A multipole representation for the response of an arbitrary receiving antenna is derived that allows the formulation of probe-corrected spherical near-field scanning simply in terms of conventional vector spherical harmonics. Both the representation and formulation are free of rotational and translational addition theorems. A sampling theorem derived for Legendre functions is used to evaluate the resulting orthogonality integrals by direct summation in a computer time proportional to (ka)3.

Cumulative atomic multipole representation of the molecular charge distribution and its basis set dependence

A simple procedure to decompose the theoretical molecular charge distribution into cumulative atomic multipoles supplementing any population analysis scheme has been described and tested for a number of molecules in extended basis sets. This approach may be applied to describe local charge distributions in neutral as well as charged systems and also leads to a simplified point-charge model conserving the local anisotropy of the atomic charge distribution in molecules. Such an approach may be useful in estimating intermolecular interactions, representing the molecular environment in solvent effect or enzyme catalytic activity studies, evaluation of molecular electrostatic potentials or tracing the quality of basis set functions.

Multipole Model of the Molecular Asymmetry in Thermotropic and Lyotropic Liquid Crystals. Volume and Surface Effects

Abstract The molecular asymmetry of mesogenic molecules is defined as a deviation of the corresponding molecular property from spherical and axial symmetry. Three types of asymmetry-electrical, steric and biphilic, are considered using a multipole representation. Two basic molecular shapes are included-rod-like and plate-like. The first three multipoles-monopole, dipole, and quadrupole, are involved. In the steric case a monopole does not exist. The effective shape of a molecule includes not only steric, but also electrical intermolecular repulsive interactions. The biphilic asymmetry is determined by the distribution of the hydrophilic and hydrophobic groups along the molecule. Three types of generalized fields—electrical, deformational, and biphilic, are also defined and their volume and surface sources are discussed. The effects of the molecular asymmetry on the intermolecular interactions in thermotropic (rod-like and disc-like) and lyotropic liquid crystals are qualitatively discussed in an analogy t...

Comment on "Maxwell's equations in the multipolar representation"

It is shown that Haller's interpretation of the unitary transformation from the minimalcoupling to the multipolar form of the Hamiltonian in nonrelativistic quantum electrodynamics not only conflicts with the Maxwell-Lorentz equations but also ascribes a special status to the minimal-coupling Hamiltonian which is not warranted by the general method of canonical quantization. The correct interpretation of the transformation as the quantum analog of a classical canonical transformation is reiterated. It is also shown that the transformation is equivalent to a gauge transformation of the electromagnetic potentials.

Maxwell's equations in the multipolar representation

Arguments are given in support of a remark made by Mandel, that Maxwell's equations are not consistent with the multipolar form of the Hamiltonian for the electromagnetic field interacting with nonrelativistic electrons.

Singlet excitons in crystalline naphthalene, antracene, tetracene and pentacene

The energies and oscillator strengths of exciton transitions in crystalline naphthalene, anthracene, tetracene and pentacene are calculated using second quantized boson theory. The lattice sums of Coulomb exciton transfer interactions consist of an Ewald sum of molecular point dipole-dipole interations and a direct sum of nondipolar interactions calculated from PPP wavefunctions using the atomic—multipole representation of transition charge densities. The calculated exciton energies and oscillator strengths are compared with available experimental data. For anthracene, inclusion of the nondipolar interactions leads to substantially better agreement between theory and experiment. For tetracene and pentracene, the factor group splittings of the lowest transition are determined primarily by crystal induced mixing with higher transitions.

Multipole expansion calculation of slow viscous flow about spheroids of different sizes

The multipole representation technique of Gluckman, Pfeffer & Weinbaum for slow, viscous, axisymmetric motion has been applied to a system of two spheroids. The two particles may have different shapes and volumes. The limitations of this method have been studied through calculations of the drag forces and velocities of particles with aspect ratios from 0·1 to 10 and relative volumesVfrom 1 to 103. The multipole-expansion convergence is quite slow for large relative volumes and requires on the order of$4 \times V^{\frac{1}{3}}$multipoles. Wild fluctuations in the drag as the number of multipoles increases were found for large oblate spheroids. Relative velocities are given quite accurately for separations of the centres greater than 1·02 times the minimum separation. Comparisons are made with previous results and approximate theories for two spheres, for non-identical particles at large separation, and for a sphere near a large spheroid.

The magnetic field of Mercury

Direct observations of the magnetic field of Mercury and its magnetosphere, formed by the interaction of the solar wind, were performed twice by the Mariner 10 spacecraft, in March 1974 and again in March 1975. From these data, it is clear that there exists an intrinsic magnetic field of the planet, sufficiently strong at present to deflect the solar wind flow around the planet and to form a detached bow shock-wave in the super-Alfvénic solar wind. Four methods have been used to analyze the magnetic field data and derive quantitative values for the description of the planetary field: (i). Comparison of bow shock and magnetopause relative positions at Mercury to those at the Earth (cf. Slavin and Holzer, 1979). (ii). Direct spherical harmonic analysis of the data. (iii). Modelling of the magnetosphere by an image dipole and infinite 2-D current sheet in addition to the planetary field. (iv). Scaling of a mathematical model for the terrestrial magnetosphere. The results obtained yield dipole moments / fr( g 1 0) 2+(h 1 1) 2+(g 1 1) 2 ranging from 2.4–5.1 × 10 22gauss cm 3, with the lower values associated with certain models using partial quadrupole ( g 2 0) and octapole ( g 3 0) terms to improve the least-squares fit of models to observations. Because the data set is incomplete, in the mathematical sense, no unique representation of the planetary field multipolar representation can be derived by method (ii). The use of only one of the five quadrupole moment terms and one of the eight octapole moment terms corresponds to a displacement of the dipole along its axis. These terms, used in methods (iii) and (iv), yield equivalent offsets of the dipole by approximately 0.2 R M. The selection of only those higher-order terms possessing axial symmetry cannot be justified. Thus, the large offset may reflect the limitations of the models used to represent the external current systems. Because of the relatively short radial excursion of the data, the g 2 0 and g 3 0 terms can also be spatially aliased with the g 1 0 term. Analyses by method (ii) of subsets of data from the third encounter, taken near closest approach, yield a convergent series of dipole moment values which are believed to best represent the intrinsic planetary field. These provide a mean moment of 330 (±18) γ R M 3 = 4.8 × 10 22 gauss cm 3 at a tilt angle of 14 ± 5° and a longitude of 148 ± 21°. This means that the surface field of Mercury is about 1% that of the Earth, while the moment is 6 × 10 −4 that of the Earth. The polarity sense is the same as the Earth's. The origin of the field cannot be uniquely determined. It may be due to an active dynamo, a remanent magnetic field or a combination of both. Consideration of remanence as the source leads to some difficulties, because there is insufficient definitive knowledge of planetary interior structure and thermal state to eliminate this source. Success in attempting to explain the field as due to an active dynamo has encouraged these efforts. Therefore, Mercury may join the Earth and Jupiter as an example of a planet possessing an internal fluid region with a convecting motion which regeneratively maintains the magnetic field. The sources of convective energy may be radiogenic decay and heat release, gravitational settling and differentiation or precessional torques.

An optimized potential function for the calculation of nucleic acid interaction energies I. Base stacking

An optimized potential function for base-stacking interactions is constructed. Stacking energies between the complementary pairs of a dimer are calculated as a function of the rotational angle and separation distance. Using several different sets of atomic charges, the electrostatic component in the monopole-monopole approximation (MMA) is compared to the more refined segmented multipole-multipole representation (SMMA); the general features of the stacking minima are found to be correctly reproduced with IEHT or CNDO atomic charges. The electrostatic component is observed to control the location of stacking minima.The MMA, in general, is not a reliable approximation of the SMMA in regions away from minima; however, the MMA is reliable in predicting the location and nature of stacking minima.The attractive part of the Lennard-Jones 6-12 potential is compared to and parameterized against the expressions for the second-order interaction terms composed of multipole-bond polarizability for the polarization energy and transition-dipole bond polariz abilities for approximation of the dispersion energy. The repulsive part of the Lennard-Jones potential is compared to a Kitaygorodski-type repulsive function; changing the exponent from its usual value of 12 to 11.7 gives significantly better agreement with the more refined repulsive function.Stacking minima calculated with the optimized potential method are compared with various perturbation-type treatments. The optimized potential method yields results that compare as well with melting data as do any of the more recent and expensive perturbation methods.

A group-theoretical formulation of geophysical elastodynamics

The methods described in an earlier paper, which are based on the theory of the irreducible representations of the three-dimensional rotation group, are applied in an analysis of rotationally invariant, self-adjoint systems of differential equations and boundary conditions. Particular emphasis is placed on the tensor field Helmholtz equation and the dynamical elastic equation for both homogeneous and inhomogeneous media, and with boundary conditions for both finite and infinite media. Consideration of the appropriate Green dyadic leads to a representation of the source field in terms of an infinite set of multipoles, each of which may in turn be related to a geometrical configuration of tensor point sources. A new formula is derived which describes the effect on an arbitrary tensor field of a translation of the origin, and thus ensures that the theory summarized above is applicable to a source region at a general location. Geophysical applications include the calculation of the excitation of the modes of free oscillation of the Earth by earthquake sources, and, more generally, the multipolar representation of seismic sources.

Transformation of multipolar source fields under a change of reference frame

Simple and convenient formulae are derived which describe the transformation of a multipolar expansion under an arbitrary proper rotation of the reference frame. When combined with the corresponding formulae for a translation, these results show how multipolar representations of source fields transform under any proper displacement of the reference frame. Particular emphasis is placed on the seismic source problem; however, these results find applications in many other physical problems.

Charge overlap effects and the validity of the multipole results for first-order molecule-molecule interaction energies. Formalism and an application to H2-H2

For the interaction of two molecules, the first-order coulomb energy, E e (1), is expressed as a sum of non-expanded partial wave components which exhibit the orientation dependence of the energy explicitly. By comparing these non-expanded results with the analogous multipolar results for the first-order energy, the effects of charge overlap on the multipole representation of the energy can be investigated as a function of intermolecular separation and orientation, and the validity of the multipole results for E e (1) assessed. The interaction of two ground-state hydrogen molecules is considered as a specific example of this approach for studying the validity of the multipole expansion as a representation of molecular first-order Coulomb energies. All calculations are presented as a function of four molecular wave functions of varying degrees of sophistication in order to assess the sensitivity of (1) the representation of charge overlap effects and (2) the calculated quadrupole and hexadecapole moments o...

Diatomic generalized x-ray scattering factors: Results from Hartree–Fock electron density functions

Generalized x-ray scattering factors based on multipole representation of the atoms have been calculated from Hartree–Fock Roothaan one-electron molecular density functions of first row diatomic hydrides (BH to FH) and the 14 electron series BF, N2, (1Σ)CO, and (3Π)CO. The effect of different truncation of the mutipole expansions is discussed for NH. Results for the other molecules from expansions up to quadrupolar terms on each center are discussed. The monopole scattering factors are virtually the same as Hartree–Fock atomic form factors except at low sinϑ/λ values. The dipole and quadrupole scattering factors are rather specific to the particular molecule for the valence region, but core polarization features are similar. Static charge properties of the molecule are accurately given by the generalized scattering factors. However, charge properties associated with the individual pseudoatoms are strongly dependent on truncation of the multipole expansion. The coherent x-ray scattering intensities of the molecule can be calculated from the generalized x-ray scattering factors with a maximum relative error no larger than 4×10−5.

On the multipole representation of the interaction Hamiltonian of a particle with a static or dynamic field

On the multipole representation of the interaction Hamiltonian of a particle with a static or dynamic field

A multipole representation of earthquake source mechanisms

abstract The static displacement field in an elastic half-space caused by a general, finite disturbance is derived in the form of an infinite series (multipole) expansion. The relationship between this theory and dislocation theory is discussed with the aid of examples. A multipole source model is derived for the strain-step field associated with the nuclear blast HANDLEY set off at the Nevada Test Site on March 26, 1970. The results suggest that there was significant fault movement associated with the detonation.