Time-averaged position

Liquid crystals, commonly referred to as the fourth state of matter, are materials which are intermediate in character between the solid and liquid states. Unlike normal isotropic liquids, they show some time-averaged positional orientation of the molecules, but they retain many of the properties of liquids, such as the ability to flow. In recent decades, liquid crystals have played an increasingly important part in our lives. Probably their most familiar application is in the information displays which provide the visual interface with microprocessor-controlled instrumentation. Liquid crystal displays have superseded more traditional display technology, such as light-emitting diodes and cathode ray tubes, for many appliances principally because of the advantages of visual appeal, low power consumption, and their ability to facilitate the miniaturisation of devices into which they are incorporated. They are encoun-... 

Figure 2 shows the schematic structure in the paraelectric (T > Tn) and an-tiferroelectric (T < Tn) phases, hi the paraelectric phase the time-averaged position of the H atoms hes in the middle of an O - H...0 bond, whereas in the antiferroelectric phase, the protons locahze close to one or the other O atom. Prior to the recent NMR work [20-25], the largely accepted model of the phase transition was that the phase transition involved only the ordering of the H atoms in the O - H...0 bonds, and no changes in the electronic structure of the C4 moieties were considered to take place. The NMR results show that, in addition to the order/disorder motion of the H atoms, the transition also involves a change in the electronic charge distribution and symmetry of the C4 squares. 

Fig. 6.99. The contact-adsorbed ions in a hexagonal array at the IHP. Ions are considered spatially distributed in cells, with the centers of the cells corresponding to the time-average positions of the ions. (Reprinted from J. O M. Bockris. M. Gamboa-Aldeco, and M. Szklarczyk, J. Electroanal. Chem. 339 355, copyright 1992, Fig. 51, with permission from Elsevier Science.)...

In an MD study of methane sorption and diffusion in silicalite, Nicholas et al. (67) identified favorable sites for sorption. From the MD calculations, the time-averaged position of the center of mass of the methane molecule was plotted. Energy minimization calculations were then performed, locating the methane molecule at positions where the MD calculations predicted they spent the most time. Each channel intersection region was found to contain two sites that are minima for methane-zeolite interactions. These two sites are separated by a translation parallel to the straight channel... 

In many types of observations, for example, in the determination of X-ray structures, the molecule is approximated as a rigid body and is represented by an iconic (static) model. Of course, even supposedly rigid molecules are far from that Their atoms vibrate about time-averaged positions. Nevertheless, the choice of a static model under the rigid-body approximation is reasonable because the corresponding molecule occupies only a single minimum on the potential-energy hypersurface. 

Electron correlation The adjustment of electron motion to the instantaneous (as opposed to time-averaged) positions of all the electrons in a molecular entity. 

The time-average positions for Mg, F , Cs, and I can be seen in Fig. 2.64. For the arrangement is octahedral but for F" there is only a small preference for octahedral coordination. On the other hand, Cs and I" are firmly octahedral. As one goes outward past 400 pm, the preferential orientation is gone except for Li" " and this seems to form a second shell. It must be again stressed that the numbers are all time-averaged (coordination) numbers and have only a tenuous relation to the time-dependent hydration numbers. 

Spectroscopic methods, molten salts, 702 Spectroscopy detection of stmctnral nnits in liquid silicates, 747 and structure near an ion, 72 Standard partial gram ionic entropies, absolute, II Thermodynamics, applied to heats of solvation, 51 of ions in solution, 55 Time average positions of water near ions. 163 Tools, for investigating solvation, 50 Transformation, chemical, involving electrons, 8 Transition metals... 

It is of interest to compare in more detail the average atomic positions obtained from the fully solvated crystal and aqueous solution simulations.92 Figure 40 shows the magnitude of the difference between the time-averaged positions of protein atoms in solution and in the crystal the C atom results are plotted in the lower portion and those for the end atoms of sidechains are shown in the upper portion. Since the time average is limited to a 12-ps period, a substantial error is likely to be associated with this comparison e.g., C" atom differences less than 1.5 A and sidechain atom differences less than 2.0 A are not considered significant. With this criterion, the only significant mainchain differences occur for C atoms involved in crystal contacts and external loops, i.e., residues 24 to 29, residues 12 to 15, and the carboxy-terminal residues (see Fig. 17). For sidechain atoms, crystal contacts account... 

Fig. 3 depicts the time-averaged position of phenyl acetates in the cavity of a-cyclodextrin, determined by the above method. In these time-averaged conformations, the centers of the aromatic rings of p-nitrophenyl acetate, phenyl acetate, and nj-nitrophenyl acetate, respectively, are at the heights of 2.2, 1.9 and 1.7 A with respect to the plane comprised of the 6 H-3 atoms of a-cyclodextrin. As shown in Table 1, the calculated values of the anisotropic shielding effects of the aromatic rings of the phenyl acetates on both the H-3 and H-5 atoms agree fairly well with the observed values. 

The most important multicolour display technology in current use involves liquid crystal displays (LCDs), which overtook CRTs in popularity around the mid-2000s. Liquid crystals (LCs), commonly referred to as the fourth state of matter, are materials that are intermediate in character between the crystalline solid and liquid states.Unlike normal isotropic liquids in which the molecules essentially adopt a randomised orientation, liquid crystals show some time-averaged positional orientation of the molecules. In this sense, they resemble solid crystalline materials, although they retain most of the properties of liquids, notably the ability to flow. They are formed most commonly from molecules with rod-like geometry, which are referred to as calamitic. These molecules may orientate in various ways to form different types of LC phases [mesophases). There are three main mesophase types smectic, nematic and chiral nematic. In the smectic mesophase, as illustrated in Figure 11.1(a), the molecules are arranged in raft-like layers with their molecular axes parallel. These layers can pass over each other as the material flows. In the... 

Fig. 6.3. Motions and crystalline arrangements of trapped charged particles. The left figure is a photo of a cluster of 32 charged micro-particles stored in a Paul trap. The two other figures are results from a numerical simulation of a 1000-ion Coulomb crystal confined in a linear quadrupole trap. In the left and the center panel the micro-motion of the particles in the oscillatory electric field can be seen. The amplitude increases proportionally to the distance from the center. The time-averaged positions plotted in the right part for a selected sample shows that the ions remain well-localized. This is the basis for defining an effective translational temperature of the ion cluster by subtracting the periodic oscillation from the overall motion.

Coulomb crystal confined in a linear quadrupole trap published by Schiffer et al As in the picture on the left, the lines in the middle image indicate the micro-motion. Other than in the left picture, here the ions are stored in a linear quadrupole and the image shows the motion in a plane orthogonal to the axis of the rod system. As a consequence the direction of the rf held induced micro-motion is radial in front of the rods and tangential in between them. More details on the micro-motion can be foimd in Chapter 3 (see for example Fig. 3.1). The time-averaged positions plotted in the right part for a selected sample of ions shows that they remain weU-localized. As discussed already in Chapter 3 and in more detail below, this observation is the basis for dehning an effective translational temperature of the ion cluster. 

Figure 5.23 A model study of the organization of water during the unstacking of DNA base pairs (adenine and thymine). Each dot is a time-averaged position of the O atom of a water molecule. The conformation shown corresponds to an opening of thymine toward the major groove [adapted from E. Giudice, P. Vrnai, and R. Lavery, Chem. Phys. Chem. 2, 673 (2001)].

Thermal expansion is a result of the asymmetry of the energy-bond curve, which produces anharmonic oscillations on the part of the atoms in the lattice. Thus the time-average position of the atoms is displaced from the equilibrium position at the bottom of the bond-energy well. This displacement increases with temperature, giving rise to the coefficient of linear expansion. 

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