Chirality fittingness of an orbit governed by a coset representation. Integration of point-group and permutation-group theories to treat local chirality and prochirality

1990 ◽  
Vol 112 (9) ◽  
pp. 3390-3397 ◽  
Author(s):  
Shinsaku Fujita
Keyword(s):  
Permutation Group ◽  
Point Group ◽  
Coset Representation ◽  
Chirality Fittingness

scholarly journals Application of GAP (groups, algorithms and programming) system to stereoisograms for characterizing RS-stereoisomers of cubane derivatives

2021 ◽  
Vol 87 (2) ◽  
pp. 207-270
Author(s):  
Shinsaku Fujita ◽  
Keyword(s):  
Permutation Group ◽  
Point Group ◽  
Gap Functions ◽  
Programming System ◽  
Cycle Index ◽  
Index Method ◽  
Type V ◽  
Partial Cycle

The PCI (Partial-Cycle-Index) method of Fujita’s USCI (Unit-Subduced-CycleIndex) approach has been applied to symmetry-itemized enumerations of cubane derivatives, where groups for specifying three-aspects of symmetry, i.e., the point group for chirality/achirality, the RS-stereogenic group for RS-stereogenicity/RS-astereogenicity, and the LR-permutation group for sclerality/ascrelarity are considered as the subgroups of the RS-stereoisomeric group . Five types of stereoisograms are adopted as diagrammatical expressions of , after combined-permutation representations (CPR) are created as new tools for treating various groups according to Fujita’s stereoisogram approach. The use of CPRs under the GAP (Groups, Algorithms and Programming) system has provided new GAP functions for promoting symmetry-itemized enumerations. The type indices for characterizing stereoisograms (e.g., for a type-V stereoisogram) have been sophisticated into RS-stereoisomeric indices (e.g., for a cubane derivative with the composition ). The type-V stereoisograms for cubane derivatives with the composition are discussed under extended pseudoasymmetry as a new concept.

scholarly journals Standard mark table and standard USCI-CF table of the point group D6h. Unification of candidates derived from different D6h-skeletons

2021 ◽  
Vol 87 (3) ◽  
pp. 481-525
Author(s):  
Shinsaku Fujita ◽  
Keyword(s):  
Point Group ◽  
Gap Function ◽  
Combinatorial Enumeration ◽  
Cycle Index ◽  
Single Standard ◽  
Partial Cycle ◽  
Unit Subduced Cycle Index ◽  
Chirality Fittingness ◽  
Usci Approach ◽  
Cycle Indices

Combined-permutation representations (CPRs) for characterizing -skeletons (a benzene skeleton, a Haworth-projected skeleton, a superphane skeleton, and a coronene skeleton) are constructed by starting from respective sets of generators, where the permutation of each generator is combined with a mirror-permutation of 2-cycle to treat both achiral and chiral substituents under the GAP system. Thereby, the CPR of degree 8 (= 6 + 2) for the benzene skeleton, the CPR of degree 14 (= 12 + 2) for the Haworth-projected skeleton, the CPR of degree 14 (= 12 + 2) for the superphane skeleton, the CPR of degree 14 (= 12 + 2) for the coronene skeleton are generated to give primary mark tables (tables of marks) based on these CPRs. These primary mark tables generated by the GAP system are different in the sequence of subgroups from each other, although they stem from the same point group . They are unified into a single standard mark table by means of a newly-devised GAP function MarkTableforUSCI. Moreover, another newly-devised GAP function constructUSCITable is employed to construct a standard USCI-CF (unit-subduced-cycle-index-with-chirality-fittingness) table concordantly. After a set of PCI-CFs (partial cycle indices with chirality fittingness) is calculated for each skeleton, symmetry-itemized combinatorial enumeration is conducted by means of the PCI method of Fujita’s USCI approach (S. Fujita, Symmetry and Combinatorial Enumeration in Chemistry, Springer-Verlag, Berlin-Heidelberg, 1991).

scholarly journals The Valence-Bond (VB) Model and Its Intimate Relationship to the Symmetric or Permutation Group

Molecules ◽  
2021 ◽  
Vol 26 (15) ◽  
pp. 4524
Author(s):  
Marco Antonio Chaer Nascimento
Keyword(s):  
Permutation Group ◽  
Point Group ◽  
Valence Bond ◽  
Fundamental Difference ◽  
Wave Functions ◽  
Group Symmetry ◽  
Chemical Bonds ◽  
Permutation Symmetry ◽  
Independent Entity ◽  
The Many

VB and molecular orbital (MO) models are normally distinguished by the fact the first looks at molecules as a collection of atoms held together by chemical bonds while the latter adopts the view that each molecule should be regarded as an independent entity built up of electrons and nuclei and characterized by its molecular structure. Nevertheless, there is a much more fundamental difference between these two models which is only revealed when the symmetries of the many-electron Hamiltonian are fully taken into account: while the VB and MO wave functions exhibit the point-group symmetry, whenever present in the many-electron Hamiltonian, only VB wave functions exhibit the permutation symmetry, which is always present in the many-electron Hamiltonian. Practically all the conflicts among the practitioners of the two models can be traced down to the lack of permutation symmetry in the MO wave functions. Moreover, when examined from the permutation group perspective, it becomes clear that the concepts introduced by Pauling to deal with molecules can be equally applied to the study of the atomic structure. In other words, as strange as it may sound, VB can be extended to the study of atoms and, therefore, is a much more general model than MO.

Crystal Structure Analysis of Cu-Zr Alloys by Convergent Beam Diffraction

1981 ◽  
Vol 39 ◽  
pp. 354-355
Author(s):  
A. F. Marshall ◽  
J. W. Steeds ◽  
D. Bouchet ◽  
S. L. Shinde ◽  
R. G. Walmsley
Keyword(s):  
Crystal Structure ◽  
Point Group ◽  
Intermetallic Phase ◽  
Three Dimensional ◽  
Crystal Structure Analysis ◽  
Ion Milling ◽  
Composition And Structure ◽  
Unit Cells ◽  
Sputter Deposited ◽  
Convergent Beam

Convergent beam electron diffraction is a powerful technique for determining the crystal structure of a material in TEM. In this paper we have applied it to the study of the intermetallic phases in the Cu-rich end of the Cu-Zr system. These phases are highly ordered. Their composition and structure has been previously studied by microprobe and x-ray diffraction with sometimes conflicting results.The crystalline phases were obtained by annealing amorphous sputter-deposited Cu-Zr. Specimens were thinned for TEM by ion milling and observed in a Philips EM 400. Due to the large unit cells involved, a small convergence angle of diffraction was used; however, the three-dimensional lattice and symmetry information of convergent beam microdiffraction patterns is still present. The results are as follows:1) 21 at% Zr in Cu: annealed at 500°C for 5 hours. An intermetallic phase, Cu3.6Zr (21.7% Zr), space group P6/m has been proposed near this composition (2). The major phase of our annealed material was hexagonal with a point group determined as 6/m.

Friedel'S law breakdown and interpretation of stacking fault images in tetrahedral semiconductors

1987 ◽  
Vol 45 ◽  
pp. 318-319
Author(s):  
Rob. W. Glaisher ◽  
A.E.C. Spargo
Keyword(s):  
Relative Phase ◽  
Point Group ◽  
Binary Compound ◽  
Stacking Fault ◽  
Fold Axis ◽  
Atom Pair ◽  
Tetrahedral Semiconductors ◽  
Group Symmetries ◽  
Thin Crystal ◽  
Phase Differences

Images of <11> oriented crystals with diamond structure (i.e. C,Si,Ge) are dominated by white spot contrast which, depending on thickness and defocus, can correspond to either atom-pair columns or tunnel sites. Olsen and Spence have demonstrated a method for identifying the correspondence which involves the assumed structure of a stacking fault and the preservation of point-group symmetries by correctly aligned and stigmated images. For an intrinsic stacking fault, a two-fold axis lies on a row of atoms (not tunnels) and the contrast (black/white) of the atoms is that of the {111} fringe containing the two-fold axis. The breakdown of Friedel's law renders this technique unsuitable for the related, but non-centrosymmetric binary compound sphalerite materials (e.g. GaAs, InP, CdTe). Under dynamical scattering conditions, Bijvoet related reflections (e.g. (111)/(111)) rapidly acquire relative phase differences deviating markedly from thin-crystal (kinematic) values, which alter the apparent location of the symmetry elements needed to identify the defect.

Symmetry breaking in convergent-beam diffraction

1989 ◽  
Vol 47 ◽  
pp. 480-481
Author(s):  
D.J. Eaglesham
Keyword(s):  
Symmetry Breaking ◽  
Point Group ◽  
X Ray Diffraction ◽  
Multiple Diffraction ◽  
Space Groups ◽  
Atomic Displacements ◽  
Small Probe ◽  
Phase Sensitive ◽  
Convergent Beam

Convergent Beam Electron Diffraction is now almost routinely used in the determination of the point- and space-groups of crystalline samples. In addition to its small-probe capability, CBED is also postulated to be more sensitive than X-ray diffraction in determining crystal symmetries. Multiple diffraction is phase-sensitive, so that the distinction between centro- and non-centro-symmetric space groups should be trivial in CBED: in addition, the stronger scattering of electrons may give a general increase in sensitivity to small atomic displacements. However, the sensitivity of CBED symmetry to the crystal point group has rarely been quantified, and CBED is also subject to symmetry-breaking due to local strains and inhomogeneities. The purpose of this paper is to classify the various types of symmetry-breaking, present calculations of the sensitivity, and illustrate symmetry-breaking by surface strains.CBED symmetry determinations usually proceed by determining the diffraction group along various zone axes, and hence finding the point group. The diffraction group can be found using either the intensity distribution in the discs

TEM study of lanthanum aluminate

1990 ◽  
Vol 48 (4) ◽  
pp. 1060-1061
Author(s):  
C.Y. Yang ◽  
Z.R. Huang ◽  
Y.Q. Zhou ◽  
C.Z. Li ◽  
W.H. Yang ◽  
...  
Keyword(s):  
Domain Boundary ◽  
Point Group ◽  
Optical Microscope ◽  
Dark Field ◽  
Zone Axis ◽  
Superconducting Film ◽  
Ion Milling ◽  
Lanthanum Aluminate ◽  
Ring Diameter ◽  
Diffraction Patterns

Lanthanum aluminate(LaAlO3) single crystal as a substrate for high Tc superconducting film has attracted attention recently. We report here a transmission electron microscopy(TEM) study of the crystal structure and phase transformation of LaAlO3 by using Philips EM420 and EM430 microscopes. Single crystals of LaAlO3 were investigated first by optical microscope. Stripe-shaped domains of mm size are clearly seen(Fig.1a), and 90° domain boundary is also obvious. TEM specimens were prepared by mechanical grinding and polishing followed by ion-milling.Fig.lb shows μm size stripe domains of LaAlO3. Convergent beam electron diffraction patterns (CBED) from single domain were taken.Fig. 2a and Fig. 2c are [001] zone axis patterns which show a 4mm symmetry, and the (200) dark field of this zone axis gives 2mm symmetry(fig.2b). Therefore the point group of this crystal is either 4/mmm or m3m. The projection of the first order Laue zone(FOLZ) reflections on zero layer (fig. 2c) shows that the unit cell is face centered. A tetragonal unit ceil is chosen, with a=0.532nm and c=0.753nm, c being determined from the FOLZ ring diameter.

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