Order symmetry

The chemistry of propylene is characterized both by the double bond and by the aHyUc hydrogen atoms. Propylene is the smallest stable unsaturated hydrocarbon molecule that exhibits low order symmetry, ie, only reflection along the main plane. This loss of symmetry, which implies the possibiUty of different types of chemical reactions, is also responsible for the existence of the propylene dipole moment of 0.35 D. Carbon atoms 1 and 2 have trigonal planar geometry identical to that of ethylene. Generally, these carbons are not free to rotate, because of the double bond. Carbon atom 3 is tetrahedral, like methane, and is free to rotate. The hydrogen atoms attached to this carbon are aUyflc. 

Anisotropic dipole interaction in two-dimensional lattices with a no higher than the second-order symmetry axis leads to the fact that the function J(k) does not go to zero at any k. Actually, in this case / = 1, P h, 12) > Pi(h, 12) in Eq. (2.2.14) and... 

One might suppose it would be easier, and perhaps more profitable, to examine, observe, and experiment, than to turn one s eyes inwards with the hope of discovering exactly "the road followed by the Great Architect of the Universe in the creation of the world." But the alchemical method found it easier to begin by introspection. The alchemist spun his universe from his own ideas of order, symmetry, and simplicity, as the spider spins her web from her own substance. 

Figure 11.4. The symmetry elements of the five 2D lattices. For each pair the lattice is represented on the left by the points defining one unit cell and on the right are the symmetry elements. Different orientations of the symbols are used to differentiate nonequivalent axes of the same order. Symmetry symbols in parentheses are redundant, that is, indicate those that arise automatically from those preceding them.

In 1984 a very unusual metallic alloy was discovered [21]. The structure of this alloy was characterized by the simultaneous presence of the five-order symmetry and the long-range ordering of atoms (this combination is fully impossible from the point of view of classical crystallography). The active discussion, scepticism, and refusal from the side of representatives of traditional crystallography were initiated by this discovery. The accusations of both violation of the first principles and false-scientificy in treating of the obtained results were... 

When electron-phonon interactions are taken into account, the regular conducting chains are found to be unstable they undergo a low-temperature phase transition which results in both a modulation of the chains and the opening of an energy gap at the Fermi level. This is the so-called Peierls transition, at T = TP, a second-order symmetry-breaking transition that is characteristic of the quasi-one-dimensional conductors [2,3]. 

We have found that the structure of the empty (8,8) CNT at the ground state is the Kekule structure with tripling of the translational period and two essentially different bonds (Fig. 1). The structure of C20 is shown in Fig. 2. It has Di symmetry with three orthogonal second order symmetry axes, one of which is shown in Fig. 2 by the dashed line. 

Merrifield-Simmons bond order -> symmetry descriptors (O Merrifield-Simmons index) Merrifield-Simmons index -> symmetry descriptors mesomeric effect -> electronic substituent constants... 

It is necessary to number the boron skeleton for each cluster systematically, so as to permit the unambiguous naming of the substituted derivatives. For this purpose, the boron atoms of closo structures are considered to occupy planes disposed sequentially, perpendicular to the axis of highest order symmetry. (If there are two such axes, the longer , in terms of the greater number of perpendicular planes crossed, is chosen.)... 

In order to analyze the interaction energy component, the symmetry-adapted perturbation theory (SAPT) [12] calculations were performed. SAPT have been used to analyze the interaction energies in terms of electrostatic, induction, dispersion, and exchange interaction components. The SAPT interaction energy (Dint) has been analyzed up to the second-order symmetry adapted perturbation theory the electrostatic energy (Dekt) consisting of and... 

Merrifield-Simmons bond order symmetry descriptors ( Merrifield-Simmons index)... 

Chiral A geometric figure, or group of points is chiral if it is nonsuperimposable on its mirror image [82]. A chiral object lacks all of the second order (improper) symmetry elements, a mirror plane), i center of symmetry), and S rotation-reflection axis). In chemistry, the term is (properly) only applied to entire molecules, not to parts of molecules. A chiral compound may be either racemic or nonracemic. An object that has any of the second order symmetry elements i.e., that is superimposable on its mirror image) is achiral. It is inappropriate to use the adjective chiral to modify an abstract noun one cannot have a chiral opinion and one cannot execute a chiral resolution or synthesis. 

Lao, K. U., and Herbert, J. M. (2012). Breakdown of the single-exchange approximation in third-order symmetry-adapted perturbation theory, Phys. Chem. A 116, pp. 3042-3047, doi 10.1021/jp300109y. 

Chinese gallotannin see Tannins Chirality the necessary and sufficient condition for optical activity (rotation of the plane of polarized light). C. means handedness (from the Greek Kelp = hand). Chiral molecules have no second order symmetry element (center, plane or axis of symmetry) and exist in two mirror-image forms (enantiomers) which cannot be rotated in such a way as to coincide. Most chiral compounds contain an asymmetrically substituted C-atom, i.e. a tetrahedral C-atom with 4 different substituents [E.L.EIiel, S.H.Wilen L.N.Mander Stereochemistry of Organic Compounds, Wiley Sons New York, 1994]... 

Both the metal-metal a and ct orbitals should be filled for dinuclear Pd(II) complexes (Fig. 20). On the basis of these considerations alone, no attractive metal-metal interaction is expected there is no Pd-Pd bond [87]. Second-order, symmetry-allowed mixing of the Pd d 2 orbital with the and the 5s orbital, however, perturbs the molecular orbital diagram based only on d orbital interactions [88-91]. In 2010, an evaluation of the bonding interactions between the Pd centers in acetate-bridged, dinuclear Pd(II) complex 36 was reported (Fig. 21) [88]. On the basis of DFT calculations, the authors predicted a weak attractive interaction between the Pd centers in 36 and computed a Pd-Pd bond order of 0.11. 

An amorphous solid is characterized by the lack of long-range order symmetry operators (translational, orientational, and conformational order) found in crystalline solid. The absence of long-range order can be ascribed to a random distribution of molecular units. Individual molecules are randomly oriented to one another and exist in a variety of conformational states. The molecular pattern of an amorphous solid is often depicted as that of a frozen liquid with the viscosity of a solid having many internal degrees of freedoms and conformational diversities (disorder). An amorphous solid, at the molecular level, has properties similar to liquids but at the macroscopic level, it has properties of solids. 

The fourth order symmetry axis (Af) has the same effect with the tetra-gyroid (A ). Both complex operations require the existence of a second order gyre. For this reason, the tetra-gyroid is noted by A4. 

The superior category has no singular directions and possesses more higher-order symmetry axis, where the cube belongs. 

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