Calculations of Molecular Quadratic Hyperpolarizabilities

Response theory and perturbation theory approaches are the two classes of methods to compute the hyperpolarizabilities. Within each class, several levels of approximation have been worked out, providing more and more accurate 

These effects can be accounted for by employing the numerical finite field (FF) procedure in combination with post-Hartree-Fock (HF) levels of calculation, following the expression below (given for the diagonal components of the 0 tensor)  

This approximation is based on the assumption that frequency dispersion and electron correlation effects can be treated independently, which has been shown to be suitable for push-pull ir-conjugated compounds in off-resonant conditions [53]. Response calculations can also be performed at correlated levels to provide frequency-dependent quantities [54]. Though they have been mostly applied to small compounds, applications to NLO phores have also been reported [55]. 

Correlated dynamic NLO responses can also be computed using the time-dependent density functional theory (TDDFT) method, whose basic equations are similar to those of TDHF. However, conventional exchange - correlation (XC) functionals suffer from severe drawbacks when computing the NLO properties of extended systems [56], as well as when evaluating the /I contrast in molecular NLO switches. The values collected in Table 8.1 illustrate the performance of XC functionals to provide a balanced description of the static HRS hyperpolarizabilities of the two tautomeric forms of compound 11, compared to reference MP2 calculations. 

Another way to calculate fi with different levels of electron correlation consists in using the Sum-Over-States (SOS) expression [57, 58], which is derived from the many-body perturbation theory. Within this approach, P is partitioned into contributions from various pairs of excited states  

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Di Bella. S.. Marks, T.J., Ratner, M.A. Environmental elects on nonlinear optical chromophore performance. Calculation of molecular quadratic hyperpolarizabilities in solvating media. J. Am. Chem. Soc. 116, 4440-4445 (1994)... 

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