Local-field corrections

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. 

N Is the number of molecules per unit volume (packing density factor), fv Is a Lorentz local field correction at frequency v(fv= [(nv)2 + 2]/3, v = u) or 2u). Although generally admitted, this type of local field correction Is an approximation vdilch certainly deserves further Investigation. IJK (resp Ijk) are axis denominations of the crystalline (resp. molecular) reference frames, n(g) Is the number of equivalent positions In the unit cell for the crystal point symmetry group g bjjj, crystalline nonlinearity per molecule, has been recently Introduced 0.4) to get general expressions, lndependant of the actual number of molecules within the unit cell (possibly a (sub) multiple of n(g)). 

For ions in crystals of high symmetry, as in the case of our reference octahedral ABe center, the correction factor is Eioc/Eo = (n + 2)/3 (Fox, 2001), where n is the refractive index of the medium. Although this correction factor is not strictly valid for centers of low symmetry, it is often used even for these centers. Thus, assuming this local field correction and inserting numerical values for the different physical constants, expression (5.21) becomes... 

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)... 

F(o>i) are the local field corrections for a"3wave of frequency Generally, one utilizes the Lorentz approximation for the local field in which case (1,4)... 

The measurement of x of solutions can be used to determine the microscopic nonlinearities Y of a solute, provided Y of the solvent is known. This measurement also provides information on the sign of y and (hence x of the molecules if one knows the sign of Y for the solvent (5,7) Under favorable conditions one can also use solution measurements to determine if Y is a complex quantity. The method utilizes two basic assumptions (i) the nonlinearities of the solute and the solvent molecules are additive, and (ii) Lorentz approximation can be used for the local field correction. Under these two assumptions one can write the x of the solution to be... 

The macroscopic polarization of the phase is given by equations 1 and 2, where Di is the number density of the ith conformation, jlj is the component of the molecular dipole normal to the tilt plane when the ith conformation of the molecule is oriented in the rotational minimum in the binding site, ROFj is the "rotational orientation factor", a number from zero to one reflecting the degree of rotational order for the ith conformation, and e is a complex and unmeasured dielectric constant of the medium (local field correction). 

Note that these results are local field corrected... 

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. 

The need for a local field correction in Raman spectra was first suggested by Woodward and George [39] who, however, made no attempt to present a quantitative expression for the magnitude of the effect. Starting from Onsager s theory, Pivovarov derived an expression for the ratio between polarizability derivatives in solution and in vacuo (and then Raman intensities) [34,35] ... 

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. 

In the relations between the macroscopic susceptibilities y , y and the microscopic or molecular properties a, ft, y, local field corrections have to be considered as explained above. The molecule experiences the external electric field E altered by the polarization of the surrounding material leading to a local electric field E[oc. In the most widely used approach to approximate the local electric field the molecule sits in a spherical cavity of a homogenous media. According to Lorentz the local electric field [9] is... 

Wortmann, R. and Bishop D.M., Effective polarizabilities and local field corrections for nonlinear optical experiments in condensed media. J.Chem.Phys. (1998) 108 1001—1007. 

Hiis polarizability is measured by electric-field-induced second-harmonic generation (EFISHG). Again, local field corrections for the optical fields do not yield the second-order polarizability j8 of the free molecule but rather the solute polarizability /3 which contains a contribution induced by the static... 

It follows with (18)-(21) that the units for the quantities are Cm" V "mole i.e. those of an nth-order polarizability per mole. Therefore, we refer to as an nth-order molar polarizability of the constituent J. These quantities have to be calculated on the basis of a specific molecular model and appropriate local field corrections have to be taken into account. To simplify the notation, we will drop the index J in the following. A summation according to (99) is implied if the system consists of more than one constituent. 

Let us initially consider electric fields that are spatially uniform. This still allows us to consider light since the wavelengths of light will in all cases contain many interatomic distances and the fields may be considered uniform. The assumed uniformity does imply, however, that the net field within the material is also uniform, though certainly a field varies over the atomic cell even if the applied field is uniform the field at any one bond is affected by the polarization of a neighboring bond, Such variations are called local field corrections and have been treated by a number of authors, among them, Decarpigny and Lannoo (1976), Lannoo (1977), Louie, Chelikowsky, and Cohen (1975), and Hanke and Sham... 

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