Local-Field Correction Factors

The next step to include electron-electron correlation more precisely historically was the introduction of the (somewhat misleading) so-called local- field correction factor g(q), accounting for statically screening of the Coulomb interaction by modifying the polarizability [4] ... 

Figure 3. Local-field correction factor calculations. 

To determine the behavior of g(q) for large q, we performed measurements of iS lq, ) of Li for 1.1 a.u. < q < 2.6 a.u. and performed for each spectrum a fit of the g(g)-modified c° to the experimental data. Figure 10 shows the result of this semi-empirical determination of g(q) together with the shape of the local-field correction factor after Farid et al. [7] calculated for different values ofz solid line (z = 0.1), dashed line (z = 0.5) and dash-dotted line (z = 0.7). One clearly sees that the curve for the surprisingly small value of z = 0.1 fits our experimental findings best. 

We have shown for the case of Li that the step in the occupation number function is surprisingly small z 0.1 and provided semi-empirically obtained values for the local-field correction factor. For the case of Al, we showed the additional cancellation of self-energy and vertex correction. 

This expression is exactly coincident with Equation (5.21), which leads to Smakula s formula. Equation (5.22), after inserting numerical values and the local field correction factor for centers of high symmetry. 

In the weak coupling limit, as is the case for most molecular systems, each molecule can be treated as an independent source of nonrlinear optical effects. Then the macroscopic susceptibilities X are derived from the microscopic nonlinearities 3 and Y by simple orientationally-averaged site sums using appropriate local field correction factors which relate the applied field to the local field at the molecular site. Therefore (1,3)... 

When a non-centrosymmetric solvent is used, there is still hyper-Rayleigh scattering at zero solute concentration. The intercept is then determined by the number density of the pure solvent and the hyperpolarizability of the solvent. This provides a means of internal calibration, without the need for local field correction factors at optical frequencies. No dc field correction factors are necessary, since in HRS, unlike in EFISHG, no dc field is applied. By comparing intercept and slope, a hyperpolarizability value can be deduced for the solute from the one for the solvent. This is referred to as the internal reference method. Alternatively, or when the solvent is centrosymmetric, slopes can be compared directly. One slope is then for a reference molecule with an accurately known hyperpolarizability the other slope is for the unknown, with the hyperpolarizability to be determined. This is referred to as the external reference method. If the same solvent is used, then no field correction factor is necessary. When another solvent needs to be used, the different refractive index calls for a local field correction factor at optical frequencies. The usual Lorentz correction factors can be used. 

In principle, the electric fields to be inserted in Eq.(7) are the electric fields at the location of the molecule. Instead of the local electric fields oc the external fields E are usually used. Therefore, local field correction factors have to account for the electric field screening of the surrounding material when going from the macroscopic susceptibilities to the molecular hyperpolarizabilities as shown below. 

The rough surface of the nanorod is assumed to be a random collection of noninteracting hemispheroids of height a, radius b and volume V = A/3Uab. The local field correction factor within the spheroid L(co) is then expressed by Equation (16.25) ... 

Note that the local-field correction factor n[s(n -1) -i-1] varies fi om n to n as s varies from 0 to 1. For 9,10-diphenylanthracene (DPA), the correction factor was given as n[(0.128)(n - 1) +1], which lies between n and n. This agrees with the observed data of fluorescence lifetimes of DPA in various solvents. 

In the above equations is the component of the molecular first-order hyperpolarizability tensor p along the principal molecular axis, and are angular averages which describe the degree of polar order,/"/ " are the local field correction factors at [Pg.125]

Now consider an absorbing molecule dissolved in the linear medium we have been discussing. If the molecular polarizability is different from the polarizability of the medium, the local electric field inside the molecule (Pfoc) will differ from the field in the medium (E j). The ratio of the two fields ( Eioc flE J), or local-field correction factor (/), depends on the shape and polarizability of the molecule and the refractive index of the medium. One model for this effect is an empty spherical cavity embedded in a homogeneous medium with dielectric constant e. For high-frequency fields (e = r ), the electric field in such a cavity is given by... 

Fig. 3.6 The effective electric field acting on a molecule in a polarizable medium (shaded rectangles) is Eiq fErtj d where is the field in the medium and / is the local-field correction factor, hi the cavity-field model (A) Ei c is the field that would be present if the molecule were replaced by an empty cavity (Ecav), in the Lorentz model (B) Ei c is the sum of E av and the reaction field (Ereaa) resulting from polarization of the medium by induced dipoles within the molecule (P)...

Figure 3.7 shows the local-field correction factors given by Eqs. (3.35) and (3.36). The Lorentz correction is somewhat larger and may tend to overestimate the contribution of the reaction field, because the cavity-field expression agrees better with experiment in some cases (Fig. 4.5). 

With the local-field correction factor, the relationships between the energy density and irradiance in the medium (p(v) and /(v)) and the amplitude of the local field (l <,c(o)l) become ... 

We have assumed that the refractive index (n) and the local-field correction factor (/) are essentially constant over the spectral region of the absorption band, so that the ratio njf can be extracted from the integral in Eq. (4.16a). As discussed in Chap. 3, / depends on the shape and polarizability of the molecule, and usually... 

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]...

Solvatochromic effects on the transition energies of molecules in solution often can be related phenomenologically to the solvent s dielectric constant and refractive index. The analysis is similar to that used for local-field correction factors (Sect. 3.1.5). Polar solvent molecules around the chromophore will be ordered in response to the chromophore s ground-state dipole moment (/taa), and the oriented... 

In the simplest situation for a molecular chromophore, the magnitude and direction of the shift depends oti the dot product of the local electric field vector (Eext =fEapp where E pp is the applied field and/is the local-field correction factor) with the vector difference between the chromophore s permanent dipole moments in the excited and ground states (Ap) ... 

Pierce and Boxer [102] have described Stark effects on IV-acetyl-L-tryptophanamide and the single tryptophan residue in the protein melittin. They separated effects on the Lj, and bands by taking advantage of the different fiuorescence anisotropy of the two bands (Sect. 5.6). In agreement with Fig. 4.27, the La band exhibited a relatively large Afi of approximately 6 fT>ehye, where/is the unknown local-field correction factor. The Aji for the Lb band was much smaller. 

The refractive index data for n and n, calculated from n and (Eq. 11), may also be used to calculate the order parameter (5) in the chiral nematic phase using the Haller [56] technique with the Vuks local field correction factor [57] from... 

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