Dynamics in the Linear Response Domain

Thus far, the strengths 5(cu) or detailed energies with transition moment (N D 0) describe the gross features of the spectra as they are accessible, e.g. by photoabsorption measurements. There is much more information carried in the normal modes of a many-body system. One can thus look one step further into it by considering the whole transition density 

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An alternative approach to DS study is to examine the dynamic molecular properties of a substance directly in the time domain. In the linear response approximation, the fluctuations of polarization caused by thermal motion are the same as for the macroscopic rearrangements induced by the electric field [27,28], Thus, one can equate the relaxation function < )(t) and the macroscopic dipole correlation function (DCF) V(t) as follows ... 

We first consider the intermolecular modes of liquid CS2. One of the details that two-dimensional Raman spectroscopy has the potential to reveal is the coupling between intermolecular motions on different time scales. We start with the one-dimensional Raman spectrum. The best linear spectra are based on time domain third-order Raman data, and these spectra demonstrate the existence of three dynamic time scales in the intermolecular response. In Fig. 3 we have modeled the one-dimensional time domain spectrum of CS2 for 3 cases (A) a single mode represented by the sum of three Brownian oscillators, (B) three Brownian oscillators, and (C) a distribution of 20 arbitrary Brownian oscillators. Case (A) represents the fully coupled, or isotropic case where the liquid is completely homogeneous on the time scales of the simulation. Case (B) deconvolutes the linear response into the three time scales that are directly evident in the measured response and is in the limit that the motions associated with each of the three timescales are uncoupled. Case (C) is an example where the liquid is represented by a large distribution of uncoupled motions. 

Several simulation runs follow for the two cases. The runs are represented by their initial conditions and a few selected profiles in order to show the progress of the dynamics. The spatial domain was divided into 100 segments so that A.x = 1 x 10. The dimensionless time increment for these runs was At = 1 x 10. At each time interval, the linearized response converged within 5 iterations to the nonlinear response. Figure 8.23 shows the dynamics of a startup from a cold reactor with a single steady state. For this run, the system exhibits uniform asymptotic stability. 

We only consider static response properties in this chapter, which arise from fixed external field. Their dynamic counterparts describe the response to an oscillating electric field of electromagnetic radiation and are of great importance in the context of non-linear optics. As an entry point to the treatment of frequency-dependent electric response properties in the domain of time-dependent DFT we recommend the studies by van Gisbergen, Snijders, and Baerends, 1998a and 1998b. 

Chapter 3 presented the Bayesian spectral density approach for the parametric identification of the multi-degree-of-freedom dynamical model using the measured response time history. The methodology is applicable for linear models and can also be utilized for weakly nonlinear models by obtaining the mean spectrum with equivalent linearization or strongly nonlinear models by obtaining the mean spectrum with simulations. The stationarity assumption in modal/model identification for an ambient vibration survey is common but there are many cases where the response measurements are better modeled as nonstationary, e.g., the structural response due to a series of wind gusts or seismic responses. In the literature, there are very few approaches which consider explicitly nonstationary response data, for example, [226,229]. Meanwhile, extension of the Bayesian spectral density approach for nonstationary response measurement is difficult since construction of the likelihood function is nontrivial in the frequency domain. Estimation of the time-dependent spectrum requires a number of data sets, which are associated with the same statistical time-frequency properties but this is impossible to achieve in practice. 

The Bayesian time-domain approach presented in this chapter addresses this problem of parametric identification of linear dynamical models using a measured nonstationary response time history. This method has an explicit treatment on the nonstationarity of the response measurements and is based on an approximated probability density function (PDF) expansion of the response measurements. It allows for the direct calculation of the updated PDF of the model parameters. Therefore, the method provides not only the most probable values of the model parameters but also their associated uncertainty using one set of response data only. It is found that the updated PDF can be well approximated by an appropriately selected multi-variate Gaussian distribution centered at the most probable values of the parameters if the problem is... 

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