Symmetry defined

Substitutiofi products are superimpouble. There is a plane of symmetry defined by the atoms H-C(2)-D. 

A phenomenological description of the differential cross-section for emission of photoelectrons into solid angle O in the lab frame can be written, assuming random molecular orientation and an axis of cylindrical symmetry defined by the photon polarization, as... 

There are several forms of rotational symmetry. The simplest is cyclic symmetry, involving rotation about a single axis (Fig. 4—24a). If subunits can be superimposed by rotation about a single axis, the protein has a symmetry defined by convention as Gn (C for cyclic, n for the number of subunits related by the axis). The axis itself is described as an w-fold rotational axis. The a/3 protomers of hemoglobin (Fig. 4-23) are related by C2 symmetry. A somewhat more complicated rotational symmetry is dihedral symmetry, in which a twofold rotational axis intersects an w-fold axis at right angles. The symmetry is defined as DTO (Fig. 4—24b). A protein with dihedral symmetry has 2n protomers. 

Many spectroscope and magnetic studies have been concerned with empirical correlations between these parameters and features of structural and chemical interest in the molecules. It should be noticed, however, that these symmetry-based parameters are global (like /HDvv which is discussed earlier), referring to the field of all ligands as a whole. [The same is true of recent more comprehensive symmetry-defined parameters proposed by Donini et al. (17).] Being based on the minimum assumptions of ligand field theory, and hence, for some, preferred as more basic, these parameters lack possibilities for immediate chemical relevance and appeal. 

The question in the title can be reformulated by asking how much can be dug out of an analogy between broken symmetry in dissipative structures (such as the ripple marks generated by wind, i.e., an external perturbation, in an otherwise flat surface of sand) and broken symmetry defined as phenomena of condensed matter systems of the kind observed near the critical points. The value of Anderson s discussion is to be seen more in the deepening of the question itself than in the answer that cannot yet be final, and for the moment, according to the author, appears to be more on the negative side. 

The molecule has two planes of symmetry defined by the three carbons of each CH3CCH3 unit. 

Translational symmetry requires that any matter located at xea + yeb + zec must also be replicated exactly at the coordinates (x + l a ) ea + (y + m b ) eb + z + n c ) ecr where /, m, and n are integers (positive, negative, or zero). This translation symmetry defines the unit cell and the unit cell axes a, b, c. In the least symmetric case, the contents of the unit cell (atoms, ions, molecules, trapped solvents, proteins) may not have any symmetry at all. 

If the degree of symmetry increases—for example, if the solid has an additional plane of symmetry defined by the axes X2 and Xi—the matrix of director cosines will be... 

Molecular orbital calculations using 4-3IG and 6-3IG basis sets led to the conclusion that the minimum energy structure for C4Ht was the bisected cyclopropylcarbinyl cation 61 (Table 12) with a plane of symmetry defined by the formally cationic carbonA second structure (64) was also found as an energy minimum that was subtly different from 61, but possessed an unsymmetrical cyclopropyl ring with one corner bent toward the... 

Figure 2.19 The divisions of the vertices of the regular orbit of point symmetry, defining the great rhombicosidodecahedron, into decoration sets about the rotational axes points [C5], row b, [C3], row c and [C2], row d, on the unit sphere.

To accoimt for steric effects in molecule-receptor interactions, the weighted information indices by volume have been proposed [Ray et al, 1985]. These molecular descriptors are calculated in the same way as the indices of neighbourhood symmetry defined above using the atomic van der Waals volumes to get the probabilities of the equivalence classes. In other words, the van der Waals voliunes of the atoms belonging to each equivalent class are summed to give a molecule subvolume, then divided by the total molecule volume. For example, the weighted information content by volume is defined as ... 

The combination of rotational and translational symmetry defines the space group of the crystal. It is shown that 235 space groups exist, but only 65 allow the handedness of the molecule to be preserved, and so only 65 can occur in macromolecular crystallography. The space groups are numbered, but are commonly referred to by their symbols, such as P212121. The most common in macromolecular crystallography are P212121, PI, P21 and C2. 

Many protein crystals exist with more than one molecule per asymmetric unit. These molecules are sometimes related by noncrystallographic symmetry (pseudosymmetry), that is, additional symmetry (such as a twofold rotation axis) that is not part of the symmetry defined by the space group. This feature can be very useful in finding the molecular structure using rotation functions. Additionally, it is also possible for a compound to crystallize in different forms with different packing (polymorphism). If the molecular transforms of the components of the crystal are known, it is possible, by the methods described above, to determine their positions and orientations in the respective unit cells. 

Table 15. Limiting symmetry defining the nnmber of independent lattiee orientations. The (idealized) symmetries of the lattiee and of the family structure are given. The limiting symmetry corresponds to the lower of the two. For mixed-rotation polytypes the family stmcture is defined only within the Pauling model and the limiting symmetry by definition coincides with the symmetry of the lattice.

Use to denote the parameter vector for identification. It includes the model parameters 9m and the parameters that determine the elements of the upper right triangular part of the prediction-error covariance matrix (symmetry defines fhe lower friangular part of this matrix). 

In the special case of modal identification, they are the elements of the upper right triangle (diagonal inclusive) of the Nm x Nm submatrix of S/o corresponding to the lowest Nm modes. Symmetry defines the lower triangle. 

The elements of the upper right triangular part (diagonal inclusive) of (symmetry defines the lower triangular part of this matrix) or the parameters to be used to determine this matrix. 

Defined as zero when the FPF bisectors lie in the P2NSi plane. The N(PF2)2 unit has Cj symmetry. ) Defined as zero when one Si-H bond is on the skeletal plane. 

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