Correlation functions, time-dependent

Time correlation function Time-dependent quantity characterizing the correlation between dynamical events occurring at different times in a system. 

The frequency-time correlation function is dependent on the frequency and the force constants of the vibrational mode whose dephasing is being considered. They are determined by fitting the vibrational bond energies to a Morse potential of the following form ... 

Diffusion. - Distribution of the diffusivitity of fluid in a horizontally oriented cylinder was demonstrated by NMR imaging in two papers on a granular flow system and in the earth s magnetic field. Correlation time (ic) and diffusion coefficient (D = Xc) imaging (CTDCI) was applied to a granular flow system of 2 mm oil-filled sphere rotated in a half-filled horizontal cylinder, ie. to an Omstein-Uhlenbeck process with a velocity autocorrelation function. Time dependent apparent diffusion coefficients are measured, and Tc... 

Time correlation function Time correlation functions are of great value for the analysis of dynamical processes in condensed phases. A time correlation function C(t) is obtained when a time-dependent quantity A(t) is multiplied by itself (auto-correlation) or by another time-dependent quantity B(t ) evaluated at time / (cross-correlation) and the product is averaged over some equilibrium ensemble. For example, the self-diffusion coefficient can be obtained... 

Abbreviations WF (wave function), EC (electron correlation), TD (time-dependent), AO (atomic orbital). 

Approach to Improving Approximate Exchange-Correlation Potentials Time-Dependent Density-Functional Theory Calculations of Molecular Excitation Spectra. 

We need for negative times because this determines the initial condition j/(0), which, for this example, is stochastic. Temporally speaking, the correlation function < /(t)> depends on , 6i /(s), 8 (s), s < t, and, in turn, the fluctuation Si /(s) depends on , 6 (t), and 6 (t) at previous times T < s. Equation (148) is not solvable in general, so we invoke a closure ansatz called first-order smoothing which amounts to throwing away the bracketed fluctuation term in Eq. (148), and then becomes (Bourret, 1964, 1965, 1966 Zwanzig, 1964)... 

Thachuk M and Schatz G C 1992 Time dependent methods for calculating thermal rate coefficients using flux correlation functions J. Chem. Phys. 97 7297-313... 

Here, I(co) is the Fourier transform of the above C(t) and AEq f is the adiabatic electronic energy difference (i.e., the energy difference between the v = 0 level in the final electronic state and the v = 0 level in the initial electronic state) for the electronic transition of interest. The above C(t) clearly contains Franck-Condon factors as well as time dependence exp(icOfvjvt + iAEi ft/h) that produces 5-function spikes at each electronic-vibrational transition frequency and rotational time dependence contained in the time correlation function quantity <5ir Eg ii,f(Re) Eg ii,f(Re,t)... 

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 ... 

A second method is to use a perturbation theory expansion. This is formulated as a sum-over-states algorithm (SOS). This can be done for correlated wave functions and has only a modest CPU time requirement. The random-phase approximation is a time-dependent extension of this method. 

Often such correlation functions are time dependent, and measure how the correlation between two quantities changes over time. They may be normalized by the corresponding static (i.e. t = to) limit. 

The above reasoning applies also to the correlation function of the torque. It follows from the stationarity of the random processes J(t) and M(t) that their correlation functions depend solely on difference of times arguments... 

Note in particular that the exchange-correlation functional that emCTges here does not involve the kinetic energy. From the perspective of the DFT literature, (3.16) is a formulation of the Hohenberg-Kohn functional that is constructed to ensure that the functional derivatives required for variational minimization actually exist. We return to these issues in Sect. 3.3. Also note that in the time-dependent case the external potential V(r, )is often considered to be explicitly... 

G. C. Schatz and M. A. Ratner (1993) Quantum Mechanics in Chemistry (Prentice-Hall, Englewood Cliffs, NJ). An advanced text emphasizing molecular symmetry and rotations, time-dependent quantum mechanics, collisions and rate processes, correlation functions, and density matrices. 

---