Translation defined

Crystal lattices can be depicted not only by the lattice translation defined in Eq. (7.2), but also by the performance of various point symmetry operations. A symmetry operation is defined as an operation that moves the system into a new configuration that is equivalent to and indistinguishable from the original one. A symmetry element is a point, line, or plane with respect to which a symmetry operation is performed. The complete ensemble of symmetry operations that define the spatial properties of a molecule or its crystal are referred to as its group. In addition to the fundamental symmetry operations associated with molecular species that define the point group of the molecule, there are additional symmetry operations necessary to define the space group of its crystal. These will only be briefly outlined here, but additional information on molecular symmetry [10] and solid-state symmetry [11] is available. 

Here, Xy ique i real space coordinate within a region of density that is repeated elsewhere in the as)nnmetric unit after a rotation and/or translation defined by the transformation T. Also these equations can be substituted in the Fourier summation of Eq. 1, effectively further reducing the number of imknowns in real space down to the number of independent grid points within the fraction of unique density. 

Several types of symmetry operations can be distinguished in a crystalline substance. Purely translational operations, such as the translations defining the crystal lattice, are represented by I 1, n3, with nu n2, n3 being integers. 

The expansion Eq. (5), taken in its entirety, is evidently valid regardless of the origin or the orientation of the chosen axis. But it is essential to keep in mind that the various integrals depend on the choice of the reference system. To see this we perform for instance the translation defined by ... 

First, for every statement expressed using the SMO language, there are formal semantics associated with it that describe forward and reverse translation of schemas. The reverse translation defines, for each statement, the inverse action that effectively undoes the translation. The only SMO statements that lack these forward and reverse translations are the CREATE TABLE and DROP TABLE operations logical formalism for these statements is impossible, since one is effectively stating that a tuple satisfies a predicate in the before or after state, but that the predicate itself does not exist in the other state. The work on PRISM describes quasi-inverses of such operations for instance, if one had copied the table before dropping it, one could recover the dropped information from other sources. PRISM offers some support for allowing a user to manually specify such inverses. 

Table 13 shows the features of compound tessellations (3, 6 [ 3, 6 ] to r = lOOA [Eqn. (12) assuming a = 5.3 A], each of which describes a coincidence-site lattice (CSL) (Ranganathan 1961) the multiplicity n of its mesh is termed coincidence index or Z factor and corresponds to the order of the subgroup of translation defining the two-dimensional CSL with respect to the hp lattice. As shown in Table 13, the minimal value of the E factor for the hp lattice is 7 (see also Pleasants et al. 1996). 

In the translation defined to mbC in [6] a different rule to translate op is presented, but such translation does not permit that (op) and p take simultaneously the value 0, while the semantic valuation for mbC permits v(op) = v(p) = 0. Our definition fix this problem. 

We now define the effect of a translational synnnetry operation on a fiinction. Figure Al.4.3 shows how a PHg molecule is displaced a distance A X along the X axis by the translational symmetry operation that changes Xq to X = Xq -1- A X. Together with the molecule, we have drawn a sine wave symbolizing the... 

That is, the effect of a translational operation is detennined solely by the vector with components (kyj yj ) which defines the linear momenUim. 

Sun Y-P and Saltiel J 1989 Application of the Kramers equation to stiibene photoisomerization in / -alkanes using translational diffusion coefficients to define microviscosity J. Phys. Chem. 93 8310-16... 

The quantities in this fomuila are defined as in equation Bl.5,32. but with the laser parameters translated into more convenient tenns is the average power at the indicated frequency is the laser pulse duration ... 

Now let us consider tire implications of tliese results for energy transfer. First we recognize tliat tliere is no directed energy transfer of tire fonn considered in the incoherent case. Molecules in tire dimer cannot be recognized as well defined separate entities tliat can capture and translate excitation from one to anotlier. The captured excitation belongs to tire dimer, in otlier words, it is shared by botli molecules. The only counteriDart to energy migration... 

The time-dependent Schrddinger equation governs the evolution of a quantum mechanical system from an initial wavepacket. In the case of a semiclassical simulation, this wavepacket must be translated into a set of initial positions and momenta for the pseudoparticles. What the initial wavepacket is depends on the process being studied. This may either be a physically defined situation, such as a molecular beam experiment in which the paiticles are defined in particular quantum states moving relative to one another, or a theoretically defined situation suitable for a mechanistic study of the type what would happen if. .. 

As was said in the introduction (Section 2.1), chemical structures are the universal and the most natural language of chemists, but not for computers. Computers woi k with bits packed into words or bytes, and they perceive neither atoms noi bonds. On the other hand, human beings do not cope with bits very well. Instead of thinking in terms of 0 and 1, chemists try to build models of the world of molecules. The models ai e conceptually quite simple 2D plots of molecular sti uctures or projections of 3D structures onto a plane. The problem is how to transfer these models to computers and how to make computers understand them. This communication must somehow be handled by widely understood input and output processes. The chemists way of thinking about structures must be translated into computers internal, machine representation through one or more intermediate steps or representations (sec figure 2-23, The input/output processes defined... 

The earlier sections have only considered the way atoms are bonded to each other in a molecule (topology) and how this is translated into a computer-readable form. Chemists define this arrangement of the bonds as the constitution of a molecule. The example in Figure 2-39, Section 2.5.2.1, shows that molecules with a given empirical formula, e.g., C H O, can have several different structures, which are called isomers [lOOj. Isomeric structures can be divided into constitutional isomers and stereoisomers (see Figure 2-67). 

We next solve the secular equation F — I = 0 to obtain the eigenvalues and eigenvectors o the matrix F. This step is usually performed using matrix diagonalisation, as outlined ii Section 1.10.3. If the Hessian is defined in terms of Cartesian coordinates then six of thes( eigenvalues will be zero as they correspond to translational and rotational motion of th( entire system. The frequency of each normal mode is then calculated from the eigenvalue using the relationship ... 

The rotational temperature is defined as the temperature that describes the Boltzmann population distribution among rotational levels. For example, for a diatomic molecule, this is the temperature in Equation (5.15). Since collisions are not so efficient in producing rotational cooling as for translational cooling, rotational temperatures are rather higher, typically about 10 K. 

For example, a polypeptide is synthesized as a linear polymer derived from the 20 natural amino acids by translation of a nucleotide sequence present in a messenger RNA (mRNA). The mature protein exists as a weU-defined three-dimensional stmcture. The information necessary to specify the final (tertiary) stmcture of the protein is present in the molecule itself, in the form of the specific sequence of amino acids that form the protein (57). This information is used in the form of myriad noncovalent interactions (such as those in Table 1) that first form relatively simple local stmctural motifs (helix... 

Translating odor modifiers into consumer products results in forms, such as soHds, Hquids, and aerosols, for a market defined as products "for the nose." This includes products that cover up or eliminate odors, perfume the home, or cleanse the air. Such products thus defined were reported to have sales in 1992 of just under 2 biUion. The categories of this market can be broken out as traditional air fresheners, cat Utter products, aroma care, air purification, and disinfectant in both consumer and industrial appUcations. 

Once a plant is built, the conditions of agitation, aeration, oxygen transfer, and heat transfer are more or less set, and sterilization cycles are defined. Those environmental conditions achievable in plant-scale equipment should be scaled down to the pilot plant and laboratoiy equipment (shaken flasks) to insure that results can be translated. 

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