The Dynamic Contribution

The events taking place in the RCs within the timescale of ps and sub-ps ranges usually involve vibrational relaxation, internal conversion, and photo-induced electron and energy transfers. It is important to note that in order to observe such ultrafast processes, ultrashort pulse laser spectroscopic techniques are often employed. In such cases, from the uncertainty principle AEAt Ti/2, one can see that a number of states can be coherently (or simultaneously) excited. In this case, the observed time-resolved spectra contain the information of the dynamics of both populations and coherences (or phases) of the system. Due to the dynamical contribution of coherences, the quantum beat is often observed in the fs time-resolved experiments. 

The dynamic RIS formalism is used to calculate the rate of first passage from non-excimer-forming conformations to excimer-formlng conformations in seven aromatic polyesters with different flexible spacers between the aromatic rings. The equilibrium chain statistics provides a good description of the relative excimer population for these polyesters, even at times where the dynamic contribution is significant. 

A non-dynamical contribution to the correlation energy corresponds to the change in energy produced by a density transformation. On the other hand, the dynamical contribution arises from a wavefunction change at fixed density. Thus, the two following decompositions of the correlation energy (labelled as I and II) are possible [41] ... 

Accordingly, the modifications to the KS operator are twofold (i) a static contribution through the static multipole moments (here charges) of the solvent molecules and (ii) a dynamical contribution which depends linearly on the electronic polarizability of the environment and also depends on the electronic density of the QM region. Due to the latter fact we need within each SCF iteration to update the DFT/MM part of the KS operator with the set of induced dipole moments determined from Eq. (13-29). We emphasize that it is the dynamical contribution that gives rise to polarization of the MM subsystem by the QM subsystem. 

Consideration of Fig. 1 suggests that /qo for a realistic V x) may be reasonably approximated by this model (suppose V x) decreases like do(x)), but that the integrals with k =0 are greatly overestimated so that the ratio in (6-70) gives a poor lower bound to Our claim therefore is that S , Eq. (6-46), may make a significant contribution to the AOM parameter but that one can expect Si, Eq. (6-47) to be unimportant as far as is concerned. It then follows that the crucial determinant of ei will be the dynamic contribution to v, Eq. (6-22), to which we must now turn. 

On the other hand, using the formalism of Sects. 6.3.1 and 6.3.2 it is clear that it is the dynamic contribution to which is important, and this has a negative sign because the metal s- (and p-)orbitals have smaller ionization potentials than the metal d-orbitals so that E - is negative. 

The result analogous to the dynamic contribution (37) is written down immediately on substituting transition electric dipoles by and electric... 

The intensity of a Bragg reflection depends on the size of the anisotropic displacement parameters of those atoms that contribute to the reflection [6] and so individual parameters can be extracted from crystallographic data. As extracted they are referred to the crystallographic axes and are not necessarily simply oriented with respect to significant molecular directions. These parameters are of some interest in the study of molecular vibrations with neutrons because of the dynamical contribution to their value (but not the disorder contribution). The dynamic contribution plays an important role in determining the observed transition intensities, through the Debye-Waller factor ( 2.5.1.2). 

Although we are unable to use the master equation approach to examine the dynamics of relaxation from the liquid-like state because a statistical method that can account for the dynamical contribution from the huge number of liquid-like minima has not yet been developed, we are able to illustrate the type of processes that would operate by examining relaxation from the highest-energy minima in our sample (Fig. 1.14). The probability... 

The dynamic contribution to the damping matrix is also slightly modified because now appears. We have... 

Perez-Hemandez and Blum" have recently obtained essentially analytic expressions for in the MSA, and quantitative results are given by Veri-cat et al. The MSA results are found to be qualitatively similar to those described above, with decreasing as the ionic concentration is increased. Vericat et al. also compare the MSA results for with experimental values of figoL foi" aqueous solutions. They show that if one is willing to treat the dipole moment of water as an adjustable parameter, rough agreement can be obtained. However, in view of the simplicity of the model, the likely inaccuracy of the MSA (cf. Section III.D), and the failure to account for the dynamic contributions to Esol> it is difficult to reach any conclusions from such comparisons. 

As one moves along a given streamline, the dynamic contribution to fluid pressure, the magnitude of the tangential velocity component, and the relation between r and 6 are illnstrated in Table 8-2. 

TABLE 8-2 Numerical Evaluation of the Tangential Velocity Component and the Dynamic Contribution to Eluid Pressure p (not dynamic pressure IP) as One Traverses a Streamline with f = —0.002 around a Solid Sphere... 

If the chemical shifts are also anisotropic, as for both carbons and protons in unsaturated moieties, then their resonances will be shifted, because their shift tensors will no longer average to zero. In the case of very slow overall rotation, they will also be anisotropically broadened in proportion to Bq. (This should not be confused with the dynamic contributions that shift anisotropy makes to relaxation. As equation (4.16) shows, these are proportional to the square of the static interaction.) Shift anisotropy is most marked for heavier nuclei, e.g. for attached atoms. The resulting lineshapes can be analysed to reveal the ranges of orientation of the liquid crystal relative to Bn. 

Simulation of minerals using both LD and MD requires the calculation of the total interaction energy, Uiatt, and the force on each atom, F,. The dynamical contribution is evaluated via the equations of motion ... 

With increasing anisotropy, lOSA approximation breaks very early which indicates, of course, the importance of the dynamical contribution of rotation to the vibrational transition. One should expect that this contribution will show up in a substantial modification of the Landau-Teller exponent. Once it is realized that the Ehrenfest-Landau-Teller semiclassical exponential factor is proportional to the square of the Fourier component of the external time-dependent perturbation, and that the respective generalized Landau-Teller exponent can be recovered from the classical exponent, the strategy of finding the most efficient energy-transfer pathway becomes clear one should simply look for a mode which will provide the largest high-frequency Fourier components of the time-dependent perturbation that simulates a collision. 

Here, g can be assumed to be equal to the emissivity e. For the dynamic contribution, the relationship suggested by DeWasch and Froment (1972) may be used ... 

It is worth noting that even for a damped system, what matters in structural analysis is the maximum displacement max u, which causes the maximum strain and, hence, the maximum stress, required for proportioning the structural members. As a rule of thumb, this max u is taken for equal to a combination of the maximum displacements max u, of the three or four first modes of the structure, which necessitates that the corresponding synchronous modal velocities become zero, li, = 0. Thus, the dynamic contribution of the modal accelerations to account for... 

To derive expressions for the dynamic contribution we must make some assumptions as to the relevant variables to include in these terms. It seems reasonable given the experiments of Zwetkoff [4] and Miesowicz [S] to assume that at any material point T and L are functions of n, Vv and w, evaluated at that point at that instant. However, since the gradient of the angular velocity is not included in our list of kinematic variables, it immediately follows from EQN (18) that... 

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