Spectral density classical

Let us consider the quasi-classical formulation of impact theory. A rotational spectrum of ifth order at every value of co is a sum of spectral densities at a given frequency of all J-components of all branches... 

It may be of interest to observe that at zero temperature this last expression of the Robertson and Yarwood spectral density (in which we consider the fluctuation of the slow mode as classical in contrast with the RY paper) reduces (unsatisfactorily ) to a single Dirac delta peak ... 

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...

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. 

Classical moment expressions. Spectral moments expressed in the Heisenberg notation can be immediately interpreted in terms of classical physics. For a discussion of classical moments, we consider the moments Af of the spectral density, J co), which are related to the moments, y , of the absorption coefficient, a(co), according to Eq. 5.8. By combining that equation with Eq. 5.16, we get at once... 

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... 

Thus we see that the first moment of the spectral density multiplied by h is the reorganization energy (i.e., one half of the Stokes shift magnitude), whereas the time dependence of the first moment of p(w) corresponds to the fluorescence Stokes shift. Thus the time dependence of S t) is determined entirely by the spectral density. At high temperature [i.e., when p(w) contains frequencies less than 2kBT], S(t) becomes the classical correlation function [36] used by many previous authors [7-10], This follows from... 

As a consequence, the classical spectral density taking into account both direct and indirect dampings become... 

As it appears, the classical spectral density (174) is very similar to the semiclassical approached (144) obtained above and it is equivalent to that given by Eq. (151). 

The equilibrium FDT is usually written in a form that involves frequency-dependent quantities such as generalized susceptibilities and spectral densities [30-35]. We show below how this theorem can be formulated in the time domain. Our arguments do not reduce to a simple Fourier transformation of the usual frequency-domain formulation. Instead they are developed from the very beginning in the time domain, and they use only the various time-dependent quantities entering into play. The corresponding formulations of the FDT, which are established in the whole range of temperatures, allow in particular for a discussion of both the classical limit and the zero-temperature case [36]. 

Room temperature simulations at the classical level yielded values of all AX° and quantities. Quantal results for G were also obtained from path integral simulations (in conjunction with Eq. 41) or semiclassical results based on spectral densities from the classical simulations [36]. The calculated classical G and 7. values and the AG° value were found to conform well to Eq. 112, and Gi t]) and Gf t]) functions were close to parabolic in form. Due to systematic deficiencies in the aqueous solvent model (a nonpolarizable force field and finite Coulombic cut-off, ca. 10 A), corrections to / were made on the basis of reference dielectric continuum models, and then G was corrected accordingly on the basis of Eq. 112. 

With this approximate expression for iLw, (62) becomes the quantum-classical evolution equation for the spectral density function,... 

Here C is the gas phase (uncoupled) flux autocorrelation function, Zbath is the bath partition function, J(co) is the bath spectral density (computed as described above from a classical molecular dynamics computation), Bi and B2 are combinations of trigonometric functions of the frequency a> and the inverse barrier frequency, and Anally ... 

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