scholarly journals The Symmetry of Linear Molecules

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.

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

2008 ◽  
Vol 73 (6-7) ◽  
pp. 937-944 ◽  
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.

scholarly journals Electron and Positron Scattering by Non-Central Potentials: Matrix Elements and Symmetry Properties in the First Born Approximation

2021 ◽  
Author(s):  
Marcos Barp ◽  
Felipe Arretche
Keyword(s):  
Fourier Transform ◽  
Born Approximation ◽  
Scattering Amplitude ◽  
Point Group ◽  
Computational Effort ◽  
Positron Scattering ◽  
Symmetry Properties ◽  
Linear Molecules ◽  
The Fourier Transform ◽  
Symmetry Relations

The Fourier transform of Cartesian Gaussian functions product is presented in the light of positron scattering. The calculation of this class of integrals is crucial in order to obtain the scattering amplitude in the first Born approximation framework for an ab initio method recently proposed. A general solution to the scattering amplitude is given to a molecular target with no restriction due to symmetry. Moreover, symmetry relations are presented with the purpose of identifying terms that do not contribute to the calculation for the molecules in the D∞h point group optimizing the computational effort. Keywords — Positron and electron scattering, Fourier transform of the Gaussian product theorem, McMurchie-Davidson procedure, Obara-Saika procedure, linear molecules .

Hydrothermal Synthesis and Solid-State Laser Refrigeration of Ytterbium-Doped Potassium Lutetium Fluoride (KLF) Microcrystals

2020 ◽  
Author(s):  
Xiaojing Xia ◽  
Anupum Pant ◽  
Xuezhe Zhou ◽  
Elena Dobretsova ◽  
Alex Bard ◽  
...  
Keyword(s):  
Hydrothermal Synthesis ◽  
Solid State ◽  
Information Science ◽  
Point Group ◽  
Solid State Laser ◽  
Lanthanide Ions ◽  
Group Symmetry ◽  
Luminescence Spectra ◽  
Crystalline Phases ◽  
State Laser

Fluoride crystals, due to their low phonon energies, are attractive hosts of trivalent lanthanide ions for applications in upconverting phosphors, quantum information science, and solid-state laser refrigeration. In this article, we report the rapid, low-cost hydrothermal synthesis of potassium lutetium fluoride (KLF) microcrystals for applications in solid-state laser refrigeration. Four crystalline phases were synthesized, namely orthorhombic K<sub>2</sub>LuF<sub>5</sub> (Pnma), trigonal KLuF<sub>4</sub> (P3<sub>1</sub>21), orthorhombic KLu<sub>2</sub>F<sub>7</sub> (Pna2<sub>1</sub>), and cubic KLu<sub>3</sub>F<sub>10</sub> (Fm3m), with each phase exhibiting unique microcrystalline morphologies. Luminescence spectra and emission lifetimes of the four crystalline phases were characterized based on the point-group symmetry of trivalent cations. Laser refrigeration was measured by observing both the optomechanical eigenfrequencies of microcrystals on cantilevers in vacuum, and also the Brownian dynamics of optically trapped microcrystals in water. Among all four crystalline phases, the most significant cooling was observed for 10%Yb:KLuF<sub>4</sub> with cooling of 8.6 $\pm$ 2.1 K below room temperature. Reduced heating was observed with 10%Yb:K<sub>2</sub>LuF<sub>5</sub>

scholarly journals Multiplicity, Parity and Angular Momentum of a Cooper Pair in Unconventional Superconductors of D4h Symmetry: Sr2RuO4 and Fe-Pnictide Materials

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1435
Author(s):  
Victor G. Yarzhemsky
Keyword(s):  
Angular Momentum ◽  
Cooper Pair ◽  
Point Group ◽  
Group Symmetry ◽  
Cooper Pairs ◽  
Group Approach ◽  
Angular Momentum Projection ◽  
Space Group ◽  
Unconventional Superconductors ◽  
Even Pairs

Sr2RuO4 and Fe-pnictide superconductors belong to the same point group symmetry D4h. Many experimental data confirm odd pairs in Sr2RuO4 and even pairs in Fe-pnictides, but opposite conclusions also exist. Recent NMR results of Pustogow et al., which revealed even Cooper pairs in Sr2RuO4, require reconsideration of symmetry treatment of its SOP (superconducting order parameter). In the present work making use of the Mackey–Bradley theorem on symmetrized squares, a group theoretical investigation of possible pairing states in D4h symmetry is performed. It is obtained for I4/mmm , i.e., space group of Sr2RuO4, that triplet pairs with even spatial parts are possible in kz direction and in points M and Y. For the two latter cases pairing of equivalent electrons with nonzero total momentum is proposed. In P4/nmm space group of Fe- pnictides in point M, even and odd pairs are possible for singlet and triplet cases. It it shown that even and odd chiral states with angular momentum projection m=±1 have nodes in vertical planes, but Eg is nodal , whereas Eu is nodeless in the basal plane. It is also shown that the widely accepted assertion that the parity of angular momentum value is directly connected with the spatial parity of a pair is not valid in a space-group approach to the wavefunction of a Cooper pair.

Raman-Spektren von SnJ4, GeJ4, TiBr4 und TiJ4 in flüssiger und kristalliner Phase / Raman Spectra of SnJ4, GeJ4, TiBr4, and TiJ4 in the Liquid and Crystalline Phase

1970 ◽  
Vol 25 (12) ◽  
pp. 1374-1381 ◽  
Author(s):  
W. Kiefer ◽  
H. W. Schrötter
Keyword(s):  
Raman Spectra ◽  
Ruby Laser ◽  
Detection System ◽  
Point Group ◽  
Visible Region ◽  
Group Symmetry ◽  
Pure Liquid ◽  
Electronic Detection ◽  
Polarization Studies ◽  
First Time

The Raman spectra of four molecules absorbing in the visible region (SnJ4, GeJ4, TiBr4, and TiJ4) are presented. They were excited with a quasi-continuous ruby laser and recorded with a special electronic detection system. Except for TiJ4, complete Raman spectra of crystal powder pellets could be obtained for the first time. The assignment reported by previous authors was confirmed by accurate polarization studies of solutions or pure liquid. The assignment is also in the solid state possible on the basis of Td point group symmetry. The fundamental vibrations of TiJ4 in solutions are: ν1 (A1) =162, ν2 (E) =51, ν3 (F2) =319 and ν4 (F2) Y = 67 cm-1

Gas Phase Spin Relaxation of ZX3Y Molecules in the Dipolar Approximation

1975 ◽  
Vol 53 (7) ◽  
pp. 723-738 ◽  
Author(s):  
B. C. Sanctuary ◽  
R. F. Snider
Keyword(s):  
Kinetic Theory ◽  
Gas Phase ◽  
Magnetic Relaxation ◽  
Dipolar Coupling ◽  
Point Group ◽  
Group Symmetry ◽  
Subsequent Paper ◽  
Point Group Symmetry ◽  
Proper Treatment ◽  
Gas Kinetic Theory

The gas kinetic theory of nuclear magnetic relaxation of a polyatomic gas, as formulated in the previous paper, is evaluated for ZX3Y molecules relaxing via a dipolar coupling Hamiltonian. Stress is given to a proper treatment of point group symmetry, here C3v, and the possibility of molecular inversion is included. The detailed formula for the spin traces is however restricted to X nuclei with spin 1/2. A subsequent paper uses these results to elucidate the structure of the high density dependence of T1 forCF3H.

scholarly journals Spin-reversal energy barriers of 305 K for Fe2+ d6 ions with linear ligand coordination

Nanoscale ◽  
2017 ◽  
Vol 9 (30) ◽  
pp. 10596-10600 ◽  
Author(s):  
Lei Xu ◽  
Ziba Zangeneh ◽  
Ravi Yadav ◽  
Stanislav Avdoshenko ◽  
Jeroen van den Brink ◽  
...  
Keyword(s):  
Solid State ◽  
Magnetic Anisotropy ◽  
Point Group ◽  
Group Symmetry ◽  
Point Group Symmetry ◽  
State Matrix ◽  
Magnetic Anisotropy Energy ◽  
Spin Reversal ◽  
Ligand Coordination ◽  
Li Ion

A remarkably large magnetic anisotropy energy of 305 K is computed by quantum chemistry methods for divalent Fe2+ d6 substitutes at Li-ion sites with D6h point-group symmetry within the solid-state matrix of Li3N.

scholarly journals Crystal structures of isotypic poly[bis(benzimidazolium) [tetra-μ-iodido-stannate(II)]] and poly[bis(5,6-difluorobenzimidazolium) [tetra-μ-iodido-stannate(II)]]

2014 ◽  
Vol 70 (10) ◽  
pp. 178-182 ◽  
Author(s):  
Iwan Zimmermann ◽  
Tony D. Keene ◽  
Jürg Hauser ◽  
Silvio Decurtins ◽  
Shi-Xia Liu
Keyword(s):  
Hydrogen Bonds ◽  
Point Group ◽  
Van Der Waals Interactions ◽  
Group Symmetry ◽  
Layered Perovskite ◽  
Out Of Plane ◽  
Related Compounds ◽  
Perovskite Type ◽  
Inorganic Layers ◽  
Perovskite Type Structure

The isostructural title compounds, {(C7H7N2)2[SnI4]}n, (1), and {(C7H5F2N2)2[SnI4]}n, (2), show a layered perovskite-type structure composed of anionic {[SnI4]2−}nsheets parallel to (100), which are decorated on both sides with templating benzimidazolium or 5,6-difluorobenzimidazolium cations, respectively. These planar organic heterocycles mainly form N—H...I hydrogen bonds to the terminal I atoms of the corner-sharing [SnI6] octahedra (point group symmetry 2) from the inorganic layer, but not to the bridging ones. This is in contrast to most of the reported structures of related compounds where ammonium cations are involved. Here hydrogen bonding to both types of iodine atoms and thereby a distortion of the inorganic layers to various extents is observed. For (1) and (2), all Sn—I—Sn angles are linear and no out-of-plane distortions of the inorganic layers occur, a fact of relevance in view of the material properties. The arrangement of the aromatic cations is mainly determined through the direction of the N—H...I hydrogen bonds. The coherence between organic bilayers along [100] is mainly achieved through van der Waals interactions.

Phase Transitions, Superconductivity, and Ferromagnetism in Fullerene Systems

MRS Bulletin ◽  
1994 ◽  
Vol 19 (11) ◽  
pp. 28-30 ◽  
Author(s):  
C.N.R. Rao ◽  
Ram Seshadri
Keyword(s):  
Solid State ◽  
Alkaline Earth ◽  
Point Group ◽  
Group Symmetry ◽  
High Electron ◽  
Chemical Transformations ◽  
Disordered Solids ◽  
High Electron Affinity ◽  
Centers Of Mass ◽  
Image Position

By virtue of their unique structures, fullerenes exhibit novel chemical transformations. Particularly pertinent to this article are the interesting properties exhibited by fullerenes in the solid state. These molecules are spherical or near-spherical in shape. Molecules with high point-group symmetry, which are not bound strongly in the solid state, tend to crystallize into structures with long-range periodicity of the molecular centers of mass, but the molecular orientations are random or even dynamically disordered. When dynamically disordered, themolecules rotate about some preferred axis. C60 and C70 satisfy the criteria for such orientationally disordered solids and exhibit rich phase behavior in the solid state. Since C60 has high electron affinity, it forms anion salts with alkali and alkaline-earth metals as well as with strong organic donor molecules. With tetrakis dimethylaminoethylene (TDAE), which is a very powerful electron donor, C60 forms a 1:1 solid that is ferromagnetic. C60-TDAE is the molecular organic ferromagnet with the highest Tc (of 16 K) known to date. Some of the alkali and alkaline-earth fullerides, on the other hand, show superconductivity, with transition temperatures going up to 33K. We shall briefly examine some of these solid-state properties.

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