Threefold symmetry

Similar to the case without consideration of the GP effect, the nuclear probability densities of Ai and A2 symmetries have threefold symmetry, while each component of E symmetry has twofold symmetry with respect to the line defined by (3 = 0. However, the nuclear probability density for the lowest E state has a higher symmetry, being cylindrical with an empty core. This is easyly understand since there is no potential barrier for pseudorotation in the upper sheet. Thus, the nuclear wave function can move freely all the way around the conical intersection. Note that the nuclear probability density vanishes at the conical intersection in the single-surface calculations as first noted by Mead [76] and generally proved by Varandas and Xu [77]. The nuclear probability density of the lowest state of Aj (A2) locates at regions where the lower sheet of the potential energy surface has A2 (Ai) symmetry in 5s. Note also that the Ai levels are raised up, and the A2 levels lowered down, while the order of the E levels has been altered by consideration of the GP effect. Such behavior is similar to that encountered for the trough states [11]. 

Figure 16.4 The division of the surface of an icosahedron into asymmetric units, (a) One triangular face is divided into three asymmetric units into which an object is placed. These are related by the threefold symmetry axis.

Subunits VP2 and VP3 from different pentamers alternate around the threefold symmetry axes like subunits B and C in the plant viruses (Figure 16.12b). Since VP2 and VP3 are quite different polypeptide chains, they cannot be related to each other by strict symmetry, or even by quasi-symmetry in the original sense of the word. To a first approximation, however, they are related by a quasi-sixfold symmetry axis, since the folded structures of the cores of the subunits are very similar. 

The threefold symmetry of rotation about the C—C bond of ethane disappears when substituents are introduced on both of the carbon... 

The symmetry of the anions and the surface stmcture tetrahedral bisulfates bond very strongly to threefold symmetry sites of (111), but weakly to (100) and (110) surfaces. Halides adsorb more strongly on (100) surfaces. 

Scheme 11 The dihalopyrazinotetrathiafulvalenes and analogues involved in threefold symmetry salts...

Furthermore, electrocrystallization of the diodo derivatives DIPS and DIPSe afforded original salts with threefold symmetry and varying stoichiometries, such as (DIPS)3(PF6)(PhCl)i.i5 or (DIPSe)3(PF6)i.33 (CH202)1.2 [91]. hi these salts, the halogen bond is further enhanced as... 

Fig. 12 The threefold symmetry rigid structure adopted by the DIPSe salts with PF6", AsFg or SbFg , showing the two types of channels (denoted A and B) occupied by the anions and solvent molecules...

Paoletti et al. used a mixed aza oxo macrocycle (53) to form (yu3-C03) carbonate species on absorption of atmospheric C02. The crystal structure showed a trimer with threefold symmetry and six-coordinate zinc centers.461 This was described as C02 fixation however, three equivalents of zinc complex are required for each C02 molecule and so it is not a catalytic process. 

Similar to the case without consideration of the GP effect, the nuclear probability densities ofAi and A 2 symmetries have threefold symmetry, while each component of E symmetry has twofold symmetry with respect to the line defined by (1 0. However, the nuclear probability density for the lowest E state... 

A very productive strategy for the synthesis of glass-forming materials is the use of highly branched rigid structures. As a suitable center for starburst molecules with a threefold symmetry, triarylamine or benzene are used most frequently. Due to the large number of starburst molecules described in the literature, we divide this class into two subgroups, compounds based on the triarylamine and the benzene centers. 

STM has also been used to examine the dynamics of potential-dependent ordering of adsorbed molecules [475-478]. For example, the reversible, charge-induced order-disorder transition of a 2-2 bipyridine mono-layer on Au(lll) has been studied [477]. At positive charges, the planar molecule stands vertically on the surface forming polymeric chains. The chains are randomly oriented at low surface charge but at higher potentials organize into a parallel array of chains, which follow the threefold symmetry of the Au(l 11) substrate as shown in Fig. 34. Similar results were found for uracil adsorption on Au(lll) and Au(lOO) [475,476]. 

Figure 8.18. A schematic view of ATCase based on electron micrographs, viewed along the threefold symmetry axis. The outer equilateral triangle has an edge of 145 A. The (almost) inscribed solid triangle with edge 95 A is rotated by 60° relative to the large triangle.

Fig. 34a,b. SFM-micrographs of the monolayers on a HOPG b mica prepared by spincasting of 14-ABG-PS solutions in cyclopentane (c = 0.1 mg/ml). Individual molecules in (a) aligned parallel to the substrate and bent at characteristic angles of 60° and 120° to foUow the threefold symmetry of graphite. The worm-like molecules In (b) woimd aroimd each other and resulted in a felt-like structure [86]... 

Figure 4.7 shows top-down views of the fee (001), (111), and (110) surfaces. These views highlight the different symmetry of each surface. The (001) surface has fourfold symmetry, the (111) surface has threefold symmetry, and the (110) has twofold symmetry. These three fee surfaces are all atomically flat in the sense that on each surface every atom on the surface has the same coordination and the same coordinate relative to the surface normal. Collectively, they are referred to as the low-index surfaces of fee materials. Other crystal structures also have low-index surfaces, but they can have different Miller indices than for the fee structure. For bcc materials, for example, the surface with the highest density of surface atoms is the (110) surface. 

Fig. 1.11. The nascent Si(lll) surface and its reconstruction, (a) The nascent Si(lll) surface has a threefold symmetry, with nearest-neighbor atomic di.stance 3.84 A. (b) The Si( 111) surface reconstructs immediately at room temperature to a metastable Si(lll)-2XI surface, which has a lower symmetry. Two rows of dangling bond states are formed One is filled, another is empty.

By inspecting Fig. 1.10, it is obvious that the most natural cleavage plane of a Si crystal is the (111) plane or its equivalent, namely, (iTl), (111), etc. Right after cleaving, on each of the surface Si atoms, there is a broken bond, or a dangling bond, that is perpendicular to the (111) surface. Each of the dangling bond orbitals is half filled, that is, has only one electron. The nascent Si(lll) surface is thus metallic and exhibits a threefold symmetry, as shown in Fig. 1.11 (a). However, because of the large number of unsaturated bonds, such a surface is unstable. It reconstructs even at room temperature, and loses its threefold symmetry. 

Fig. 13.3. FIM image of a W tip immediateiy after etching. The tip is etched from a single-crystal W wire with (111) orientation. The threefold symmetry is visible on a large scale. Locally, severe dislocations are observed. (Courtesy of U. Staufer.)...

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