Macroscopic susceptibilities

Consider an isotropic medium that consists of independent and identical microscopic cln-omophores (molecules) at number density N. At. sth order, each element of the macroscopic susceptibility tensor, given in laboratory Cartesian coordinates A, B, C, D, must carry s + 1 (laboratory) Cartesian indices (X, Y or Z) and... 

Thus, since intramolecular bonding interactions in the solid are much stronger than relatively weak i n termo1ecu1 ar van der Vaals interactions, each molecular unit is essentially an independent source of nonlinear response, arrayed in an acentric cystal structure, and coupled to its neighbors mainly through weak local fields. In the rigid lattice. gas approximation, the macroscopic susceptibility X is expressed as... 

The next step is to connect the macroscopic susceptibility to individual molecular moments and finally to the number of unpaired electrons. From classical theory, the corrected or paramagnetic molar susceptibility is related to the permanent paramagnetic moment of a molecule, y. by ... 

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)... 

However, even when all these effects are included in the solvation model, the calculated quantities are still microscopic and cannot be directly compared with their macroscopic manifestation, i.e. the macroscopic susceptibilities determined experimentally. 

The tensorial coefficients are now the nth-order macroscopic susceptibilities x,rl>. They are tensors of rank n + 1 with 3("+1) components. The prefactors in Equations (2.165)-... 

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. 

For the macroscopic polarization analogous expressions can be obtained by replacing the tensors fi and y by the corresponding macroscopic susceptibilities and 3. ... 

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... 

While the above discussion clearly highlights the importance of including solvent effects in the calculations, the calculated properties cannot be compared directly with experimental results. This is mainly caused by the many different conventions used for representing hyperpolarizabilities and susceptibilities. However, the calculated properties can be combined with appropriate, calculated Lorentz/Onsager local field factors to obtain macroscopic susceptibilities that can be compared with experimental results. For water, we used this to calculate the refractive index and the third harmonic generation (THG) and the electric field-induced second harmonic (EFISH) non-linear susceptibilities. The results are collected in Table 3-11. 

Since the non-linear susceptibility is generally complex, each resonant term in the summation is associated with a relative phase, y , which describes the interference between overlapping vibrational modes. The resonant macroscopic susceptibility associated with a particular vibrational mode v, Xr, is related to the microscopic susceptibility also called the molecular hyperpolarizability, fiy, in the following way... 

Combining Equations (8>-(10) yields the following expression for the resonant macroscopic susceptibility... 

Macroscopic susceptibilities and molecular polarizabilities 155 Experimental determination of molecular second-order polarizabilities 161... 

The different symmetry properties considered above (p. 131) for macroscopic susceptibilities apply equally for molecular polarizabilities. The linear polarizability a - w w) is a symmetric second-rank tensor like Therefore, only six of its nine components are independent. It can always be transformed to a main axes system where it has only three independent components, and If the molecule possesses one or more symmetry axes, these coincide with the main axes of the polarizability ellipsoid. Like /J is a third-rank tensor with 27 components. All coefficients of third-rank tensors vanish in centrosymmetric media effects of the molecular polarizability of second order may therefore not be observed in them. Solutions and gases are statistically isotropic and therefore not useful technically. However, local fluctuations in solutions may be used analytically to probe elements of /3 (see p. 163 for hyper-Rayleigh scattering). The number of independent and significant components of /3 is considerably reduced by spatial symmetry. The non-zero components for a few important point groups are shown in (42)-(44). 

In this section, we investigate the relations between the macroscopic susceptibilities and the molecular polarizabilities. Consistent microscopic interpretations of many of the non-linear susceptibilities introduced in Section 2 will be given. Molar polarizabilities will be defined in analogy to the partial molar quantities (PMQ) known from chemical thermodynamics of multicomponent systems. The molar polarizabilities can be used as a consistent and general concept to describe virtually all linear and non-linear optical experiments on molecular media. First, these quantities will be explicitly derived for a number of NLO susceptibilities. Physical effects arising from will then be discussed very briefly, followed by a survey of experimental methods to determine second-order polarizabilities. 

Now we study in detail how the macroscopic susceptibilities are related to the molecular properties. A thorough understanding of these relations is essential for both the rational design of molecular NLO materials as well as the experimental determination of the molecular electric properties. Models for the interpretation of macroscopic susceptibilities in terms of molecular dipole moments and polarizabilities usually assume additive molecular contributions (Liptay et al., 1982a,c). Thus, an nth-order susceptibility can be represented by (99) as a sum of terms that are proportional to concentrations Cj (moles per cubic metre, mol m ) of the different constituents J of the medium. 

In the following we present explicit relations for the molar polarizabilities for a number of important macroscopic susceptibilities. These equations will be used subsequently as a basis for the experimental determination of molecular polarizabilities. 

The linear and non-linear polarizabilities of organic molecules are usually determined from measurements of macroscopic susceptibilities of liquid solutions. Classical examples are the measurements of the refractive index, n, or the relative permittivity of pure organic liquids and their interpretation by the well-known Lorentz-Lorenz and Clausius-Mosotti equations. These... 

Macroscopic susceptibilities and molecnlar polarizabilities 155 Experimental determination of molecnlar second-order polarizabilities Optimization of second-order polarizabilities applications to real molecules a Systems and one-dimensional rr systems 168 Two-dimensional (2D) NLO-phores ID and 2D architectnre 196 Conclnsion 206 Acknowledgements 208 References 208... 

The different symmetry properties considered above (p. 131) for macroscopic susceptibilities apply equally for molecular polarizabilities. The linear polarizability a( w w) is a symmetric second-rank tensor hke Therefore, only six of its nine components are independent. It can always be transformed to a main axes system where it has only three independent components,... 

Azo-benzene molecules are widely recognized as attractive candidates for many nonlinear optical applications. A highly deformable distribution of the ic-electron gives rise to very lar molecular optical nonlinearitics, Phdto-isomerization of azo molecules allows linear and nonlinear macroscopic susceptibilities to be easily modified, giving an opportunity to optically control the nonlinear susceptibilities. In this chapter, we will discuss third-order nonlinear optical effects related to photoisornerization of azo-dye polymer optical materials. 

As it is seen from Eqs. (1) and (2) the values of molecular hyperpolarizabilities and macroscopic susceptibilities, within a given system of units, depend directly on the conventions used. Therefore, it is important when comparing data coming from different determinations or with theoretical calculations to ensure what kind of conventions were used. 

Another scheme to calculate and interpret macroscopic nonlinear optical responses was formulated by Mukamel and co-workers [112 114] and incorporated intermolecular interactions as well as correlation between matter and the radiation field in a consistent way by using a multipolar Hamiltonian. Contrary to the local field approximation, the macroscopic susceptibilities cannot be expressed as simple functionals of the single-molecule polarizabilities, but retarded intermolecular interactions (polariton effects) can be included. 

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