Symmetry relationships

Preference for the chair transition state is a consequence of orbital-symmetry relationships Hoffmann, R. Woodward, R.B. J. Am. Chem. Soc., 1965, 87, 4389 Fukui, K. Fujimoto, H. Tetrahedron Lett, 1966, 251. 

The ligands of a tetrahedral complex occupy the comers of a tetrahedron rather than the comers of a square. The symmetry relationships between the d orbitals and these ligands are not easy to visualize, but the splitting pattern of the d orbitals can be determined using geometry. The result is the opposite of the pattern found in octahedral... 

Photocycloaddition of Alkenes and Dienes. Photochemical cycloadditions provide a method that is often complementary to thermal cycloadditions with regard to the types of compounds that can be prepared. The theoretical basis for this complementary relationship between thermal and photochemical modes of reaction lies in orbital symmetry relationships, as discussed in Chapter 10 of Part A. The reaction types permitted by photochemical excitation that are particularly useful for synthesis are [2 + 2] additions between two carbon-carbon double bonds and [2+2] additions of alkenes and carbonyl groups to form oxetanes. Photochemical cycloadditions are often not concerted processes because in many cases the reactive excited state is a triplet. The initial adduct is a triplet 1,4-diradical that must undergo spin inversion before product formation is complete. Stereospecificity is lost if the intermediate 1,4-diradical undergoes bond rotation faster than ring closure. 

Higher-order terms have been neglected in this small-x expansion that is valid in the intermediate regime. This expression obeys exactly the symmetry relationships, and it obeys the reduction condition to leading order. (See Eq. (68) for a more complete expression that obeys the reduction condition fully.)... 

G. C. Chapuis, Symmetry relationships between crystal structures and their practical applications. In Modem Perspectives in Inorganic Crystal Chemistry (E. Parthe, ed.), p. 1. Kluwer, 1992. 

The situation becomes more complex in three-dimensional structures if we lay the palms of our two hands together, the mirror plane lies exactly between them, so that the hands are chiral. Whether or not a molecule is chiral depends upon symmetry relationships ... 

The unit cell considered here is a primitive (P) unit cell that is, each unit cell has one lattice point. Nonprimitive cells contain two or more lattice points per unit cell. If the unit cell is centered in the (010) planes, this cell becomes a B unit cell for the (100) planes, an A cell for the (001) planes a C cell. Body-centered unit cells are designated I, and face-centered cells are called F. Regular packing of molecules into a crystal lattice often leads to symmetry relationships between the molecules. Common symmetry operations are two- or three-fold screw (rotation) axes, mirror planes, inversion centers (centers of symmetry), and rotation followed by inversion. There are 230 different ways to combine allowed symmetry operations in a crystal leading to 230 space groups.12 Not all of these are allowed for protein crystals because of amino acid asymmetry (only L-amino acids are found in proteins). Only those space groups without symmetry (triclinic) or with rotation or screw axes are allowed. However, mirror lines and inversion centers may occur in protein structures along an axis. 

Modifications separated by a second-order transition can never be coexistent. One typical second-order transition, the displacive structural transition, is characterized by the distortion of bonds rather than their breaking, and the structural changes that occur are usually small. Typically, there is continuous variation in the positional parameters and the unit cell dimensions as a function of temperature. The structural changes in the system occur gradually as the system moves away from the transition point. As well as a structural similarity, a symmetry relationship... 

Photochemical cycloadditions provide a method that is often complementary to thermal cycloadditions with regard to the types of compounds that can be prepared. The theoretical basis for this complementary relationship between thermal and photochemical modes of reaction lies in orbital symmetry relationships, as discussed in Chapter 13 of Part... 

Symmetry relationship of cross peak locations has been used for improving the quality of correlation spectra for quite a long time in NMR spectroscopy both for diagonally [21] and laterally symmetric spectra [22]. Such data processing procedures have their advantage but may introduce artifacts and remove real information and therefore should be used with caution [10]. 2Q-HoMQC spectra can be symmetrized directly using the appropriate symmetry function [33], but most commercial software do not provide such option. Also, fine structure of the direct correlation peaks in 2Q-H0MQC spectra is antisymmetric in the SQ dimension which requires extra attention. 

X(ABCD) and X(FFCD) the centers X are clearly chirotopic and achirotopic, respectively (see Section 1.1.3.5.). However, this possibility is barred, as the permutation characteristics of these units are preserved in chiral models such as X(AFFG), with three chiral ligands. Here we have a pseudoasymmetric center and X is chirotopic. This observation emphasizes again the necessity to separate the concepts of stereogenicity and symmetry relationships. 

Mislow and Siegel6 presented a penetrating analysis of these problems. However, they underestimated the capacity of logical description systems based on permutation characteristics, such as the C1P system, to make symmetry relationships among stereoisomers apparent (see Section 1.1.5.3.4.). 

The monolayer structures for tryptophan are consistent with one and two molecules per unit cell, respectively, for Cu(100) and Cu(l 11). The structures observed for D- and L-tryptophan are related by mirror inversion which is consistent with the symmetry relationship between the two molecules. A mixture of the optical isomers, DL-tryptophan does not form an ordered monolayer, thus there is no segregation or cooperative interaction between the different isomers. 

The coefficients A defined in Eq. 4.18 satisfy certain symmetry relationships [323, 391], From parity considerations, it follows that X +2.2 + L must be odd. Moreover, because the dipole operator is Hermitian, the expansion coefficients A are all real. The symmetry property... 

Symmetry. For the translational and rototranslational bands, Eq. 5.73 is usually a good description of the observed asymmetry. In that case, the use of the BC and K0 profiles is straightforward because these have the same symmetry, Eq. 5.73. In the next Chapter we will see that profiles of the rotovibrational bands are of a symmetry which is different from that relationship [295], In that case, new line profiles must be constructed that satisfy the correct symmetry relationship [295, 62, 48], These will be discussed next Chapter. 

All systems studied in this way until now have a spherical distribution of the non-bonding electrons, so no account need be taken of the symmetry relationships between the ground state and the possible transition states. Consequently, the size of the reaction centre, the coordination number and the effective size and packability of the ligands determine whether the mode of activation is associative or dissociative. There is, as yet, no suitable evidence to decide whether the nucleophilicity of the entering group plays a significant part in deciding the mode of activation. 

The cylindrically averaged Fourier transform of the sevenfold and six-fold triple-stranded structures are shown in Figure 15. The Fourier transform of the six-fold triple-stranded model illustrates the symmetry of the system by the total absence of intensity on layer lines with index i 3 n, where n is an integer. The Fourier transform of the seven-fold triple-stranded structure shows that in destroying this precise symmetry relationship intensity occurs on all layer lines which are orders of the 2.27 nm spacing. This reinforces the concept of an indigenous triple-stranded structure which is perturbed slightly by the interaction of solvent. 

Orbital-symmetry relationships for thermal and photochemical concerted cycloadditions to the benzene ring were published in 1969 by Bryce-Smith [81], From correlations between low-lying excited states of starting materials and products, it was concluded that photochemical ortho addition is allowed ... 

Based on some interesting reactions in certain inorganic crystalline compounds, Kohlschutter [9,10] proposed that the nature and properties of the products obtained take place on the surface or within the solid state. Indeed, he coined the term topochemistry for such reactions in the solid state. However, systematic investigations of photoinduced reactions in crystals began from 1964 onward by Schmidt and Cohen [11], Their studies on the 2tt + 2tt photoreaction of cinnamic acid derivatives in the crystalline state and correlation with the molecular organization in these crystals led to what are now known as Topochemical Principles. The most important conclusions reached by them are as follows (1) The necessary conditions for the reactions to take place are that the reactive double bonds are parallel to one another and the center-to-center distance be within 4.1 A (2) there is one-to-one correspondence between the stereochemistry of the photoproduct and the symmetry relationship between the reactants. The centrosymmet-ric relationship (called the a-form) leads to centrosymmetric cyclobutane (anti-HT), whereas the mirror symmetric arrangements (called the (5-form) produce mirror symmetric dimer (yy -HH). 

Usually NLEs are studied for a chiral auxiliary of a given absolute configuration by variation of its ee between 0 and 100%. The graph is presented in a standard way in Scheme 3, which is irrespective of the absolute configuration of the chiral auxiliary and product. Sometimes experiments are performed with both enantiomers of the chiral auxiliary. In this case the results may be plotted as in Scheme 4 or 5. Symmetry relationships relate the curves in Schemes 4 and 5, since experiments performed with enantiomeric catalysts are strictly mirror image. 

The four different periodic tables account for the observed elemental diversity and provide compelling evidence that the properties of atomic matter are intimately related to the local properties of space-time, conditioned by the golden parameter r = l/. The appearance of r in the geometrical description of the very small (atomic nuclei) and the very large (spiral galaxies) emphasizes its universal importance and implies the symmetry relationship of self-similarity between all states of matter. This property is vividly illustrated by the formulation of r as a continued fraction ... 

Orbitals are identified by their spontaneous symmetry relationships (<73 and its two symmetric counterparts resemble sp2 hybrids on boron, see Fig. 1). 

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