Hyperpolarizabilities, second

Koch H, Hattig C, Larsen H, Olsen J, Jorgensen P, Fernandez B, Rizzo A (1999) The effect of intermolecular interactions on the electric properties of helium and argon. II. The dielectric, refrac-tivity, Kerr, and hyperpolarizability second virial coefficients. J Chem Phys 111 10108-10118... 

Hattig C, L opez Cacheiro J, Fernandez B, Rizzo A (2003) Ab initio calculation of the refractivity and hyperpolarizability second virial coefficients of neon gas. Mol Phys 101 1983-1995... 

Keywords First-order hyperpolarizability second-order hyperpolarizability sum-over-states method ... 

Two triphenylamino-substituted chromophores with and without hydroxyl end, named TIOH and Tl (Figure 6.7), respectively, were synthesized and incorporated into hybrid organic-inorganic materials derived from 3-glydoxypropyltrimethoxysi-lane, tetraethoxysilane, and 3-aminopropyltriethoxysilane [54]. These stilbene-type chromophores were characterized by elemental analysis by NMR, FT-IR, UV-vis spectra, and TGA. The hyperpolarizabilities were characterized through solvato-chromic method. Both chromophores possessed higher thermal stability and competitive hyperpolarizabilities. Second harmonic generation was observed on poled films. The nonlinear coefficient of the samples was established at 41.2 pm/V for TIOFJ doped film and at 24.8 pm/V for Tl doped film. 

Fernandez et al calculated the frequency dependent interaction-induced second hyperpolarizability of two argon atoms. Subsequently, they evaluated the dielectric, the refractivity, the Kerr and the hyperpolarizability second virial coefficients. They obtained the interaction-induced mean and the anisotropy of the dipole polarizability and the mean second hyperpolarizability of Ai2 in the range of internuclear separations defined by 5 < R/ao < 30 at the CCSD level of theory, keeping the ten innermost MO frozen. They tested several basis sets at the experimental bond length of the argon dimer. Their final choice for the calculations was a d-aug-cc-pVTZ-33211 basis. 

The interaction-induced dipole polarizability and second hyperpolarizability of two neon atoms was reported by Hattig et al They subsequently used the calculated values along with an accurate potential for Nc2 to estimate the refractivity and hyperpolarizability second virial coefficients of gaseous neon. The calculation of ctint, Aai t and yjnt was performed at the CCSD level of theory with a d-aug-cc-pVQZ-33211 basis set. The R-dependence of the interaction-induced electric properties was obtained at a range of internuclear separations defined by 3 < R/ao < 20. 

Table 3.2 Polarizability a, first hyperpolarizability / , second hyperpolarizabUity 7 and dipole moment ytt of 1 calculated at the HF/6-31G(d) level of theory for the structures optimized using PM3 and B3LYP/6-31G(d) methods...

The susceptibility tensors give the correct relationship for the macroscopic material. For individual molecules, the polarizability a, hyperpolarizability P, and second hyperpolarizability y, can be defined they are also tensor quantities. The susceptibility tensors are weighted averages of the molecular values, where the weight accounts for molecular orientation. The obvious correspondence is correct, meaning that is a linear combination of a values, is a linear combination of P values, and so on. 

Gaussian can also predict some other properties dependent on the second and h er derivatives of the energy, such as the polarizabilities and hyperpolarizabilities. These depend on the second derivative with respect to an electric field, and are included automatically in every Hartree-Fock frequency calculation. 

Many ab initio packages use the two key equations given above in order to calculate the polarizabilities and hyperpolarizabilities. If analytical gradients are available, as they are for many levels of theory, then the quantities are calculated from the first or second derivative (with respect to the electric field), as appropriate. If analytical formulae do not exist, then numerical methods are used. 

Frequency-dependent polarizability a and second hyperpolarizability y corresponding to various third-order nonlinear optical processes have been... 

Dispersion coefficients for second hyperpolarizabilities using coupled cluster cubic response theory... 

In the next section we derive the Taylor expansion of the coupled cluster cubic response function in its frequency arguments and the equations for the required expansions of the cluster amplitude and Lagrangian multiplier responses. For the experimentally important isotropic averages 7, 7i and yx we give explicit expressions for the A and higher-order coefficients in terms of the coefficients of the Taylor series. In Sec. 4 we present an application of the developed approach to the second hyperpolarizability of the methane molecule. We test the convergence of the hyperpolarizabilities with respect to the order of the expansion and investigate the sensitivity of the coefficients to basis sets and correlation treatment. The results are compared with dispersion coefficients derived by least square fits to experimental hyperpolarizability data or to pointwise calculated hyperpolarizabilities of other ab inito studies. 

The matrices F, G, F-, H, A", B, and C which appear in the expression for the second hyperpolarizability in Eq. (30) are defined as partial derivatives of the quasienergy Lagrangian taken at zero field strengths and hence are frequency-independent. 

The coupled cluster expression for frequency-dependent second hyperpolarizability, Eq. (30), can now be expanded in a Taylor series in its frequency arguments around its static limit as ... 

An alternative compact expansion with coefficients which are independent of the optical process can be derived for the isotropic parallel average of the second hyperpolarizability 7 defined as [13]... 

Expressions for the sixth- through tenth-order coefficients are given in the Appendix. In ESHG experiments with the optical field polarized perpendicular to the static electric field, the measured second hyperpolarizability is [13]... 

The component y is also related to the second hyperpolarizability measured in dc-Kerr experiments... 

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