Stochastic coordinates

Dynamics on longer time scales determines spectral line shapes and requires more coarse-grained models rooted in a stochastic approach. For semirigid systems the relevant set of stochastic coordinates can be restricted to the set of orientational coordinates RS10W = Cl, which can be described, in turn, in terms of a simple formulation for a diffusive rotator, characterized by a diffusion tensor D [16], i.e. 

This appendix is devoted [60] to the calculation of the correlation function, at a given time, of the two stochastic coordinates that are, the position Q, and its indefinite integral S over time, that is, ( Q(0)S(0)). 

Let us consider the two usual Langevin equations (161) and (162), dealing with the Brownian oscillator together with the definition of the stochastic coordinate S... 

On the other hand, let us look at the stochastic variable S(f) given by Eq. (M.3). It must be emphasized that this stochastic variable, the time derivative of which is the stochastic coordinate Q(t), is given by an indefinite integral which is... 

Usually we are interested in the (auto)correlation function G(r) of an observable (i.e., a function of some stochastic coordinates). In the following we will consider either rotational correlation functions (i.e., involving the spherical harmonics Ty ,(ii])) or momentum correlation functions (i.e., involving the components of L,) for the first rotator (body 1), identified as the solute molecule... 

A common feature of all the stochastic models considered here is the presence of several important decay times, usually at least as many as the number of stochastic coordinates included in the system, but even more are found under certain conditions. To display the multiexponential decay of a process one can use different representations. First of all, such evidence can be obtained by plotting the correlation function G t) versus t. Also a representation in the frequency domain by spectral densities y(w) versus (o can be useful. Cole-Cole plots may also have a certain usefulness, but they do not give much more information. We have chosen to give only time domain representations here, largely for reasons of space. A few spectral densities are shown in our initial reports. If a more... 

Kramers solution of the barrier crossing problem [45] is discussed at length in chapter A3.8 dealing with condensed-phase reaction dynamics. As the starting point to derive its simplest version one may use the Langevin equation, a stochastic differential equation for the time evolution of a slow variable, the reaction coordinate r, subject to a rapidly statistically fluctuating force F caused by microscopic solute-solvent interactions under the influence of an external force field generated by the PES F for the reaction... 

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. 

The crystal structures in Chapters 5 and 6 were determined by x-ray diffraction, and the papers illustrate Pauling s approach to this experimental technique, including his most notable methodological contributions—the coordination method (SP 42) and the stochastic method (SP 47). In its day, SP 47 was a tour de force in the determination of a complex crystal structure. SP 46 contains Pauling s famous discovery of two quite different crystal structures giving the same x-ray diffraction pattern, which violated the then-current conventional wisdom in x-ray crystallography. 

In Eq. (13), the vector q denotes a set of mass-weighted coordinates in a configuration space of arbitrary dimension N, U(q) is the potential of mean force governing the reaction, T is a symmetric positive-definite friction matrix, and , (/) is a stochastic force that is assumed to represent white noise that is Gaussian distributed with zero mean. The subscript a in Eq. (13) is used to label a particular noise sequence For any given a, there are infinitely many... 

This is the same equation of motion that is satisfied by the original coordinate qa(t), except that the stochastic driving term is absent. The relative dynamics is therefore deterministic. We have chosen the notation accordingly and left out the index a in the definition (41) of Aq (although, of course, we cannot expect the relative dynamics to remain noiseless in the full nonlinear system). Although noiseless, the relative dynamics is still dissipative because Eq. (43) retains the damping term. 

In the stochastic approach, the Markovian random process is usually used for the description of the solvent, and it is assumed that the velocity relaxation is much faster than the coordinate relaxation.74 Such a description is applicable at long time intervals which considerably exceed the characteristic times of the electron... 

Instead of using MD, the X variables may also be sampled stochastically. In the hybrid CMC/MD approach, Metropolis Monte Carlo69 is used to evolve the X variables and molecular dynamics is used to evolve the atomic coordinates. The Metropolis Monte Carlo criteria leads to the generation of a canonical ensemble of the ligands when the following transition probability is used... 

The formulation outlined above allows for a simple stochastic implementation of the deterministic differential equation (35). Starting with an ensemble of trajectories on a given adiabatic PES W, at each time step At we (i) compute the transition probability pk k, (h) compare it to a random number ( e [0,1], and (iii) perform a hop if pt t > C- In Ih se of a pure A -level system (i.e., in the absence of nuclear dynamics), the assumption (37) holds in general, and the stochastic modeling of Eq. (35) is exact. Considering a vibronic problem with coordinate-dependent however, it can be shown that the electronic... 

A model of the ZFS coupling removing the restriction of its constant amplitude and allowing both processes, the stochastic variations of the internal coordinates and the rotational diffusion, to modulate the ZFS interaction was proposed by Westlund and co-workers (13,85,88,91). According to this model, the ZFS interaction provided the coupling between the electron spin variables, the stochastic time-dependent distortion coordinates and the reorientational degrees of freedom by the expression ... 

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