Third order polarizability

In the studies discussed thus far, the potential has been maintained at the PZC in order to minimize the additional dc electric field contribution to the polarizability which can arise through the third order polarizability x(3) [24, 126] where... 

Xjj is the linear polarizability, f the first-order hyperpolarizability (or second-order polarizability), and the second-order hyperpolarizability (or third-order polarizability), which is the main focus of this chapter. 

Femtosecond optical heterodyn-detected optical Kerr effect spectroscopy and low-frequency Raman spectroscopy were used to study the molecular dynamics of selenophene <1998JCP10948>. Femtosecond Kerr effect spectroscopy was also used to examine the third-order polarizabilities of furan, thiophene, and selenophene, which was found to increase from furan to thiophene to selenophene <1996CPL(263)215>. 

To add to the confusion noted for conventions of polarizabilities, both cgs and recommended SI units for linear and non-linear optical polarizabilities coexist in the literature. We strongly advocate the use of SI units. The SI unit of the electric dipole moment is Cm (Cohen and Giacomo, 1987). Thus, consistent SI units of an nth-order polarizability are defined as C m(mV )" = C m " V ", cf. (34)-(37). Conversions from the SI to the esu system for the dipole moment, the first-, second-, and third-order polarizability, are given in (38)-(41). 

Some other cross-conjugated structures of the type [131] have been characterized for their third-order polarizabilities (Bosshard et al., 1996). Computations have been performed for the tetra-substituted type [98] (Nalwa et al., 1995 Tomonari et al., 1997) at the semi-empirical and ab initio level. 

Table 12 Tensor elements y of third-order polarizability, for all crystallographic classes, on the assumption of total symmetry. The tensor elements are expressed by their subscripts only...

An axially symmetric molecule is characterized by its linear polarizability in the principal axes a x and a y = a" and a" = af/. It is a good approximation to assume that its second- and third-order polarizability tensors each have only one component and respectively, which is parallel to the z principal axis of the molecule. For linear and nonlinear optical processes, the macroscopic polarization is defined as the dipole moment per unit volume, and it is obtained by the linear sum of the molecular poiarizabilities averaged over the statistical orientational distribution function G(Q). This is done by projecting the optical fields on the molecular axis the obtained dipole is projected on the laboratory axes and orientational averaging is performed. The components of the linear and nonlinear macroscopic polarizabilies are then given by ... 

In the original papers by Hameka et al., the quantity y is called susceptibility, but since they calculated pure molecular response it should be rather referred to as the third-order polarizability or the second-order hyperpolari /ability. 

The molecular third-order polarizability has three contributions [33] ... 

In the case of isotropic liquids, the instantaneous coefficients have to be averaged over all direction in order to obtain the orientationally averaged third-order polarizability, showing the statistical contributions of all molecules ... 

It is important to realize that experimentally, for amorphous materials, only the averaged third-order polarizability can be measured, so the information about the different components of y,j7./ is lost. 

In order to assess the accuracy of the dynamic SDMRG technique, we have performed a series of model calculations we computed the dynamic linear polarizabilities (0(0 )) and third-order polarizabilities (7(0 , 01,0 )) corresponding to the third harmonic generation (THG), for the Hubbard and J7—V polyene-like chains of up to 20 sites with and without dimerizations (note that the dimerization parameter S reflects the degree of carbon-carbon bond-length alternation along the polyene chain nonzero S leads to alternating t and V values while zero 5 means uniform t and V). 

In Fig. 6, we present the dependence of 7 on chain length for various values of U/t for the uniform (Fig. 6a) and dimerized (Fig. 6b) chains. There are some very interesting differences between the Hubbard and the t/ — F chains. The behaviour of the 17 — F chains is similar to the behaviour of the Hubbard chains for U > 2F. In this regime, which is the SDW regime, the third-order polarizability of the 17 — F model is larger than that in the Hubbard model for corresponding chain lengths both with and without dimerization. 

Figure 5 Plot of the In of average third-order polarizability (7) in units of 10 esu versus log of the chain length L in A, for Hubbard chains, with three different values of Uft for (a) 5 = 0 and (b) 5 = 0.09.

However, when U = 2V, which is the CDW/BOW (Bond Order Wave) transition point, we find that the Hubbard chains have higher third-order polarizability than their counterparts in the f/ — F model. This trend is observed only for 7 and is not seen for the polarizability a. 

The linear, second, and third order polarizabilities are usually denoted a, p, and y, respectively... 

Using eqs F21 and F24, we obtain the following 8-term expression for the third-order polarizability (symmetrized with respect to aji, (02, and 0)3 permutations)... 

The third-order polarizability calculated from this measurement is an orientationally averaged quantity, (y), which can be expressed as [113]... 

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