Second spectral moments

Comparison of formulae (2.51) and (2.64) allows one to understand the limits and advantages of the impact approximation in the theory of orientational relaxation. The results agree solely in second order with respect to time. Everything else is different. In the impact theory the expansion involves odd powers of time, though, strictly speaking, the latter should not appear. Furthermore the coefficient /4/Tj defined in (2.61) differs from the fourth spectral moment I4 both in value and in sign. Moreover, in the impact approximation all spectral moments higher than the second one are infinite. This is due to the non-analytical nature of Kj and Kf in the impact approximation. In reality, of course, all of them exist and the lowest two are usually utilized to find from Eq. (2.66) either the dispersion of the torque (M2) or related Rq defined in Eq. (1.82) ... 

This leads to expressions of the zeroth and first spectral moments that may be computed in seconds, even on computers of small capacity. However, few systems of practical interest are actually classical and better approximations are often needed. 

For a computation of the zeroth and first moments, this lowest-order Wigner-Kirkwood correction is often sufficient, but the second and higher spectral moments require additional dynamical corrections [177]. 

Model correlation functions. Certain model correlation functions have been found that model the intracollisional process fairly closely. These satisfy a number of physical and mathematical requirements and their Fourier transforms provide a simple analytical model of the spectral profile. The model functions depend on the choice of two or three parameters which may be related to the physics (i.e., the spectral moments) of the system. Sears [363, 362] expanded the classical correlation function as a series in powers of time squared, assuming an exponential overlap-induced dipole moment as in Eq. 4.1. The series was truncated at the second term and the parameters of the dipole model were related to the spectral moments [79]. The spectral model profile was obtained by Fourier transform. Levine and Birnbaum [232] developed a classical line shape, assuming straight trajectories and a Gaussian dipole function. The model was successful in reproducing measured He-Ar [232] and other [189, 245] spectra. Moreover, the quantum effect associated with the straight path approximation could also be estimated. We will be interested in such three-parameter model correlation functions below whose Fourier transforms fit measured spectra and the computed quantum profiles closely see Section 5.10. Intracollisional model correlation functions were discussed by Birnbaum et a/., (1982). 

Classical and semiclassical moment expressions. The expressions for the spectral moments can be made classical by substituting the classical distribution function, g(R) = exp (— V(R)/kT), for the quantum expressions. Wigner-Kirkwood corrections are known which account to lowest order for the static quantum corrections, Eq. 5.44 [177, 292]. For the second and higher moments, dynamic quantum corrections must also be made [177]. As was mentioned in the previous Chapter, such semiclassical corrections are useful in supplementing quantum computations of the spectral moments at large separations where the quantum effects are small the computational effort of quantum calculations, which is substantial at large separations, may thus be avoided. 

Second moments have also been computed, both from first principles and on the basis of the classical multipole-induction model. These are found to be in close agreement with measurements where these exist. Second moments are of a special interest in connection with modeling of three-parameter line profiles from three spectral moments [52]. In analyses based on classical expressions, the second moment is expressible in terms of the first moment specified above, multiplied by 2kT/h. 

Spectral moments can also be computed from classical expressions with Wigner-Kirkwood quantum corrections [177, 189, 317] of the order lV(H2). For the quadrupole-induced 0223 and 2023 components of H2-H2, at the temperature of 40 K, such results differ from the exact zeroth, first and second moments by -10%, -10%, and +30% respectively. For the leading overlap-induced 0221 and 2021 components, we get similarly +14%, +12%, and -56%. These numbers illustrate the significance of a quantum treatment of the hydrogen pair at low temperatures. At room temperature, the semiclassical and quantum moments of low order differ by a few percent at most. Quantum calculations of higher-order moments differ, however, more strongly from their classical counterparts. 

H2-H2 rototranslational spectra. For the significant A1A2AL induction components, Table 4.11, values of the various spectral functions have been computed at frequencies from 0 to 1800 cm-1 and for temperatures from 40 to 300 K, Fig. 6.3 [282]. As a test of these line shape computations, the zeroth, first and second spectral moments have been computed in two independent ways by integration of the spectral functions with respect to frequency, Eq. 3.4, and also from the quantum sum formulae, Eqs. 6.13, 6.16, and 6.21. Agreement of the numerical results within 0.3% is observed for the 0223, 2023 components, and 1% for the other less important components. This agreement indicates that the line shape computations are as accurate as numerical tests with varying grid widths, etc., have indicated, namely about 1% see Table 6.2 as an example (p. 293). 

We have not attempted to exhibit in great detail the effects of the rotational excitations on the induced dipole components B and those of vibrational excitation on the interaction potential because this was done elsewhere for similar systems [151, 63,295,294], The significance of the j,f corrections is readily seen in the Tables and need not be displayed beyond that. The vibrational influence is displayed in Fig. 6.20 first and second spectral moments are strongly affected, especially at high temperatures, similar to that which was seen earlier for H2-He [294], Fig. 6.23. The close agreement of the measurements of the rotovibrational collision-induced absorption bands of hydrogen with the fundamental theory shown above certainly depends on proper accounting for the rotational dependences of the induced dipole moment, and of the vibrational dependences of the final translational states of the molecular pair. 

Atomic spectral moments can be expressed in terms of the averages of the products of the relevant operators. Let Oi,C>2, --,Ok be the operators of interactions in definite shells or the operators of electronic transitions between definite shells in the second quantization form. The average of... 

Figure 3 Franck-Condon weighted density of energy gaps between the donor and acceptor electronic energy levels. The parameters (A ) and indicate the first and second spectral moments, respectively. FCWD(O) shows the probability of zero energy gap entering the ET rate (Eq. [2]).

The second spectral moments of absorption and emission lines... 

The Stokes shift and two second spectral moments fully define the parameters of the model. In addition, they should satisfy Eqs. [73] and [74]. The latter feature establishes the condition of model consistency that is important for mapping the model onto condensed-phase simulations that we discuss below. 

The second relation coimects the intramolecular and solvent reorganization energies to the second spectral moments of absorption, ((5v) )abs, and emission, ((5v) )abs, spectral lines... 

In Figure 7, the absorption and emission second spectral moments are plotted against the Stokes shift obtained for coumarin-153 optical dye in 40 molecular solvents of varying polarity. The slopes of ((5v) ) versus Vst have opposite signs for absorption and emission transitions in contrast to the prediction of equation (39). 

The understanding of optical experiments demonstrating different optical widths (second spectral moments) for absorption and emission spectral lines requires an extension of the Marcus-Hush model that would incorporate different reorganization energies for forward and backward charge transfers... 

The ratio of second and zeroth moment, M2/M0, defines some average squared width of the spectrum. Spectral moments are related to the induction operator and interaction potential by well-known sum formulas [200, 215, 235, 292, 318, 319, 326, 350, 351, 422] that permit a comparison of the measurements with the fundamental theory [196,208,209,293,296, 307,316, 335, 357]. 

G. Briganti, D. Rocca, and M. Nardone. Interaction induced light scattering First and second spectral moments in the superposition approximation. Molec. Phys., 59 1259-1272 (1986). 

In order to exclude dynamically induced effects the analysis of the H-NMR lineshape has to rely on other spectral parameters, which are invariant to the motionally induced averaging processes, such as the spectral moments [47]. Specifically, the first and second moments, Mv,i and Mv,2> which can be calculated from a H-NMR lineshape F v), according to ... 

However, calculation of m2 and m4 involves practical difficulties, as sample spacing and signal processing. For this work, the authors made trials to determine the second and fourth spectral moments, calculating the variation (dz/dx) and (d z/dx ) of the measured points by different methods, as described below. The profile file measured by the MAHR Concept Perthometer PGK equipment was used. The file contains 8064 points, 0.695 pm spaced. Because, in this equipment, the data contained in the file is supplied as measured, an additional filtering software has to be developed and it is described in Annex I. This user filtering process can be omitted if the equipment supplies the profile already filtered. 

As discussed in [91], the shape of a static spectrum determines significantly the spectral transformation as frequency exchange increases. In particular, spectral narrowing will take place only if the second moment of the spectrum is finite. In our case... 

The second problem of interest is to find normal vibrational frequencies and integral intensities for spectral lines that are active in infrared absorption spectra. In this instance, we can consider the molecular orientations, to be already specified. Further, it is of no significance whether the orientational structure eRj results from energy minimization for static dipole-dipole interactions or from the competition of any other interactions (e.g. adsorption potentials). For non-polar molecules (iij = 0), the vectors eRy describe dipole moment orientations for dipole transitions. 

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