Force autocorrelation function

From this expression we see that the friction cannot be determined from the infinite-time integral of the unprojected force correlation function but only from its plateau value if there is time scale separation between the force and momentum correlation functions decay times. The friction may also be estimated from the extrapolation of the long-time decay of the force autocorrelation function to t = 0, or from the decay rates of the momentum or force autocorrelation functions using the above formulas. 

In the canonical ensemble (P2) = 3kBTM and p M. In the microcanonical ensemble (P2) = 3kgT i = 3kBTMNm/(M + Nm) [49]. If the limit M —> oo is first taken in the calculation of the force autocorrelation function, then p = Nm and the projected and unprojected force correlations are the same in the thermodynamic limit. Since MD simulations are carried out at finite N, the study of the N (and M) dependence of (u(t) and the estimate of the friction coefficient from either the decay of the momentum or force correlation functions is of interest. Molecular dynamics simulations of the momentum and force autocorrelation functions as a function of N have been carried out [49, 50]. 

P. Espanol and I. Zuniga, Force autocorrelation function in Brownian motion theory, J. Chem. Phys. 98, 574 (1993). 

We mention this result here in order to assert that the spectral distribution of B(jf is the Fourier transform of the (force) autocorrelation function 0(t). In view of Eqn. (5.45), we can restate this result in terms of the velocity t>(/). The spectral distribution of the velocity autocorrelation function is directly related to the Fourier transform of 0 j), the force autocorrelation function. Thus, we see that the classical equation of motion when properly averaged over many particles provides insight into the relation between transport kinetics and particle dynamics [R. Becker (1966)]. 

The friction on a tagged particle is expressed in terms of the time-dependent force-force autocorrelation function. Although the bare, short-time part of the friction that arises from binary collisions can be calculated from kinetic theory, the long-time part needs the knowledge of the solvent and the solute dynamics and the coupling between them. The solvent dynamic quantities... 

The immediate question, then, is whether this scenario reflects what actually happens. Do the INM theories really work There is, in fact, some evidence on this score (45,52). If we compare the vibrational friction predicted by INM theory, Equation (20), with that revealed by an exact molecular-dynamics evaluation of the force autocorrelation function, Equations (4) and (13), we see some reasonably impressive agreement (Fig. 3) (52). Not only is the few hundred cm-1 spectral range of the friction predicted quite nicely, but the basic form of the response is as well. Each example shows that the friction diminishes as the frequency rises, beginning with a sharp drop from its maximum value at co = 0 and gradually going over to a much slower decay, behavior captured nicely by the INM formulas. 

SIZE EFFECTS IN ENCOUNTER AND REACTION DYNAMICS written as a force autocorrelation function,... 

Though Eq. (27) is exact in a perturbative sense, it is demanding to calculate the quantum mechanical force autocorrelation function 5(f) even for small molecular systems. Hence, many computational schemes have been developed to approximate the quantum mechanical force autocorrelation function. 

A different approach to the calculation ofthe rates kf i, Eq. (13.40), and the related rates k and ks is to evaluate the force autocorrelation function associated with the interaction (13.13) and (13.14) and the coiTesponding force... 

Difficulties associated with disparity of timescales may still be encountered even within this approach, in cases where the frequency of the impurity molecule is much larger than the cutoff (Debye) frequency, md, of the host. Note that the rate (13.26) is given by the Fourier transform of the force autocorrelation function, taken at the frequency of the impurity oscillator. The time dependence of this correlation function reflects the time dependence of... 

This result is akin to Eq. (13.22), which relates the vibrational relaxation rate of a single harmonic oscillator of frequency m to the Fourier transform of the force autocorrelation function at that frequency. In the Markovian limit, where (cf. Eq. (8.60)) Z(Z) = 2y<5(Z), we recover Eq. (14.78). 

To establish a relationship between the Hamiltonian equation (10) and the actual enzymatic system one performs a molecular dynamics simulation to obtain the force F(t) acted upon the reaction coordinate. Then the force autocorrelation function , which is proportional to the friction kernel y(t), is related to the parameters of the fictitious medium of Equation (10) through... 

These considerations also show [cf. Eq. (11.5.37d)] that the random-force autocorrelation function and thereby Aq) are also even functions of q that is,... 

The potential of mean force will typically have two wells, corresponding to reactants and products, separated by a barrier. To set the notation, we denote the location of the reactant well, the barrier, and the product well by qa, q, and qb, respectively. One usually expects that the dynamics will be governed by the behavior of the system around the barrier top. Thus the standard procedure (48,49) for generating a GLE is to restrict the system to the barrier top q = q and determine the force autocorrelation function of all other degrees of freedom. The force is just VV, so by using molecular dynamics constrained to the barrier top one can compute the force autocorrelation function (VV(f) W(0)). One then models the true dynamics in terms of a GLE in which the time-dependent friction function is determined through the fluctuation dissipation relation, Eq. (5). 

The VTST method has been recently implemented for the model SN2 reaction CL + CH,C1 -+ C1CH3 + CL in water. An extensive numerical study of this model was described by Bergsma et al. and Gertner et al. in Refs. 48 and 49. They used the reactive flux formulation to estimate the numerically exact rate constant. They also tested the GLE model for the reaction dynamics. As mentioned in Sec. II.C they represented the time-dependent friction function as the force autocorrelation function when the reacting system is clamped at the top of the barrier of the potential of mean force. 

Velocity or Force Autocorrelation Function in Molecular Dynamic Simulations.112... 

DETERMINATION DIFFUSION COEFFICIENT OF BROWNIAN PARTICLES USING VELOCITY OR FORCE AUTOCORRELATION FUNCTION IN MOLECULAR DYNAMIC SIMULATIONS... 

The friction coefficient C of the Brownian particle was determined by means of the stochastic force autocorrelation function (FACT) [4] ... 

Espanol, Pep. Zuniga, Ignacio. Force autocorrelation function in brownian motion theory. The Journal of Chemical Chem. Physics Phys., 1993, 98(1), 574-580. Yuan-Hui, Li. Gregory, Sandra. Diffusion of ions in sea water and in deep-sea sediments. Geochimica et Cosmochimica Acta, 1974, 38(5), 703-714. 

Two parallel simulations for 108 particles interacting with Lennard-Jones plus Axllrod-Teller potentials have been performed. The first calculation utilized the CMD method in which the forces were explicitly evaluated at each time step. In the second run the two body forces were determined in the standard way and the LMTS method described above was applied to the three body forces. Both runs were started from the same initial particle positions and velocities and both were continued for 1650 time steps. A comparison of the properties obtained from the two calculations is given in Table 1. In addition to the properties listed in Table I, radial distribution functions, velocity, speed, and force autocorrelation functions, and atomic mean squared displacements (from which diffusion coefficients were obtained) were calculated. For all of these properties, the LMTS values were within 0.1% of the values obtained by the CMD method. Figure 4 shows the per cent deviation in the instantaneous total energy of the two calculations. 

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