Hyperpolarizability first

The much better performance of the diagonal Fade approximants compared to the Taylor expansion is not unexpected since similar convergence trends were found previously for the expansion of the first hyperpolarizabilities and u>) of ammonia. The good... 

In Table 3 we have listed the results of a basis set and correlation study for the hyperpolarizability dispersion coefficients. In a previous investigation of the basis set effects on the dispersion coefficients for the first hyperpolarizability (3 of ammonia [22] we found quite different trends for the static hyperpolarizability and for the dispersion coefficients. While the static hyperpolarizability was very sensitive to the inclusion of diffuse functions, the dispersion coefficients remained almost unchanged on augmentation of the basis set with additional diffuse functions, but the results obtained with the CC2 and CCSD models, which include dynamic electron correlation, showed large changes with an increase of the... 

Acentricity greatly enhances the y-value (see 92 vs 91 and 90 or 101 vs 99 and 100, Fig. 8). Such a trend had been predicted for certain ranges of compounds by theory [137] however when the first hyperpolarizability, which determines second-order nonlinear optical properties, is maximized, y is predicted to be zero [138]. 

These structure-function relationships provide extremely useful guidance for the future rational design of molecules and polymers with even higher optical nonlinearities. For non-centrosymmetric molecules such as 95, very high first hyperpolarizabilities /3 that determine the second-order nonfinear optical properties were also measured [140]. 

The tensor of the static first hyperpolarizabilities P is defined as the third derivative of the energy with respect to the electric field components and hence involves one additional field differentiation compared to polarizabilities. Implementations employing analytic derivatives in the Kohn-Sham framework have been described by Colwell et al., 1993, and Lee and Colwell, 1994, for LDA and GGA functionals, respectively. If no analytic derivatives are available, some finite field approximation is used. In these cases the P tensor is preferably computed by numerically differentiating the analytically obtained polarizabilities. In this way only one non-analytical step, susceptible to numerical noise, is involved. Just as for polarizabilities, the individual tensor components are not regularly reported, but rather... 

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). 

The ethynyl-linked complexes 105 were prepared and explored as potential building blocks for nonlinear optical (NLO) materials.129 Spectroscopic and cyclic voltammetry data indicate a small but real interaction between the ferrocenyl donor group and the borabenzene unit, increasing in the order RuHyper-Rayleigh scattering revealed small values for the first hyperpolarizability / , which increases in the same order. 

Sum Over States [3] (SOS) approaches constitute one of the most commonly used class of methodologies for theoretical estimation of hyperpolarizabilities. The strength of this approach is related to the fact that for many compounds of interest, only a few excited states make a major contribution. The simplest scheme, proposed by Oudar and Chemla [4—5] to analyze variations of p among push-pull conjugated materials, restricts the summation to a unique excited state. In this resulting two-state approximation (TS A), the static longitudinal electronic first hyperpolarizability, Pl, is given by ... 

Figures 1-3 display the basis set dependence of the average polarizability, (a), the polarizability anisotropy, Aa, and the vector component of the first hyperpolarizability,...

Fig. 3. Basis set dependence of the vector component of the first hyperpolarizability (in a.u., 1.0 a.u. offirst hyperpolarizability = 3.2063 X 10 = 8.641 X 10 esu) for the eight reference molecules...

Ab initio summation over states/SCI for static and dynamic first hyperpolarizabilities of small molecules... 

We have considered scalar, vector, and matrix molecular properties. A scalar is a zero-dimensional array a vector is a one-dimensional array a matrix is a two-dimensional array. In general, an 5-dimensional array is called a tensor of rank (or order) s a tensor of order s has ns components, where n is the number of dimensions of the coordinate system (usually 3). Thus the dipole moment is a first-order tensor with 31 = 3 components the polarizability is a second-order tensor with 32 = 9 components. The molecular first hyperpolarizability (which we will not define) is a third-order tensor. 

Two of the most important nonlinear optical (NLO) processess, electro-optic switching and second harmonic generation, are second order effects. As such, they occur in materials consisting of noncentrosymmetrically arranged molecular subunits whose polarizability contains a second order dependence on electric fields. Excluding the special cases of noncentrosymmetric but nonpolar crystals, which would be nearly impossible to design from first principles, the rational fabrication of an optimal material would result from the simultaneous maximization of the molecular second order coefficients (first hyperpolarizabilities, p) and the polar order parameters of the assembly of subunits. (1)... 

Calculations of Higher Polarizabilities and Semi-empirical Calculations.—O Hare and Hurst77 have calculated the first hyperpolarizabilities of some first-row diatomics by the uncoupled method. They find that a, and to a greater extent 0, are very sensitive to the basis set. Table 8 shows the effect of basis set on a for LiCl. Gupta et al.7 have... 

---