Spectral moment first

The Compton scattering cannot be neglected, but it is independent of molecular structure. Then, fitting experimental data to formulas from gas phase theory, the concentration of excited molecules can be determined. Another problem is that the undulator X-ray spectrum is not strictly monochromatic, but has a slightly asymmetric lineshape extending toward lower energies. This problem may be handled in different ways, for example, by approximating its spectral distribution by its first spectral moment [12]. 

Furthermore, in all phases studied the first spectral moment Mi of the H NMR spectra can be calculated and the weighted mean splitting of the H NMR spectrum can be obtained, which is proportional to the average chain orientational order parameter of the lipid, using ... 

Fig. 16. Temperature dependence of first spectral moment Afi for H NMR spectra of d62-DPPC (solid symbols) and d62-DPPC with 28.5 mol% cholesterol (open symbols) at ambient pressure (O), 1000 bar (O) and 2100 bar (A) (after Ref. 55).

Fig. 20. (a) Temperature dependence (top) and (b) pressure dependence (bottom) of the first spectral moment Mi of d62-DPPC and dg2-DPPC-GD mixtures at 55 °C. 

For He-Ar spectral moments have been computed from first principles, using advanced quantum chemical methods [278] details may be found in Chapters 4 and 5. We quote the results of the ab initio calculations of the moments in Table 3.1, columns 4 and 6. The agreement with measurement is satisfactory in view of the experimental uncertainties. We... 

Table 3.3. Spectral moments of the neon-argon liquid mixture along the coexistence curve measurement [107] compared with binary values calculated from first principles. (Calculated ternary moments are given in Table 3.2 above.)...

Also shown in the figures are the theoretical temperature variations of the spectral moments (the curves). These are obtained from first principles in the case of H2-He and H2-H2 [279, 282] measurement and theory are in very close agreement. 

Figure 3.42 shows the measurements (dots, etc.), at various temperatures of the spectral moment yo of H2-H2 pairs for the fundamental band, v = 0 —> v = 1, at woi = 4161.1 cm-1. Also shown are computations of that quantity from first principles (curve) [281]. The agreement is well within the experimental uncertainties. Figures 3.43 and 3.44 show similarly the spectral moments of H2-He pairs. Again, measurement and... 

Fig. 3.43. Spectral moment 70 of the H2-He fundamental band as function of temperature after [151]. Various measurements are shown ( , o, x, ). The curve is computed from first principles.

Table 3.7 also lists ternary spectral moments for a few systems other than H2-H2-H2. For the H2-He-He system, the pairwise-additive dipole moments are also known from first principles. The measured spectral moments are substantially greater than the ones calculated with the assumption of pairwise additivity - just as this was seen in pure hydrogen. For the other systems listed in the Table, no ab initio dipole surfaces are known and a comparison with theory must therefore be based on the approximate, classical multipole model. 

The theory of collision-induced absorption developed by van Kranendonk and coworkers [405] and other authors [288, 289, 81, 126, 125] has emphasized spectral moments (sum formulae) of low order. These are given in closed form by relatively simple expressions which are readily evaluated. Moments can also be obtained from spectroscopic measurements by integrations over the profile so that theory and measurement may be compared. A high degree of understanding of the observations could thus be achieved at a fundamental level. Moments characterize spectral profiles in important ways. The zeroth and first moments, for example, represent in essence total intensity and mean width, the most striking parameters of a spectral profile. 

Equations 5.37 and 5.38 are the desired expressions for the zeroth and first binary spectral moments. 

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]. 

Induced dipole autocorrelation functions of three-body systems have not yet been computed from first principles. Such work involves the solution of Schrodinger s equation of three interacting atoms. However, classical and semi-classical methods, especially molecular dynamics calculations, exist which offer some insight into three-body dynamics and interactions. Very useful expressions exist for the three-body spectral moments, with the lowest-order Wigner-Kirkwood quantum corrections which were discussed above. 

Constant acceleration approximation. An approximation introduced to the time-dependent intermolecular correlation function G, which was commonly referred to as the constant acceleration approximation (CAA), was used to compute the line shapes of collision-induced absorption spectra of rare gas mixtures, but the computed profiles were found to be unsatisfactory [286], It does not give the correct first spectral moment. 

Anisotropic potentials. The anisotropy of the interaction potentials may be taken into account in the computation of spectral moments. For the zeroth and first moments, the anisotropy of the interaction affects the pair distribution function, g(R,Qi,Q2), which thus becomes dependent on the orientations of molecules 1 and 2. A perturbation treatment based on the assumption of small anisotropy was given later for an estimate of the effects of the anisotropy of H2-He and H2-H2 pairs [293], Moments of more strongly anisotropic molecules (N2, CO2) were recently considered [67, 122],... 

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. 

Figures 3.42 through 3.44 (pp. 122 and 123) compare the spectral moments yo, yi, computed from first principles (solid curves) for the fundamental band of hydrogen, of the systems H2-H2 and H2-He, with the existing measurements (dots, circles, squares, etc.). The agreement is well within the experimental uncertainties of such measurements.

Hi-Ar-Ar rototranslational spectra. Tables 6.3 and 6.4 show the roto-translational spectral moments computed from first principles, with the advanced TT3 H2-Ar [228] and the HFD-C Ar-Ar interaction poten-... 

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. 

Explicit combinatorial expressions are known for the first few coefficients of (f>(B, x) [2, 21, 33, 39, 40]. We will skip these results because in the subsequent paragraph the spectral moments are discussed at due length. Using the Newton identities [24] it is easy to compute the coefficients of the characteristic polynomial from spectral moments and vice versa. 

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]).

If both the forward (absorption) and backward (emission) optical transitions are available, their first spectral moments determine the reorganization energies of quantum vibrations, Xy, and of the classical nuclear motions of the donor-acceptor skeleton and the solvent, Xj-i ... 

Also, the solvent-induced Stokes shift between the absorption and emission first spectral moments is... 

The first six spectral moments of the edge-adjacency matrix E are expressed as linear combinations of the occurrence numbers of fragments listed in Figure 1.5 ... 

At higher densities the shapes of most CILS spectra vary with density as it had been noticed since the first experimental studies [307, 308]. This fact reveals the presence of three-body and possibly higher CILS components. The onset of discernible many-body spectral components is best dealt with in the form of a virial expansion of the spectral moments [208,209,326] that at least in principle permits the separation of the binary, ternary, and so on spectral... 

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). 

F. Barocchi, M. Zoppi, and M. Neumann. First-order quantum corrections to depolarized interaction induced light scattering spectral moments Molecular dynamics calculation. Phys. Rev. A, 27 1587-1593 (1983). 

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