Internal field factor

Relative and ab.solute Raman scattering cross sections have been measured by a number of groups and were summarized by Schrotter and Klockner (1979). The formula for the total differential Raman scattering cross section of a vibrational-rotational band has been given in Eq. 2.4-6 the internal field factor L is equal to 1 for ga.ses, see also Sec. 3.5.4. 

The various authors make claims for the accuracy of their methods—in particular relating to the quality of the basis sets and the accuracy of the geometry that has been used—but it is difficult to believe that the theoretical value of (gas, electronic, 1.17 eV) has been settled to better than about 500 au. Provided a fixed geometry is assumed (whatever it may be) the quantity that is being calculated is clearly defined, but it excludes vibrational effects and the effects of intermolecular interactions. It has usually been assumed for frequency doubling, where two optical frequency fields have to interact, that the vibrational effects will be small and in the gas phase the internal field factors do not differ greatly from unity. Even if they are by no means the whole story, the quantities, (gas, electronic, 0 eV) and... 

Smaller molecules and inert gas atoms have been extensively studied using EFISH in the gas phase (see for example Miller and Ward, Ward and Miller, Shelton ). Shelton and Rice provide a comprehensive list of gaseous EFISH measurements on small molecules up to 1994. The only such result reported for molecules with donor/ acceptor substitution on a benzene ring appears to be that obtained by Kaatz et alP for pNA in 1998. In this experiment a gas mixture containing 0.075 mole fraction of pNA was used to obtain an EFISH measurement at 1064 nm at one temperature. The y (—2(B (B,a),0) of eqn (4.15) was estimated from a THG experiment and taken as the intercept on a two point plot of y versus /T. The value of was calculated from the slope. The linearity of such plots has been confirmed in the work on smaller molecules. The gas phase method differs from that used for solutions in that the extrapolation to infinite dilution is not made since the molecular density in the gas is very much smaller. Also the internal field factors are close to unity. It is usually possible to make measurements over a sufficiently wide range of temperatures to obtain the quantity (jifi/k) from the plot of F versus l/F. In the case of pNA the value of the dipole was chosen as 6.87 D. 

The internal field factors are now in general to be adjusted to take account of the differing environments of the molecules of the two species. In particular the reaction field introduced in the Onsager field will be different, even in very dilute solution, in the cavity surrounding the polar solute as compared to that of a non-polar solvent. 

Table 4 the product has been calculated from eqns (7.24) and (7.25) using the additional data from Table 7. The direct third order term, has been neglected. The published value obtained from )i and jSj as given in the papers is given in the seventh coluttm. In rows 1 and 2 a dipole of 6.2D has been assumed (see text). In the final coluttm the published values that have been obtained from simplified equations such as (7.24) and (7.25) have been reduced by the ratio of the published / eqn (7.25) results of Teng and Garito. Values in brackets are unaltered, except in the case of Wortmann et al. where the published result has been multiplied by the missing 2a> internal field factor. TG Teng and Garito Bur Burland et al.f Boss Bosshard et Stah Stahlein et al. Wort Wortman et al.f° Pal Paley et al.- KS Kaatz and Shelton, Hyper Rayleigh measurement. The pi values are in atomic units and the value in the last column has been adjusted, where appropriate, by the same factors used for the macroscopic quantities in Table 4.

I) Lorentz-Lorenz, Onsager internal field factors (QM(g), Continuum)... 

In case (I), the theory that has been used to describe the EFISH analysis in earlier sections, the isolated molecule calculation provides the pz value and the internal field factors adjust the fields to allow for the polarization on the cavity surface. The elfect of reaction fields due to the additional polarization on the cavity walls induced by the permanent and induced dipoles on the central molecule is implicitly included in the low frequency Onsager field factor through the dielectric constant values. The choice of the high frequency dielectric constant ( o) in this formulation is rather ill defined and no account is taken of changes in the static or dynamic polarizabilities of the molecule as a result of the surrounding fields. 

Since pNA and most of the chromophores of interest have large dipole moments an important feature of the continuum models is the introduction of the reaction field. The pNA molecule at the centre of the cavity in the continuum induces a polarization on the surface of the cavity, which produces the reaction field acting on the central molecule. This reaction field changes the dipole moment of the pNA molecule via the linear polarizability. A self consistent procedure is required in which the effects of the reaction field and also the effects of the applied macroscopic fields modified by the internal field factors are included in a self-consistent determination of the molecular response within a specified quantum mechanical model. 

In comparing solution calculations, however, there are still differences in the choice of the quantity that is used to characterise the response of the solution. A distinction is made between the property of a solvated pNA molecule and the effective hyperpolarizability of pNA in solution. In the former case the modifications of the pNA molecule have been calculated in the solution without any applied electric field present. In the second case the applied electric field has been present while the structure of the pNA and its immediate surroundings have been optimized. In both cases the calculated molecular quantity has to be used with internal field factors to recover the macroscopic response—but the choice of field factors may be different in the two cases. 

Table 14). The differences in the mq values obtained by these two methods may be caused by some uncertainty of the internal field factor in Bi (Eq.(85 ), p. 181). The experimental data on poly(alkyl isocyanate)s and poly(7-benzyl-L-glutamate) yield d = 0 which is in good agreement with the axially symmetrical structure of the main chain of these polymers. It is shown in F. 77 that for cellulose carbanUate the agreement between the theoretical curve and exp imental points is attained at d = 65°. Evidently, in this case the side group of the monomer unit containing polar bonds provides an important contribution to the dipole moment of the monomer unit. 

The relation between the molecular polarizabilities to the bulk susceptibilities depends on the molecular density, N, and the local internal field factor F. Particularly, it relates to the molecular orientation. For example, the second NLO coefficient is expressed as... 

Because of the incidence of the internal field factor, this is not the value of the molecular dipole relaxation. Depending upon the internal field assumption a variety of relationships between theoretical and effective relaxation times have been defined. Relaxation times for dipole orientation at room temperature are between 10 s for small dipoles diluted in a solvent of low viscosity and more than 10 s for large dipoles in a viscous medium such as polymers (polyethylene) or dipole relaxations in crystals (the relaxation associated with pairs of lattice vacancies). 

This is the Lorenz internal field factor which is 0 in a vacuum (e - 1) and greater than one in a real media. 

It differs from Rgoan for gases (Equation 2.2-18) by the internal field factor [(/i 2)/3) ... 

Po t) will not be a simple exponential decay function, but would take the form (p t) = exp —(t/ii) as discussed above for PEA. Similarly (pp t) may not be a simple exponential decay function. In addition the distribution of environments r will lead to a spectrum of values so that the P process would be predicted to be very broad on the log //fo) plot. If we ignore the internal field factors [22] Eq. (6) gives ... 

Setting the internal field factor equal to unity, then (A.7) yields... 

These changes reflect the molecular reorganization that takes place at transitions between different liquid crystal phases. The interpretation of the dielectric properties of smectic phases can be carried out using Eq. (34). Differences in orientational order in smectic phases are accounted for through the appropriate orientational order parameters given in Eq. (29), while other influences of the translational smectic order will affect the internal field factors and short range dipole-dipole interactions. For strongly... 

P(co) is an internal field factor and A t) is a time-correlation function which represents the fluctuations of the macroscopic dipole moment of the volume V in time in the absence of an applied electric field. Equations (44) and (45) are a consequence of applying linear-response theory (Kubo-Callen-Green) to the case of dielectric relaxation, as was first described by Glarum in connexion with dipolar liquids. For the special case of flexible polymer chains of high molecular weight having intramolecular correlations between dipoles but no intermolecular correlations between dipoles of different chains we may write... 

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