Autocorrelation function/time

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... 

An important property of the time autocorrelation function CaU) is that by taking its Fourier transform, F CA(t) a, one gets a spectral decomposition of all the frequencies that contribute to the motion. For example, consider the motion of a single particle in a hannonic potential (harmonic oscillator). The time series describing the position of the... 

White J. A., Velasco S., Calvo Hernandez A., Luis D. On the quantum time autocorrelation function, Phys. Lett. A130, 237-9 (1988). 

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. 

Dynamics of Reorientation. The approach of Impey, Madden and McDonald is used to consider the dynamics of reorientation for water molecules. The time autocorrelation function of the second Legrendre polynomial P2 of the angle subtended by the intramolecular H-H bond vector at time t with respect to its position at time t = 0 is calculated ... 

In considering the statistical nature of the scattered light, it is useful to use the concept of the time autocorrelation function which is defined by... 

It follows from Bochner s theorem that the normalized time autocorrelation function C(t) is the characteristic function of a probability distribution G(co) so that... 

With this consideration die relaxation equation will give rise to a set of coupled equations involving the time autocorrelation function of the density and the longitudinal current fluctuation, and also there will be cross terms that involve the correlation between the density fluctuation and the longitudinal current fluctuation. This set of coupled equations can be written in matrix notation, which becomes identical to that derived by Gotze from the Liouvillian resolvent matrix [3]. 

With the above-mentioned simplifications, the equation of motion for the density time autocorrelation function reduces to the following form of a nonlinear equation of motion for a damped oscillator ... 

In the following we present the axioms or basic postulates of quantum mechanics and accompany them by their classical counterparts in the Hamiltonian formalism. We begin the presentation with a brief summary of some of the mathematical background essential for the developments in the following. It is by no means a comprehensive presentation, and the reader is supposed to have some basic knowledge about quantum mechanics that may be obtained from any of the many introductory textbooks in quantum mechanics. The focus here is on results of particular relevance to the subjects of this book. We consider, for example, a derivation of a formal expression for the flux density operator in quantum mechanics and its coordinate representation. A systematic way of generating any representation of any combination of operators is set up, and is of immediate usage for the time autocorrelation function of the flux operator used to determine the rate constants of a chemical process. 

Let us then derive an expression for the matrix element of the flux operator in the coordinate representation, an expression we need in order to develop the time autocorrelation function of the flux operator in the coordinate representation. We use the axiom for the matrix element of the momentum operator in the coordinate representation, and obtain... 

The time autocorrelation functions of primary interest to the various spectroscopic probes now available are the orientational acf s, for example,... 

As a demonstration of the effectiveness of this approach, 500 Lennard-Jones particles were simulated using the RPUT technique during this simulation, the velocity time autocorrelation function was computed along with the normalized time autocorrelation function for and for u i. Each of the simulations was run 50-100 times longer than the time it takes for the correlation... 

Figure 18 Various time autocorrelation functions. The curve with the triangles is for X = Usa, the circles are for X = Uji with = 2.15 x 10, and the squares represent X = p. C is the appropriate normalization constant, and x is the fundamental unit of time.

These elements are slowly varying function of nuclear coordinates and generally treated as constants in accordance with the applicability of the Condon approximation in a diabatic electronic basis [7,78]. The quantity C/(f) = ( I /(0) I /(0)> is the time autocorrelation function of the wave packet (WP) initially prepared on the /th electronic state and, I /(0 = I /(0). 

Hence, the autocorrelation function always decays from the mean of the squared static intensity to the square of the mean intensity [see Fig. 8.20(b)]. In the simplest case of a single relaxation mode, the time autocorrelation function decays like a single exponential with correlation time r ... 

Hence, the time autocorrelation function provides a direct means to determine the diffusion coefficient D in dilute monodisperse solutions ... 

In the case of a more general polydispersity, the time autocorrelation function corresponds to the sum of many modes ... 

Silvestrelli et al. (1997) used Car-Parrinello molecular dynamics to obtain the IR spectrum of D2O. They did so by determining the time autocorrelation function for the dipole moment of their cell and then relating this function to the absorption coefficient as a function of frequency. Their computed spectrum and comparison to experiment are displayed in Fig. 11, taking into account corrections introduced by the authors. The essential features of the experimental spectrum, particularly the low-frequency peak, are reproduced well. The authors were then able to assign specific modes of the spectrum. 

For periodically replicated nonconducting molecular systems, the complex dielectric permittivity s(co) can be computed from the time autocorrelation function (ACF) of the total dipole moment [64] during the course of a simulation run as follows ... 

The scattered light is detected by a photomultipler tube (PMT). The area that is seen by the PMT encloses the spot of the laser beam on the film. The PMT signal is fed either to a correlator or to a spectrum analyzer. The correlator consists of a fast analogue-to-digital converter and a minicomputer, programmed to calculate the time autocorrelation function of the signal. The correlation function, calculated in this way, is proportional to the time autocorrelation function of the fluctuations in the scattered intensity. 

The output of the spectrum analyzer is proportional to the power spectrum of the fluctuations in the scattered intensity. Although the power spectrum of the fluctuations contains the same information as the time autocorrelation function, each has its own merits, making it necessary to use them both to cover a broad spectral range (see Section VII.C.4). 

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