Multipole expansion for magnetic structures: A generation scheme for a symmetry-adapted orthonormal basis set in the crystallographic point group

We report the release within the Bilbao Crystallographic server [1] of a webpage providing detailed quantitative information on a representative set of published magnetic structures. Under the name of MAGNDATA (www.cryst.ehu.es/magndata) more than 140 entries are available. Each magnetic structure has been saved making use of magnetic symmetry, i.e. Shubnikov magnetic groups for commensurate structures, and magnetic superspace groups for incommensurate ones. This ensures a unified communication method and a robust and unambiguous description of both atomic positions and magnetic moments. The origin and main crystallographic axes of the parent phase are usually kept, with the cost of often using a non-standard setting for the magnetic symmetry. The magnetic point group is also given, so that the allowed macroscopic tensor properties can be derived. The fact that magnetic structures are being described according to various methods, often with ambiguous information, has forced an elaborate interpretation and transformation of the original data. For this purpose the freely available internet tools MAXMAGN [1] and ISODISTORT [2] have been our essential tools. Most of the analyzed structures happen to possess maximal magnetic symmetries within the constraints imposed by the magnetic propagation vector, and the relevant model could be derived in a straightforward manner using MAXMAGN [1]. In a few cases a lower symmetry is realized, but then it corresponded to one isotropy subgroup of one or several irreducible representations (irreps) of the paramagnetic grey space group, and ISODISTORT [2] could be applied to model the structure. Although the structure description is done using magnetic groups, the active irrep(s) are also given in most cases. The entries of the collection can be retrieved in a cif-like format, which is supported by internet tools as STRCONVERT [1] and ISOCIF [2], the visualization program VESTA [3], and some refinement programs (JANA2006, FULLPROF). Each entry also includes Vesta files that allow the visualization of a single magnetic unit cell.

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The Symmetry of Linear Molecules

10.20944/preprints201804.0142.v1 ◽

2018 ◽

Author(s):

Katy L. Chubb ◽

Per Jensen ◽

Sergei N. Yurchenko

Keyword(s):

Eigenvalue Problems ◽

Point Group ◽

Basis Set ◽

Group Symmetry ◽

Linear Molecule ◽

Numerical Application ◽

Set Functions ◽

Symmetry Properties ◽

Linear Molecules ◽

Representation Transformation

A numerical application of linear-molecule symmetry properties, described by the D∞h point group, is formulated in terms of lower-order symmetry groups Dnh with finite n. Character tables and irreducible representation transformation matrices are presented for Dnh groups with arbitrary n-values. These groups are subsequently used in the construction of symmetry-adapted ro-vibrational basis functions for solving the Schrödinger equations of linear molecules as part of the variational nuclear motion program TROVE. The TROVE symmetrisation procedure is based on a set of "reduced" vibrational eigenvalue problems with simplified Hamiltonians. The solutions of these eigenvalue problems have now been extended to include the classification of basis-set functions using ℓ, the eigenvalue (in units of ℏ) of the vibrational angular momentum operator L ^ z . This facilitates the symmetry adaptation of the basis set functions in terms of the irreducible representations of Dnh. 12C2H2 is used as an example of a linear molecule of D∞h point group symmetry to illustrate the symmetrisation procedure.

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A Note on the Symmetry Properties of Löwdin's Orthogonalization Schemes

Collection of Czechoslovak Chemical Communications ◽

10.1135/cccc20080937 ◽

2008 ◽

Vol 73 (6-7) ◽

pp. 937-944 ◽

Cited By ~ 2

Author(s):

András T. Rokob ◽

Ágnes Szabados ◽

Peter R. Surján

Keyword(s):

Point Group ◽

Symmetry Operator ◽

Basis Set ◽

Symmetry Relation ◽

Symmetry Properties ◽

The Matrix

We point out that the well-known symmetry properties of the symmetrically and canonically orthogonalized vectors hold only under certain conditions on the overlapping vectors. In particular, the matrix of the transformation induced by the symmetry operator must be unitary. This requirement is not fulfilled if Cartesian d or f functions are used in the basis set. If such functions are present, canonically orthogonalized orbitals do not transform according to representations of the molecular point group; nor do Löwdin orthogonalized vectors preserve symmetry relation of the original vectors.

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Fundamentals of an orthonormal basis set molecular orbital theory

Chemical Physics Letters ◽

10.1016/0009-2614(72)80012-5 ◽

1972 ◽

Vol 12 (4) ◽

pp. 579-582 ◽

Cited By ~ 31

Author(s):

K.R. Roby

Keyword(s):

Molecular Orbital ◽

Orthonormal Basis ◽

Basis Set ◽

Molecular Orbital Theory ◽

Orbital Theory

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MAGNDATA: towards a database of magnetic structures. II. The incommensurate case

Journal of Applied Crystallography ◽

10.1107/s1600576716015491 ◽

2016 ◽

Vol 49 (6) ◽

pp. 1941-1956 ◽

Cited By ~ 10

Author(s):

Samuel V. Gallego ◽

J. Manuel Perez-Mato ◽

Luis Elcoro ◽

Emre S. Tasci ◽

Robert M. Hanson ◽

...

Keyword(s):

Magnetic Moments ◽

Parent Phase ◽

Point Group ◽

Quantitative Information ◽

Isotropy Subgroup ◽

Group Symmetry ◽

Published Data ◽

Point Group Symmetry ◽

Magnetic Structures ◽

Incommensurate Structures

A free web page under the nameMAGNDATA, which provides detailed quantitative information on more than 400 published magnetic structures, has been made available at the Bilbao Crystallographic Server (http://www.cryst.ehu.es). It includes both commensurate and incommensurate structures. In the first article in this series, the information available on commensurate magnetic structures was presented [Gallego, Perez-Mato, Elcoro, Tasci, Hanson, Momma, Aroyo & Madariaga (2016).J. Appl. Cryst.49, 1750–1776]. In this second article, the subset of the database devoted to incommensurate magnetic structures is discussed. These structures are described using magnetic superspace groups,i.e.a direct extension of the non-magnetic superspace groups, which is the standard approach in the description of aperiodic crystals. The use of magnetic superspace symmetry ensures a robust and unambiguous description of both atomic positions and magnetic moments within a common unique formalism. The point-group symmetry of each structure is derived from its magnetic superspace group, and any macroscopic tensor property of interest governed by this point-group symmetry can be retrieved through direct links to other programs of the Bilbao Crystallographic Server. The fact that incommensurate magnetic structures are often reported with ambiguous or incomplete information has made it impossible to include in this collection a good number of the published structures which were initially considered. However, as a proof of concept, the published data of about 30 structures have been re-interpreted and transformed, and together with ten structures where the superspace formalism was directly employed, they form this section ofMAGNDATA. The relevant symmetry of most of the structures could be identified with an epikernel or isotropy subgroup of one irreducible representation of the space group of the parent phase, but in some cases several irreducible representations are active. Any entry of the collection can be visualized using the online tools available on the Bilbao server or can be retrieved as a magCIF file, a file format under development by the International Union of Crystallography. These CIF-like files are supported by visualization programs likeJmoland by analysis programs likeJANAandISODISTORT.

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Symmetry vibronic wavefunction basis set and Hamiltonian representation for XY3Z molecules near a doubly degenerate Jahn-Teller electronic state in the C3v point group

Chemical Physics Letters ◽

10.1016/0009-2614(94)00683-0 ◽

1994 ◽

Vol 226 (1-2) ◽

pp. 63-72

Author(s):

H. Cardy ◽

B. Aboab ◽

M. Roche

Keyword(s):

Electronic State ◽

Point Group ◽

Basis Set ◽

Jahn Teller

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MAGNDATA: towards a database of magnetic structures. I. The commensurate case

Journal of Applied Crystallography ◽

10.1107/s1600576716012863 ◽

2016 ◽

Vol 49 (5) ◽

pp. 1750-1776 ◽

Cited By ~ 28

Author(s):

Samuel V. Gallego ◽

J. Manuel Perez-Mato ◽

Luis Elcoro ◽

Emre S. Tasci ◽

Robert M. Hanson ◽

...

Keyword(s):

Magnetic Moments ◽

Parent Phase ◽

Point Group ◽

Quantitative Information ◽

Isotropy Subgroup ◽

Published Data ◽

Cell Orientation ◽

Space Group ◽

Magnetic Structures ◽

Magnetic Symmetry

A free web page under the name MAGNDATA, which provides detailed quantitative information on more than 400 published magnetic structures, has been developed and is available at the Bilbao Crystallographic Server (http://www.cryst.ehu.es). It includes both commensurate and incommensurate structures. This first article is devoted to explaining the information available on commensurate magnetic structures. Each magnetic structure is described using magnetic symmetry, i.e. a magnetic space group (or Shubnikov group). This ensures a robust and unambiguous description of both atomic positions and magnetic moments within a common unique formalism. A non-standard setting of the magnetic space group is often used in order to keep the origin and unit-cell orientation of the paramagnetic phase, but a description in any desired setting is possible. Domain-related equivalent structures can also be downloaded. For each structure its magnetic point group is given, and the resulting constraints on any macroscopic tensor property of interest can be consulted. Any entry can be retrieved as a magCIF file, a file format under development by the International Union of Crystallography. An online visualization tool using Jmol is available, and the latest versions of VESTA and Jmol support the magCIF format, such that these programs can be used locally for visualization and analysis of any of the entries in the collection. The fact that magnetic structures are often reported without identifying their symmetry and/or with ambiguous information has in many cases forced a reinterpretation and transformation of the published data. Most of the structures in the collection possess a maximal magnetic symmetry within the constraints imposed by the magnetic propagation vector(s). When a lower symmetry is realized, it usually corresponds to an epikernel (isotropy subgroup) of one irreducible representation of the space group of the parent phase. Various examples of the structures present in this collection are discussed.

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Expansion techniques for collisionless stellar dynamical simulations

Proceedings of the International Astronomical Union ◽

10.1017/s1743921315007851 ◽

2014 ◽

Vol 10 (S312) ◽

pp. 227-230

Author(s):

Yohai Meiron

Keyword(s):

Graphics Processing Units ◽

Multipole Expansion ◽

Basis Set ◽

Field Methods ◽

The Past ◽

The Poisson Equation ◽

Radial Structure ◽

Dynamical Simulations ◽

Self Consistent Field ◽

Graphics Processing

AbstractWe present ETICS, a collisionless N-body code based on two kinds of series expansions of the Poisson equation, implemented for graphics processing units (GPUs). The code is publicly available and can be used as a standalone program or as a library (an AMUSE plugin is included). One of the two expansion methods available is the self-consistent field (SCF) method, which is a Fourier-like expansion of the density field in some basis set; the other is the multipole expansion (MEX) method, which is a Taylor-like expansion of the Green's function. MEX, which has been advocated in the past, has not gained as much popularity as SCF. Both are particle-field methods and optimized for collisionless galactic dynamics, but while SCF is a “pure” expansion, MEX is an expansion in just the angular part; thus, MEX is capable of capturing radial structure easily, while SCF needs a large number of radial terms.

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Construction of Dirac Delta Function from the Discrete Orthonormal Basis of the Function Space

Himalayan Physics ◽

10.3126/hj.v6i0.18360 ◽

2017 ◽

pp. 54-57

Author(s):

Hari Prasad Lamichhane

Keyword(s):

Function Space ◽

Orthonormal Basis ◽

Delta Function ◽

Hamiltonian Operator ◽

Dirac Delta Function ◽

Basis Set ◽

One Dimensional ◽

Dirac Delta ◽

Simple Basis ◽

Infinite Potential Well

Orthonormal basis of the function space can be used to construct Dirac delta function. In particular, set of eigenfunctions of the Hamiltonian operator of a particle in one dimensional infinite potential well forms a non-degenerate discrete orthonormal basis of the function space. Such a simple basis set is suitable to study closure property of the basis and various properties of Dirac delta function in Physics graduate lab.The Himalayan Physics Vol. 6 & 7, April 2017 (54-57)

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Geometry, and Normal Modes of Vibration (3N-6) for Di and Tetra-Rings Layer (6, 0) Linear (Zigzag) SWCNTs; A DFT Treatment

Baghdad Science Journal ◽

10.21123/bsj.2019.16.3(suppl.).0726 ◽

2019 ◽

Vol 16 (3(Suppl.)) ◽

pp. 0726

Author(s):

Kubba Et al.

Keyword(s):

Density Functional ◽

Point Group ◽

Normal Modes ◽

Dft Method ◽

Basis Set ◽

Single Wall Carbon Nanotubes ◽

Equilibrium Geometry ◽

Vibration Frequencies ◽

Bending Vibrations ◽

Modes Of Vibration

Density Functional Theory (DFT) method of the type (B3LYP) and a Gaussian basis set (6-311G) were applied for calculating the vibration frequencies and absorption intensities for normal coordinates (3N-6) at the equilibrium geometry of the Di and Tetra-rings layer (6, 0) zigzag single wall carbon nanotubes (SWCNTs) by using Gaussian-09 program. Both were found to have the same symmetry of D6d point group with C--C bond alternation in all tube rings (for axial bonds, which are the vertical C--Ca bonds in rings layer and for circumferential bonds C—Cc in the outer and mid rings bonds). Assignments of the modes of vibration IR active and inactive vibration frequencies (symmetric and asymmetric modes) based on the image modes applied by the Gaussian 09 display. The whole relations for the vibration modes were also done including nCH stretching, nC--C stretching, δCH, δring (δC--C--C) deformation in plane of the molecule) and gCH, gring (gC--C--C) deformation out of plane of the molecule. The assignment also included modes of puckering, breathing and clock-anticlockwise bending vibrations.

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