Models translational motion

Outer sphere relaxation arises from the dipolar intermolecular interaction between the water proton nuclear spins and the gadolinium electron spin whose fluctuations are governed by random translational motion of the molecules (106). The outer sphere relaxation rate depends on several parameters, such as the closest approach of the solvent water protons and the Gdm complex, their relative diffusion coefficient, and the electron spin relaxation rate (107-109). Freed and others (110-112) developed an analytical expression for the outer sphere longitudinal relaxation rate, (l/Ti)os, for the simplest case of a force-free model. The force-free model is only a rough approximation for the interaction of outer sphere water molecules with Gdm complexes. 

To simulate the particle-particle collision, the hard-sphere model, which is based on the conservation law for linear momentum and angular momentum, is used. Two empirical parameters, a restitution coefficient of 0.9 and a friction coefficient of 0.3, are utilized in the simulation. In this study, collisions between spherical particles are assumed to be binary and quasi-instantaneous. The equations, which follow those of molecular dynamic simulation, are used to locate the minimum flight time of particles before any collision. Compared with the soft-sphere particle-particle collision model, the hard-sphere model accounts for the rotational particle motion in the collision dynamics calculation thus, only the translational motion equation is required to describe the fluid induced particle motion. In addition, the hard-sphere model also permits larger time steps in the calculation therefore, the simulation of a sequence of collisions can be more computationally effective. The details of this approach can be found in the literature (Hoomans et al., 1996 Crowe et al., 1998). 

Equations (1-3) are widely used for protein dynamics analysis from relaxation measurements. The primary goals here are (A) to measure the spectral densities J(co) and, most important, (B) to translate them into an adequate picture of protein dynamics. The latter goal requires adequate theoretical models of motion that could be obtained from comparison with molecular dynamics simulations (see for example Ref. [23]). However, accurate analysis of experimental data is an essential prerequisite for such a comparison. 

Given the character of the water-water interaction, particularly its strength, directionality and saturability, it is tempting to formulate a lattice model, or a cell model, of the liquid. In such models, local structure is the most important of the factors determining equilibrium properties. This structure appears when the molecular motion is defined relative to the vertices of a virtual lattice that spans the volume occupied by the liquid. In general, the translational motion of a molecule is either suppressed completely (static lattice model), or confined to the interior of a small region defined by repulsive interactions with surrounding molecules (cell model). Clearly, the nature of these models is such that they describe best those properties which are structure determined, and describe poorly those properties which, in some sense, depend on the breakdown of positional and orientational correlations between molecules. 

Because simulated water is a classical liquid, the computed power spectrum which describes the translational motions, is bound to disagree with that of real water. Figure 37, shows that the power spectrum has peaks at 44 cm-1 and 215 cm-1, whereas for real water they occur at 60 cm-1 and 170 cm-1. A similar discrepancy exists between simulated and real water rotational power spectra (compare the simulated water frequencies 410 cm-1, 450 cm-1 and 800-925 cm-1 with the accepted experimental values 439 cm-1, 538 cm-1 and 717 cm-1). In this model localization of the molecules around their momentary orientations is only marginal. 

A = solv, pol. The striking similarity in the spectra and their components is evident, justifying the use of OKE data to model SD in this liquid. It should be noted that the two experiments have very (tifierent dynamical origins in some liquids such as, for example, water, where SD is strongly dominated by rotational dynamics, " whereas OKE probes mainly translational motions due to the very small molecular polarizability anisotropy. ... 

To interpret our data, we have developed a simple model which takes into account the orientation of the n donor molecules in the first solvent shell with respect to the acceptor. The donor molecules can be sorted into three types. Donors that are in an optimal position to quench the acceptor molecules (Da), donors that have first to rotate (Z) ) and those which need translational motion (Dc). Hence, the rate constant, kq, for the fluorescence decay of an acceptor with a given solvent configuration is the sum of the individual ET rate constants with the donors of the first solvent shell... 

Before discussing other results it is informative to first consider some correlation and memory functions obtained from a few simple models of rotational and translational motion in liquids. One might expect a fluid molecule to behave in some respects like a Brownian particle. That is, its actual motion is very erratic due to the rapidly varying forces and torques that other molecules exert on it. To a first approximation its motion might then be governed by the Langevin equations for a Brownian particle 61... 

The MD simulations show that second shell water molecules exist and are distinct from freely diffusing bulk water. Freed s analytical force-free model can only be applied to water molecules without interacting force relative to the Gd-complex, it should therefore be restricted to water molecules without hydrogen bonds formed. Freed s general model [91,92] allows the calculation of NMRD profiles if the radial distribution function g(r) is known and if the fluctuation of the water-proton - Gd vector can be described by a translational motion. The potential of mean force in Eq. 24 is obtained from U(r) = -kBT In [g(r)] and the spectral density functions have to be calculated numerically [91,97]. 

As opposed to the adiabatic limit, we assume in the sudden approximation that the internal motion is slow compared to the external (i.e., translational) motion. Most familiar is the rotational sudden approximation which is frequently exploited in energy transfer studies in full collisions (Pack 1974 Secrest 1975 Parker and Pack 1978 Kouri 1979 Gianturco 1979 ch.4). Its application to photodissociation is straightforward and will be outlined below for the model discussed in Section 3.2. 

The superscript 0 on the diffusivities listed above refers to the fact that these are for dilute solutions. In a concentrated system the rate of rotation will be slowed down considerably because of steric hinderance from nearest neighbors. The nature of the entanglements from other rods onto a test rod is such that the translational motion perpendicular to the rod axis becomes highly constrained. The translation along the chain axis, on the other hand, is for the most part unaffected. The steric interactions imposed by the neighboring rods on a single test rod can be modeled by placing such a rod within a tube of radius ac... 

The TSM for elementary processes in condensed systems [58] employs the traditional tenets of Eyring s model [20,23], although it follows from the nature of the condensed state that the driving force of the process is ensured by thermofluctuation excitations of the medium, and not by the translational motion of the particles. In the transition state, an AC reacts with the neighboring particles. The parameter of their interaction - differs from the parameter of interaction between the initial reactant i which the AC is formed from and the neighboring particle j. 

In solution things are more complex. The reaction partners are no longer free in their translational motion as they are in the gas phase they have to move in a condensed medium, and their motion is governed by other physical phenomena which for economy of exposition we shall not consider in detail. It is sufficient to recall that the physical models for the most important terms, Brownian motions, diffusion forces, are expressed in their basic form using a continuum description of the medium. 

Thermodynamics deals with relations among bulk (macroscopic) properties of matter. Bulk matter, however, is comprised of atoms and molecules and, therefore, its properties must result from the nature and behavior of these microscopic particles. An explanation of a bulk property based on molecular behavior is a theory for the behavior. Today, we know that the behavior of atoms and molecules is described by quantum mechanics. However, theories for gas properties predate the development of quantum mechanics. An early model of gases found to be very successftd in explaining their equation of state at low pressures was the kinetic model of noninteracting particles, attributed to Bernoulli. In this model, the pressure exerted by n moles of gas confined to a container of volume V at temperature T is explained as due to the incessant collisions of the gas molecules with the walls of the container. Only the translational motion of gas particles contributes to the pressure, and for translational motion Newtonian mechanics is an excellent approximation to quantum mechanics. We will see that ideal gas behavior results when interactions between gas molecules are completely neglected. 

Some systems cannot be well described by translational motion, so instead they require a model based on rotational diffusion. The commonly used model is one where the nuclei and radical form a bound complex, then this complex rotates to modulate dipolar coupling.74 Here, the overall correlation time consists of the rotational correlation time of the solvent complex, xT, and the exchange rate of molecules in and out of the complex, tm, where 1 /tc 1 /tr I 1 /tm. The form of this spectral density function is simpler4,25 ... 

Barker and Grimson (1991) modeled the flow of deformable particles after a free-draining floe whose shape, orientation, and internal structure ranged between the extremes of an extended chain and a folded globule. They interpreted the unhindered motions of free-flowing, deformable droplets to result from an unbalanced force imposed by the flow field, resulting in rotations around the particles center of mass this rotation is superimposed on the steady translational motion. 

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