Motion, time scale

The physical parameters that determine under what circumstances the BO approximation is accurate relate to the motional time scales of the electronic and vibrational/rotational coordinates. 

The SCRF models assume that solvent response to the solute is dominated by motions that are slow on the solute electronic motion time scales, i.e., Xp Telec. Thus, as explained in Section 2.1, the solvent sees the solute electrons only in an averaged way. If, in addition to the SCRF approximation, we make the usual Bom-Oppenheimer approximation for the solute, then we have xs Xelect-In this case the solute electronic motion is treated as adjusting adiabatically both to the solvent motion and to the solute nuclear motion. 

Scheme 1 The motional time scale of proteins covers over 14 orders of magnitude, starting from the slowest one (formation of aggregates requires typically minutes to hours) up to the fastest event side-chain rotation < ps). NMR-based techniques capable of capturing dynamics at different time scales are shown boxed and positioned approximately at their relevant range of time scales.

DNA/Conditions Technique Motions Time Scale Comments Ref. 

Figure 8.2 Motional time scale in biomembranes and NMR windows. Drawings and microscopy image depict the spatial scale at which events may occur...

One possibility is to determine the time scale for diffusion around a spherical particle by relaxation experiments, since transverse relaxation rates are sensitive to slow motional time scales. The 2H relaxation time T2 was measiued in a quadrupolar CPMG experiment, in which the delay between pulses, r, in the pulse sequence was varied. With increasing r, the experiment increasingly allows for diffusion as a slow motional mode to contribute to the relaxation rate, and to decrease T2. With this principle, from the dependence of R2 = T2 against the diffusion coefficient D is extracted, as shown in fig. 15, where the slope is proportional to D [52], Diffusion coefficients of the phospholipid POPC were determined at different temperatures [52], the diffusion of the inner and outer monolayer of DPPC on silica could be separated [53], and recently a diffusion coefficient was determined for lipid bilayers on polymeric support [54],... 

To optimize force fields for long time scale motions Aliev et al. propose a new robust approach to use NMR spin-lattice relaxation times Ti of both backbone and sidechain carbons. This allows a selective determination of both overall molecular and intramolecular motional time scales. In addition they use motionally averaged experimental/ coupling constants for torsional FF parameters. The force constants in the FFs and the correlation times are fitted in an Arrhenius-type of equation. 

Thus the average velocity decays exponentially to zero on a time scale detennined by the friction coefficient and the mass of the particle. This average behaviour is not very interesting, because it corresponds to tlie average of a quantity that may take values in all directions, due to the noise and friction, and so the decay of the average value tells us little about the details of the motion of the Brownian particle. A more interesting... 

Due to the conservation law, the diffiision field 5 j/ relaxes in a time much shorter than tlie time taken by significant interface motion. If the domain size is R(x), the difhision field relaxes over a time scale R Flowever a typical interface velocity is shown below to be R. Thus in time Tq, interfaces move a distanc of about one, much smaller compared to R. This implies that the difhision field 6vj is essentially always in equilibrium with tlie interfaces and, thus, obeys Laplace s equation... 

This is no longer the case when (iii) motion along the reaction patir occurs on a time scale comparable to other relaxation times of the solute or the solvent, i.e. the system is partially non-relaxed. In this situation dynamic effects have to be taken into account explicitly, such as solvent-assisted intramolecular vibrational energy redistribution (IVR) in the solute, solvent-induced electronic surface hopping, dephasing, solute-solvent energy transfer, dynamic caging, rotational relaxation, or solvent dielectric and momentum relaxation. 

For very fast reactions, as they are accessible to investigation by pico- and femtosecond laser spectroscopy, the separation of time scales into slow motion along the reaction path and fast relaxation of other degrees of freedom in most cases is no longer possible and it is necessary to consider dynamical models, which are not the topic of this section. But often the temperature, solvent or pressure dependence of reaction rate... 

The dependence of k on viscosity becomes even more puzzling when the time scale of motion along the reaction coordinate becomes comparable to that of solvent dipole reorientation around the changing charge distribution... 

Many of the fiindamental physical and chemical processes at surfaces and interfaces occur on extremely fast time scales. For example, atomic and molecular motions take place on time scales as short as 100 fs, while surface electronic states may have lifetimes as short as 10 fs. With the dramatic recent advances in laser tecluiology, however, such time scales have become increasingly accessible. Surface nonlinear optics provides an attractive approach to capture such events directly in the time domain. Some examples of application of the method include probing the dynamics of melting on the time scale of phonon vibrations [82], photoisomerization of molecules [88], molecular dynamics of adsorbates [89, 90], interfacial solvent dynamics [91], transient band-flattening in semiconductors [92] and laser-induced desorption [93]. A review article discussing such time-resolved studies in metals can be found in... 

Genberg L, Richard L, McLendon G and Miller R J D 1991 Direct observation of global protein motion in hemoglobin and myoglobin on picosecond time scales Science 251 1051-6... 

The method of molecular dynamics (MD), described earlier in this book, is a powerful approach for simulating the dynamics and predicting the rates of chemical reactions. In the MD approach most commonly used, the potential of interaction is specified between atoms participating in the reaction, and the time evolution of their positions is obtained by solving Hamilton s equations for the classical motions of the nuclei. Because MD simulations of etching reactions must include a significant number of atoms from the substrate as well as the gaseous etchant species, the calculations become computationally intensive, and the time scale of the simulation is limited to the... 

The first requirement is the definition of a low-dimensional space of reaction coordinates that still captures the essential dynamics of the processes we consider. Motions in the perpendicular null space should have irrelevant detail and equilibrate fast, preferably on a time scale that is separated from the time scale of the essential motions. Motions in the two spaces are separated much like is done in the Born-Oppenheimer approximation. The average influence of the fast motions on the essential degrees of freedom must be taken into account this concerns (i) correlations with positions expressed in a potential of mean force, (ii) correlations with velocities expressed in frictional terms, and iit) an uncorrelated remainder that can be modeled by stochastic terms. Of course, this scheme is the general idea behind the well-known Langevin and Brownian dynamics. 

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