Vibration time

We next consider the effective force balance for all >N variables, while treating the system as an unconstrained system. For simplicity, we consider the case in which the crossover from ballistic motion to diffusion occurs at a timescale much less than any characteristic relaxation time for vibrations of the hard coordinates, so that the vibrations are overdamped, but in which the vibrational relaxation times are much less than any timescale for the diffusion of the soft coordinates. In this case, we may assume local equilibration of all 3N momenta at timescales of order the vibration time. Repeating the analysis of the Section V.A, while now treating all 3N coordinates as unconstrained, yields an effective force balance... 

Adiabatic passage schemes are particularly suited to control population transfer between states, since the adiabatic following condition assesses the stability of the dynamics to fluctuations in the pulse duration and intensity [3]. The time evolution of the wave function does not depend on the dynamical phase, and is therefore slow in comparison with the vibrational time scale. This fact guarantees that the time variation of the observables during the controlled dynamics will be slow. Adiabatic methods can therefore be of great utility to control dynamic observables that do not commute with the Hamiltonian. We are interested in the control of the bond length of a diatomic molecule [4]. 

If the infrared band profile of single vibrational transition is given by Lorentzian, the band profile corresponds to the reorientational and vibrational time-correlation functions of exponential form, the relaxation time is expressed by,... 

The far-infrared spectrum of Fe(EtPhDtc)3 as a function of pressure shows that the intensity of the band assigned to the 2 V2 state increases relative to that assigned to the 6A t state upon increasing pressure (93b) (38). For the Fe(n-Pr2 -Dtc)3 complex, the Fe—S vibration at 367 cm"1 was assigned to the low-spin-state (6i4j) isomer. On the basis of these results, the spin-state equilibrium was adopted as the true model with a spin-state interconversion rate lower than the vibrational time scale ( 10" 3 sec). 

In principle, a great deal of information concerning intramolecular electron transfer is available from IT absorption band measurements. Optical electron transfer is rapid on the vibrational time-scale and, as illustrated in Figure 7, the optical transition is a vertical process in the Franck— Condon sense. 

Atd > A t s the time atoms take to move through distance, %, is longer than the characteristic vibration time, Atd. Thus, they can be trapped by or unable to escape from the potential, (x ) and the configuration is in static (solid) state. 

The motions associated with the degrees of freedom of the solvent molecules involve different time scales. In particular, typical vibration times being of the order of 10-14-10-12s, it is clear that the orientational component of the solvent polarization cannot instantaneously readjust to follow the oscillating solute , so that a nonequilibrium solute-solvent system has to be considered. 

To capture the relaxation times change from molecular vibration times to values prototypical near the glass transition temperature, other metrics of fragility have been used that exploits the shape of the normalized excess entropy data in a reduced-7), form. This includes the ratio F of VFT temperature, where the relaxation time diverges, to the glass transition temperature, To/Tg [18,107] or... 

J. Stare, J. Mavri, Numerical solving of the vibrational time-independent Schroedinger equation in one and two dimensions using the variational method. J. Comput. Phys. Commun. 143, 222-240 (2002)... 

In this case, because the bath includes intramolecular coordinates, the evaluation of this formula, which involves products of translational and vibrational time-correlation functions, is quite complicated (12). For a polyatomic solute with a large enough number of vibrational modes, we argue that the time-correlation functions for translations decay on the time scale of the inverse of the characteristic frequencies of the translational bath, which is much slower than the decay of time-correlation functions for the intramolecular vibrations. Therefore we can replace the translational time-correlation functions by their initial values, which we evaluate classically. The upshot is that the rate constant can be written as (12)... 

Fig. 10. Dependence of the packing factor of highly-loaded coarse-dispersion composites on vibration time. Numbers on the curves correspond to numbers of composites in Table 2...

Figure 2.2d shows the simplest type of nondissociative electron capture into discrete states of AB- that will occur between energies E1 and Ei, resulting in a vibrationally excited AB- molecular ion. If the capture process remains an isolated event, the electron will be ejected by autodetachment (Auger process) within a time comparable with a vibration time. 

For real molecules, however, as opposed to hard sphere molecules, the collisions are not instantaneous, rather, they take place over a finite time usually longer than vibrational times. The effect of such slow collisions is to make the momentum exchange between AB and C take place smoothly over many vibration cycles and thus relatively independently... 

The nature of the analyte interactions with liophilic ions could be electrostatic attraction, ion association, or dispersive-type interactions. Most probably all mentioned types are present. Ion association is essentially the same as an ion-pairing used in a general form of time-dependent interionic formation with the average lifetime on the level of 10 sec in water-organic solution with dielectric constant between 30 and 40. With increase of the water content in the mobile phase, the dielectric constant increases and approaches 80 (water) this decrease the lifetime of ion-associated complexes to approximately 10 sec, which is still about four orders of magnitude longer than average molecular vibration time. 

Hie instantaneous structure structure I) can be thought of as a snapshot of the network with an exposure time which is short compared with the molecular vibration times, that is, Ti = s (this is precisely the order of magnitude of the time s s used in computer simulation exf riments ). 

Even though the triplet appears localized, at least on a vibrational time scale, the apparent delayed fluorescence does indicate substantial mobility on the time scale of phosphorescence decay. We have examined these decays and found them to be quite nonlinear. With the exoeption pf the g-naphthyl polymer, the first half lives are about 1-2 x 10 seconds. 

PSZ,PSZ and SUS 304 mixture,and SUS 304 wet materials were laminated into a die,and then were compacted by a vibration pressing method. 120 sec is taken to reach a fixed pressure after the die was set in vibration plates. The forming vibration time after a fixed pressure was 60 sec,amplitude was 0.9 mm. 

The rate constant of proton dissociation is extremely sensitive to its environment. Except for very strong acids (pK < 0), no dissociation will take place unless water molecules are in the immediate vicinity to act as proton acceptors. What is more, these water molecules must be free to react with the dissociating proton at a time scale comparable with the vibration time (30/sec). During such a short period, the water molecules are practically fixed in space and only those molecules that are at the... 

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