Helical chain,

We have extended the linear combination of Gaussian-type orbitals local-density functional approach to calculate the total energies and electronic structures of helical chain polymers[35]. This method was originally developed for molecular systems[36-40], and extended to two-dimensionally periodic sys-tems[41,42] and chain polymers[34j. The one-electron wavefunctions here are constructed from a linear combination of Bloch functions c>>, which are in turn constructed from a linear combination of nuclear-centered Gaussian-type orbitals Xylr) (in ihis case, products of Gaussians and the real solid spherical harmonics). The one-electron density matrix is given by... 

FIGURE 6.18 Poly(Gly-Pro-Pro), a collagen-like right-handed triple helix composed of three left-handed helical chains. (Adaptedfrom Miller Scheraga, H. A., 1976, Calculation of the... 

Figure 16.1 Structures of various allotropes of selenium and the structure of crystalline tellurium (a) the Seg unit in a- fi- and y-red selenium (b) the helical Se chain along the c-axis in hexagonal grey selenium (c) the similar helical chain in crystalline tellurium shown in perspective and (d) projection of the tellurium structure on a plane perpendicular to the c-axis.

The symmetrical thiosulfoxide isomer of tetrasulfane (HS)2S=S is also of Cs symmetry and by 123 kJ mol" less stable than the chain-like ground state of C2 symmetry (at the MP2/6-31GV/HF/4-31G level of theory) [51]. However, at the much higher G3(MP2) level the energy difference is only 93 kJ mol [52]. Similarly, the two hexasulfane isomers (HSS)2S=S (Cs symmetry) and (HS)(HSSS)S=S (Q symmetry) have recently been calculated at the G3X(MP2) level to be by 53 and 54 kJ mol respectively, less stable than the helical chain of C2 symmetry [52]. The decreasing energy difference be-... 

Fig. 17.5 Pi Hared chiral structure of Cu(pzc)2AgRe04 (a) and helical chain (b).

Trigonal selenium is variously called metallic gray or black selenium and occurs in lustrous hexagonal crystals, which melt at 220.5 °C. Its structure, which has no sulfur analogue, consists of infinite, unbranched helical chains. Its density, 4.82 g cm , is the highest of any form of the element. Trigonal selenium is a semiconductor (intrinsic p-type with a rather indirect transition at about 1.85 eV [5]), and its electronic and photoelectric properties are the basis for many industrial uses of this element. 

Mercuric sulfide (HgS) is dimorphic. The more common form, cinnabar (red a-form), has a distorted RS, trigonal structure which is unique among the monosulfides, for the crystal is built of helical chains in which Hg has two nearest neighbors at 2.36 A, two more at 3.10 A, and two at 3.30 A. Bulk a-HgS is a large-gap semiconductor (2.1 eV), transparent in the red and near IR bands. The rare, black mineral metacinnabarite is the 3-HgS polymorph with a ZB structure, in which Hg forms tetrahedral bonds. Upon heating, 3-HgS is converted to the stable a-form. The ZB structure of HgS is stabilized under a few percent admixture of transition metals, which replace Hg ions in the lattice. 

Figure 9. The cable model for the structure of concentrated calcium pectate gels. Egg-box dimers link single-chains segments (top left) and are themselves ed together by larger aggregates of either egg-box or 3i helical chains (lower right)...

The symbols -A for 32 screw axes mark the axes of the helical chains. The slight tilting of the tetrahedra relative to the direction of view (c axis) vanishes in /3-quartz (stereo image)... 

In contrast to 1, the related pure host 7 may be obtained in crystalline form 68). The crystal structure of 7 is built via helical chains of alternating intra- and inter-molecular H-bonding through the carboxyl functions. This structure supplies the information that the carboxyl groups are therefore already positioned in an appropriate way to facilitate analogous H-bonding in the known inclusions of 7. As discussed later (Sect. 4.2.2), these are exclusively salt-type associates and as such, intimately interact with the carboxyl groups. Hence one may infer that displacement of the carboxyl functions from position 2 in 1 to position 8 in 7 reduces the ability of inclusion formation. Similar reasons such as the solid-solubility differences observed in the classical naphthalene/chloronaphthalene systems (alpha- vs. beta-substituted derivatives, cf. Ref. 28 may also be applied here. 

Iso tactic poly(methyl methacrylate) (it-PMMA) can form a stereocomplex with st-PMMA. Recent X-ray studies 179) of this material indicate that the two polymer chains probably interact to form a double helical structure. The it-PMMA chain forms the inner helix and is surrounded by the st-PMMA helical chain which winds around it. If subsequent work confirms this model, this material would constitute a most unusual inclusion compound involving only one monomeric substance. 

Figure 17 Schematic representation of heterogeneous portions of curdlan hydrogel (left) (A) liquid-like portion, (B) portion of intermediate mobility, and (C) triple-helical cross-links in the solid-like portion and crystallites as additional cross-links, and branched glucans (triple helical chains) (right). From Ref. 117 with permission.

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