Vibrational dynamics spectral density

Since the stochastic Langevin force mimics collisions among solvent molecules and the biomolecule (the solute), the characteristic vibrational frequencies of a molecule in vacuum are dampened. In particular, the low-frequency vibrational modes are overdamped, and various correlation functions are smoothed (see Case [35] for a review and further references). The magnitude of such disturbances with respect to Newtonian behavior depends on 7, as can be seen from Fig. 8 showing computed spectral densities of the protein BPTI for three 7 values. Overall, this effect can certainly alter the dynamics of a system, and it remains to study these consequences in connection with biomolecular dynamics. 

The linear response theory [50,51] provides us with an adequate framework in order to study the dynamics of the hydrogen bond because it allows us to account for relaxational mechanisms. If one assumes that the time-dependent electrical field is weak, such that its interaction with the stretching vibration X-H Y may be treated perturbatively to first order, linearly with respect to the electrical field, then the IR spectral density may be obtained by the Fourier transform of the autocorrelation function G(t) of the dipole moment operator of the X-H bond ... 

Similar methods have been used to integrate thermodynamic properties of harmonic lattice vibrations over the spectral density of lattice vibration frequencies.21,34 Very accurate error bounds are obtained for properties like the heat capacity,34 using just the moments of the lattice vibrational frequency spectrum.35 These moments are known35 in terms of the force constants and masses and lattice type, so that one need not actually solve the lattice equations of motion to obtain thermodynamic properties of the lattice. In this way, one can avoid the usual stochastic method36 in lattice dynamics, which solves a random sample of the (factored) secular determinants for the lattice vibration frequencies. Figure 3 gives a typical set of error bounds to the heat capacity of a lattice, derived from moments of the spectrum of lattice vibrations.34 Useful error bounds are obtained... 

Very few experiments have been performed on vibrational dynamics in supercritical fluids (47). A few spectral line experiments, both Raman and infrared, have been conducted (48-58). While some studies show nothing unique occurring near the critical point (48,51,53), other work finds anomalous behavior, such as significant line broadening in the vicinity of the critical point (52,54-60). Troe and coworkers examined the excited electronic state vibrational relaxation of azulene in supercritical ethane and propane (61-64). Relaxation rates of azulene in propane along a near-critical isotherm show the three-region dependence on density, as does the shift in the electronic absorption frequency. Their relaxation experiments in supercritical carbon dioxide, xenon, and ethane were done farther from the critical point, and the three-region behavior was not observed. The measured density dependence of vibrational relaxation in these fluids was... 

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 dynamics of different modes of molecular librations (hindered rotations) and intramolecular vibrations in supercritical water can now be analyzed in terms of velocity autocorrelation functions for the corresponding projections (Eqns. 22-27) (Kalinichev and Heinzinger 1992, 1995 Kalinichev 1993). The velocity autocorrelation functions calculated for the quantities Qi (Eqns. 25-27) are shown in Figure 19 for two extreme cases of high-density and low-density supercritical water. The Fourier transforms of these functions result in the spectral densities of the corresponding vibrational modes. They are shown in Figure 20 for the supercritical thermodynamic states listed in Table 5. 

A Computer-oriented algorithm for nonstationary random vibrations of polygonal Kirchhoff-plates has been developed. A fast and accurate BEM for calculation of undamped frequency response function has been used. Light hysteretic damping has been introduced subsequently, and the modified spectral Priestley-formulation has been applied in order to calculate evolutionary output power spectral density functions. It is hoped that this solution strategy will be of interest for workers in the field of probabilistic structural dynamics. 

Abstract Photoinduced processes in extended molecular systems are often ultrafast and involve strong electron-vibration (vibronic) coupling effects which necessitate a non-perturbative treatment. In the approach presented here, high-dimensional vibrational subspaces are expressed in terms of effective modes, and hierarchical chains of such modes which sequentially resolve the dynamics as a function of time. This permits introducing systematic reduction procedures, both for discretized vibrational distributions and for continuous distributions characterized by spectral densities. In the latter case, a sequence of spectral densities is obtained from a Mori/Rubin-type continued fraction representation. The approach is suitable to describe nonadiabatic processes at conical intersections, excitation energy transfer in molecular aggregates, and related transport phenomena that can be described by generalized spin-boson models. 

The effective-mode construction can thus be employed both for a discrete set of vibrational modes (e.g., in a polyatomic molecule) and for typical system-bath type situations where the spectrum of bath modes is dense. In the latter case, the environment and its coupling to the electronic subsystem are entirely characterized by a spectral density. Approximate spectral densities can be constructed from few effective modes, representing a simplified realization of the true environmental spectral density that is designed to give a faithful representation of the dynamics on short time scales [32,33]. Thus, even a highly structured, multi-peaked spectral density can be reduced to an effective, simplified spectral density on ultrafast time scales. Importantly, the procedure converges, as has recently been shown in Ref. [34]. 

The most sophisticated deterministic dynamic analysis of structures requires that the load should be applied in time domain. This is one of the major challenges in the reliability analysis for seismic loading. The classical random vibration-based approaches were used in the past for this purpose however, they did not provide information acceptable to the deterministic community. The classical random vibration-based approaches have numerous limitations including the loads which are applied in the form of power spectral density functions essentially appropriate for linear structural behavior, the uncertainty in the linear or nonlinear structural behavior which may need to be incorporated in approximate ways, several performance-enhancing features currently introduced in structures which cannot be incorporated in appropriate ways, etc. The most severe weakness is that the seismic loading cannot be applied in time domain. 

The study of dynamics of a real polymer chain of finite length and containing some conformational defects represents a very difficult task. Due to the lack of symmetry and selection mles, the number of vibrational modes is enormous. In this case, instead of calculating the frequency of each mode, it is more convenient to determine the density of vibrational modes, that is, the number of frequencies that occur in a given spectral interval. The density diagram matches, apart from an intensity factor, the experimental spectmm. Conformational defects can produce resonance frequencies when the proper frequency of the defect is resonating with those of the perfect lattice (the ideal chain), or quasi-localized frequencies when the vibrational mode of the defect cannot be transmitted by the lattice. The number and distribution of the defects may be such... 

The molecular dynamic technique has been validated for water structures through comparison of calculated properties with experimental thermodynamic water data, such as the density maximum, the high heat capacity, and diffraction patterns (Stillinger and Rahman, 1974) as well as the hydrate infrared (vibrational) spectral data by Bertie and Jacobs (1977, 1982). With acceptable comparisons of many computed and experimental properties of water structures, there is little doubt that a substance similar to water has been simulated. 

INS spectroscopy avoids this problematic feedback effect, in the sense that it measures nuclear motion directly. This provides a simple relation between the amplitude of the motion, the cross section of the nuclei and the vibrational frequency to determine the intensity and shape of the spectral lines. The electron density is, of course, used to determine the potential energy surface that controls the nuclear motion but it is not involved in the calculation of the spectra. The combined use of INS spectroscopy, ab initio calculations and ACLIMAX can provide a good test of the molecular model, assessing the quality of structure and dynamics. 

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