Second-order polarizability

Cyvin, S. J., Rauch, J. E. and Decius, J. C. (1965) Theory of hyper-Raman effects (nonlinear inelastic light scattering) selection rules and depolarization ratios for the second-order polarizability. 

The proportionality constants a and (> are the linear polarizability and the second-order polarizability (or first hyperpolarizability), and x(1) and x<2) are the first- and second-order susceptibility. The quadratic terms (> and x<2) are related by x(2) = (V/(P) and are responsible for second-order nonlinear optical (NLO) effects such as frequency doubling (or second-harmonic generation), frequency mixing, and the electro-optic effect (or Pockels effect). These effects are schematically illustrated in Figure 9.3. In the remainder of this chapter, we will primarily focus on the process of second-harmonic generation (SHG). 

In order to describe second-order nonlinear optical effects, it is not sufficient to treat (> and x<2) as a scalar quantity. Instead the second-order polarizability and susceptibility must be treated as a third-rank tensors 3p and Xp with 27 components and the dipole moment, polarization, and electric field as vectors. As such, the relations between the dipole moment (polarization) vector and the electric field vector can be defined as ... 

From nonlinear light scattering in methane. Maker 26O) deduced values for the second-order polarizability. 

The resulting equations show that the first-order polarizabilities atJ depend on the second moments of the distribution, while the second-order polarizabilities pijk are functions of both the second and third moments of the polarizable body, as in... 

Physically, the higher-order ( i.e., nonlinear) terms such as 6 relate to the potential well anharmonicity. Miller has suggested that to a first approximation the second-order polarizability is directly proportional to the linear polarizability (first-order) times a parameter defining the anharmonic potential (14,15). This relationship works best for inorganic materials. In organic molecules the relationship becomes complex because the linear polarizability and the anharmonicity are not necessarily independent variables (see tutorial by D. N. Beratan). 

Quantum-chemical basis for second-order polarizabilities. 136... 

Optimization of second-order polarizabilities applications to real molecules. 168... 

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. 

Perturbation theoretical expressions 136 Molecular design and two- and three-level contributions Molecular orbital calculations for second-order polarizabilities of rr systems 141... 

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

An introduction to the phenomena of NLO will be given first (Section 2), followed by the evaluation of molecular second-order polarizabilities by theoretical models that both allow their rationalization and the design of promising molecular structures (Section 3). It will be necessary to develop different models for molecular symmetries, but the approach will remain the same. NLO effects and experiments used for the determination of molecular (hyper)polarizabilities will be dealt with in Section 4. Finally, experimental investigations will be dealt with in Section 5, followed by some concluding remarks. 

The number of independent components may be further reduced by intrinsic or Kleinman symmetry (cf. p. 131). Intrinsic permutation symmetry, /3 , = holds for the second-order polarizability /3 ,(-2w Kleinman symmetry, i.e. permutation symmetry in all Cartesian indices (cf. p. 131), generally holds only in the limit w—>0. 

The molecular polarizabilities can be interpreted quantum mechanically by using the methods of time-dependent perturbation theory. Under the influence of the electric fleld, the molecular ground state ( g)) is changed by admixture of excited states ( /), m). ..). Collections of such expressions are available in the literature (Ward, 1965 Orr and Ward, 1971 Bishop, 1994b). A comprehensive treatment has also been given by Flytzanis (1975). Here, we only quote the results for the linear optical polarizability a(-a) a)) and the second-order polarizability /3(-2a) o), co). The linear optical polarizability may be represented by the sum of two-level contributions (45). 

The second-order polarizability can be written as a sum of two-level contributions and a double sum of three-level contributions in the form (48) (Wolff et al, 1997),... 

MOLECULAR ORBITAL CALCULATIONS FOR SECOND-ORDER POLARIZABILITIES OF tt SYSTEMS... 

The equations derived for two- and three-level contributions to the second-order polarizability can be used to derive design strategies for the optimization of second-order molecular polarizabilities. As shown in the section above, the NLO response of molecules is controlled by three molecular parameters transition dipoles between ground and excited states, yu. and ju. ", dipole differences between ground and excited states, AjU,, and transition energies between ground and excited states, hoj/g and Here we concentrate on TT-conjugated systems and the analysis of the transition dipoles and dipole differences associated with tttt transitions. 

Fig. 1 Left Transition dipole moment p," and dipole difference Ap in units of (el). Right First- and second-order polarizabilities, nd in units of and...

Expressions (46) and (53) allow us also to analyse the impact of the LCAO parameter c on the first-order and second-order polarizabilities, a and /3. We assume that the excitation energy does not vary with c as discussed above and... 

In the non-linear case, the Fourier component induced by an external optical field can be represented in terms of an effective second-order polarizability (—2[Pg.152]

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... 

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. 

The quadratic effect of an externally applied field on the refractive index n is described by the third-order susceptibility (- ) w,0,0) (Kerr susceptibility). The two independent components Yilzz and x ixx can be interpreted in terms of molar polarizabilities. The results for 2 symmetric molecules with only one significant component of the second-order polarizability are expressed in (113) and (114),... 

The square of the transition dipole occurs in the two-level contributions and to the first-order and second-order polarizability and is important for their optimization. 

EXPERIMENTAL DETERMINATION OF MOLECULAR SECOND-ORDER POLARIZABILITIES... 

Despite these shortcomings it will become clear that in the one-dimensional NLO-phores treated in this section, which display a wide range of seemingly disparate chemical structures, the crude model works surprisingly well. Thus, as a consequence of the validity of the two-state model, their second-order polarizabilities in principle reduce to p-nitroaniline . The reader may even gain the impression that the efforts to improve on the hyperpolarizabilities of even the simplest and most easily accessible -n systems (like p-nitroaniline) have been futile. It is true that an efficiency-transparency trade-off exists At a given wavelength of absorption (related to A ) a maximum value for the second-order molecular polarizability per volume element exists which is not tremendously different from that of very basic unoptimized rr systems. However, for applications like the electro-optical effect, a bathochromic shift of the UV-visible absorption is tolerable so that to strive for maximum hyperpolarizabilities is a viable quest. Furthermore, molecular structures with the same intrinsic second-order polarizabilities may differ substantially in their chemical stabilities and their abilities to be incorporated into ordered bulk structures. 

Therefore, we have quoted second-order polarizabilities in terms of /3o of p-nitroaniline in dioxane (A a = 354 nm, /3o = 13.5 Cm T convention, relative to quartz dn = 0.5 pm/V at 1064 nm). p-Nitroaniline is a truly one-dimensional NLO-phore with one significant component as has been verified experimentally by depolarized EFISHG (Wortmann et al, 1993). If different standards and conventions are taken into account, the values measured by different groups are quite consistent. Note that the intrinsic /3o of p-nitroaniline depends on the solvent, even when normalized for the solvatochromic shift of the CT absorption. We have chosen the lowest intrinsic /3o it is higher by a factor of 1.6 in very polar solvents (see p. 183). Also note that /3 values from HRS measurements of molecules with several significant tensor elements will not allow a true comparison of... 

---