Polarizability second-hyper

Where a, P and y are the linear polarizability, the first- and second-hyper polarizabilities, respectively, and are represented by second, third and fourth rank tensors, respectively, and is a static polarizability. 

Here, E is the strength of the applied electric field (laser beam), a the polarizability and / and y the first and second hyper-polarizabilities, respectively. In the case of conventional Raman spectroscopy with CW lasers (E, 104 V cm-1), the contributions of the / and y terms to P are insignificant since a fi y. Their contributions become significant, however, when the sample is irradiated with extremely strong laser pulses ( 109 V cm-1) created by Q-switched ruby or Nd-YAG lasers (10-100 MW peak power). These giant pulses lead to novel spectroscopic phenomena such as the hyper-Raman effect, stimulated Raman effect, inverse Raman effect, coherent anti-Stokes Raman scattering (CARS), and photoacoustic Raman spectroscopy (PARS). Figure 3-40 shows transition schemes involved in each type of nonlinear Raman spectroscopy. (See Refs. 104-110.)... 

FIGURE 6 Effect of bond alternation and ligand electronegativity on the second hyper-polarizability of the phosphonitrilic polymer. 

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. 

Just as a is the linear polarizability, the higher order terms p and y (equation 19) are the first and second hvperpolarizabilities. respectively. If the valence electrons are localized and can be assigned to specific bonds, the second-order coefficient, 6, is referred to as the bond (hyper) polarizability. If the valence electron distribution is delocalized, as in organic aromatic or acetylenic molecules, 6 can be described in terms of molecular (hyper)polarizability. Equation 19 describes polarization at the atomic or molecular level where first-order (a), second-order (6), etc., coefficients are defined in terms of atom, bond, or molecular polarizabilities, p is then the net bond or molecular polarization. 

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 PCM calculation are performed by using the CPHF formalism [51] for the static case, and to the TD-CPHF formalism for the frequency dependent case [52]. There are also calculations at higher levels of the QM theory which have not been fully analyzed. The formulas are quite complex, and we refer the interested readers to the two source papers. What is worth remarking here is that (hyper)polarizability values are quite sensitive to the cavity errors. In passing from 7I ) (i.e. a, the polarizability tensor) to 7 1 (i.e. /9, the first hyperpoljirizability) and to 7 (i.e. 7, the second hyperpolarizability) the problem of cavity errors become worse and worse. 

The second category includes the response properties which describe the effects of any external applied field on a molecular system. This category includes the electronic and vibrational dipole (hyper)polarizabilities, both static and frequency dependent, magnetic and chiro-optic properties, etc.. Response properties are essential for a deeper understanding of molecular behaviors, and they represent the basis for an ever increasing number of technical applications. 

The obtained electronic (hyper)polarizabilities were compared with values published previously by Campbell et al. [70, 71] which were computed using an uncoupled approximation to coupled-perturbed HF theory, with the 6-3IG basis set. Their values were very different from those obtained in Ref. [60], e.g. for the first hyperpolarizabilities they reported Pyyy —7,000 au, Pyyy 15,000 au, about an order of magnitude larger than in Ref. [61]. For the second hyperpolarizability Yxxxx 320 X 10 au, Vyyyy 540 X 10 au, yzzzz —320 x 10 au. These large differences may be interpreted as an additional indication that HF theory is unreliable for the hyperpolarizabiUties of Li C6o-... 

The (monochromatic) electric fields are characterized by Cartesian directions indicated by the Greek letters and by circular optical frequencies, ( i, ( 2, and 0)3. The induced dipole moment oscillates at (0 = 2i cOi. and are such that the p and y values associated with different NLO processes converge towards the same static value. The 0 superscript indicates that the properties are evaluated at zero electric fields. Eq. (2) is not the unique phenomenological expression defining the (hyper)polarizabilities. Another often-applied expression is the analogous power series expansion where the 1/2 and 1 /6 factors in front of the second- and third-order terms are absent. 

Chou and Jin have addressed the importance of the vibrational contributions to the polarizability and second hyperpolarizability within the two-level and the two-band models. Their study adopts the sum-over-state (SOS) expressions of the (hyper)polarizabilities expressed in terms of vibronic states and includes two states and a single vibrational normal mode. Moreover, the Herzberg-Teller expansion is applied to these SOS formulas including vibrational energy levels without employing the Plac-zek s approximation. Thus, this method includes not only the vibrational contribution from the lattice relaxation but also the contribution arising... 

---