Hyperpolarizabilities ratios

A crude experimental estimate for the A s coefficient is obtained from Sheltons fit of ESHG measurements for the hyperpolarizability ratio 7 /7i to the expression 3(1 +A a 22)- Using the dispersion expansions (79) and (89), we obtain for the ratio 7 /7x up to second order in the frequencies the expansion... 

B1.3.2.5 THE MICROSCOPIC HYPERPOLARIZABILITY TENSOR, ORIENTATIONAL AVERAGING, THE KRAMERS-HEISENBERG EXPRESSION AND DEPOLARIZATION RATIOS... 

Figure 3. ixxxx/ lo calculated from a Huckel-like bond alternated chain as a function of the number of sites (N). The ratio of the coupling between p-orbitals in single vs. multiple bonds determines the saturation of the 7/CV70 plot (here the ratio is 0.79 to model a polyene). 70 is the hyperpolarizability of an isolated double bond. 

In this study we have described theoretical calculations, syntheses, optical spectra, ground-state dipole moment measurements, and measurements of molecular second-order hyperpolarizability coefficients (/J) for new stilbene and azobenzene derivatives containing a methylsulfonyl group as the electron acceptor. We have shown that theoretical calculations can be used to predict the ratio of molecular hyperpolarizabilities between similar compounds, and that these gas phase calculations underestimate /J, probably as a result of the valence basis set used in the calculations. 

In Figure 9 we plotted the ratio of the magnitude (square root of the sum of the squares of all components) of the vector part and the septor part y 3 / )93 vs. P xxIPzzz for molecules with only two independent hyperpolarizability components... 

However, for most other ratios of P-xxIPzzz octopolar contribution to the nonlinearity is dominant. The dipolar contribution to the hyperpolarizability completely vanishes when P xx/Pzzz = 1 the molecule is purely octopolar. This situation occurs for molecule b with purely octopolar D h symmetry and nonvanishing hyperpolarizability components zzz = -zxx = —xxz = xzx. The hyperpolarizability tensor then reduces to... 

The photoinduced susceptibility shown in Equation 11.14a is the sum of two terms one with exp(-2Dt) (relaxation of the first-order parameter A ) decay and the second with exp(-12D ) (relaxation of the third-order pammeter A3) decay. Hence, the first very rapid decay may contain the fast exp(-12D ) contribution. However, as can be seen from Figure 11.14, the relative magnitude of this initial very fast decay does not depend on the optimization of the intensity ratio between the writing beams. So, this first rapid decay may not be due to the decay of the third-order parameter A3. In addition, because the hyperpolarizability P of DRl is different in the ds and in the trans state, the first very rapid decay also contains a contribution connected with the hferime of the metastable ds form, which is due to molecules coming back to the trans form without any net orientation. A better model would have to account for a distribution of diffusion constants for molecules embedded with various free volumes, which may explain the multiexponential behavior of the decay. 

By using simple scaling laws for the polarizability and first hyperpolarizability as a function of the molecular size, Hurst and Munn [93] addressed, within the point dipole approximation, the relationship between nd molecular elongation and found that very large can be obtained for compact molecules, provided that the ratio between the polarizability and the molecular volume is large. 

A similar reasoning can be applied to linear polarizabilities, a, and second-order hyperpolarizabilities, y, for the donor-acceptor Tr-conjugated molecules. The two-state models for a and y are given in Appendix A. Similarly as in the case of jS, the lowest CT excited state gives also a significant contribution to the values of a and y for the analyzed molecules [11, 14, 17-19, 38, 40], Also, the behaviour of a and y as a function of the solvent polarity seems to reflect the tendencies for jS. However, the a values appear to be less sensitive to the solvent effects than jS and y [14, 19, 26, 31, 36, 38], Recently, for the case of static vector components of jS and scalar part of y, it has been shown that the calculated ratio of is approximately equal to the ratio for a variety of the donor-acceptor... 

We are now in a position to compare directly with experiment it has always been my philosophy to calculate as nearly as possible what is actually measured, in this case the ratios of hyperpolarizabilities for dc-SHG and differences for dc-Kerr. Tables 5 and 6 show the high degree of agreement between theory and experiment [22,24-26], In Table 6 the fitted experimental values are those obtained by smoothing out the experimental errors by essentially... 

In Section 3.1.2. reference was made to the scaling of accurate static hyperpolarizabilities by a ratio of the SCF dynamic/static values. Judging from Eqs. (61) and (62) this is the correct thing to do (rather than simple additivity) if the frequency is small and the constant A is independent of electron correlation. 

A similar analysis can be done on the first hyperpolarizability also here vacuum results are in good accord with previous calculations on conjugated systems, for example for NH2(CHCH)2N02 Kirtman etnd Champagne find a ratio of 2.20 with a RHF/6-31G calculation to be compared with our value of 2.03. Once more, solvent effects lead to large increases in the relative importance of the vibrational contribution with respect to the electronic one multiplying the gas phase value of by factors firom 1.5 (in the first term of series I) to 2 (in the second term of series II). 

Table XIV. TDHF/AMl static first hyperpolarizability (in a.u.) and the corresponding packing ratios for MNA ID clusters containing from two to ten dimers along die three crystallographic axes. The infinite extr olated ratios are reported at the bottom of the Table.

Table XVII. TDHF/AMl static first hyperpolarizabilities ( in a. u.) and the corresponding calculated and estimated packing ratios for MNA clusters. The error made on the estimated ratios by using the multiplicative scheme is given in the last column.

Using these crystal packing ratios, effective properties of the MNA dimer in the MNA crystal can be computed (Table XXI). For example, for the first hyperpolarizability, it reads as ... 

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