Hyperpolarizability, dynamic second

The static and dynamic linear responses, a(0 0) and a( co co), correspond to the so-called static and dynamic polarizabilities, respectively. At second order in the fields, the responses are named first hyperpolarizabilities whereas second hyperpolarizabilities correspond to the third-order responses. Different phenomena can be distinguished as a function of the combination of optical frequencies. So, p(0 0,0), p(—co co,0), p(0 o), — ea), and p(— 2co co,co) are associated with the static, dc-Pockels (dc-P), optical rectification (OR), and second harmonic generation (SHG) processes whereas y(0 0,0,0), y(- ( ( ,0,0), y( 2co co,( ,0), y( co co, — ca, ), and y(— 3 , , ) describe the static, dc-Kerr, electric-field-induced second harmonic generation (EFISHG), degenerate four-wave mixing (DFWM),... 

The dynamic second hyperpolarizability of Sis and Si4 clusters has been calculated using the CCSD cubic response method and the aug-cc-pVTZ basis set . At A,= 1064nm, their y//( —3co a),a),a)) value amounts to... 

Using time-dependent density functional cubic response theory, a scheme has been designed to analyze the static and dynamic second hyperpolarizabilities in terms of y densities as well as in terms of contributions from natural bond orbitals (NBOs) and natural localized molecular orbitals (NLMOs). This approach, which has been implemented for both hybrid and nonhybrid TDDFT schemes and which is based on Slater-type basis functions, constitutes an extension of previously proposed schemes limited to the static responses. 

Lan, Y.-Z., Feng, Y.-L., Wen, Y.-H., Teng, B.-T. (2008). Dynamic second-order hyperpolarizabilities of Sis and Si4 clusters using coupled cluster cubic response theory. Chemical Physics Letters, 461(1-3), 118-121. 

In Fig. 3, the values for the electron-number-related static hyperpolarizability fiJN312 obtained for these ionic chromophores (open symbols) have been compared with the same values for the best dipolar, neutral chromophores reported so far (diamonds).31 32 These chromophores, with a reduced number of electrons N equal to 20, have dynamic first hyperpolarizabilities approaching 3000 x 10 30 esu at a fundamental wavelength of 1.064 pm, in combination with a charge transfer (CT) absorption band around 650 nm. It is clear that at this point, the neutral NLOphores surpass the available ionic stilbazolium chromophores for second-order NLO applications, however, only at the molecular level. The chromophore number density that can be achieved in ionic crystals is larger than the optimal chromophore density in guest-host systems. 

Table 5.3 Statistically averaged static and dynamic third harmonic generation mean second hyperpolarizabilities of methanol in vacuum or in methanol solution. Results are in a.u. and are based on the use of the M2P2BM water potential...

When Pauling published his famous paper in 1927, it was several decades before the invention of the laser. This was an event which would transform scientific research in the second half of the century, not least of all for the investigation of hyperpolarizabilities. If, in Eq. (10), electric fields associated with light are introduced (i.e. dynamic fields) then the F3 factor becomes F F F where co, are the frequencies of the fields, some of which... 

Theoretical calculations have been made on stilbene which are relevant to photoisomerization dynamics. MNDO calculations of stilbene potential energy properties shows no evidence of a doubly excited "phantom" state but a singly excited state with adiabatic rotation around the central ethylene bond has only a small barrier on this path23T Calculations of dipole moments, optical spectra, and second order hyperpolarizability coefficients of some mono- and disubstituted stilbene molecules allows the design of useful nonlinear optical molecules 38. 

Electron correlation plays a role in electrical response properties and where nondynamical correlation is important for the potential surface, it is likely to be important for electrical properties. It is also the case that correlation tends to be more important for higher-order derivatives. However, a deficient basis can exaggerate the correlation effect. For small, fight molecules that are covalently bonded and near their equilibrium structure, correlation tends to have an effect of 1 5% on the first derivative properties (electrical moments) [92] and around 5 15% on the second derivative properties (polarizabilities) [93 99]. A still greater correlation effect is possible, if not typical, for third derivative properties (hyperpolarizabilities). Ionic bonding can exhibit a sizable correlation effect on hyperpolarizabilities. For instance, the dipole hyperpolarizability p of LiH at equilibrium is about half its size with the neglect of correlation effects [100]. For the many cases in which dynamical correlation is not significant, the nondynamical correlation effect on properties is fairly well determined with MP2. For example, in five small covalent molecules chosen as a test set, the mean deviation of a elements obtained with MP2 from those obtained with a coupled cluster level of treatment was 2% [101]. 

Table 5. The static and dynamic average polarizability and the second hyperpolarizability of the nitrogen molecule in the aug-cc-pVTZ basis set. The HF, the DFT Potential/Kernel combinations and the CCSD methods were used. Reproduced from [42]...

For neon there has been a good dynamic calculation carried out by Jaszunski et al. [87] along the lines of their hyperpolarizability calculation for FH [47] which was mentioned at the end of Section 3.1.2. that is using a QRF with MCSCF reference functions. At X = 5145 A they estimate Aq to be 2.670 a.u., which can be compared with experimental values of 1.25 0.07 a.u. at this frequency [93] and 4.10 3.00 a.u. at X = 6328 A [91] and with our static estimate of 3.08 a.u. [86] (their static value is 2.687 a.u.). It is clear that both experiments must be re-evaluated (the second one because of the large error... 

In the same way as in the two previous sections, the first hyperpolarizabilities associated to three different nonlinear effects were calculated the static response and two dynamic effects, the dc-Pockels response characterized by P(-(b o),0) and the second harmonic generation response, p(-2 to, ). Again, in the dynamic case ffl=1.16 eV. The results are listed in Tables XIV, XV, and XVI for the ID clusters, and in Tables XVII, XVIII, and XIX for the 2D and 3D clusters. Similarly to the dipole moment case, only the major tensor component, Paaa> is considered. 

Electric moments, polarizabilities, and hyperpolarizabilities for BH were calculated for the first time [23], as were field and field gradient polarizabilities [24]. Spectroscopic properties were calculated for BH using the coupled electron pair approximation. The potential curve for BH was calculated at 22 points and Rq was found to be 1.23115 A and p to be 1.244 D [21]. The radiative lifetime of the A state of BH was calculated from second-order polarization propagator calculations [25], and the singlet-triplet separation in BH was calculated using ab initio MO methods. The latter, described as the singlet-triplet separation, was found to be 31.9 kcal/mol [26]. Finally, the possible dynamical pathways in the system BH + H+ were probed [27]. 

By combining classical samplings with quantum chemistry semiempirical TDHF calculations the impact of dynamic fluctuations on the first hyperpolarizability of helical strands has been evidenced . In particular, these fluctuations are responsible for relative variations of 20% in the hyper-Rayleigh responses in both pyridine-pyrimidine (py-pym) and hydrazone-pyrimidine (hy-pym) strands. Dynamical disorder has an even more important impact on the electric field-induced second harmonic generation responses, whose variations can reach 2 (py-pym) or 5 (hy-pym) times their mean value. These results demonstrate that geometrical fluctuations have to be taken into account for a reliable description of the second-order NLO properties in flexible structures such as helical strands. This work has also highlighted the relationships between the nature of the unit cell and the helical conformation of foldamers and their second-order NLO responses. In particular, the value of the hyper-Rayleigh depolarization ratio, which is characteristics of octupolar symmetry, is related to the helix periodicity, of three unit cells per turn in both compounds. 

A recent review on the nonlinear optical response and ultrafast dynamics in has emphasized on the difficulty of quantum chemistry methods to predict accurate second hyperpolarizabilities of Cgo over the whole frequency range, going from the static limit and small-frequency region to the resonant regions. 

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