Time correlation functions motion

All of these time correlation functions contain time dependences that arise from rotational motion of a dipole-related vector (i.e., the vibrationally averaged dipole P-avejv (t), the vibrational transition dipole itrans (t) or the electronic transition dipole ii f(Re,t)) and the latter two also contain oscillatory time dependences (i.e., exp(icofv,ivt) or exp(icOfvjvt + iAEi ft/h)) that arise from vibrational or electronic-vibrational energy level differences. In the treatments of the following sections, consideration is given to the rotational contributions under circumstances that characterize, for example, dilute gaseous samples where the collision frequency is low and liquid-phase samples where rotational motion is better described in terms of diffusional motion. 

If the rotational motion of the molecules is assumed to be entirely unhindered (e.g., by any environment or by collisions with other molecules), it is appropriate to express the time dependence of each of the dipole time correlation functions listed above in terms of a "free rotation" model. For example, when dealing with diatomic molecules, the electronic-vibrational-rotational C(t) appropriate to a specific electronic-vibrational transition becomes ... 

We discuss the rotational dynamics of water molecules in terms of the time correlation functions, Ciit) = (P [cos 0 (it)]) (/ = 1, 2), where Pi is the /th Legendre polynomial, cos 0 (it) = U (0) U (it), u [, Is a unit vector along the water dipole (HOH bisector), and U2 is a unit vector along an OH bond. Infrared spectroscopy probes Ci(it), and deuterium NMR probes According to the Debye model (Brownian rotational motion), both... 

Zatsepin V. M. Time correlation functions of one-dimensional rotational Brownian motion in n-fold periodical potential. Theor. and Math. Phys. 

TOWARDS THE HYDRODYNAMIC LIMIT STRUCTURE FACTORS AND SOUND DISPERSION. The collective motions of water molecules give rise to many hydrodynamical phenomena observable in the laboratories. They are most conveniently studied in terms of the spatial Fourier ( ) components of the density, particle currents, stress, and energy fluxes. The time correlation function of those Fourier components detail the decay of density, current, and fluctuation on the length scale of the Ijk. 

While between the swaps the motion of the system is somewhat realistic, it is important to emphasize that the swaps between two temperatures are nonphysical. This therefore destroys the sequencing of dynamical events (that would be required to calculate, for example, time correlation functions) and renders the dynamics and kinetics artificial. 

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. 

Consider a general system described by the Hamiltonian of Eq. (5), where = Huif) describes the interaction between the spin system (7) and its environment (the lattice, L). The interaction is characterized by a strength parameter co/i- When deriving the WBR (or the Redfield relaxation theory), the time-dependence of the density operator is expressed as a kind of power expansion in Huif) or (17-20). The first (linear) term in the expansion vanishes if the ensemble average of HiL(t) is zero. If oo/z,Tc <5c 1, where the correlation time, t, describes the decay rate of the time correlation functions of Huif), the expansion is convergent and it is sufficient to retain the first non-zero term corresponding to oo/l. This leads to the Redfield equation of motion as stated in Eq. (18) or (19). In the other limit, 1> the expan-... 

The spectrum of scattered light contains dynamical information related to translational and internal motions of polymer chains. In the self-beating mode, the intensity-intensity time correlation function can be expressed (ID) as... 

This damping function s time scale parameter x is assumed to characterize the average time between collisions and thus should be inversely proportional to the collision frequency. Its magnitude is also related to the effectiveness with which collisions cause the dipole function to deviate from its unhindered rotational motion (i.e., related to the collision strength). In effect, the exponential damping causes the time correlation function <% I Eq ... 

Both Pecora (16) and Komarov and Fisher (17) adapted van Hove s space-time correlation function approach for neutron scattering (18) to the light-scattering problem to calculate the spectral distribution of the light scattered from a solution. Using a molecular analysis, Pecora assumed the scattering particles to be undergoing Brownian motion, and predicted a Lorentzian line shape for the spectral distribution of the... 

Gs(r, t) and Gd(r, t) are called the Van Hove self and distinct space-time correlation functions.18 It clearly follows that probability distribution describing the event that a molecule is at the origin at t = 0 and at the point r at the time t. Gs(r, t) is consequently the probability distribution characterizing the net displacement or diffusion of a particle in the time t. Gd(r, t) on the other hand is a probability distribution describing the event that a molecule is at the origin at t = 0 and a different molecule is at the point r at the time t. Gd(r, t) describes the correlated motion of two molecules. It should be noted that the initial value of Gs(r, t) is... 

The present reduced density operator treatment allows for a general description of fluctuation and dissipation phenomena in an extended atomic system displaying both fast and slow motions, for a general case where the medium is evolving over time. It involves transient time-correlation functions of an active medium where its density operator depends on time. The treatment is based on a partition of the total system into coupled primary and secondary regions each with both electronic and atomic degrees of freedom, and can therefore be applied to many-atom systems as they arise in adsorbates or biomolecular systems. 

The persistence of the fluctuating local fields before being averaged out by molecular motion, and hence their effectiveness in causing relaxation, is described by a time-correlation function (TCF). Because the TCF embodies all the information about mechanisms and rates of motion, obtaining this function is the crucial point for a quantitative interpretation of relaxation data. As will be seen later, the spectral-density and time-correlation functions are Fourier-transform pairs, interrelating motional frequencies (spectral density, frequency domain) and motional rates (TCF, time domain). 

In a dynamic light scattering experiment, the measured intensity-intensity time-correlation function g<2)(tc), where tc is the delay time, is related to the normalized electric field correlation function g(1)frc), representative of the motion of the particles, by the Siegert relation [18] ... 

Here B is an optical constant, or is the total polarizability of the particle, and n is the number of components in each particle. The indexes i and j refer to components of the same particle. If the assumption of independent particles was not made, then the indexes could refer to components of any two particles, and the autocorrelation expression could not be written as a simple sum of contributions from individual particles. The spatial vector r(r) refers to the center of mass of the particle. R(r). In the case of a nonspherical particle (arbitrary shape), Eq. (I0) would describe the coupled motion of the center of mass and the relative arrangement of the components of the particle. For spherical particles, translational and rotational motion arc uncoupled and we have a simplified expression for the electric field time correlation function ... 

A review of this nature comes to a halt rather than a conclusion. The time-correlation-function technique is proving not merely fashionable but profitable, though there remain problems more easily discussed by analysis in the frequency variable. A fair vocabulary of functions giving consistent description of simple motions is now available, and we can proceed to compound them to describe more realistic situations. The solution of the internal field problem for the Onsager model opens a new field for calculation, and the massive computations of the molecular dynamidsts offer well-defined systems for the testing of more speculative theories. This field of research is in a period of most interesting and fruitful development. 

There are, of course, many different kinds of approximations involved, when deriving explicit expressions from a theoretical model. Many of these are, of course, also required for deriving results from the MD simulation. However, the MD simulations depends only on the most fundamental and highly reliable approximations, which need no examination. The remaining approximations, which the MD simulation can examine, are the ones concerning the structure and dynamics of the liquid. In some sense, these are system specific and therefore very difficult to treat. The approximations involve assumptions of the form of time correlation functions and radial distribution functions, or whether different motions are correlated or not. 

Experimentalists often rely on motional models, based on hydrodynamics, in order to interpret their liquid state spectra. MD simulations, can be considered as model-free in the sense that they do not assume the molecular motion to be in any specific regime. MD can be used to evaluate the motional models and even replace them. MD simulations can be used to calculate both the correlation times and the whole correlation functions. This is useful in those cases when correlation times cannot be deduced from measurements of other isotopes in the same molecule or when there is no method available at all. Correlation functions give information about intermolecular interactions and reveal cases when several motional modes are contributing to relaxation mechanisms at slightly different time scales. This can be observed as multiple decay rates. Time correlation functions from MD simulations can be Fourier transformed to power spectra if needed to provide line shapes and frequencies. 

---