Traceless unit

Such order can be described in terms of the preferential alignment of the director, a unit vector that describes the orientation of molecules in a nematic phase. Because the molecules are still subject to random fluctuations, only an average orientation can be described, usually by an ordering matrix S, which can be expressed in terms of any Cartesian coordinate system fixed in the molecule. S is symmetric and traceless and hence has five independent elements, but a suitable choice of the molecular axes may reduce the number. In principle, it is always possible to diagonalize S, and in such a principal axis coordinate system there are only two nonzero elements (as there would be, for example, in a quadrupole coupling tensor). In the absence of symmetry in the molecule, there is no way of specifying the orientation of the principal axes of S, but considerable simplification is obtained for symmetric molecules. If a molecule has a threefold or higher axis of symmetry, its selection as one of the axes of the Cartesian coordinate system leaves only one independent order parameter, with the now familiar form ... 

Tables III-XVII give calculated permanent moments. Selected comparisons with experimental values or calculations of others are also listed. All values are in atomic units, and traceless rather than Cartesian forms are distinguished with Greek letters, 6 (quadrupole) and G (octupole). Coordinates for the atomic centers are listed. These specify the geometry used, which were equilibrium geometries, and implicitly the multipole expansion center (x = 0, y = 0, z = 0). The moments are given at both the SCF level and at the well-correlated level of coupled-cluster theory [95-102]. ACCD [103-106] was the particular coupled-cluster approach, and the moments were evaluated by expectation [102] with the cluster expansion truncated at single and double substitutions.

One-dimensional quadrupole echo NMR lineshape analysis of powder samples is particularly informative when fast, discrete jumps occur between sites of well-defined geometry as, for example, in a phenyl group undergoing two-site exchange. In this case, the characteristic Pake-pattern is transformed into an axially asymmetric lineshape with an apparent asymmetry parameter r] 9 0 (see Equation (6.2.3)) [1-8]. The asymmetric lineshapes, shown on the left in Fig. 6.2.2, can be derived by considering how the individual components of the principal EFG tensor become averaged by the discrete jumps. Within the molecular frame, and in units of as defined by Equation (6.2.2), the static axially symmetric tensor consists of the components = 1, = — 1/2, and V y = — 112. This traceless tensor satisfies the... 

Problem 8-19. Stresslet Drift. In this problem we consider the drift of a stresslet that is due to the presence of a plane boundary. As we have noted (Eq. 8-128), the stresslet is specified by a second-order physical tensor S)j that is symmetric (Sy = Sji) and traceless (SijSij = 0). The drift velocity n, is proportional to S)j for creeping flows, and will be a function of the unit normal , describing the orientation of the plane. 

Because of chain inextensibility, the shear rate of any slip system is not dependent on the normal-stress component in the chain direction (Parks and Ahzi 1990). This renders the crystalline lamellae rigid in the chain direction. To cope with this problem operationally, and to prevent global locking-up of deformation, a special modification is introduced to truncate the stress tensor in the chain direction c. Thus, we denote by S° this modification of the deviatoric Cauchy stress tensor S in the crystalline lamella to have a zero normal component in the chain direction, i.e., by requiring that 5 c,c = 0, where c,- and c,- are components of the c vector (Lee et al. 1993a). The resolved shear stress in the slip system a can then be expressed as r = where R is the symmetrical traceless Schmid tensor of stress resolution associated with the slip system a. The components of the symmetrical part of the Schmid tensor / , can be defined as = Ksfw" + fs ), where if and nj are the unit-vector components of the slip direction and the slip-plane normal of the given slip system a, respectively. 

Sulfur-based linker units have been developed that utilize the reactivity of sulfur in a multitude of different forms and oxidation states. i ° The simplest linker units are the thioether-based linkers, and initially conditions for traceless cleavage of aliphatic... 

Finally, homolytic cleavage via a radical mechanism has also proven a powerful cleavage technique. Such cleavage is traceless and can be mediated by AIBN and tributyltin hydride (Table 1.14, Entry 14) or AlBN/tils(trimethylsilyl)silane (Table 1.14, Entry 15). Tellurium linker units are cleavable via the same mechanisms (Table 1.14, Entry... 

Alkoxyaniline units have been used as traceless linkers for the solid-phase combinatorial synthesis of 2-arylbenzoxazoles, 2-arylbenzothioazoles, and 2-arylbenzimidazoles [40] (Scheme 18). This method has an added advantage as the products can be released from the solid support only by exposing it to an air atmosphere without any kinds of oxidants under neutral conditions. The solid support could be easily recovered and recycled after cleavage. 

The notation used for the elastic dumbbell model is shown in Figure 9. There are n dumbbells per unit volume, dissolved in a solvent with viscosity rj. The imposed velocity distribution for the solution is given by v = Vq + [k i ], in which Vq is independent of position, k = (Vv) is a position-independent traceless tensor, and r is the position vector such a velocity distribution is referred to as homogeneous since the velocity gradients are constant throughout the fluid. 

The synthetic route developed by Craig et al. examines a traceless cleavage model for resin-bound o-quinodimethane (753) and the transformation into naphthalene-derivative 759,2,3-naphthoquinone (755) and 3-(trichloro-methyl)-isoquinoline (757) (Scheme 112). The linker unit synthesized via a three-step procedure contains no visible dien structure to undergo Diels-Alder reactions but can be seen as a precursor for a solid-supported dien. 

Related isoflavones have been prepared, using SPOS, by Harikrishnan and Showalter. Resin-bound salicyl aldehyde derivatives 336, immobilized via a traceless silyl linker unit, were prepared and treated with a Grignard reagent to introduce the first point of diversity. Subsequent re-oxidation with 2-iodoxybenzoic acid (IBX), and deprotection of the methoxymethyl (MOM) group provided ketone 338. Treatment of ketone 338 with amide acetals promoted an unusual cyclization, which concurrently installed R to give 339. Final traceless cleavage with CsF or TBAF releases isoflavones 340 (Scheme 3.59). 

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