Threefold axis

The symmetry argument actually goes beyond the above deterniination of the symmetries of Jahn-Teller active modes, the coefficients of the matrix element expansions in different coordinates are also symmetry determined. Consider, for simplicity, an electronic state of symmetiy in an even-electron molecule with a single threefold axis of symmetry, and choose a representation in which two complex electronic components, e ) = 1/v ( ca) i cb)), and two degenerate complex nuclear coordinate combinations Q = re " each have character T under the C3 operation, where x — The bras e have character x. Since the Hamiltonian operator is totally symmetric, the diagonal matrix elements e H e ) are totally symmetric, while the characters of the off-diagonal elements ezf H e ) are x. Since x = 1, it follows that an expansion of the complex Hamiltonian matrix to quadratic terms in Q. takes the form... 

The crystal stmcture of the calcium fluoroapatite has two different crystallographic sites for the Ca " ion. The Ca(I) site has a threefold axis of symmetry and is coordinated to six oxygen ions at the vertices of a distorted trigonal prism. The Ca(Il) ions are located at the corners of equilateral... 

The triaLkoxy(aryloxy)boranes are typically monomeric, soluble in most organic solvents, and dissolve in water with hydrolysis to form boric acid and the corresponding alcohol and phenol. Although the rate of hydrolysis is usually very fast, it is dependent on the bulk of the alkyl or aryl substituent groups bonded to the boron atom. Secondary and tertiary alkyl esters are generally more stable than the primary alkyl esters. The boron atom in these compounds is in a trigonal coplanar state with bond hybridization. A vacantp orbital exists along the threefold axis perpendicular to the BO plane. 

Figure 16.6 A T = 3 icosahedral virus structure contains 180 subunits in its protein shell. Each asymmetric unit (one such unit is shown in thick lines) contains three protein subunits A, B, and C. The icosahedral structure is viewed along a threefold axis, the same view as in Figure 16.5. One asymmetric unit is shown in dark colors.

Figure 16.10 The arms of all 60 C subunits in tomato bushy stunt virus form an internal framework, (a) Configuration of interdigitated arms from the three C subunits, viewed down a threefold axis. The p strands are shown as arrows, (b) Cutaway view of the virus particle, emphasizing the framework function of the C-subunit arms. These arms are shown as chains of small balls, one per residue. The region where three arms meet and interdigitate is shown schematically in (a). The main part of each subunit is represented by large balls. Only about one hemisphere of these is drawn, but all the C-subunit arms are included.

The other tetrahalides can all readily be made by direct reactions of the elements. Crystalline SeCU, TeCU and -SeBr4 are isotypic and the structural unit is a cubane-like tetramer of the same general type as [Me3Pt(/Z3-Cl)]4 (p. 1168). This is illustrated schematically for TeCU in Fig. 16.13d each Te is displaced outwards along a threefold axis and thus has a distorted octahedral environment. This can be visualized as resulting from repulsions due to the Te lone-pairs directed towards the cube centre and, in the limit, would result in the separation into... 

To treat the reagents and products as indistinguishable, one must make the total (electronic + nuclear) wave function symmetric under a cyclic exchange of nuclei, which is equivalent to making it symmetric under rotations 2n/3, 4n/3. about the threefold axis of symmetry. Mead showed that, because the electronic wave function >]> is antisymmetric under 2ti/3> must be symmetrized... 

Fig. 6.1 The fundamental structural unit found in the Chevrel phases (cluster MoeXg full circles Mo atoms) displayed in three ways to emphasize different views of the connectivity. In (a) an octahedron of molybdenums (Mo-Mo = 2.7 A) is encased in a cube of chalcogens (Mo-S 2.45 or Mo-Se 2.6 A). Scheme (b) exhibits the same cluster as consisting of an octahedron with its triangular faces capped by chalcogenides. In (c), the cluster has been reoriented so that a threefold axis is vertical. (Reproduced from [10])...

Also known for some time is a phase transition at low temperature (111K), observed in studies with various methods (NQR, elasticity measurement by ultrasound, Raman spectrometry) 112 temperature-dependent neutron diffraction showed the phase transition to be caused by an antiphase rotation of adjacent anions around the threefold axis ([111] in the cubic cell) and to lower the symmetry from cubic to rhombohedral (Ric). As shown by inelastic neutron scattering, this phase transition is driven by a low-frequency rotatory soft mode (0.288 THz 9.61 cm / 298 K) 113 a more recent NQR study revealed a small hysteresis and hence first-order character of this transition.114 This rhombohedral structure is adopted by Rb2Hg(CN)4 already at room temperature (rav(Hg—C) 218.6, rav(C—N) 114.0 pm for two independent cyano groups), and the analogous phase transition to the cubic structure occurs at 398 K.115... 

There is a second long interface stretching between the threefold and fourfold axes, involving both hydrophobic and hydrophilic interactions. Close to the threefold axis is an intersubunit salt bridge between Asp-139 of subunit I and His-128 in III, which links the N-terminal end of helix D (III) to a position near the kink in helix D (I). Further along the interface, N-terminal residues 6-12 of subunit III make several interactions with the C-helix of subunit I, including several which are mediated... 

Figure 6.7 Residues close to the threefold axes in (a) BfLF (b)Sma-l (c) EcFTN (d)EcBFR. Viewed along the threefold axis from the inside surface. Reprinted from Harrison et al., 1998, by courtesy of Marcel Dekker, Inc.

ACs. Moreover, the glue atoms beyond triacontahedra, in fact, center prolate rhombohedron (a cube elongated along a threefold axis), the recognized interstitial cavity among primitive cubic packed triacontahedra (also shown in Fig. 18). 

Defects in Si with lower symmetry than tetrahedral can have several crystallographically equivalent orientations. For example, a trigonal center can have its threefold axis along any of the (111) axes. In the absence of an applied stress, centers with different orientations are degenerate. 

We can assume (for the moment) that a carbon free radical is planar with a threefold axis of symmetry as far as the remaining bonds to carbon are concerned. Quantum mechanics predicts that delocaliza-... 

Monomeric TNF is biologically inactive the active form is a homotrimer in which the three monomers associate non-covalently about a threefold axis of symmetry, forming a compact bellshaped structure. X-ray crystallographic studies reveal that each monomer is elongated and characterized by a large content of antiparallel P pleated sheet, which closely resembles subunit proteins of many viral caspids (Figure 9.4). 

Figure 1. (a) A schematic representation of the overall organization of the molecule of human ceruloplasmin. Domains 2,4, and 6 contain mononuclear copper centers, while the trinuclear copper cluster is located at the interface of domains 1 and 6. (b) An a-carbon ribbon diagram of the human ceruloplasmin molecule viewed along the pseudo threefold axis highlighting the triplication of the structure. Domains 1, 3, and 5 are depicted by striped motifs, whereas domains 2, 4, and 6 are dark shaded. The copper... 

Because of symmetry considerations, in both compounds the Fe(II) ion lies on the threefold axis and has an inversion centre. It is in an octahedral environment formed by six crystallographically related N-4 coordinating 1-tetrazole moieties. The almost perfect Oh symmetry for the FeN6 core is therefore present in the high-spin and low-spin state. Three bis(tetrazole)alkane ligands, in a bent syn conformation, link the Fe(II) centres to form reg-... 

Figure 2.44 (afi) Two different and isosteric orientations of triplets of trails planar chains of sPS as in structure of a-form (Figure 2.24). (c) Atoms of backbone are statistically distributed in six positions around threefold axis, whereas atoms of phenyl rings are in same positions as in... 

For information about point groups and symmetry elements, see Jaffd, H. H. Orchin, M. Symmetry in Chemistry Wiley New York, 1965 pp. 8-56. The following symmetry elements and their standard symbols will be used in this chapter An object has a twofold or threefold axis of symmetry (C2 or C3) if it can be superposed upon itself by a rotation through 180° or 120° it has a fourfold or sixfold alternating axis (S4 or Sh) if the superposition is achieved by a rotation through 90° or 60° followed by a reflection in a plane that is perpendicular to the axis of the rotation a point (center) of symmetry (i) is present if every line from a point of the object to the center when prolonged for an equal distance reaches an equivalent point the familiar symmetry plane is indicated by the symbol a. 

Fig. 10.8. S urface representation of the tricorn protease with the ribbon model of one subunit superimposed. The two orthogonal views are along the molecular two-fold and threefold axis, respectively. The six solid spheres indicate the active-site positions.

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