Nonlinear response function

S. Tanaka, V. Chernyak, and S. Mukamel, Time-resolved X-ray spectroscopies nonlinear response functions and Liouville space pathways. Phys. Rev. A 63(6), 063405 (2001). 

The analytical exploitation of the full dynamic range of a detection principle invariably encompasses nonlinear portion of the concentration response function. The use of cubic spline functions for the description of this relationship is discussed after a short introduction to the theoretical principles of spline approximations. 

If a nonlinear dose-response function has been determined, it can be used with the expected exposure to estimate a risk. If an RfD or RfC is calculated, the hazard can be expressed as a Hazard Quotient (HQ), defined as the ratio of an exposure estimate over the RfD or RfC, i.e., HQ = Exposure/(RfD or RfC). 

Calibration curves were fitted, and EC50 values were derived using the nonlinear regression package pro Fit 5.5 (QuantumSoft, Zurich, Switzerland). The results of the calibration curve measurements were fitted to a sigmoidal dose-response function of the following form with a slope faetor of 1 ... 

How well the LRA describes SD depends both on the type of perturbation in in-termolecular interactions and on the strength and range of interactions within the solvent. Its breakdown has been observed in simulation studies of reasonably realistic solute-solvent systems, so it has to be used with caution. When the LRA valid, it can be veiy useful in analyzing the SD mechanism, given that much more is known about the properties of TCFs than about nonlinear response functions. 

In applying the technique of deconvolution, we take as known the spectrometer response function. It seems reasonable that the more accurately we know this function, the more accurate will be the deconvolved result. Although the nonlinear methods described in Chapter 4 are more tolerant of error, they too require a knowledge of the response function. 

The form of the response function to be fitted depends on the goal of modeling, and the amount of available theoretical and experimental information. If we simply want to avoid interpolation in extensive tables or to store and use less numerical data, the model may be a convenient class of functions such as polynomials. In many applications, however, the model is based an theoretical relationships that govern the system, and its parameters have some well defined physical meaning. A model coming from the underlying theory is, however, not necessarily the best response function in parameter estimation, since the limited amount of data may be insufficient to find the parameters with any reasonable accuracy. In such cases simplified models may be preferable, and with the problem of simplifying a nonlinear model we leave the relatively safe waters of mathematical statistics at once. 

In a strict sense parameter estimation is the procedure of computing the estimates by localizing the extremum point of an objective function. A further advantage of the least squares method is that this step is well supported by efficient numerical techniques. Its use is particularly simple if the response function (3.1) is linear in the parameters, since then the estimates are found by linear regression without the inherent iteration in nonlinear optimization problems. 

INTRODUCTION. A standard and universal description of various nonlinear spectroscopic techniques can be given in terms of the optical response functions (RFs) [1], These functions allow one to perturbatively calculate the nonlinear response of a material system to external time-dependent fields. Normally, one assumes that the Born-Oppenheimer approximation is adequate and it is sufficient to consider the ground and a certain excited electronic state of the system, which are coupled via the laser fields. One then can model the ground and excited state Hamiltonians via a collection of vibrational modes, which are usually assumed to be harmonic. The conventional damped oscillator is thus the standard model in this case [1]. 

Prof. Fleming, the expressions you are using for the nonlinear response function may be derived using the second-order cumulant expansion and do not require the use of the instantaneous normal-mode model. The relevant information (the spectral density) is related to the two-time correlation function of the electronic gap (for resonant spectroscopy) and of the electronic polarizability (for off-resonant spectroscopy). You may choose to interpret the Fourier components of the spectral density as instantaneous oscillators, but this is not necessary. The instantaneous normal mode provides a physical picture whose validity needs to be verified. Does it give new predictions beyond the second-order cumulant approach The main difficulty with this model is that the modes only exist for a time scale comparable to their frequencies. In glasses, they live much longer and the picture may be more justified than in liquids. 

S. Mukamel I would like to make a comment regarding interference effects in quantum and classical nonlinear response functions [1, 2]. Nonlinear optical measurements may be interpreted by expanding the polarization P in powers of the incoming electric field E. To nth order we have... 

However, classical nonlinear response functions do involve interference between two or more terms. The second-order quantum and classical response functions are given by... 

The stability matrix carries the necessary information related to the vicinity of the trajectory and provides an efficient numerical procedure for computing the response function. It plays an important role in the field of classical chaos the sign of its eigenvalues (related to the Lyapunov exponents) controls the chaotic nature of the system. Interference effects in classical response functions have a different origin than their quantum counterparts. For each initial phase-space point we need to launch two trajectories with very close initial conditions. [For 5(n) we need n trajectories.] The nonlinear response is obtained by adding the contributions of these trajectories and letting them interfere. 

A mixed quantum classical description of EET does not represent a unique approach. On the one hand side, as already indicated, one may solve the time-dependent Schrodinger equation responsible for the electronic states of the system and couple it to the classical nuclear dynamics. Alternatively, one may also start from the full quantum theory and derive rate equations where, in a second step, the transfer rates are transformed in a mixed description (this is the standard procedure when considering linear or nonlinear optical response functions). Such alternative ways have been already studied in discussing the linear absorbance of a CC in [9] and the computation of the Forster-rate in [10]. 

A similar translation scheme from the full quantum approach to a mixed quantum classical description has been used recently in Ref. [26-29] to calculate infrared absorption spectra of polypeptides within the amide I band (note that the translation scheme has been also used in the mentioned references to compute nonlinear response functions). 

The perpendicular slice through the phase portrait provides the stroboscopic phase portrait or Poincare section (e.g. [3]). This is in the case of the harmonic oscillator one point in the phase portrait. A further powerful method for the analysis of nonlinear dynamical systems is the determination of the Fourier spectrum of the response function >2. 

The p yR < p , > operator corresponds to the energy contribution that we previously called Uee. This operator changes during the iterative solution of the equation. VR, is said to be the response function of the reaction potential. It is important to note that this term induces a nonlinear character to Equation (1.107). Once again, in passing from the basic electrostatic model to more advanced formulations other contributions are collected in this term. The constant energy terms corresponding to f/1 and to nuclear repulsion are not reported in Equation (1.107). 

Mikkelsen). Linear and nonlinear response functions have been implemented at the MCSCF level by Mikkelsen et al. [13], for a spherical cavity, and by Cammi et al. [14] and by Frediani et al. [15] for the PCM solvation models. 

All the linear and nonlinear optical properties introduced above are therefore expressed in terms of linear, quadratic and cubic response functions. They can be computed with high efficiency using analytical response theory [9] with a variety of electronic structure models [8],... 

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