Dimerized chain

For linear chains we solve eqn (3.1) using open boundary conditions. The trial solution is 

The molecular-orbital functions, 0/3(r), are constructed in exact analogy to the Bloch functions of the last section. Thus, we have. 

These molecular-orbital functions are particle-in-a-box solutions, and not surprisingly, the molecular-orbital states satisfy the following condition under the operation of the inversion operator i, 

The unit cell for a dimerized chain is shown in Fig. 3.3. There are two sites per unit cell, and two different hybridization integrals, td = t(l + ( ) and tg = t(l — 5), representing the double (short) and single (long) bonds, respectively. 

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Figure A3.10.10 STM image (55 x 55 mn ) of a Si(lOO) surface exposed to molecular bromine at 800 K. The dark areas are etch pits on the terraces, while the bright rows that run perpendicular to the terraces are Si dimer chains. The dimer chains consist of Si atoms released from terraces and step edges during etching [28],...

The dimer chains of Ca -ATPase can also be observed by freeze-fracture electron microscopy [119,165,166,172-174], forming regular arrays of oblique parallel ridges on the concave P fracture faces of the membrane, with complementary grooves or furrows on the convex E fracture faces. Resolution of the surface projections of individual Ca -ATPase molecules within the crystalline arrays has also been achieved on freeze-dried rotary shadowed preparations of vanadate treated rabbit sarcoplasmic reticulum [163,166,173,175]. The unit cell dimensions derived from these preparations are a = 6.5 nm b = 10.7 nm and 7 = 85.5° [175], in reasonable agreement with earlier estimates on negatively stained preparations [88]. 

The cytoplasmic domains reconstructed from negatively stained [90] and from frozen-hydrated samples [91,177] have similar shapes. Both include the protruding lobe and the bridge region that links the Ca " -ATPase molecules into dimers. The intramembranous peptide domains of the two ATPase molecules which make up a dimer spread apart as they pass through the bilayer toward the luminal side of the membrane, establishing contacts with the Ca -ATPase molecules in the neighboring dimer chains. The lateral association of dimer chains into extended crystal lattice is... 

Fig. 4. Tentative allocation of probe binding sites within the three-dimensional structure of Ca -ATPase derived from vanadate-induced E2-type crystals. The top picture is the projection view of the Ca -ATPase down the x-axis, revealing the pear-shaped contours of ATPase molecules. The maximum length of the cytoplasmic domain to the tip of the lobe is =r65A. In the middle and bottom pictures the same structure is viewed down the x-axis, revealing the gap between the bridge and the bilayer surface and the connections between ATPase molecules in neighboring dimer chains. The proposed binding sites for lAEDANS and FITC are indicated. The bottom right picture is the same structure viewed down the y-axis. Adapted from Taylor et al. [90].

Antibody A52 with its epitope at residues 657-672 [129,139,274,275] inhibited the vanadate-induced crystallization of Ca " -ATPase and decreased the stability of preformed Ca " -ATPase crystals [285]. The vanadate-induced crystals arise by the association of the ATPase monomers into dimers (type A interaction), the dimers into dimer chains (type B interaction), and the dimer chains into 2-dimensional arrays (type C interaction). It is suggested that antibody A52 interferes with type B interactions, preventing the formation of dimer chains, without exerting major effect on the concentration of Ca -ATPase dimers in the membrane. The simplest interpretation of the destabilization of Ca -ATPase crystals by mAb A52 is that binding of the antibody to its antigenic site physically blocks the interaction between ATPase molecules [285]. Considering the large bulk of the antibody, such interference is not unexpected, yet only a few of the antibodies that bind to the Ca -ATPase in native sarcoplasmic reticulum interfered with crystallization. 

This note is organized as follows. In section II, the RVA is presented using the example of the two-legspin ladders. In section III, the RVA method is applied to the study of the dimerized chain described by the Extended Peierls-Hubbard model. A nice similarity between the two-leg ladder and this system is pointed out and comparisons with DMRG calculations are made. We conclude in section IV and give some reasonable perspectives. 

Figure 9. The set of local configurations considered in this work for the dimerized chain a) the F-LC b) the D-LC c) the Ctj-LC and d) the TT-LC made by combining two localized triplet symbolized by the T on the figure.

The crystal structure of the e-form is similar to that of the y-form, but there is a decisive difference between them the former is comprised of heterochiral supramolecular ID dimer chains (Figure 19), while the latter consists of... 

In addition to the chiral interactions of enantiomers, interactions of dissimilar chiral species are fundamentally important. As described above, adenine becomes chiral upon adsorption and forms homochiral dimer chains on Cu(110). These chains are tilted 19.5° with respect to the [001] surface direction. This tilt makes the sites right next to an enantiomorphous supramolecular chain chiral Consequently, the interactions of inherently chiral molecules with both chain types are energetically nonequivalent. This... 

Some excited states are partially spin-degenerate because of partial spin degeneracy of corresponding states of the dimerized chain. 

Here k is the wave vector of an electron in the it band and a is the spacing between the nearest C atoms on the polymer backbone. The energy dispersion given by Eq. (2.2) is shown by the dashed line in Fig. 2.2(c). Since each C atom gives only one electron, the n band is half filled and the r-PA should be a metal. However as was first shown by Pieirls, degeneracy of the band in the one dimensional (ID) metals is removed by spontaneous distortion of the ID chain. In the case of r-PA the distortion consists of dimerization shown in Fig. 2.2(b). Dimerization opens up a gap ( 1.4 eV) and the r-PA becomes a semiconductor. The band structure of the dimerized chain is shown by the solid lines in Fig. 2.2(c). 

The band structure of the dimerized chain is shown by the solid curves in Fig. 2.2(c). A gap in the band opens and the polymer now becomes a ID semiconductor. The bandgap... 

Distortion of the chain leading to the dimerization is induced by the electron-phonon interaction. The interaction arises due to the dependence of the transfer integral (given by Eq. (2.1)) on the spacing between the atoms which changes due to the vibrations. Since the displacement un of the nth C atom due to the dimerization is small, the transfer integral for the dimerized chain can be expressed as... 

Figure 4 Formation of a tetramerized chain definition of the two distortion parameters A z and 8 1 z, z being the chain axis, for a simple dimerized chain A = 8 = 0 (see also Fig. 3). (After Ref. 15.)...

At some critical temperature, a structural distortion to dimerized chains occurs together with the opening of a gap in the magnetic excitations. Such a distortion is attributable to a 2kF spin-Peierls transition, just as in MEM(TCNQ)2 (cf. Section III.A.7). A 2kF transition results into chain tetramerization for p = 5 and into chain dimerization for p = 1, as also shown in Fig. 5. For instance, the spin-Peierls transition takes place at T sp = 395 K in KTCNQ and at TsP = 381 K in RbTCNQ [67] (a second form, called RbTCNQ II, also exists for the Rb salt [69], but it is not considered here). By analogy with the case of polymers, the dimerized phase at low temperature is also called a bond-ordered-wave (BOW) phase [47]. 

Such an effect is not due to a trivial Joule heating of the samples. From a detailed experimental study based mainly on electrical [68], optical [70], and x-ray [71] measurements, this effect has been attributed to the motion of charged soliton-like defects existing in the TCNQ chains. A soliton in a dimerized chain is expected to have the following general form [47] ... 

After the authors of Ref. 47, the motion of charged solitons in dimerized chains requires a relatively low excitation energy, and it could take place according to the following scenario. At low electric field, this motion is hindered by the three-dimensional interactions between molecular chains. The current is then due to a few defects more or less free from these interactions. As the field is increased, more and more charged solitons are driven to motion and by a cooperative effect of these mobile solitons, a critical field th is finally attained at which value the three-dimensional order is lost. The solitons are then able to move in one chain almost independent of the others, and this new degree of freedom is believed to... 

Among the inorganic open-framework compounds, the family of phosphates is a large one [3]. A large variety of open-framework metal phosphates of different architectures have been synthesized in the last few years. They include one-dimensional (ID) linear chain and ladder structures, two-dimensional (2D) layer structures and three-dimensional (3D) channel structures [4]. In the linear chain and ladder structures, four-membered metal phosphate units of the type M2P2O4 share comers and edges respectively. Zero-dimensional four-membered zinc phosphates have been synthesised and characterized recently [5]. Several open-framework metal carboxylates have also been reported [6] and the presence of a hierarchy of zinc oxalates covering the monomer, dimer, chain, honeycomb-layer and 3D structures has indeed been established [7]. 

The notation CL u,v,w) represents the skeleton of the cluster, which is formed by (1) bonding u unit dimers into a finite unit chain, (2) connecting V unit chains into a rectangular unit plane, and (3) piling up w unit planes to form a three-dimensional cluster. Any tetrahedrally bonded cluster can be formed by polymerization of unit dimers, chains, and planes. Therefore, in a wide sense, Si-based polymers include oligomers, polymers, clusters, and even crystalline silicon. 

CO2. CHj, CO. H . propene. isobutene, dimethyl ketene. acrolein, diallyl ether, allyl methacrylate monomer and dimer, chain fragments including anhydnde structures from side-group cydization. and an aliphatic ketone in the vidnity of unsaturation CO , SO , phenol, bisphenol S, hydroquinone. fragments of backbone, benzene sulfonic acid... 

CO, CHj. CO3. H 0. phenol, p-cresol. p-ethylphenol, p-isopropylphenol, p-vmylphenol, bisphenol A. cyclic dimer, chain fragments including some ester structures... 

NbClj 4 TaCls Dimer M2CI10 (octahedral) PaClj 7 UQs 6 Pentagonal bipyramidal U2CI10 dimer chain... 

The magnetic gap in Fig. 3, for example, becomes the singlet-triplet splitting associated with breaking the a-bond. The epr intensity I(T), or the static susceptibility yield AEp = 0.27 + 0.01 ev in the range 110 < T < 370 K. The usual occurrence of g 2 doublets in Fig. 7, which probably involve misfits in dimerized chains, does not complicate the TSE intensity in the resolved fine-structure lines. The reason is that magnetic dipole-dipole interactions of the two unpaired electrons... 

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