Geometry-dependence

In contrast to the point charge model, which needs atom-centered charges from an external source (because of the geometry dependence of the charge distribution they cannot be parameterized and are often pre-calculated by quantum mechanics), the relatively few different bond dipoles are parameterized. An elegant way to calculate charges is by the use of so-called bond increments (Eq. (26)), which are defined as the charge contribution of each atom j bound to atom i. 

Dinur U and A T Hagler 1995. Geometry-Dependent Atomic Charges Methodology and Application to Alkcmes, Aldehydes, Ketones and Amides. Journal of Computational Chemistry 16 154-170. 

Laminar and Turbulent Flow, Reynolds Number These terms refer to two distinct types of flow. In laminar flow, there are smooth streamlines and the fuiid velocity components vary smoothly with position, and with time if the flow is unsteady. The flow described in reference to Fig. 6-1 is laminar. In turbulent flow, there are no smooth streamlines, and the velocity shows chaotic fluctuations in time and space. Velocities in turbulent flow may be reported as the sum of a time-averaged velocity and a velocity fluctuation from the average. For any given flow geometry, a dimensionless Reynolds number may be defined for a Newtonian fluid as Re = LU p/ I where L is a characteristic length. Below a critical value of Re the flow is laminar, while above the critical value a transition to turbulent flow occurs. The geometry-dependent critical Reynolds number is determined experimentally. 

The energy release rate (G) represents adherence and is attributed to a multiplicative combination of interfacial and bulk effects. The interface contributions to the overall adherence are captured by the adhesion energy (Go), which is assumed to be rate-independent and equal to the thermodynamic work of adhesion (IVa)-Additional dissipation occurring within the elastomer is contained in the bulk viscoelastic loss function 0, which is dependent on the crack growth velocity (v) and on temperature (T). The function 0 is therefore substrate surface independent, but test geometry dependent. 

Molecular geometry depends not only on the constituent atoms, but also on the total number of electrons. Molecules with identical formulas but with varying numbers of electrons may prefer different geometries. 

Finally, by using what is known about the geometry dependence of the mi f functions (i.e., that m, f is strongly peaked near geometries Qo where the anion and neutral surfaces approach most closely), it is possible to further simplify the semi-classical equation for Rx... 

The lowest energy structures were found to be the above carbon and above bond geometries (Fig. 6d and e, respectively). This is a function of the anisotropy of the electron density around the iodine and the geometry dependent polarization of the dihalogen. 

Rhodium complexes facilitate the reductive cydization of diyne species in good yield, although the product olefin geometry depends on the catalysts used. Moderate yields of -dialkylideneclopentane 169 resulted if a mixture of diyne 146 and trialkylsilane was added to Wilkinson s catalyst ClRh[PPh3]3 (Eq. 33) [101]. If, however, the diyne followed by silane were added to the catalyst, a Diels-Alder derived indane 170 was produced (Eq. 34). Cationic Rh complex, (S-BINAP)Rh(cod) BF4, provides good yields of the Z-dialkylidenecyclopentane derivatives, although in this case, terminal alkynes are not tolerated (Eq. 35) [102]. 

Developed inner surfaces are represented to give clever illustrations of sorption sites geometry. Depending on the temperature range and the molecular size of sorbed... 

DFT method combined with a cluster model approach was compared regarding its suitability for describing both structures and energy profiles. This study shows that the relative stability and geometry depend on the cluster sizes in agreement with previous studies [15] but shows that the energy barrier heights of the reaction processes are not affected. 

Further evidence for the above-mentioned mechanism of HOMO elevation by group 14 elements is provided by studies of thioethers. The decrease in oxidation potential of silyl ethers as compared to ethers is not realized in the case of a-silylthioethers whereas a-stannyl substituents in thioethers cause a considerable cathodic shift in oxidation potential. Moreover, the effect is geometry-dependent. Values for substituted cyclic dithianes 15 are summarized in Table 21. The difference between Si and Sn in this case is illustrative. The lone nonbonding pair in the 3p orbital of sulfur is much too low in energy compared to... 

G. D. Hawkins, C. J. Cramer, and D. G. Truhlar, Parameterized model for aqueous free energies of solvation using geometry-dependent atomic surface tensions with implicit electrostatics, J. Phys. Chem. B 101 7147 (1997) erratum to be published. 

This effect resembles the traditional Casimir effect, which describes the attraction between two parallel metallic mirrors in vacuum. Here, however, the fluctuating (bosonic) electromagnetic fields are replaced by fermionic matter fields. Furthermore, the Casimir energy is inferred from the geometry-dependent part of the density of states, and its sign is not fixed, but oscillates according to the relative arrangement and distances of the cavities. 

Let us now invert the logic and define the Casimir energy as the energy resulting from the geometry-dependent part of the density of states (d.o.s.) - a concept that is closely related to the shell correction energy in nuclear physics ... 

Now the Casimir energy can be extracted from the geometry-dependent part of the density of states as a simple integral... 

In this way, the geometry-dependent Casimir fluctuations can be extracted from the m//W/ /e-scattcri ng part of the scattering matrix. The determinant of the n-spherc/disk S-matrix can be separated into a product of the determinants of the 1-sphere/disk S-matrices S E, a,i), where a, are the radii of the single scatterers, and the ratio of the determinant of the multi-scattering matrix M(k) and its complex conjugate (A. Wirzba., 1999) ... 

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