Line shape spectral density

Time-dependent correlation functions are now widely used to provide concise statements of the miscroscopic meaning of a variety of experimental results. These connections between microscopically defined time-dependent correlation functions and macroscopic experiments are usually expressed through spectral densities, which are the Fourier transforms of correlation functions. For example, transport coefficients1 of electrical conductivity, diffusion, viscosity, and heat conductivity can be written as spectral densities of appropriate correlation functions. Likewise, spectral line shapes in absorption, Raman light scattering, neutron scattering, and nuclear jmagnetic resonance are related to appropriate microscopic spectral densities.2... 

Time scales. For an understanding of spectral line shapes of induced absorption, at not too high gas densities, it is useful to distinguish three different times associated with collisions, namely the average time between collisions, the duration of a molecular fly-by and the duration of the spectroscopic interaction. 

We note that the emitted power, Eq. 2.62, can be written as a spectral density, J(v), also called line shape, with the help of the d function, as in... 

For most treatments, the spectral density, J(a>), Eq. 2.86, also referred to as the spectral profile or line shape, is considered, since it is more directly related to physical quantities than the absorption coefficient a. The latter contains frequency-dependent factors that account for stimulated emission. For absorption, the transition frequencies ojp are positive. The spectral density may also be defined for negative frequencies which correspond to emission. 

We note that in a classical formula Planck s constant does not appear. Indeed, the zeroth moment Mo of the spectral density, J (o), does not depend on h, as the combination of Eqs. 5.35 and 5.38 shows. On the other hand, the classical moment y of the absorption profile, a(cu), is proportional to /h because the absorption coefficient a depends on Planck s constant see the discussions of the classical line shape below, p. 246. In a discussion of classical moments it is best to focus on the moments Mn of the spectral density, J co), instead of the moments, yn, of the spectral profile. 

Detailed balance. Classical line shapes are symmetric so that all classical, odd spectral moments M of the spectral function vanish. The odd moments of actual measurements are, however, non-vanishing because measured spectral density profiles satisfy the principle of detailed balance, Eq. 5.73. This problem of classical relationships may be largely alleviated by symmetrizing the measured profile prior to determining the moments, using the inverse Egelstaff procedure (P-4) discussed on p. 254 this generates a close approximation to the classical profile from the measurement and use of classical formulae is then justified. 

Summarizing, it may be said that virial expansions of spectral line shapes of induced spectra exist for frequencies much greater than the reciprocal mean free time between collisions. The coefficients of the density squared and density cubed terms represent the effects of purely binary and ternary collisions, respectively. At the present time, computations of the spectral component do not exist except in the form of the spectral moments see the previous Section for details. 

Starting from the assumption that the only known information concerning the line shape is a finite number of spectral moments, a quantity called information is computed from the probability for finding a given spectral component at a given frequency this quantity is then minimized. Alternatively, one may maximize the number of configurations of the various spectral components. This process yields an expression for the spectral density as function of frequency which contains a small number of parameters which are then related to the known spectral moments. If classical (i.e., Boltzmann) statistics are employed, information theory predicts a line shape of the form... 

By using special forms of the so-called spectral density J(w) it is possible to treat memory effects in QMEs. The spectral density J(w) contains information on the frequencies of the environmental modes and their coupling to the system. Tanimura and coworkers [18,20,26] were the first to do calculations along the lines described here using spectral densities of Drude shape. This spectral densities lead to bath correlation functions with purely exponential... 

The above experimental developments represent powerful tools for the exploration of molecular structure and dynamics complementary to other techniques. However, as is often the case for spectroscopic techniques, only interactions with effective and reliable computational models allow interpretation in structural and dynamical terms. The tools needed by EPR spectroscopists are from the world of quantum mechanics (QM), as far as the parameters of the spin Hamiltonian are concerned, and from the world of molecular dynamics (MD) and statistical thermodynamics for the simulation of spectral line shapes. The introduction of methods rooted into the Density Functional Theory (DFT) represents a turning point for the calculations of spin-dependent properties [7],... 

The spectral line shape is concentration dependent, but remains motionally narrowed until relatively high polymer concentrations are reached (Figure 1-B,C and reference 27). Loop and tail segment density near the surface may be large, but falls off rapidly with distance above the surface. We therefore expected tails and all but the smallest loops (and surface extended loops) would exhibit three line spectra similar to that observed with the free, isolated polymer molecule. 

This is performed in Fig. 6. For this purpose, two spectral densities are superposed in each case, one of reference involving only some lot of weak direct damping (grayed line shapes) and the other involving both direct and indirect damping (full lines). 

Besides, examination of Eq. (201) shows that the spectral density is the sum of different components, each of them being a superposition of Lorentzians involving different half-widthes and intensities. Note that this result differs deeply from the situation without indirect damping given by Eq. (82), for which all the Lorentzians forming the line shape have the same half-width. This Ending may be verified by taking y = 0 in Eq. (201). 

Investigating molecular dynamics using NMR, in contrast to DS and LS, involves the application of several conceptually quite different techniques. For example, in spin-lattice relaxation studies one is concerned with familiar time correlation functions that are probed as spectral density point by point (Section II.D.2). In the case of line-shape analysis, usually a two-pulse echo sequence is applied, and the... 

Much effort has been put into the explanation of the spectral line shapes but it seems that the definite theory has yet to be established. In the meantime one can extract useful information from the first few moments of the spectral density, by applying the elegant theory developed by Van Kranendonk and Poll and Van Kranendonk This theory relates the first moment to the derivative of the dimer dipole moment with respect to the intermolecular distance. The zeroth moment yields information about the square of the dipole moment. As this review is not the place to go extensively into the Van Kranendonk theory, we only note that, once the intermolecular potential surface and the interaction dipole field are known, — for instance by ab initio calculations — it is relatively easy to compute the moments of the spectral density. Since these are directly observable, the experiment of pressure induced absorption may serve as a check on the correctness of ab initio calculations, not only of the interaction energy, but also of the interaction dipole. 

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