Energy transfer, average

We start from a model in which collision cross sections or rate constants for energy transfer are compared with a reference quantity such as average Leimard-Jones collision cross sections or the usually cited Leimard-Jones collision frequencies [54]... 

Given such a reference, we can classify various mechanisms of energy transfer either by the probability tiiat a certain energy transfer process will occur in a Leimard-Jones reference collision , or by the average energy transferred by one Leimard-Jones collision . 

Experimental access to the probabilities P(E ,E) for energy transfer in large molecules usually involves teclmiques providing just the first moment of this distribution, i.e. the average energy (AE) transferred in a collision. Such methods include UV absorption, infrared fluorescence and related spectroscopic teclmiques [11. 28. 71. 72, 73 and 74]. More advanced teclmiques, such as kinetically controlled selective ionization (KCSI [74]) have also provided infonnation on higher moments of P(E ,E), such as ((AE) ). 

The measurement of fluorescence intensity from a compound containing cliromophores of two spectral types is an example of a system for which it is reasonable to operate witli tire average rates of energy transfer between spectral pools of molecules. Let us consider tire simple case of two spectral pools of donor and acceptor molecules, as illustrated in figure C3.4.2 [18]. The average rate of energy transfer can be calculated as... 

For the case of intramolecular energy transfer from excited vibrational states, a mixed quantum-classical treatment was given by Gerber et al. already in 1982 [101]. These authors used a time-dependent self-consistent field (TDSCF) approximation. In the classical limit of TDSCF averages over wave functions are replaced by averages over bundles of trajectories, each obtained by SCF methods. 

For example, the measured pressure exerted by an enclosed gas can be thought of as a time-averaged manifestation of the individual molecules random motions. When one considers an individual molecule, however, statistical thermodynamics would propose its random motion or pressure could be quite different from that measured by even the most sensitive gauge which acts to average a distribution of individual molecule pressures. The particulate nature of matter is fundamental to statistical thermodynamics as opposed to classical thermodynamics, which assumes matter is continuous. Further, these elementary particles and their complex substmctures exhibit wave properties even though intra- and interparticle energy transfers are quantized, ie, not continuous. Statistical thermodynamics holds that the impression of continuity of properties, and even the soHdity of matter is an effect of scale. 

Fig. 5.3. Computed average relative energy transfer as a function of collision energy (meV) for the Ar-N2 (jt = 0) and BTT potential in IOS (solid line) and CS (broken line) approximations [208].

Treatment Energy Recovery Total Transfers Average per Facility... 

Energy transfer measurements were used, together with fluorescence and absorption spectral data of the donor and acceptor moieties, to calculate the donor-acceptor separation via the Forster equation. The average values of R obtained assuming random donor-acceptor orientations were 21.3 1.6 for (1) and 16.7 + 1.4 for (2). The average separation obtained from molecular models is 21.8 + 2.0 for (1) and 21.5 2.0 for (2). The somewhat low calculated separation between the groups of (2) may be due to nonrandom donor-acceptor orientations. 

The slope divided by the intercept gives the rate ratio kjket. Table 6.7 gives the results for compounds (7), n = 2-4 in benzene solution. The detailed mechanism of energy transfer must be able to account for the 25-fold decrease in ket in going from n = 2 to n = 4. While inspection of molecular models shows that the average distance between chromophores... 

In addition, it can be shown for the concentration range of the 4,4 -BPDC used, assuming each molecule occupies a spherical volume, the average radius of this volume is about 108 a. This calculation predicts, on the average, the probability of an excited DMT molecule having a 4,4 -BPDC molecule within the required 15 A for energy transfer to occur by the exchange mechanism, which would be spin allowed, is small. 

The rotation of the fluorophores is a factor that affects the energy transfer. Only maximal rotational freedom will permit tda estimation. There is no way to predict this factor. Therefore the dynamic averaged value of k2 is considered 2/3. This prediction induces a certain error in the calculation of distances (see Chap. 1). 

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