Dynamic spectrum Fourier transform

In order to analyze the vibrations of a single molecule, many molecular dynamics steps must be performed. The data are then Fourier-transformed into the frequency domain to yield a vibrational spectrum. A given peak can be selected and transformed back to the time domain. This results in computing the vibra-... 

A spectrum is the distribution of physical characteristics in a system. In this sense, the Power Spectrum Density (PSD) provides information about fundamental frequencies (and their harmonics) in dynamical systems with oscillatory behavior. PSD can be used to study periodic-quasiperiodic-chaotic routes [27]. The filtered temperature measurements y t) were obtained as discrete-time functions, then PSD s were computed from Fast Fourier Transform (FFT) in order to compute the fundamental frequencies. 

In the linearly-coupled version of the INM theory, which corresponds to the expansion of the dynamical variable of interest to linear order in INMs, G i) is related by Fourier transformation to the solvation influence spectrum. 

Recent advances in ultrashort laser technology has enabled us to investigate dynamics of molecules in a time domain, and furthermore, the success of a theoretical interpretation of the results of time-domain experiments by a moving wavepacket on a potential energy surface (PES) impressively demonstrated the importance of time-domain experiments [1]. On the other hand, it is well-known that a spectrum in a frequency domain and an autocorrelation function in the time domain can be transferred with each other via a Fourier transformation [2]. Therefore, it can be said that the spectrum... 

The interpretation of a spectrum from a dynamical point of view can also be applied to a spectrum containing a broad feature associated with direct and/or indirect dissociation reactions. From such spectra dynamics of a dissociating molecule can also be extracted via the Fourier transform of a spectrum. An application of the Fourier transform to the Hartley band of ozone by Johnson and Kinsey [3] demonstrated that a small oscillatory modulation built on a broad absorption feature contains information of the classical trajectories of the vibrational motion on PES, so-called unstable periodic orbits, at the transition state of a unimolecular dissociation. 

We shall shortly consider such fundamental concepts as density matrices and the superoperator formalism which are convenient to use in a formulation of the lineshape theory of NMR spectra. The general equation of motion for the density matrix of a non-exchanging spin system is formulated in the laboratory (non-rotating) reference frame. The lineshape of a steady-state, unsaturated spectrum is given as the Fourier transform of the free induction decay after a strong radiofrequency pulse. The equations provide a starting point for the formulation of the theory of dynamic NMR spectra presented in Section III. The reader who may be interested in a more detailed consideration of the problems is referred to the fundamental works of Abragam and... 

Their theory, based on the classical Bloch equations, (31) describes the exchange of non-coupled spin systems in terms of their magnetizations. An equivalent description of the phenomena of dynamic NMR has been given by Anderson and by Kubo in terms of a stochastic model of exchange. (32, 33) In the latter approach, the spectrum of a spin system is identified with the Fourier transform of the so-called relaxation function. 

Figure 5.4, one can easily understand why the interfacial electron transfer should take place in the 10-100 fsec range because this ET process should be faster than the photo-luminescence of the dye molecules and energy transfer between the molecules. Recently Zimmermann et al. [58] have employed the 20 fsec laser pulses to study the ET dynamics in the DTB-Pe/TiC>2 system and for comparison, they have also studied the excited-state dynamics of free perylene in toluene solution. Limited by the 20 fsec pulse-duration, from the uncertainty principle, they can only observe the vibrational coherences (i.e., vibrational wave packets) of low-frequency modes (see Figure 5.5). Six significant modes, 275, 360, 420, 460, 500 and 625 cm-1, have been resolved from the Fourier transform spectra of ultrashort pulse measurements. The Fourier transform spectrum has also been compared with the Raman spectrum. A good agreement can be seen (Figure 5.5). For detail of the analysis of the quantum beat, refer to Figures 5.5-5.7 of Zimmermann et al. s paper [58], These modes should play an important role not only in ET dynamics or excited-state dynamics, but also in absorption spectra. Therefore, the steady state absorption spectra of DTB-Pe, both in...

Further, an interesting question is how the kinetic energy of turbulence will be distributed according to various eddies/frequencies. Such a distribution of the energy among the eddies/frequencies is usually termed the energy spectrum. Our focus is now on the double correlation in the Karman-Howarth equation, and finally, the dynamic equation for the energy spectrum that is obtained by the Fourier transform of the double correlation is derived as... 

Fig. 1.19. Quenching of the coherent vibrational oscillations of MDMO-PPV upon photoinduced charge transfer. The AT/T dynamics for pure MDMO-PPV (continuous line) and for MDMO-PPV/PCBM (1 3 wt. ratio) (dashed line), excited by a sub-10-fs pulse, was recorded at the probe wavelength of 610 nm. The inset shows the Fourier transform of the oscillatory component of the MDMO-PPV signal, the nonresonant Raman spectrum of MDMO-PPV (excitation 1064 nm) and the resonant Raman spectrum of an MDMO-PPV/PCBM sample (excitation 457 nm). For the resonant Raman spectrum of MDMO-PPV, it was necessary to quench the strong background luminescence by adding PCBM...

When interpreting time-resolved mid-IR spectra, it is beneficial to consider the influence of rotational dynamics on the vibrational spectrum of a heteronuclear diatomic. It was shown more than 30 years ago that the vibrational absorption spectrum of a diatomic is related to its transition dipole correlation function (/z(0) /r(t)> through a Fourier transform (10) ... 

Multidimentional nonlinear infrared spectroscopy is used for identification of dynamic structures in liquids and conformational dynamics of molecules, peptides and, in principle, small proteins in solution (Asplund et al., 2000 and references herein). This spectroscopy incorporates the ability to control the responses of particular vibrational transitions depending on their couplings to one another. Two and three-pulse IR photon echo techniques were used to eliminate the inhomogeneous broadening in the IR spectrum. In the third-order IR echo methods, three phase-locked IR pulses with wave vectors kb k2, and k3 are focused on the sample at time intervals. The IR photon echo eventually emitted and the complex 2D IR spectrum is obtained with the use of Fourier transformation. The method was applied to the examination of vibrational properties of N-methyl acetamid and a dipeptide, acyl-proline-NH2.in D20. The 2D IR spectrum showed peaks at 1,610 and 1, 670 cm 1, the two frequencies ofthe acyl-proline dipeptide. Geometry and time-ordering of the incoming pulse sequence in fifth-order 2D spectroscopy is shown in Fig. 1.3. 

FIGURE 16.3 Fourier transform of dipole auto-correlation (FT-DAC) with quantum nuclear corrections. The lower temperature spectrum shows critical features (see text) which get averaged out due to dynamics at higher temperature. See Ref. [54] for details. 

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