Autocorrelation functions model

Fig. 13 shows this autocorrelation function where the time is scaled by mean square displacement of the center of mass of the chains normalized to Ree)- All these curves follow one common function. It also shows that for these melts (note that the chains are very short ) the interpretation of a chain dynamics within the Rouse model is perfectly suitable, since the time is just given within the Rouse scaling and then normalized by the typical extension of the chains [47]. 

Subtilisin, 170 active site of, 171,173 autocorrelation function of, 216, 216 potential surfaces for, 218 site-specific mutations, 184, 185, 187-188 Sugars, see Oligosaccharides Surface-constrained solvent model, 125... 

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]. 

The ARIMA analysis evaluates the autocorrelation functions to determine the order of the appropriate moving average and the need for differencing. An appropriate model is chosen and the fit to the data is constructed followed by a careful analysis of the residuals. The parameters are adjusted and the fit is checked again. The process is applied iteratively until the errors are minimized or the model fails to converge. 

Let us now consider the velocity autocorrelation function (VACF) obtained from the MCYL potential, (namely, with the inclusion of vibrations). Figure 3 shows the velocity autocorrelation function for the oxygen and hydrogen atoms calculated for a temperature of about 300 K. The global shape of the VACF for the oxygen is very similar to what was previously determined for the MCY model. Very notable are the fast oscillations for the hydrogens relative to the oxygen. 

It can be shown [4] that the innovations of a correct filter model applied on data with Gaussian noise follows a Gaussian distribution with a mean value equal to zero and a standard deviation equal to the experimental error. A model error means that the design vector h in the measurement equation is not adequate. If, for instance, in the calibration example the model was quadratic, should be [1 c(j) c(j) ] instead of [1 c(j)]. In the MCA example h (/) is wrong if the absorptivities of some absorbing species are not included. Any error in the design vector appears by a non-zero mean for the innovation [4]. One also expects the sequence of the innovation to be random and uncorrelated. This can be checked by an investigation of the autocorrelation function (see Section 20.3) of the innovation. 

Fitzgerald et al. (1984) measured pressure fluctuations in an atmospheric fluidized bed combustor and a quarter-scale cold model. The full set of scaling parameters was matched between the beds. The autocorrelation function of the pressure fluctuations was similar for the two beds but not within the 95% confidence levels they had anticipated. The amplitude of the autocorrelation function for the hot combustor was significantly lower than that for the cold model. Also, the experimentally determined time-scaling factor differed from the theoretical value by 24%. They suggested that the differences could be due to electrostatic effects. Particle sphericity and size distribution were not discussed failure to match these could also have influenced the hydrodynamic similarity of the two beds. Bed pressure fluctuations were measured using a single pressure point which, as discussed previously, may not accurately represent the local hydrodynamics within the bed. Similar results were... 

The wobble-in-cone model has been generalized by introducing three autocorrelation functions Go(t), Gi(t) and G2(t) into the expression for r(t)... 

While the simulations of the pyrazine system discussed in the previous sections of this chapter have employed a three-mode model (Model I), the semiclassical simulations we will present here are based on two different models a four-mode model and a model including all 24 normal modes of the pyrazine molecule. Let us first consider the four-mode model of the S1-S2 conical intersection in pyrazine which was developed by Domcke and coworkers [269]. In addition to the three modes considered in Model I, it takes into account another Condon-active mode (V9a). Figure 37 shows the modulus of the autocorrelation function [cf. Eq. (24)] of this model after photoexcitation to the S2 electronic state. The exact quantum results (full line) are compared to the... 

Figure 37. Modulus of the autocorrelation function for the four-mode model of pyrazine. The full line is the quantum result, and the dotted line is the semiclassical result.

Figure 42. Diabatic population (a) and modulus of the autocorrelation function (b) of the initially prepared state for Model IVa. The full tine is the quantum result, and the dashed line depicts the semiclassical mapping result. The semiclassical data have been normalized. Panel (c) shows the norm of the semiclassical wave function.

The results obtained for the three-mode Model IVb are depicted in Fig. 45. As was found for the semiclassical mapping approach, the spin-coherent state propagators can only reproduce the short-time dynamics for the electronic population. The autocorrelation function, on the other hand, is reproduced at least qualitatively correctly by the semiclassical spin-coherent state propagator. 

The concept behind the maximum entropy model is to choose the spectrum in the form of a non-negative function of frequency, which corresponds to a time series with maximum entropy whose autocorrelation function is consistent with the set of known values. 

The DLS measurements can also be used in more complicated situations, for example, (a) when interparticle interactions are important, (b) for dispersions with particles of other shapes, (c) for monitoring coagulation, and (d) when the dispersion is polydisperse. In all these cases, a significant amount of modeling is often necessary to interpret the measured autocorrelation function, and we do not consider them here. Instead, we restrict our attention to a brief discussion of item (d) above, namely, polydisperse systems, since it is concerned with a problem of more routine interest. 

The dynamic RIS model is used to calculate the conformational and first and second orientational autocorrelation functions for PE. Various sequence lengths and directions in the chain are considered. 

The reptation model (225) also appears to produce a Rouse spectrum at long times. In order to renew its configurations a chain must diffuse out of the tunnel defined by the fixed obstacles along its length. De Gennes calculates the autocorrelation function for the end-separation vector, obtaining... 

To illustrate the potential of the hybrid method in describing the role of an intramolecular bath in the decay dynamics induced by a conical intersection, we consider the model of Ref. [7,8] for the S2-S1 Cl in pyrazine. Fig. 1 shows the wavepacket autocorrelation function C(t) = ( k(O)l (t) for an increasing number of bath modes. G-MCTDH hybrid calculations for 4 core (primary) modes plus nb bath (secondary) modes are compared with reference calculations by the standard MCTDH method. 

Fig. 1. The autocorrelation function C(t) = (U (O)l I (i) is shown for a wavepacket initially prepared on the upper diabatic surface [7]. Panels (a) and (b) C(t) for the four core modes calculated by the standard MCTDH method for the model Hamiltonian Hy of Eq. (9), shown on different scales in the two panels. Panel (c) G-MCTDH calculation (bold line) as compared with standard MCTDH calculation (dotted line) for the composite system with four core modes (combined into two 2-dimensional particles

Fig. 2. Error bounds for the cumulative frequency distribution of the spectral density for the velocity autocorrelation function using jU.0, /n2, and ja evaluated for a classical model of liquid argon.29...

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