Second-Order Electronic Polarizabilities

Second order molecular electronic polarizabilities are obtained by solving for Substituting Equation (10.22) for p t) in the right-hand side of Equation 

There are, of course, as many frequency components in p 2(f) as there are possibilities of combining co and co from the optical field. Here we focus our attention on a particular example in order to explicitly illustrate the physics. By definition, the second-order induced dipole moment is 

The /th Cartesian component of which we shall henceforth write as dP is of 

The expression for may be rewritten in mai r equivalent ways by appropriate relabehng of the indices n, m, and /. In particular, one can show in a manner analogous to the preceding section that, for a molecule in a ground state, the second-order polarizability tensor component becomes 

Notice that, in general, both single-photon resonances (e.g., Qg ,—(c ) and two-photon resonances [e.g., 2jj,-(co + CD)] are involved. 

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Paul. F.. Costuas, K., Ledoux, I., Deveau, S.. Zyss, J., Halet, J.-F., Lapinte. C. Redox-switchable second-order molecular polarizabilities with electron-rich iron cr-aryl acetyhdes. Organometalhcs 21, 5229-5235 (2002)... 

The second-order molecular polarizability, /, and the third-order nonlinear susceptibility, have been measured for many compounds (see Table 5.7). Note that the f value of nitroaniline, where the centrosymmetric benzene ring carries an electron withdrawing nitro-group and an electron-donating amino-group, is larger than that of monosubstituted benzenes. 

These concepts play an important role in the Hard and Soft Acid and Base (HSAB) principle, which states that hard acids prefer to react with hard bases, and vice versa. By means of Koopmann s theorem (Section 3.4) the hardness is related to the HOMO-LUMO energy difference, i.e. a small gap indicates a soft molecule. From second-order perturbation theory it also follows that a small gap between occupied and unoccupied orbitals will give a large contribution to the polarizability (Section 10.6), i.e. softness is a measure of how easily the electron density can be distorted by external fields, for example those generated by another molecule. In terms of the perturbation equation (15.1), a hard-hard interaction is primarily charge controlled, while a soft-soft interaction is orbital controlled. Both FMO and HSAB theories may be considered as being limiting cases of chemical reactivity described by the Fukui ftinction. 

A more refined but still debated in the literature notion is Pearson s Hard and Soft Acids and Bases (HSAB) principle [9,41], which quantifies energy changes to second order according to which hard (soft) acids (electron pair acceptors) prefer to interact with hard (soft) bases (electron pair donors). Soft likes soft relates to covalent bonds being facilitated by high polarizabilities, while hard likes hard relates to a creation of predominantly electrostatic interactions. 

Prof. Fleming, the expressions you are using for the nonlinear response function may be derived using the second-order cumulant expansion and do not require the use of the instantaneous normal-mode model. The relevant information (the spectral density) is related to the two-time correlation function of the electronic gap (for resonant spectroscopy) and of the electronic polarizability (for off-resonant spectroscopy). You may choose to interpret the Fourier components of the spectral density as instantaneous oscillators, but this is not necessary. The instantaneous normal mode provides a physical picture whose validity needs to be verified. Does it give new predictions beyond the second-order cumulant approach The main difficulty with this model is that the modes only exist for a time scale comparable to their frequencies. In glasses, they live much longer and the picture may be more justified than in liquids. 

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. 

For second-order nonlinear polarization, the problem becomes more complex. As can be seen in Figure 13 the anharmonic polarization shows the largest deviation from the linear polarization with large distortion values. Therefore, if the material is not polarizable (i.e., if the electrons can only be perturbed a small distance from their equilibrium positions), then the anharmonicity will not be manifested. For large second-order nonlinearities we need a material that offers both a large linear... 

The structure/property relationships that govern third-order NLO polarization are not well understood. Like second-order effects, third-order effects seem to scale with the linear polarizability. As a result, most research to date has been on highly polarizable molecules and materials such as polyacetylene, polythiophene and various semiconductors. To optimize third- order NLO response, a quartic, anharmonic term must be introduced into the electronic potential of the material. However, an understanding of the relationship between chemical structure and quartic anharmonicity must also be developed. Tutorials by P. Prasad and D. Eaton discuss some of the issues relating to third-order NLO materials. 

Under some simplifications associated with the symmetry of fullerenes, it has been possible to perform calculations of type Hartree-Fock in which the interelec-tronic correlation has been included up to second order Mpller-Plesset (Moller et al. 1934 Purcell 1979 Cioslowski 1995), and calculations based on the density functional (Pople et al. 1976). However, given the difficulties faced by ab initio computations when all the electrons of these large molecules are taken into account, other semiempirical methods of the Hiickel type or tight-binding (Haddon 1992) models have been developed to determine the electronic structure of C60 (Cioslowski 1995 Lin and Nori 1996) and associated properties like polarizabilities (Bonin and Kresin 1997 Rubio et al. 1993) hyperpolarizabilities (Fanti et al. 1995) plasmon excitations (Bertsch et al. 1991) etc. These semiempirical models reproduce the order of monoelectronic levels close to the Fermi level. Other more sophisticated semiempirical models, like the PPP (Pariser-Parr-Pople) (Pariser and Parr 1953 Pople 1953) obtain better quantitative results when compared with photoemission experiments (Savage 1975). 

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