Empirical line shapes

A great deal more could be said about models - to understand behavior like strong correlation, Coulomb blockade, and actual line shapes, it is necessary to use a number of empirical parameters, and a quite sophisticated form of density functional theory that deals with both static and dynamic correlation at a high level. Often this can be done only within a very simple representation of the electrons - something like the Hubbard model [51-53], which is very common in this situation. 

Collision-induced dipoles manifest themselves mainly in collision-induced spectra, in the spectra and the properties of van der Waals molecules, and in certain virial dielectric properties. Dipole moments of a number of van der Waals complexes have been measured directly by molecular beam deflection and other techniques. Empirical models of induced dipole moments have been obtained from such measurements that are consistent with spectral moments, spectral line shapes, virial coefficients, etc. We will briefly review the methods and results obtained. 

Fitting line shapes. In the next Chapter, we will discuss various approaches to computing spectral line shapes. Such computations require as input a reliable model of the interaction potential and of the dipole components. Once a profile is computed on the basis of an imperfect empirical dipole moment, the comparison with spectroscopic measurements may reveal certain inconsistencies which one may more or less successfully correct by small adjustments of the free parameters. After a few iterations, one may thus arrive at an empirical model that is consistent with a spectroscopic measurement [39], If measurements at various temperatures exist, the dipole model must reproduce all measured spectra equally well. 

Table 4.2. Empirical induced dipole moments of systems involving molecules m and Is stand for moment analysis and line shape analysis, respectively.

Other quantum line shape computations of absorption by dissimilar atomic pairs based on empirical or ab initio dipole models have been known for some time [39, 44, 76, 251, 330, 332, 361, 365, 386], Such studies are of interest for the analysis of measurements, for predicting... 

It is, therefore, interesting to point out that in a recent molecular dynamics study, shapes of intercollisional dips of collision-induced absorption were obtained. These line shapes are considered a particularly sensitive probe of intermolecular interactions [301]. Using recent pair potentials and empirical pair dipole functions, for certain rare-gas mixtures spectral profiles were obtained that differ significantly from what is observed... 

The H MAS NMR spectra resemble infrared spectra one broad peak is attributed to hydrogen bonded silanols and one sharper peak to the isolated silanols. As in infrared, it is very difficult to derive the exact line shapes on a theoretical basis. Therefore, Haukka derived the line shapes for simulating the spectra empirically from the best fit to the data. 

The dispersion phenomenon has been quantitatively approached by three models. Initially, Albrecht s theory (Tang and Albrecht, 1970) was applied to the finite segments of the polymer. Then, in the case of materials such as trans-Vk, use of an empirical distribution function P N) for the conjugation length made it possible to exactly reproduce the line shapes and line intensities resulting from excitation with different laser lines ... 

N. Meinander and G. C. Tabisz. Moment analysis and line shape calculations in depolarized induced light scattering Modeling empirical pair polarizability anisotropy. J. Quant. Spectres. Rad. Transfer, 55 39-52 (1986). 

In a final step of spectral evolution from NCA calculations for each computed transition, an appropriate profile function needs to be chosen. Usually, Gaussian or Lorentzian line shapes with an empirical half band width of 10 cm are assumed. The spectra are then generated by plotting the sum of all band intensities against the wavenumbers. 

Here, t is in the first instance a constant fit parameter, which will be analysed later. A further empirical result is the shape of the M lines for a vanishing A- B spacing, they have a Gaussian line shape and for a large A- B spacing, they are approximately Lorentzian. 

Fig. 14.9 Comparison of the measured J°values ( ) of sample C with those calculated (+) using the s values obtained from the Jp(f) line-shape analyses and the curve calculated from the empirical functional form log( J°) = a - - b/x + cjx djx x being the temperature in °C) best fitted to the calculated values. Note As the Jp(t) curve at 134.1°C is not available for analysis to obtain the s value, the J° value at 134.1°C used in the least-squares fitting is the experimental value itself.

FIGURE 13. Top PE spectra of the normal alkanes nonane C9H20 and hexatriacontane C36H74 taken from Reference 27. Bottom Calculated contours of the C2s-band systems of nonane and polyethylene using the EBO model and the empirical line shape functions G(x) and L(x) (equations 65). The intensities are in arbitrary units. The different contours have been calculated with the half-widths I or (7-indicated in the Figure... 

The broadening Fj is proportional to the probability of the excited state k) decaying into any of the other states, and it is related to the lifetime of the excited state as r. = l/Fj . For Fjt = 0, the lifetime is infinite and O Eq. 5.14 is recovered from O Eq. 5.20. Unfortunately, it is not possible to account for the finite lifetime of each individual excited state in approximate theories based on the response equations (O Eq. 5.4). We would be forced to use a sum-over-states expression, which is computationally intractable. Moreover, the lifetimes caimot be adequately determined within a semiclassical radiation theory as employed here and a fully quantized description of the electromagnetic field is required. In addition, aU decay mechanisms would have to be taken into account, for example, radiative decay, thermal excitations, and collision-induced transitions. Damped response theory for approximate electronic wave functions is therefore based on two simplifying assumptions (1) all broadening parameters are assumed to be identical, Fi = F2 = = r, and (2) the value of F is treated as an empirical parameter. With a single empirical broadening parameter, the response equations take the same form as in O Eq. 5.4 with the substitution to to+iTjl, and the damped linear response function can be calculated from first-order wave function parameters, which are now inherently complex. For absorption spectra, this leads to a Lorentzian line-shape function which is identical for all transitions. 

The width of the Gaussian function, G(v), is the Doppler width, wj), and that of the Lorentzian function, L( v — v ), is the collision width, lUc. This generalized line shape, introduced by Voigt (1912), has no analytical expression, but it can easily be computed numerically using the convolution represented in Eq. (3.6.8). Its shape is shown in Fig. 3.6.1. An empirical expression relating wd and Wc to the total Voigt width, wm, has been introduced by Whiting (1968),... 

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