Cavity field factor

In Equation (2.183) new surface charges, qex, have been introduced these charges can be described as the response of the solvent to the external field (static or oscillating) when the volume representing the molecular cavity has been created in the bulk of the solvent. We note that the effects of qex in the limit of a spherical cavity coincide with that of the cavity field factors historically introduced to take into account the changes induced by the solvent molecules on the average macroscopic field at each local position inside the medium more details on this equivalence will be given in Section 2.7.4. 

The OWB equations obtained in this semiclassical scheme analyse the effective polarizabilities in term of solvent effects on the polarizabilities of the isolated molecules. Three main effects arise due to (a) a contribution from the static reaction field which results in a solute polarizability, different from that of the isolated molecules, (b) a coupling between the induced dipole moments and the dielectric medium, represented by the reaction field factors FR n, (c) the boundary of the cavity which modifies the cavity field with respect the macroscopic field in the medium (the Maxwell field) and this effect is represented by the cavity field factors /c,n. 

In the case of a static field, the macroscopic relative permittivity e° has to be used in (82) for the cavity field factor, while the optical relative permittivity extrapolated to infinite wavelength e can be applied to estimate the static polarizability a(0 0) in (84). In this way the Onsager-Lorentz factor for a pure dipolar liquid is obtained (87). 

Hie polarizability a(-o) a)) is involved in several linear optical experiments including refractive index measurements. Equation (93) shows that the solute molecule experiences a local field which is larger than the macroscopic field by the cavity field factor/ " and by the reaction field factor For typical media the magnitude of the product is of the order of 1.3-1.4. In the case of... 

Among the few determinations of of molecular crystals, the CPHF/ INDO smdy of Yamada et al. [25] is unique because, on the one hand, it concerns an open-shell molecule, the p-nitrophenyl-nitronyl-nitroxide radical (p-NPNN) and, on the other hand, it combines in a hybrid way the oriented gas model and the supermolecule approach. Another smdy is due to Luo et al. [26], who calculated the third-order nonlinear susceptibility of amorphous thinmultilayered films of fullerenes by combining the self-consistent reaction field (SCRF) theory with cavity field factors. The amorphous namre of the system justifies the choice of the SCRF method, the removal of the sums in Eq. (3), and the use of the average second hyperpolarizability. They emphasized the differences between the Lorentz Lorenz local field factors and the more general Onsager Bbttcher ones. For Ceo the results differ by 25% but are in similar... 

Here, N stands for the molecular packing density, h = 3el(2e+ 1) is the cavity field factor, e = (ej + 2e )/3 is the averaged dielectric constant, F is the Onsager reaction field, < //> and are the principal elements of the molecular polarizability tensor. 

Recently, Sharma has proposed some extension of the Maier-Meier approach to the case of nematogens with antiparallel dipole-dipole correlations of the molecules. He treated a polar LC material as a mixture of unpaired molecules with a finite dipole moment /u. and antiparallel pairs with zero dipole moment. The molecules interact with each other through a combination of the generalized Maier-Saupe pseudopotential for nematic mixtures and a reaction field energy term calculated from an extension of the Maier-Meier theory. Additionally, it was assumed that a dipole with dipole moment fi is embedded in a spherical cavity of dielectric permittivity n, which is surrounded by a medium of average dielectric permittivity e. In that case the expressions for the cavity field factor h and the reaction field factor / are given by h + n ), /= (e - rt")/[2rre a (2e-i-n )] and the left sides of... 

Fig. 4.5 Dipole strength of the long-wavelength absorption band of bacteriochlorophyll-a, calculated by Eq. (4.16a) from absorption spectra measured in solvents with various refractive indices. Three treatments of the local-field correction factor (/) were used down triangles, f= 1.0 (no correction) filled circles, f is the cavity-field factor empty circles, f is the Lorentz factor. The dashed lines are least-squares fits to the data. Spectra measured by Connolly et al. [148] were converted to dipole strengths as described by Alden et al. [4] and Knox and Spring [5]...

Table 7.4 reports a comparison between PCM and classical local (reaction + cavity) field factors for IR intensities of a series of simple aldehydes in aqueous solution [158]. Here/ + /i is obtained as the ratio between the calculated PCM IR intensity (with the account of both reaction and cavity field) and the corresponding value for the isolated molecule. PCM factors are generally different from classical formulations, and the difference is not limited to the same compound, but there is also a discrepancy in the observed trend in passing from one species to another. The largest difference between PCM and classical data is shown by the MSP equation, as reasonably expected due to the fact that MSP does not take into account any dependence on the static dielectric constant of the solvent. Such a dependence is instead present in the reaction field term of the PCM calculated data. 

Table 7.5 IR Intensities Classical fons) and PCM fp) Cavity Field Factors for Selection of Aldehydes in Aqueous Solution...

In this expression, F and h are reaction field and cavity field factors, respectively. They accoimt for the field-dependent interaction between the molecules and the environment. 

The reaction field factor g depends on the geometry of the cavity and on the dielectric permittivity e of the solvent. 

When one uses a regularly shaped cavity, such an ellipsoid or a sphere, the reaction field factors are given by analytical expressions [53], For a spherical cavity, g is given by ... 

This situation can be somewhat ameliorated by choosing a regular ellipsoid instead of a sphere for the solute cavity. In that case, Eq. (11.17) can still be solved in a simple fashion, with the reaction field factors depending on the ellipsoidal semiaxes (Rinaldi, Rivail, and Rguini 1992). However, while this is clearly an improvement on a spherical cavity, the small number of solutes that may be well described as ellipsoidal does not make this a particularly satisfactory solution. 

A third possibility that has received extensive study in the SCRF regime is one that has seen less use at the classical level, at least within the context of general cavities, and that is representation of the reaction field by a multipole expansion. Rinaldi and Rivail (1973) presented this methodology in what is arguably the first paper to have clearly defined the SCRF procedure. While the original work focused on ideal cavities, this group later extended the method to cavities of arbitrary shape. In formalism, Eq. (11.17) is used for any choice of cavity shape, but the reaction field factors f must be evaluated numerically when the cavity is not a sphere or ellipsoid (Dillet et al. 1993). Analytic derivatives for this approach have been derived and implemented (Rinaldi et al. 2004). 

The effective polarizabilities of the OWB solute are finally obtained in terms of the cavity and reaction field factors for example, for the linear effective polarizability we obtain... 

An underestimation of the reaction field should lead to an underestimation of the solvent shift in the NMR shielding. In fact, the relation between the reaction field and ctn is in this regime linear. Thereby, choosing the cavity scaling factor to be 1.1... 

Thanks to efficient recurrence formulae, multipole moments and multipole moment derivatives can be calculated at very high order with a low computational cost. The calculation of reaction field factors, however, may become computationally expensive at high order due to the increasing number of linear equations to be solved. Thus, in practice, the multipole moment expansion is cut off at a maximum value of f (/max), usually taken around 6. In order to get an order of magnitude of the error introduced by the truncation, let us consider Kirkwood s equations [5] for the free energy of a charge distribution of charges q, and r, in a spherical cavity of radius a ... 

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