Correlation functions autocorrelation

The MD simulations are used here, as they were originally developed, to calculate time-dependent properties. The primary properties in the context of NMR relaxation are the time correlation functions (autocorrelation functions). We give here a summary of a few time correlation functions we need to calculate during our analysis. 

I quantities x and y are different, then the correlation function js sometimes referred to ross-correlation function. When x and y are the same then the function is usually called an orrelation function. An autocorrelation function indicates the extent to which the system IS a memory of its previous values (or, conversely, how long it takes the system to its memory). A simple example is the velocity autocorrelation coefficient whose indicates how closely the velocity at a time t is correlated with the velocity at time me correlation functions can be averaged over all the particles in the system (as can elocity autocorrelation function) whereas other functions are a property of the entire m (e.g. the dipole moment of the sample). The value of the velocity autocorrelation icient can be calculated by averaging over the N atoms in the simulation ... 

A powerful analytical tool is the time correlation function. For any dynamic variable A (it), such as bond lengths or dihedral angles, the time autocorrelation function Cy) is defined... 

In the case where x and y are the same, C (r) is called an autocorrelation function, if they are different, it is called a cross-correlation function. For an autocorrelation function, the initial value at t = to is 1, and it approaches 0 as t oo. How fast it approaches 0 is measured by the relaxation time. The Fourier transforms of such correlation functions are often related to experimentally observed spectra, the far infrared spectrum of a solvent, for example, is the Foiuier transform of the dipole autocorrelation function. ... 

Judging by these results the angular momentum relaxation in a dense medium has the form of damped oscillations of frequency jRo = (Rctc/to)i and decay decrement 1/(2tc). This conclusion is quantitatively verified by computer experiments [45, 54, 55]. Most of them were concerned with calculations of the autocorrelation function of the translational velocity v(t). However the relation between v(t) and the force F t) acting during collisions is the same as that between e> = J/I and M. Therefore, the results are qualitatively similar. In Fig. 1.8 we show the correlation functions of the velocity and force for the liquid state density. Oscillations are clearly seen, which point to a regular character of collisions and non-Markovian nature of velocity changes. 

Thus the nth vibrational spectral moment is equal to an equilibrium correlation function, the nth derivative of the dipole moment autocorrelation function evaluated at t=0. By using the repeated application of the Heisenberg equation of motion ... 

For the analysis of the dynamical properties of the water and ions, the simulation cell is divided into eight subshells of thickness 3.0A and of height equal to the height of one turn of DNA. The dynamical properties, such as diffusion coefficients and velocity autocorrelation functions, of the water molecules and the ions are computed in various shells. From the study of the dipole orientational correlation 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]. 

Figure 8.20. Generation of a ID correlation function, fl (x), by autocorrelation of the ID electron density, Ap (y) for a two-phase topology. Each value of ft (x) is proportional to the overlap integral (total shaded area) of the density and its displaced ghost...

When l l, the above gives the so-called cross-correlation functions and the associated cross-correlation rates (longitudinal and transverse). Crosscorrelation functions arise from the interference between two relaxation mechanisms (e.g., between the dipole-dipole and the chemical shielding anisotropy interactions, or between the anisotropies of chemical shieldings of two nuclei, etc.).40 When l = 1=2, one has the autocorrelation functions G2m(r) or simply... 

We should point out here the great analogy between and the friction coefficient studied in the Brownian motion problem of Section IV (see Eq. (242)) instead of having the time autocorrelation function of the force F , we now have the time correlation function between F and Fe. 

Figure 16 Second Legendre polynomial of the CFI vector autocorrelation function for the sp3 cis-carbon (dashed lines) and the sp2 carbon in a trans-group next to a transgroup (dashed-dotted lines) for two different temperatures. The fit curves to the cis-correlation functions are a superposition of exponential and stretched exponential discussed in the text.

Another transport property of interfacial water which can be studied by MO techniques is the dipole relaxation time. This property is computed from the dipole moment correlation function, which measures the rate at which dipole moment autocorrelation is lost due to rotational motions in time (63). Larger values for the dipole relaxation time indicate slower rotational motions of the dipole... 

This effective Q,t-range overlaps with that of DLS. DLS measures the dynamics of density or concentration fluctuations by autocorrelation of the scattered laser light intensity in time. The intensity fluctuations result from a change of the random interference pattern (speckle) from a small observation volume. The size of the observation volume and the width of the detector opening determine the contrast factor C of the fluctuations (coherence factor). The normalized intensity autocorrelation function g Q,t) relates to the field amplitude correlation function g (Q,t) in a simple way g t)=l+C g t) if Gaussian statistics holds [30]. g Q,t) represents the correlation function of the fluctuat-... 

The theory of statistical mechanics provides the formalism to obtain observables as ensemble averages from the microscopic configurations generated by such a simulation. From both the MC and MD trajectories, ensemble averages can be formed as simple averages of the properties over the set of configurations. From the time-ordered properties of the MD trajectory, additional dynamic information can be calculated via the time correlation function formalism. An autocorrelation function Caa( = (a(r) a(t + r)) is the ensemble average of the product of some function a at time r and at a later time t + r. 

One finds that the coarse-graining with 200 bins is insufficient for correct representation of the autocorrelation functions. On the other hand, using 1000 bins results in autocorrelation functions that are practically indistinguishable to the exact one, independently of the energy of the state in the Q2 space. The application of coarse-graining works equally well for the cross-correlation functions as for the autocorrelation functions. For further details on the application of coarse-graining for efficiently computing the correlation functions, we refer to the discussion in Ref. [41]. 

As for the properties themselves, there are many chemically useful autocorrelation functions. For instance, particle position or velocity autocorrelation functions can be used to determine diffusion coefficients (Ernst, Hauge, and van Leeuwen 1971), stress autocorrelation functions can be used to determine shear viscosities (Haile 1992), and dipole autocorrelation functions are related to vibrational (infrared) spectra as their reverse Fourier transforms (Berens and Wilson 1981). There are also many useful correlation functions between two different variables (Zwanzig 1965). A more detailed discussion, however, is beyond the scope of this text. 

The rate coefficient kernel of eqn. (369) and the initial condition term of eqn. (370) are those given by Northrup and Hynes [ 103]. These expressions are cast into the form of correlation functions (cf. the velocity autocorrelation function) and have a close similarity to the matrix elements in quantum mechanic applications. While they are quite easy to derive and... 

The dipole autocorrelation function, C(t), is the Fourier transform of the spectral line profile, g(v). Knowledge of the correlation function is theoretically equivalent to knowledge of the spectral profile. Correlation functions offer some insight into the molecular dynamics of dense fluids. 

Density expansion. The method of cluster expansions has been used to obtain the time-dependent correlation functions for a mixture of atomic gases. The particle dynamics was treated quantum mechanically. Expressions up to third order in density were given explicitly [331]. We have discussed similar work in the previous Section and simply state that one may talk about binary, ternary, etc., dipole autocorrelation functions. 

Binary interactions. Dipole autocorrelation functions of binary systems are readily computed. For binary systems, it is convenient to obtain the dipole autocorrelation function, C(t), from the spectral profile, G(co). Figure 5.2 shows the complex correlation function of the quantum profile of He-Ar pairs (295 K) given in Figs. 5.5 and 5.6. The real part is an even function of time, 91 C(—t) = 91 C(t) (solid upper curve). The imaginary part, on the other hand, is negative for positive times it is also an odd function of time, 3 C(—t) — — 3 C(t) (solid lower curve, Fig. 5.2). For comparison, the classical autocorrelation function is also shown. It is real, positive and symmetric in time (dotted curve). In the case considered, the... 

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