Yielding representation

I apply the terminology of Bas van Fraassen here. According to the latter, the expression observable refers to unaided acts of perception only and is object-related (e.g., van Fraassen 2001). Hence, and contrasting for example Hacking 1981, the use of even the smallest lens or optical microscope yields representations. 

In this Fourier representation the Hamiltonian is quadratic and the equipartition theorem yields for the thennal... 

U(qJ is referred to as an adiabatic-to-diabatic transformation (ADT) matrix. Its mathematical sbucture is discussed in detail in Section in.C. If the electronic wave functions in the adiabatic and diabatic representations are chosen to be real, as is normally the case, U(q ) is orthogonal and therefore has n n — l)/2 independent elements (or degrees of freedom). This transformation mabix U(qO can be chosen so as to yield a diabatic electronic basis set with desired properties, which can then be used to derive the diabatic nuclear motion Schrodinger equation. By using Eqs. (27) and (28) and the orthonormality of the diabatic and adiabatic electronic basis sets, we can relate the adiabatic and diabatic nuclear wave functions through the same n-dimensional unitary transformation matrix U(qx) according to... 

The representation of a chemical reaction should include the connection table of all participating species starting materials, reagents, solvents, catalysts, products) as well as Information on reaction conditions (temperature, concentration, time, etc.) and observations (yield, reaction rates, heat of reaction, etc.). However, reactions are only Insuffclently represented by the structure of their starting materials and products,... 

For a function to transform according to a specific irreducible representation means that the function, when operated upon by a point-group symmetry operator, yields a linear combination of the functions that transform according to that irreducible representation. For example, a 2pz orbital (z is the C3 axis of NH3) on the nitrogen atom... 

Using the orthogonality of eharaeters taken as veetors we ean reduee the above set of eharaeters to Ai + E. Henee, we say that our orbital set of three Ish orbitals forms a redueible representation eonsisting of the sum of and E IR s. This means that the three 1 sh orbitals ean be eombined to yield one orbital of Ai symmetry and a pair that transform aeeording to the E representation. 

Figure 7.9 shows a schematic representation of this effect, in which the ratio of the two isotopes changes with time. To obtain an accurate estimate of the ratio of ion abundances, it is better if the relative ion yields decrease linearly (Figure 7.9) which can be achieved by adjusting the filament temperature continuously to obtain the desired linear response. An almost constant response for the isotope ratio can be obtained by slow evaporation of the sample, viz., by keeping the filament temperature as low as is consistent with sufficient sensitivity of detection (Figure 7.9). 

In this representation the FeCl2 which takes part in the first step of the reaction is not a tme catalyst, but is continuously formed from HQ. and iron. This is a highly exothermic process with a heat of reaction of 546 kj /mol (130 kcal/mol) for the combined charging and reaction steps (50). Despite the complexity of the Bnchamp process, yields of 90—98% are often obtained. One of the major advantages of the Bnchamp process over catalytic hydrogenation is that it can be mn at atmospheric pressure. This eliminates the need for expensive high pressure equipment and makes it practical for use in small batch operations. The Bnchamp process can also be used in the laboratory for the synthesis of amines when catalytic hydrogenation caimot be used (51). 

The square root of viscosity is plotted against the reciprocal of the square root of shear rate (Fig. 3). The square of the slope is Tq, the yield stress the square of the intercept is, the viscosity at infinite shear rate. No material actually experiences an infinite shear rate, but is a good representation of the condition where all rheological stmcture has been broken down. The Casson yield stress Tq is somewhat different from the yield stress discussed earlier in that there may or may not be an intercept on the shear stress—shear rate curve for the material. If there is an intercept, then the Casson yield stress is quite close to that value. If there is no intercept, but the material is shear thinning, a Casson plot gives a value for Tq that is indicative of the degree of shear thinning. 

The relations between the compositions of Pordand cements and some other common hydrauhc cements are shown in the CaO—Si02—AI2O2 phase diagram of Figure 2 (5). In this diagram, Fe202 has been combined with AI2O2 to yield the Al O content used. This is a commonly appHed approximation that permits a two-dimensional representation of the real systems. 

Figure 27-10 provides a graphical representation of Eq. (27-32) showing the weight fraction of various products as a function of the chain growth parameter Ot. This figure shows that there is a particular Ot that will maximize the yield of a desired product, such as gasohne or... 

Another source of departure from stoichiometry occurs when cations are reduced, as for example in tire reduction of zinc oxide to yield an oxygen-defective oxide. The zinc atoms which are formed in tlris process dissolve in the lattice, Zn+ ions entering interstitial sites and the coiTesponding number of electrons being released from these dissolved atoms in much the same manner as was found when phosphorus was dissolved in the Group IV semiconductors. The Kroger-Viirk representation of dris reduction is... 

A proportionality between the theoretical spall strength and the bulk modulus is obtained when a two-parameter model is chosen to represent the intermolecular potential. Other two-parameter representations of the intermolecular potential, such as the Lennard-Jones 6-12 potential, will yield a similar proportionality although the numerical coefficients will differ slightly. 

The second generation force fields for nucleic acids were designed to be used with an explicit solvent representation along with inclusion of the appropriate ions [28,29]. In addition, efforts were made to improve the representation of the conformational energetics of selected model compounds. Eor example, the availability of high level ab initio calculations on the conformational energetics of the model compound dimethylphosphate yielded... 

A statistieal representation of the yield strength for BS 220M07 is not available however, the eoeffieient of variation, Cv, for the yield strength of steels is eommonly given as 0.08 (Furman, 1981). For eonvenienee, the parameters of the Normal distribution will be ealeulated by assuming that the minimum value is —3 standard deviations from the expeeted mean value (Cable and Virene, 1967) ... 

A reahstic modeling of N2 monolayers on graphite was done in Refs. 95,96. Concerning the N2-N2 interactions in Hamiltonian (40) and Eq. (42), the X model [146-148,318], consisting of site-site Lennard-Jones and quadrupole interactions, was shown to yield a realistic representation ... 

Both space-filling and electron density models yield similar molecular volumes, and both show the obvious differences in overall size. Because the electron density surfaces provide no discernible boundaries between atoms (and employ no colors to highlight these boundaries), the surfaces may appear to be less informative than space-filling models in helping to decide to what extent a particular atom is exposed . This weakness raises an important point, however. Electrons are associated with a molecule as a whole and not with individual atoms. The space-filling representation of a molecule in terms of discernible atoms does not reflect reality, but rather is an artifact of the model. The electron density surface is more accurate in that it shows a single electron cloud for the entire molecule. 

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