Quadratic response

In a recent publication [22] we reported the implementation of dispersion coefficients for first hyperpolarizabiiities based on the coupled cluster quadratic response approach. In the present publication we extend the work of Refs. [22-24] to the analytic calculation of dispersion coefficients for cubic response properties, i.e. second hyperpolarizabiiities. We define the dispersion coefficients by a Taylor expansion of the cubic response function in its frequency arguments. Hence, this approach is... 

Figure 5 shows a logarithmic plot of the steady state value of the SHG intensity versus applied voltage. The nearly quadratic response expected for a parabola is evident. A 20% variation from quadratic behavior would result in points within the limits indicated in Figure 5. 

The second contribution in (3.14) contains the quadratic response function, (rur2) ... 

The experimental setup for observing this effect, as seen in Figure 8, is identical to that for the quadratic response, except that the PLZT plate is prepoled to saturation remanence before using. Applications include modulators and spectral filters however, no devices have yet emerged utilizing this effect. 

If the F-test is significant then there is evidence of a quadratic effect due to at least one of the variables. With the present design, however, the investigator will not be able to determine which of the variables has a quadratic effect on the response. Additional experimentation, perhaps by augmenting the current design with some star points to construct a central composite design (see section on central composite designs below), will need to be conducted to fully explore the nature of the quadratic response surface. 

H.O. Hartley, Smallest composite designs for quadratic response surfaces. Biometrics, 15 (1959) 611-624. 

Other approaches do not consider the overlap/exchange/dispersion interactions of the molecule with the surrounding molecules, but only the electric fields and gradients generated by the molecules of the medium. The linear and quadratic response to external static electric fields F and field gradients VF were first introduced by Buckingham (93). 

The intensity of the thiolanes-thiols C28-C30 on the FPD trace is exagerated because of the two sulfur atoms carried by thiolane-thiols and because of the quadratic response of the FPD detector. The mass-spectra of the C30 thiolane-thiol, displays the fragments m/z 55 (100%), m/z 87 (62%) the typical thiolane ring, a fragment m/z 451 (17%) which corresponds to the C30 thiolane, and a molecular ion M+ 484 (60%) which could be result of the adjonction of a thiol (S-H) to the C30 thiolane. The molecules eluting in doublets with the thiolane-thiols are not identified. The C20 thiophenic isoprenoids are also present, with the compound I dominating. 

Figure 11. Variation in the ratio of eCj-C alkylbenzo[fr]thiophenes (Peaks 16-18, Figs. 10 and 13) relative to eC1-C3 alkylthiophenes (Peaks 2-14, Figs. 10 and 13) with respect to depth or temperature of artificial maturation. Maximum depths of burial are used for samples from the Paris Basin. Values determined from peak height data, corrected for quadratic response of FPD. Continued on next page.

Very recently, the study of linear birefringences has been extended to BE [32], where, as for ref. [31], furan and its homologues were investigated, in this case in solutions of cyclohexane. The latter were the subject of an experimental analysis by Dennis et al. [33], In ref. [32] advantage is taken of the recent development of frequency-dependent quadratic response in the nonequilibrium PCM solvation regime [34],... 

We obtain the terms for the solvent modifications of the quadratic response functions, denoted wj J, by collecting all terms lor n = 2 in Equation (2.308)... 

In order to determine the contributions to the quadratic response equations, one has to expand the electronic wave function O > and the operator T1v to second order. The next step concerns the collection of the appropriate terms for the quadratic response equations and as for the linear response equations it is convenient to define the following effective operators ... 

Based on the MCSCF/CM quadratic response method it is possible to calculate the hyperpolarizability tensor and the two-photon absorption cross-sections. The calculated MCSCF/CM properties exhibit for all the individual tensor components substantial shifts compared with the corresponding molecular properties of the molecule in vacuum. 

In case of the frequency-dependent first hyperpolarizability, we note that the average value of the hyperpolarizability changes sign as a water molecule is transferred from vacuum to the condensed phase. This observation has also been observed experimentally. Furthermore, the effects of the polarization terms in the structured environment are important since the quadratic response calculation within the MCSCF/CM approach without the polarization interactions leads to much smaller values for the average hyperpolarizability. 

Calibration curves (n=6, six different days) were generated at eight concentration levels in the range from 2 to 500 ng/mL and 2-1,000 ng/mL for oral fluid and plasma, respectively. Coefficient of determination (r2) was >0.99 for all analytes using a 1/x weighted linear regression, except for fluvoxamine, for which a quadratic response was observed. 

A consistent study of the linear and lowest nonlinear (quadratic) susceptibilities of a superparamagnetic system subjected to a constant (bias) field is presented. The particles forming the assembly are assumed to be uniaxial and identical. The method of study is mainly the numerical solution (which may be carried out with any given accuracy) of the Fokker-Planck equation for the orientational distribution function of the particle magnetic moment. Besides that, a simple heuristic expression for the quadratic response based on the effective relaxation... 

Figure 4.25. Signal (a) and noise (b) power densities, and SNR (c) for a superparamagnetic system at the doubled excitation frequency (i.e., quadratic response). In the first two figures the vertical scales are chosen to retain the susceptibility dependencies that really matter specifically, in (a) (T/Q)z"(2U) = Q (2U)/Vm [see Eq. (4.285)] in (b) (2Q) 2 = Q,(2Q)/V H [see...

Also in response theory the summation over excited states is effectively replaced by solving a system of linear equations. Spin-orbit matrix elements are obtained from linear response functions, whereas quadratic response functions can most elegantly be utilized to compute spin-forbidden radiative transition probabilities. We refrain from going into details here, because an excellent review on this subject has been published by Agren et al.118 While these authors focus on response theory and its application in the framework of Cl and multiconfiguration self-consistent field (MCSCF) procedures, an analogous scheme using coupled-cluster electronic structure methods was presented lately by Christiansen et al.124... 

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