Time cross-correlation

A time cross-correlation function between dynamic variables A (it) and B(t) is defined in a similar way ... 

Quasienergies from Short Time Cross-Correlation Probability Amplitudes by the Filter-Diagonalization Method. 

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... 

Equation (6), which expresses the frequency spectrum as the Fourier transform of the time autocorrelation of the initial state, is the central result of this section. It shows how to relate to the observed one-photon spectrum, determined in the frequency domain, to the time evolution of a nonstationary state on the excited electronic state. It is possible to obtain similar results for other spectroscopies. For example, the Raman excitation spectrum is determined by a time cross-correlation function between two different states. This is useful to know because the time autocorrelation function, while quite informative, really only tells us where the initial state is not, while the cross-correlation function tells where the initial state is. Before we turn to these considerations in detail, we should take advantage of what is already implied by the time autocorrelation function itself. 

The one photon absorption spectrum directly determines the time autocorrelation function, which in the notation of this subsection is C,(t). It is of interest to also determine the time cross-correlation function Cfi(t) which is the overlap of the time-evolved bright state with some different probe state... 

The Raman excitation profile is proportional to oi/((to) 2 which means that one cannot determine the time cross-correlation function directly from the observed Raman excitation profile. The indirect route is to first build a model potential energy surface for both the ground and the excited electronic states. The overlap (ijj/jih(0) is calculated by propagating the initial wave packet ij/,) on the upper electronic state and a computed resonance Raman excitation profile is obtained using Eqs. (48) and (49). The parameters of the potential energy surfaces can then be adjusted in order to get a good fit of the experimental excitation profile. In Sec. IV we shall discuss a method for a direct inversion. Another approach which has been discussed is the use of the transform theory (38,41). 

The final application we discuss is one where the maximum entropy formalism is used not only to fit the spectrum but also to extract new results. Specifically we discuss the determination of the time cross-correlation function, Cf, t) (Eq. (43)), which is the Fourier transform of the Raman scattering amplitude a/((Tu) (Eq. (44)) when what is measured is the Raman scattering cross section afi(m) a/((Tii) 2. The problem is that the experiment does not appear to determine the phase of the amplitude. The application proceeds in two stages (i) Representing the Raman spectrum as one of maximal entropy, using as constraints the Fourier transform of the observed spectrum. At the end of this stage one has a parametrization of a/,( nr) 2 whose accuracy can be determined by how well it fits the observed frequency dependence, (ii) The fact that the Raman spectrum can be written as a square modulus as in Eq. (97) implies that it can be uniquely factorized into a minimum phase function... 

Figure 18 Time cross-correlation functions for three Raman transitions in iodobenzene (from the ground state to the B excited electronic state with v = 1, 2, 3 quanta in the vu vibrational mode. (Left) Computed for a harmonic B state potential and convoluted with a 125-fs-wide window function. The spectrum is computed from this cross-correlation function. (Right) The time correlation function determined from the Raman frequency spectrum (the excitation profile) via the maximum entropy formalism, as discussed in the text, using nine Lagrange multipliers kr. (From Ref. (102).)...

The direct determination of time cross correlation functions from experiments is thus possible and reasonably accurate. Typically, experimental resonant Raman excitation profiles exhibits broad features (112), and the time cross-correlation function therefore provides short time information (for about the first 100 fs) on the excursion of the wave packet out of the Franck-Condon region. This information is complementary to that... 

Figure 19 The Raman spectrum and time cross correlation function when the motion on the excited electronic state potential is anharmonic, compare to Figs. 17 and 18, which are for a harmonic approximation. (Top, a) Computed time correlation function using a wide window function (b) The maximal entropy representation of this function, determined from the spectrum. Note the clear separation of time scales due to the anharmonicity (cf. Fig. 20). (Bottom) The Raman excitation spectrum obtained from the computed time correlation function (a). The arrows are the sequence of computations (a) is determined from the dynamics. The spectrum is determined from (a). The maximum entropy cross-correlation function (b) uses only the spectrum as input.

Once these deviations are detected, it is interesting to see their time dependence. This can be characterized by the following time cross-correlation function ... 

Narevicius E, Neuhauser D, Korsch H J and Moiseyev M 1997 Resonances from short time complex-scaled cross- correlation probability amplitudes by the filter-diagonalization method Chem. Phys. Lett. 276 250... 

Cross Correlation. Considerable research has been devoted to correlation techniques where a tracer is not used. In these methods, some characteristic pattern in the flow, either natural or induced, is computer-identified at some point or plane in the flow. It is detected again at a measurable time later at a position slightly downstream. The correlation signal can be electrical, optical, or acoustical. This technique is used commercially to measure paper pulp flow and pneumatically conveyed soHds. 

The practice of estabHshing empirical equations has provided useflil information, but also exhibits some deficiencies. Eor example, a single spray parameter, such as may not be the only parameter that characterizes the performance of a spray system. The effect of cross-correlations or interactions between variables has received scant attention. Using the approach of varying one parameter at a time to develop correlations cannot completely reveal the tme physics of compHcated spray phenomena. Hence, methods employing the statistical design of experiments must be utilized to investigate multiple factors simultaneously. 

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. ... 

The relaxation data for the anomeric protons of the polysaccharides (see Table II) lack utility, inasmuch as the / ,(ns) values are identical within experimental error. Obviously, the distribution of correlation times associated with backbone and side-chain motions, complex patterns of intramolecular interaction, and significant cross-relaxation and cross-correlation effects dramatically lessen the diagnostic potential of these relaxation rates. 

The characterization of the laser pulse widths can be done with commercial autocorrelators or by a variety of other methods that can be found in the ultrafast laser literature. " For example, we have found it convenient to find time zero delay between the pump and probe laser beams in picosecond TR experiments by using fluorescence depletion of trans-stilbene. In this method, the time zero was ascertained by varying the optical delay between the pump and probe beams to a position where the depletion of the stilbene fluorescence was halfway to the maximum fluorescence depletion by the probe laser. The accuracy of the time zero measurement was estimated to be +0.5ps for 1.5ps laser pulses. A typical cross correlation time between the pump and probe pulses can also be measured by the fluorescence depletion method. 

Figure 3 Creation of the longitudinal order by cross-correlation as a function of the mixing time fm which follows the inversion of a carbon-13 doublet (due to a./-coupling with a bonded proton). The read-pulse transforms the longitudinal polarization into an in-phase doublet and the longitudinal order into an antiphase doublet. The superposition of these two doublets leads to the observation of an asymmetric doublet.

For short times, the correlation function (7(f) depends on the microscopic details of the dynamics as the system crosses from to 38. These motions take place on a molecular time scale rmoi essentially equal to the time required to move through the transition region. For times f larger than rmoi but still very small compared to the reaction time rrxn (if the crossing event is rare rrxn L> rmoi such that such an intermediate time regime exists), C(f) can be replaced by an approximation linear in time. Using the detailed balance condition k jk = h )/ h ) [33] one then obtains... 

The time delay between two signals can be estimated using different methods among them the most popular is finding the maximum of the cross-correlation function... 

A commonly used approach for computing the transition amplitudes is to approximate the propagator in the Krylov subspace, in a similar spirit to the time-dependent wave packet approach.7 For example, the Lanczos-based QMR has been used for U(H) = (E — H)-1 when calculating S-matrix elements from an initial channel (%m )-93 97 The transition amplitudes to all final channels (Xm) can be computed from the cross-correlation functions, namely their overlaps with the recurring vectors. Since the initial vector is given by xmo only a column of the S-matrix can be obtained from a single Lanczos recursion. 

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