Showing papers on "Symmetry (geometry) published in 1971"

Symmetry, topology, and aromaticity

Abstract: The 4q + 2 T-electron prerequisite for cyclic stabilization is shown to be a necessary consequence whenever orbitals of ir symmetry are constrained to interact within a pericyclic topology. Corresponding symmetryimposed rules are then derived for three other topologies in a way that permits still further extension. Finally, a topological definition of aromaticity is provided as a stimulus and guide to further experiments. or more than two decades, the Huckel rule has F helped to fashion the development of contemporary organic chemistry.2 Through the ingenuity and the diligence of organic chemists, the “aromatic character” of fully conjugated, monocyclic hydrocarbons containing 4q + 2 T electrons has now been opened to scrutiny for values of q from 0 to 7. During this process, the original structural prerequisite for the 4q + 2 rule has also been subjected to extensive variations. Among these, we note the use of dehydro derivatives (e.g., 13) and of spanning alkyl fragments (as in z 4 and 3 7 to prevent intramolecular cyclization. Distortions from coplanarity have thus become commonplace and even the otherwise continuous polyene conjugation has been interrupted (cf. 46). Indeed, only the essentially pericyclic x-electron topology has been left more or less intact.

Studies of Vibrational Surface Modes. I. General Formulation

Abstract: A general formulation is given for studies of the vibrational properties of systems which have two-dimensional periodicity and one or two surfaces. Although layered structures and other systems with interfaces fall within the scope of this formulation, the principal motivation is to provide a framework for calculating and interpreting vibrational surface properties. No assumption is made concerning crystal structure, surface orientation, the interaction between particles, or the number of particles per unit cell. Also, the treatment is applicable to reconstructed surfaces, surfaces with adsorbed impurity particles, etc., as well as unreconstructed clean surfaces, provided that the two-dimensional periodicity is preserved. A discussion is given of the properties of the vibrational modes: In general, the displacement ellipse for a given mode can have any orientation. For surfaces with "axial-inversion symmetry," however, one axis of the ellipse is always normal to the surface. If the surface has "complete reflection symmetry" with respect to a given plane, then for any two-dimensional wave vector parallel to the plane the modes will separate into two classes: one-third of the modes will be pure shear-horizontal (SH) modes, and the other two-thirds will be polarized strictly in the sagittal plane. It is possible for surface modes of one class to lie within the bulk subbands of the other class. If the crystal has either axial-inversion symmetry or a three-dimensional center of inversion, then the complex dynamical matrix can be reduced to a real, symmetric matrix of the same size. If both symmetries are present, as is the case for many surfaces of interest, then a further reduction is possible. Finally, notations are suggested for distinguishing two-dimensional vectors and for labeling symmetry points in the two-dimensional Brillouin zone associated with a surface.

On the Sign Test for Symmetry

Abstract: This article studies the effect of estimating the center of symmetry by the sample mean on the sign test for symmetry. Under the null hypothesis, the modified sign test is not distribution-free. Indeed, the asymptotic variance of the modified sign test can be quite different from that of the standard sign test.

Attitude stability of a symmetric satellite at the equilibrium points in the restricted three-body problem

Abstract: A problem of attitude motion of the smallest body for the restricted three-body problem is analyzed. Axial symmetry is assumed for the body, and attention is focused on the case in which the symmetry axis is normal to the orbit plane. For libration point satellites, results are similar to those for a satellite in orbit about a single body. However, for orbit equilibrium points lying on the line joining the two larger bodies, attitude stability results depart markedly from those for the two-body problem.

Symmetry coupling coefficients for the 0* group

Abstract: The V symmetry coupling coefficients for the octahedral double group are generated from the behaviour of a minimum number of |JM> ket vectors where the phases are determined from the way the eight irreducible representations are defined. Using the generated V symmetry coupling coefficients, the behaviour of the |JM> ket vectors under the 0* group are determined for integral and half-integral values of J up to J = 9.

The isotropic invariants of fifth‐rank cartesian tensors

Abstract: An orthonormal set of irreducible fifth-rank tensors having the required permutation symmetry is constructed. Various problems not encountered in the analogous problem for tensors of ranks two, three, four and six are discussed.

Structural relationships in compounds with Rc symmetry

Abstract: Introduction A trifluoride series BF3 (B = Ru, Co, V, Fe, Ti) is related to different perovskite-type series AB03 (A = Bi, Pb, La, Li and B= Fe, Zr-Ti, Co, AI, Nb, Ta) as being based on the same anion framework. This anion framework can be viewed as generated by linear chains of regular octahedra rotated from the ideal close-packed configuration. Experimental data for these compounds are in excellent agreement with the relationships between atomic parameters and the unit-cell dimensions developed from a study of this model possessing R~c symmetry.

Harmonic Analysis of Electron Microscope Images with Rotational Symmetry

Abstract: This paper describes a method of analysing images from electron micrographs of biological specimens believed to possess rotational symmetry. An objective analysis of the symmetry is possible because the method, which is computational, produces a rotational power spectrum of the image. We can then combine just those components which are consistent with the previously determined symmetry to produce a altered image. The method is applied to the base plate of bacteriophage T4 and to discs of tobacco mosaic virus protein. The advantages of this new approach over the well-known Markham rotation technique are discussed.

Bond length alternation in infinitely long conjugated polyacenes

Abstract: This paper concludes that whereas linear polyacenes are stable to distortions that preserve mirror symmetry with respect to the plane which is the perpendicular bisector of their interior bonds, they are unlikely to be stable with respect to distortions that do not preserve this symmetry.

Orbital symmetry rules and the mechanisms of inorganic reactions

Abstract: In the last few years symmetry arguments have been used very effectively to predict the course of chemical reactions. The Woodward—Hoffmann rules are famous examples. A complete, but simple, theory of how symmetry enters into a chemical process can be given. Use is made of group theory and secondorder quantum mechanical perturbation theory. The resulting simple equations can be reduced even further to a consideration of the symmetry of the molecular orbitals of the reactants. The relevant orbitals are the highest filled (HOMO) and the lowest empty (LU MO) with the correct symmetries to match the symmetry of the reaction coordinate. The closer in energy these orbitals are, the lower the activation energy. An orbital symmetry forbidden reaction is one where no orbitals of the right symmetry exist within a reasonable energy range of each other. In the usual case it is unnecessary to know the molecular orbital scheme of the products. For bimolecular and trimolecular reactions, the reaction coordinate must be totally symmetrical, therefore the symmetry requirement for the HOMO and LUMO is that they have a net positive overlap. For unimolecular reactions, the reaction path need not be totally symmetrical. The direct product of the HOMO and LUMO symmetries determines the symmetry of the reaction coordinate. The HOMO and LUMO also must correspond to bonds that are to be broken and bonds that are to be made; if they are bonding MOs the reverse statement holds true for antibonding MOs. Examples are given for all of these rules. The development of so-called orbital symmetry rules for chemical reactions has had a great impact on organic chemistry'. Corresponding rules for inorganic reactions have not been extensively presented or used up to now. The attemptin the literature2 have dealt only with the d orbitals of transition metal complexes. The conclusions have been neither definitive nor consistent. While d orbitals are of great importance in coordination chemistry, it is unlikely that these are the only important orbitals in chemical reactions. Also much of inorganic chemistry deals with the non-transition elements. It is necessary to include molecular orbitals made up of s and p atomic orbitals to have a complete understanding. In this article we will show in the most general way how symmetry rules for all chemical reactions can be derived3. The procedure used is to consider the variation of potential energy with

Internal-symmetry groups and their automorphisms.

Abstract: It is proposed that outer automorphisms of degenerate internal-symmetry groups must be symmetry operators themselves. In general, however, they are hidden (spontaneously broken) symmetries. Consequences of this proposition are studied. It is found that internal-symmetry groups are not arbitrary but that their intrinsic properties play an important role. The existence of discrete symmetries (such as charge conjugation) follows naturally from assuming the continuous symmetry groups (such as gauge groups). We also find that the enlargement of the isospin symmetry and parity leads directly to the chiral SU(3) \ifmmode\times\else\texttimes\fi{} SU(3), so that the existence of an "exact SU(3) limit" is in principle not allowed.

Measures of symmetry

Abstract: Three discrimination, rating, and ranking experiments were conducted to test the utility of two new single-form measures of symmetry in predicting response to plane figures. The predictive power of the new measures varied with the type of the perceptual task, prediction being better for tasks of a more clearly perceptual nature. The rest of variance in response was due to poorly defined conceptions of symmetry in many Ss, the confusion of symmetry with compactness, and the overestimation of symmetry in figures that approximate in appearance their ideally symmetric prototypes. These and previously conducted experiments lead to the conclusion that in many instances the symmetry parameter of random polygons has no significant influence on response variance and that, for figures of a given level of complexity, the only practically significant predictor of response is a measure of dispersion of shape contours.

Note on the new symmetry of the racah coefficients.

Abstract: It is shown that the ``new symmetry'' of Racah coefficients, recently derived in a paper by Minton [J. Math. Phys. 11, 3061 (1970)], does not exist.

An unconventional approach to discrete point groups of symmetry elements in quantum mechanics

Abstract: A tentatively complete set of independent and commuting symmetry operators is defined for point groups of symmetry elements. Eigenvalues of these operators are discussed and the relation of the eigenvalue description to the standard notation of irreducible representations is shown. A new useful type of symmetry projection operator is introduced.

Totally real representations and real function spaces

Abstract: 0* Introduction* Let X be a riemannian symmetric space of compact type and G its largest connected group of isometries. In his 1929 paper [1] on class 1 representation s, E. Cartan showed that the symmetry of X sends every uniformly closed G-invariant function space on X to its complex conjugate. Starting from the point of view of algebras, Mirkil and de Leeuw [4] showed that every rotation invariant function algebra on the sphere Sn(n ^ 2) was spanned by real-valued functions, hence (Stone-Weierstrass theorem) that such an algebra necessarily was all continuous functions on Sn, all continuous functions on real protective w-space, or just the constantsthat state of affairs is quite different from the case n = 1. When the rotation group S0(n + 1) contains the symmetry of S w, i.e. when n is even, Cartan's result mentioned above implies reality of such function algebras. The published Mirkil-de Leeuw argument rests rather on the fact that the spherical harmonics are real-valued.

A clarification of differences in site orientation in crystals

Abstract: Tables are presented whereby physically relevant differences in the relationship of the elements of symmetry of a Wyckoff site to those of the crystal class can be distinguished.

Symmetry. Fundamentals of Crystallography

Abstract: A. Definitions and Basic Concepts. Symmetryis the property of geometrical figures which have identical and regularly distributed parts. We shall consider only finite figures although there is, in general, no limitation to the size of a figure in point symmetry. The laws governing the uniform distribution of equal parts in a figure are given by symmetry operations or symmetry transformations. The science of symmetry shows that symmetry transformations of three-dimensional figures are simple (proper) rotation and rotation accompanied by reflection (improper or reflection rotation).*