Autocorrelation function, molecular

Arcsine distribution, 105, 111 Assumption of molecular chaos, 17 Asymptotic theory, 384 of relaxation oscillations, 388 Asynchronous excitation, 373 Asynchronous quenching, 373 Autocorrelation function, 146,174 Autocovariance function, 174 Autonomous problems, 340 nonresonance oscillations, 350 resonance oscillations, 350 Autonomous systems, 356 problems of, 323 Autoperiodic oscillation, 372 Averages, 100... 

Steele W. A. Molecular reorientation in liquids. II. Angular autocorrelation functions, J. Chem. Phys. 38, 2411-18 (1963). 

Raman intensities of the molecular vibrations as well as of their crystal components have been calculated by means of a bond polarizibility model based on two different intramolecular force fields ([87], the UBFF after Scott et al. [78] and the GVFF after Eysel [83]). Vibrational spectra have also been calculated using velocity autocorrelation functions in MD simulations with respect to the symmetry of intramolecular vibrations [82]. 

This expression is derived as the Fourier transform of a time-dependent one-particle autocorrelation function (26) (i.e. propagator), and cast in matrix form G(co) over a suitable molecular orbital (e.g. HF) basis, by means of the related set of one-electron creation (ai" ") and annihilation (aj) operators. In this equation, the sums over m and p run over all the states of the (N-1)- and (N+l)-electron system, l P > and I P " respectively. Eq and e[ represent the energy of the... 

Fluorescence intensity detected with a confocal microscope for the small area of diluted solution temporally fluctuates in sync with (i) motions of solute molecules going in/out of the confocal volume, (ii) intersystem crossing in the solute, and (hi) quenching by molecular interactions. The degree of fluctuation is also dependent on the number of dye molecules in the confocal area (concentration) with an increase in the concentration of the dye, the degree of fluctuation decreases. The autocorrelation function (ACF) of the time profile of the fluorescence fluctuation provides quantitative information on the dynamics of molecules. This method of measurement is well known as fluorescence correlation spectroscopy (FCS) [8, 9]. 

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]. 

Formally, S2 represents a decrease in the autocorrelation function caused by the motion S2=0 corresponds to completely unrestricted motion of a bond (N-H in this case), while S2=0 is expected if the bond reorientations are frozen. It was shown recently that the order parameter may be related to the statistical mechanical properties of a protein molecule [33-35] hence, changes in the NMR-derived order parameters can indicate localized contributions to overall molecular entropy. 

For a rod-like probe with its absorption transition moment direction coinciding with the long molecular axis, the rotational motion in this potential well is described by the diffusion coefficient D. The decay of the autocorrelation functions is then shown to be an infinite sum of exponential terms ... 

Abbreviations MD, molecular dynamics TST, transition state theory EM, energy minimization MSD, mean square displacement PFG-NMR, pulsed field gradient nuclear magnetic resonance VAF, velocity autocorrelation function RDF, radial distribution function MEP, minimum energy path MC, Monte Carlo GC-MC, grand canonical Monte Carlo CB-MC, configurational-bias Monte Carlo MM, molecular mechanics QM, quantum mechanics FLF, Hartree-Fock DFT, density functional theory BSSE, basis set superposition error DME, dimethyl ether MTG, methanol to gasoline. 

Fig. 39. Velocity autocorrelation function, (u(0)u(t))/(u(0)u(0)), for carbon tetrachloride liquid at 301K from a molecular dynamics study by Steinhauser and Neumann [452a],...

It shows the clear link between the change of motion of the particle and its diffusion coefficient. In Fig. 50, the velocity autocorrelation function of liquid argon at 90 K (calculated by computer simulation) is shown [451], The velocity becomes effectively randomised within a time less than lps. Further comments on the velocity autocorrelation functions obtained by computer simulation are reserved until the next sub-section. Because the velocity autocorrelation function of molecular liquids is small for times of a picosecond or more, the diffusion coefficient defined in the limit above is effectively established and constant. Consequently, the diffusion equation becomes a reasonable description of molecular motion over times comparable with or greater than the time over which the velocity autocorrelation function had decayed effectively to zero. Under... 

Fig. 50. Velocity autocorrelation function for argon at 90 K. O, From the molecular dynamics calculation of Rahman [451] ——, from the Langevin approximation, exp — mDtfkuT D is taken as 2.72 X 10 9 m2 s"1.

In more complex molecular systems, increased coupling between the translational motion and both rotational and vibrational modes occurs. It is difficult to separate these effects completely. Nevertheless, the velocity autocorrelation functions of the Lennard—Jones spheres [519] (Fig. 52) and the numerical simulation of the carbon tetrachloride (Fig. 39) are quite similar [452a]. 

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. 

Induced dipole autocorrelation functions of three-body systems have not yet been computed from first principles. Such work involves the solution of Schrodinger s equation of three interacting atoms. However, classical and semi-classical methods, especially molecular dynamics calculations, exist which offer some insight into three-body dynamics and interactions. Very useful expressions exist for the three-body spectral moments, with the lowest-order Wigner-Kirkwood quantum corrections which were discussed above. 

The shape of the vibration-rotation bands in infrared absorption and Raman scattering experiments on diatomic molecules dissolved in a host fluid have been used to determine2,15 the autocorrelation functions unit vector pointing along the molecular axis and P2(x) is the Legendre polynomial of index 2. These correlation functions measure the rate of rotational reorientation of the molecule in the host fluid. The observed temperature- and density-dependence of these functions yields a great deal of information about reorientation in solids, liquids, and gases. These correlation functions have been successfully evaluated on the basis of molecular models.15... 

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