Distance decay with

Hard-sphere models lack a characteristic energy scale and, hence, only entropic packing effects can be investigated. A more realistic modelling has to take hard-core-like repulsion at small distances and an attractive interaction at intennediate distances into account. In non-polar liquids the attraction is of the van der Waals type and decays with the sixth power of the interparticle distance r. It can be modelled in the fonn of a Leimard-Jones potential Fj j(r) between segments... 

In addition, the non-bonded forces can be divided into several regions according to pair distances. The near region is normally more important than the distant region because the non-bonded forces decay with distance. Since most of the CPU time in a MD simulation is spent in the calculation of these non-bonded interactions, the separation in pair distance results in valuable speedups. Using a 3-fold distance split, the non-bonded forces are separated in 3 regions near, medium, and fax distance zones. Thus, the Liouville operator can be express as a sum of five terms... 

Three-pulse ESEEM spectrum of perdeuterated P-carotene imbedded in Cu-MCM-41 exhibits an echo decay with an echo modulation due to deuterons. The three-pulse ESEEM is plotted as a function of time, and curves are drawn through the maximum and minima. From ratio analysis of these curves, a best nonlinear least-squares lit determines the number of interacting deuterons, the distance (3.3 0.2A), and the isotopic coupling (0.06 0.2MHz). This analysis made it possible to explain the observed reversible forward and backward electron transfer between the carotenoid and Cu2+ as the temperature was cycled (77-300 K). 

Previous studies of Vapor Cloud Explosions (VCE) have used a correlation between the mass of a gas in the cloud and equivalent mass of TNT to predict explosion overpressures. This was always thought to give conservative results, but recent research evidence indicates that this approach is not accurate to natural gas and air mixtures. The TNT models do not correlate well in the areas near to the point of ignition, and generally over estimate the level of overpressures in the near field. Experiments on methane explosions in "unconfined" areas have indicated a maximum overpressure of 0.2 bar (2.9 psio). This overpressure then decays with distance Therefore newer computer models have been generated to better simulate the effects... 

A number of distance dependence studies have been carried out using synthetic DNA duplexes. In most of these investigations, the data were found to be consistent with the superexchange mechanism since the ET and HT rates followed exponential decays with increasing distance, with /J ranging from 0.1 A 1 to 1.4 A A1461... 

For pairs of like chromophores at a fixed distance and with random and uncorrelated static orientations, the decay of emission anisotropy of the indirectly excited chromophore varies with time, tending to zero (Berberan-Santos and Valeur, 1991) in contradiction to earlier works where it was reported to be 4% of that of the directly excited chromophore. Therefore, because the probability that emission arises from the directly excited chromophore is 1/2, the decay of emission anisotropy of the latter levels off at r0/2. This can be generalized to an ensemble of n chromophores (with random and uncorrelated static orientations) the decay of emission anisotropy of the directly excited chromophore levels off at r0/n. 

For an infinitely wide beam (i.e., a beam width many times the wavelength of the light, which is a very good approximation for our purposes), the intensity of the evanescent wave (measured in units of energy per unit area per second) exponentially decays with perpendicular distance z from the interface ... 

Experimental data and theoretical considerations suggest that water pressures developed in LNG-water RPTs are significantly less than critical pressure of the LNG. Measured values have not exceeded 10-20 bar. Average overpressures in the air show a rapid decay with distance and are approximately equivalent to values expected from the detonation of a few tenths of a kilogram of TNT (see Section III,K). 

In the diagram of Fig. 7.7 we show all possible pair correlations for the finite open linear case with m = 6. The parameters are still the same h = 0.01, K=4, and Ti = 0.1. Note that there are three different nn pair correlations gj 2= 5 g, 2,3 = 4,5 and gj 4. There are two different second-n pair correlations, gj 3 and g24 (only the different ones are indicated in the diagram). There are two third-n pair correlations, gj 4 and g2 5- There is one fourth-n pair correlation gj 5 and one sixth- n pair correlation gj g. It should also be noted that, as in the infinite chain (m °o) case, the correlations decay with distance /. 

P(l), for an isolated step decays with distance I as 1//. In Section 3, we extend the... 

In general, oscillations may be oblate-prolate (H8, S5), oblate-spherical, or oblate-less oblate (E2, FI, H8, R3, R4, S5). Correlations of the amplitude of fluctuation have been given (R3, S5), but these are at best approximate since the amplitude varies erratically as noted above. For low M systems, secondary motion may become marked, leading to what has been described as random wobbling (E2, S4, Wl). There appears to have been little systematic work on oscillations of liquid drops in gases. Such oscillations have been observed (FI, M4) and undoubtedly influence drag as noted earlier in this chapter. Measurements (Y3) for 3-6 mm water drops in air show that the amplitude of oscillation increases with while the frequency is initially close to the Lamb value (Eq. 7-30) but decays with distance of fall. 

Before we move on to consider the interaction between macroscopic bodies, let us look briefly at the phenomenon of retardation . The electric field emitted by an instantaneously polarized neutral molecule takes a finite time to travel to another, neighbouring molecule. If the molecules are not too far apart the field produced by the induced dipole will reach the first molecule before it has time to disappear, or perhaps form a dipole in the opposite direction. The latter effect does, however, occur at larger separations (>5 nm) and effectively strengthens the rate of decay with distance, producing a dependence of 1/R instead of 1/R . 

Cai et al. [7e] investigated electron and hole transfer in various polynucleotide duplexes and compared them with previous results found for salmon sperm DNA, to examine the effect of base sequence on excess electron and hole transfer along the DNA 71-way at low temperature. Electron and hole transfer in DNA was found to be clearly base sequence dependent. In glassy aqueous systems (7M LiBr glasses at 77 K), excess electron-transfer rates increase in the order polydIdC-polydIdC<salmon testes DNAexcess electron and hole transfer rates increase in the order polyC-polyG<salmon testes DNATransfer distances at 1 min and distance decay constants for electron and hole transfer from base radicals to MX in polynucleotides-MX and DNA-MX at 77 K are derived and compiled in Table 3. This table clearly shows that the electron-transfer rate from donor sites decreases in... 

STOs have a number of features that make them attractive. The orbital has the correct exponential decay with increasing r, the angular component is hydrogenic, and the Is orbital has, as it should, a cusp at the nucleus (i.e., it is not smooth). More importantly, from a practical point of view, overlap integrals between two STOs as a function of interatomic distance are readily computed (Mulliken Rieke and Orloff 1949 Bishop 1966). Thus, in contrast to simple Huckel theory, overlap matrix elements in EHT are not assumed to be equal to the Kronecker delta, but are directly computed in every instance. 

In Chap. 2 and 3, the motion of two reactants was considered and a diffusion equation was derived based upon the equation of continuity and Fick s first law of diffusion (see, for instance, Chap. 2 and Chap. 3, Sect. 1.1). When one reactant (say D) can transfer energy or an electron to the other reactant (say A) over distances greater than the encounter separation, an additional term must be considered in the equation of continuity. The two-body density n (rj, r2, t) decays with a rate coefficient l(r, — r2) due to long-range transfer. Furthermore, if energy is being transferred from an excited donor to an acceptor, the donor molecular excited state will decay, even in the absence of acceptor molecules with a natural lifetime r0. Hence, the equation of continuity (42) becomes extended to include two such terms and is... 

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