Single exponential approximation

Temporal behavior of the correlation function was studied in Ref. 91 using a particular example of the correlation function of sin x(f) in a periodic potential with periodic boundary conditions. In that case the use of single exponential approximation had also given a rather adequate description. The considered... 

The single exponential approximation works especially well for observables that are less sensitive to the location of initial distribution, such as transition probabilities and correlation functions. 

The exponential approximation may lead to a significant error in the case when the noise intensity is small, the potential is tilted, and the barrier is absent (purely dynamical motion slightly modulated by noise perturbations). But, to the contrary, as it has been observed for all considered examples, the single exponential approximation is more adequate for a noise-assisted process either (a) a noise-induced escape over a barrier or (b) motion under intensive fluctuations. 

The d, b n, and cmn coefficients are given numerically as polynomials of S0 and their expansion coefficients are tabulated in Table 1 of Voids for the Maier-Saupe potential. In the single exponential approximation for gmn(t),119 the spectral densities simplify to... 

The importance of the carboxylate donors is underlined by a study of the lanthanide coordination chemistry of the similar terdentate ligand 2,6 -bis( 1 -pyrazol-3 -yl)pyridine, L24 (63). The complex structure of [Tb(L24)3][PF6]3, shown in Fig. 11, appears to be fairly robust in methanolic solution, with Horrocks analysis (q = 0.6) suggesting the 9-coordinate structure is retained the small quenching effect of outer sphere coordination explains the q-value. However, in aqueous solution, the lability of the ligands dramatically changes the luminescence. Whilst the emission decays are not exactly single exponential, approximate lifetimes in H20 and DoO suggest a solvation value of 4-5. 

Figure 33. Single-exponential approximation of the temperature dependence of the spin-lattice relaxation rate in two temperature intervals. (From Ref. 360.)...

This expression has been used to describe uniaxial restricted molecular reorientation in liquid crystals [7.11, 7.41]. The above series expansion converges rapidly generally, only the first five terms are required [7.43]. Using the anisotropic viscosity model to describe the a— and / — motion, in the single exponential approximation it is found that... 

When this holds, the kinetic equations reduce to single exponentials. Chipperfield6 demonstrates that approximate adherence to Eq. (4-25) suffices to fit 20 absorbancetime pairs spaced at equal times over the first 75 percent of the reaction with correlation coefficients better than 0.999. 

Case E, where the three rate constants are comparable, does not feature A2 s> A3. That is, the progress curve is not a single exponential. None of the approximations is very good, although fcjmp is the best. Case F, characterized by a value of k 1 that is much smaller than kss or 2, is similar. The fit for k is quite poor that for ktmp is the best, but it is hardly adequate. 

The model [39] was developed using three assumptions the conformers are in thermodynamic equilibrium, the peak intensities of the T-shaped and linear features are proportional to the populations of the T-shaped and linear ground-state conformers, and the internal energy of the complexes is adequately represented by the monomer rotational temperature. By using these assumptions, the temperature dependence of the ratio of the intensities of the features were equated to the ratio of the quantum mechanical partition functions for the T-shaped and linear conformers (Eq. (7) of Ref. [39]). The ratio of the He l Cl T-shaped linear intensity ratios were observed to decay single exponentially. Fits of the decays yielded an approximate ground-state binding... 

Hence, a series of measurements with several Tcp values will provide a data set with variable decays due to both diffusion and relaxation. Numerical inversion can be applied to such data set to obtain the diffusion-relaxation correlation spectrum [44— 46]. However, this type of experiment is different from the 2D experiments, such as T,-T2. For example, the diffusion and relaxation effects are mixed and not separated as in the PFG-CPMG experiment Eq. (2.7.6). Furthermore, as the diffusion decay of CPMG is not a single exponential in a constant field gradient [41, 42], the above kernel is only an approximation. It is possible that the diffusion resolution may be compromised. 

For 8,9,10,11-tetrahydro-BA the lifetimes measured with and without DNA are the same within experimental error ( 2 nsec). Without DNA the decay profile of trans-7,8-dihydroxy-7,8-dihydro-BP follows a single-exponential decay law. With DNA the decay profile has a small contribution from a short-lived component (x = 5 nsec) which arises from DNA complexes. This indicates that Equation 1 is not strictly valid. However, the analysis of the decay profile with DNA also indicates that the short lifetime component contributes less than 11% to the total emission observed at [POa ] 5 x 10 M. Under these conditions Equation 1 still yields a good approximate value to the association constant for intercalation. 

The required correlation function (5.13) may be approximated by the single exponential function... 

The most simple GMA is the (1,1) approximation which reproduces the moments p0 and p,. In this case, the relaxation of AM(f) is approximated by a single exponential... 

Even if we consider a single solvent, e g., water, at a single temperature, say 298K, depends on the solute and in fact on the coordinate of the solute which is under consideration, and we cannot take xF as a constant. Nevertheless, in the absence of a molecular dynamics simulation for the solute motion of interest, XF for polar solvents like water is often approximated by the Debye model. In this model, the dielectric polarization of the solvent relaxes as a single exponential with a relaxation time equal to the rotational (i.e., reorientational) relaxation time of a single molecule, which is called Tp) or the Debye time [32, 347], The Debye time may be associated with the relaxation of the transverse component of the polarization field. However the solvent fluctuations and frictional relaxation occur on a faster scale given by [348,349]... 

The NO dissociation rate constants are summarized in Table HI (50) and are smaller than those seen with NO-metmyoglobin complexes. NP2 and NP3 (A ofr 0.1 s ) release NO approximately 10 times more slowly than NPl and NP4 kofs 2-3 s ) at pH 8.0, and the NO release rate for all nitrophorins decreases as the pH is lowered to 5.0. The NO release curves cannot be fit with a single exponential, indicating two off rates at each pH, as previously noted for both recombinant and insect-derived NPl (46), and which has also been recently reported for recombinant NP2 (145). The biphasic kinetics suggest the presence of slowly interconverting conformations. The values obtained are pH and protein dependent, ranging from 2.6 to 0.05 (Table HI) (50), values that are considerably slower than found for sperm whale metmyoglobin... 

Fig. 7 Left Arrhenius plot of the approximate folding and unfolding rates of 15. Right At low temperatures, the relaxation is better fitted by a stretched exponential or biexponential the stretching exponent is shown here as a convenient measure of the deviation from single exponentiality...

In agreement with theoretical prediction, the experimental analysis shows the more positive atoms to be contracted. This is explained by the decrease in electron-electron repulsions, or, in a somewhat different language, the decreased screening of the nuclear attraction forces by a smaller number of electrons. This contraction is incorporated in Slater s rules for approximate, single exponential (and therefore nodeless), hydrogen-like orbital functions (Slater 1932). For a 2px orbital of a second-row atom, for example, the orbital function is given by... 

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