Exponential damping functions

Essential dynamics, 2, 233, 236, 242-244, 247 Euler angles, 3,168 evolutionary determinants, 4, 4, 5 evolvability, 4, 7-9,17 Ewald summation, 2, 265 Ewald summation techniques, 1, 59, 62, 75 exact exchange, 1, 26, 27 exchange repulsion, 3,179,180 excited state structure/dynamics, 1, 24 excretion see ADMET properties explicit-rl2 correlation, 5, 132,140 explicit solvent, 2, 98, 99,101,102,104-106 exponential damping functions, 3,180 extended systems, 1, 26... 

Steady state shear viscosity and primary normal stress coefficient for low density polyethylene melt T and from the Kaye-Bernstein, Kearsley, Zapas (K-BKZ) equation wim the double exponential damping function, eq4.4.13 (solid lines) and with the single exponential, eq4.4.12 (dotted Une). Data at different temperatures have been shifted to one master curve by ar T). Replotted from Laun (1978). 

The strains and at which the maxima occur in the shear stress and first normal stress difference are sometimes used as empirical measures of nonlinearity. Using the exponential damping function (Eq. 10.31) in Wagner s equation (Eq. 10.10) Osaki etal. [42] derived the following relationships ... 

In the real space the correlation function (6) exhibits exponentially damped oscillations, and the structure is characterized by two lengths the period of the oscillations A, related to the size of oil and water domains, and the correlation length In the microemulsion > A and the water-rich and oil-rich domains are correlated, hence the water-water structure factor assumes a maximum for k = k 7 0. When the concentration of surfac-... 

The effects of such collisionally induced kicks are treated within the so-called pressure broadening (sometimes called collisional broadening) model by modifying the free-rotation correlation function through the introduction of an exponential damping factor exp( -Itl/x) ... 

This damping function s time scale parameter x is assumed to characterize the average time between collisions and thus should be inversely proportional to the collision frequency. Its magnitude is also related to the effectiveness with which collisions cause the dipole function to deviate from its unhindered rotational motion (i.e., related to the collision strength). In effect, the exponential damping causes the time correlation function <% I Eq ... 

We will first consider the error bounds for a spectral density broadened by a Lorentzian slit function, Eq. (15), describing the response to an exponentially damped perturbation. In this case the broadened spectrum,... 

Fig. 1. Error bounds for the nuclear resonance line shape of crystalline CaF2, broadened by a Lorentzian slit function (i.e., the energy absorption by the coupled nuclear spins, due to an exponentially damped harmonic perturbation by a radiofrequency magnetic field).

Because there is no general microscopic theory of liquids, the analysis of inelastic neutron scattering experiments must proceed on the basis of model calculations. Recently1 we have derived a simple interpolation model for single particle motions in simple liquids. This derivation, which was based on the correlation function formalism, depends on dispersion relation and sum rule arguments and the assumption of simple exponential decay for the damping function. According to the model, the linear response in the displacement, yft), satisfies the equation... 

Although most implementations of VARPRO have concentrated on using exponentially damped sinusoids for the basis functions (i.e. Lorentzian lineshapes), there is no reason why more realistic models (e.g. Gaussian or Voigt32) should not be used. 

Actually, the spectroscopic data would more closely resemble the pattern in Figure 3.15, which is the same as the wave in Figure 3.14, except that the overall intensity of the signal decays exponentially with time. (Note that the decay does not affect the frequencies.) Such a pattern is called the modulated free induction decay (FID) signal (or time-domain spectrum). The decay is the result of spin-spin relaxation (Section 2.3.2), which reduces the net magnetization in the, y plane. The envelope (see Section 3.6.2) of the damped wave is described by an exponential decay function whose decay time is T, the effective spin-spin relaxation time. 

Linear prediction methods assume that the future time domain data can be represented as a linear combination of past time domain data, and vice versa—an assumption that is valid when the time domain data consist of exponentially damped sine functions and the signal/noise ratio is sufficiently large. FIDs from... 

A study was made of the ability of viscoelastic models to describe the measured material functions of unplasticised PVC during extrusion and to determine whether it was possible to reproduce the elastic properties of the large entrance pressure drop and small extrudate swell during the extrusion of PVC using a capillary rheometer. Models used were the Phan-Thien and Tanner model and the K-BKZ-Wagner model with a single exponential damping... 

The quantity < f m > e has been termed Raman overlap (Myers and Mathies, 1987). It is the product of the modulus of a time-dependent Franck-Condon factor between the final state and the initial wavefunction propagated on the electronic surface, i. e. I < f m > I and a damping function e which decreases exponentially with time. 

Wagner proposes a single exponential form of the damping function ... 

Fig. 5 Disjoining pressure isotherm of a Q2G2/PDADMAC film. The solid line represents an exponentially damped cosine function. The mechanically stable and unstable parts are marked (from Ref. [33])...

For all wavelength, t) shows coherent oscillations well beyond the 3 ps time mark. This contrasts with experimental results showing an exponential damping of the time correlation function within this time scale. Such a damping might be due to frictional damping or static disorder on the RCs. 

Slater and Kirkwood s idea [121] of an exponential repulsion plus dispersion needs only one concept, damping functions, see section Al.5.3.3. to lead to a working template for contemporary work. Buckingham and Comer [126] suggested such a potential with an empirical damping function more than 50 years ago ... 

Modem versions of this approach use a more elaborate exponential function for the repulsion, more dispersion terms, induction terms if necessary, and individual damping functions for each of the dispersion, and sometimes induction, terms as in equation (Al.5.37). 

The damping factor e y acts on a wave function, multiplying it by the function e Mr). The role of e 7recursion relations clearly is also analogous to that of the exponential damping factor used in conjunction with the split-operator time-propagation scheme (4). 

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