Distance dependence, linear

The original FMM has been refined by adjusting the accuracy of the multipole expansion as a function of the distance between boxes, producing the very Fast Multipole Moment (vFMM) method. Both of these have been generalized tc continuous charge distributions, as is required for calculating the Coulomb interactioi between electrons in a quantum description. The use of FMM methods in electronic structure calculations enables the Coulomb part of the electron-electron interaction h be calculated with a computational effort which depends linearly on the number of basi functions, once the system becomes sufficiently large. 

Tobi D, Elber R. Distance-dependent, pair potential for protein folding results from linear optimization. Proteins 2000 41 40-6. 

Fig. 5 Schematic representation of long distance radical cation migration in DNA. In AQ-DNA(3), irradiation of the anthraquinone group linked at the 5 -terminus leads to reaction at GG steps that are 10, 28, 46 and 55 base pairs from the charge injection site. The solid arrows indicate approximately the amount of reaction observed at each GG step. The plot shows the natural log of the normalized amount of reaction as a function of distance from the AQ. The results appear to give a linear distance dependence...

A semi-log plot of the distance dependence of strand cleavage efficiency, see Fig. 8, gives a linear relationship with a slope experimentally indistinguishable from zero. DNA has occasionally been characterized as a molecu-... 

Fig. 8 Semi-log plots of the distance dependence of reaction for DNA(8-11). There is an apparent linear relationship in each case, but the slopes differ according to the specific sequence of DNA bases...

The linear distance dependence seen for AQ-DNA(3) is not observed to be universally independent of specific DNA base sequence. This is clearly revealed by examination of AQ-DNA(4) and AQ-DNA(5). Plots of the distance dependence of strand cleavage at the GG steps in these oligomers are shown in Fig. 11. Both show stepped rather than linear behavior, and the size of... 

Fig. 35 Changes in R2n, the averaged distance between atoms n-bonds apart along the chain, during crystallization at 330 K. In the initial state of undercooled melt ( ), R2n depends linearly on n showing the random coil nature of the melt chains. With the onset of crystallization, the functional form of R2n changes considerably (o) after 6.4 ns, (A) after 12.8 ns, and (A) after 24.32 ns...

It should be realized that Equation 11 was chosen because of the unavailability of a solution for the electrostatic potential p(x) for a system of two interacting bodies in contact with solutions containing both monovalent and divalent cations. At the same time, we have solutions for the surface charge and surface potential for isolated plates, in contact with both monovalent and divalent ions, which bind to the surface to some degree. Our solution for the isolated plates also gives the distance dependence of j/(x) (30). The potential j/(x) falls off with x, the distance from the surface, more steeply than according to the linear approximation. Therefore, the linear approximation in Equation 11 is regarded as an overestimate of Ve. 

In cases such as AM Na+ +. 5mM Ca2+ (or less Ca2+) the applicability of the linear approximation (Equation 11) may be questionable because the value of the surface potential for the isolated surfaces is = — 60mV. In order to get estimates for the lower bounds of 1/2(10) and i/2(20), we took the values of k and for this case and used Equation 11. This is justified because our studies (30) indicate that the distance dependence of j/ (for the isolated surfaces using the modified Gouy Chapman equa-... 

The experimentally determined dependence of the Rb NMR chemical shielding on ionic distance is shown in Figure 9A. Upon initial inspection, it can be seen that the halides form three parallel lines. The dependence of the 87Rb shielding on the Rb-X distance is linear for all three halides. The slopes are very similar to each other (about 400 ppm/A), indicating that the sensitivity of the shielding to interionic distance is probably a property of Rb and is independent of the identity of the halide and the other cation. The three halides form three separate but parallel lines due to an offset caused by changing the identity of the halide. 

This simple and appealing result shows that, for H 1 /k, the sphere-wall interaction depends linearly on the charge densities of each surface, and decays exponentially with the separation distance. The result does not depend on whether the surfaces are considered to be constant charge density or constant potential, because the potentials of an isolated wall and sphere were used in its derivation. Phillips [13] has compared Eq. (24) with a numerical solution of the linear Poisson-Boltzmann equation, and shows that it errs by less than about 10% for xh>3 when 0.5 [Pg.257]

---