Third derivative

In Fig. 8.2b the minima and the maximum are described by (dAGn /dXj) = 0 and the inflection points by (9 AGj /dXj ) =0. If the second derivative describes the inflection point, the third derivative describes the displacement of the inflection point. Hence at the point where the two inflection points merge, (9 AGj /9x )y = 0. 

For characterization purposes the most useful form of external modulation is electromodulation, because it provides the sharpest structure (third derivative of R in bulk or thin films) and is sensitive to surface or interface electric fields. The most widely used contacdess mode of electromodulation is termed Photoreflectance (PR) 5.7.8... 

If the term involving the third derivative. Equation (B.9) in the Taylor series, is zero, then the next higher term... 

If ACp is independent of temperature, the final term in Eq. (6-16) can be neglected. Clarke and Glew expanded AH in a Taylor s series, truncating at the third derivative of ACp. and obtained Eq. (6-17). 

The nth-order property is the nth-order derivative of the energy, d EjdX" (the factor 1 /n may or may not be included in the property). Note that the perturbation is usually a vector, and the first derivative is therefore also a vector, the second derivative a matrix, the third derivative a (third-order) tensor etc. 

The first derivative is the gradient g, the second derivative is the force constant (Hessian) H, the third derivative is the anharmonicity K etc. If the Rq geometry is a stationary point (g = 0) the force constant matrix may be used for evaluating harmonic vibrational frequencies and normal coordinates, q, as discussed in Section 13.1. If higher-order terms are included in the expansion, it is possible to determine also anharmonic frequencies and phenomena such as Fermi resonance. 

The velocities v, are the first derivatives of the positions with respect to time (dr/dt) at time ti, the accelerations a are the second derivatives (d r/d ) at time are the third derivatives etc. 

If we consider an absorption band showing a normal (Gaussian) distribution [Fig. 17.13(a)], we find [Figs. (b) and (d)] that the first- and third-derivative plots are disperse functions that are unlike the original curve, but they can be used to fix accurately the wavelength of maximum absorption, Amax (point M in the diagram). 

The matrices B, G and are defined as partial third derivatives of the Lagrangian ... 

In this case the condition u(a ,0) = Ug x) and the boundary conditions are approximated exactly. For instance, one of the schemes arising in Section 1.2 is good enough for the difference approximation of the initial equation. No doubt, we preassumed not only the existence and continuity of the derivatives involved in the equation on the boundary of the domain in view (at. r = 0 or f = 0), but also the existence and boundedness of the third derivatives of a solution for raising the order of approximation of boundary and initial conditions. 

The tensor of the static first hyperpolarizabilities P is defined as the third derivative of the energy with respect to the electric field components and hence involves one additional field differentiation compared to polarizabilities. Implementations employing analytic derivatives in the Kohn-Sham framework have been described by Colwell et al., 1993, and Lee and Colwell, 1994, for LDA and GGA functionals, respectively. If no analytic derivatives are available, some finite field approximation is used. In these cases the P tensor is preferably computed by numerically differentiating the analytically obtained polarizabilities. In this way only one non-analytical step, susceptible to numerical noise, is involved. Just as for polarizabilities, the individual tensor components are not regularly reported, but rather... 

Liu et al. [23] used a third-derivative spectrophotometric method for the determination of miconazole nitrate in Pikangshuang (cream). The detection range was 60-300 pg/mL and recovery was 100.1%. 

What determines the number of rows and columns The number of rows is determined by the number of coefficients that are to be calculated. In this example, therefore, we will compute a set (sets, actually, as we will see) of seven coefficients. The number of columns is determined by the degree of the polynomial that will be used as the fitting function. The number of columns also determines the maximum order of derivative that can be computed. In our example we will use a third-power fitting function and we can produce up to a third derivative. As we shall see, coefficients for lower-order derivatives are also computed simultaneously. 

Equation 56-27 contains scaled coefficients for the zeroth through third derivative convolution functions, using a third degree polynomial fitting function. The first row of equation 56-27 contains the coefficients for smoothing, the second row contains the coefficients for the first derivative, and so forth. 

The first two compounds discussed in this section are truly sulfhydryl-reactive, using the common iodoacetyl and maleimide functionalities, respectively. The third derivative, however, is not reactive directly with sulfhydryl groups, but contains a protected sulfhydryl which, after deprotection, can be used to react with other sulfhydryl-reactive crosslinkers. 

A Sulfur K Edge X-ray Absorption Near Edge Structure (XANES) Spectroscopy method has been developed for the direct determination and quantification of the forms of organically bound sulfur in nonvolatile petroleum and coal samples. XANES spectra were taken of a number of model compounds, mixtures of model compounds, heavy petroleum and coal samples. Analysis of the third derivatives of these spectra allowed approximate quantification of the sulfidic and thiophenic components of the model mixtures and of heavy petroleum and coal samples. These results are compared with those obtained by X-ray Photoelectron Spectroscopy (XPS). 

Examination of Table I reveals that the edge of dibenzothiophene is displaced from that of dibenzyl sulfide, the first inflection energy being some 0.6 eV higher for the former compound. From previous XANES data on dibenzothiophene and dibenzyl sulfide and physical mixtures of the two, it proved possible to identify each compound in the presence of the other (3b,8). Additionally by simply measuring the heights of the third derivative features at 2469.8 eV and 2470.4 eV relative to the base line in the model compound mixtures, a calibration was established which allowed an approximate estimate of the amounts of each component in hydrocarbon samples to be obtained. 

XANES of Petroleum Residua. On the left side of Figure 1 the sulfur K edge spectra for three different petroleum residua and the asphaltene samples prepared from them are shown. While the absorption spectra all appear to be similar, differences are revealed by examining the third derivatives of the spectra, which are shown on the right side of the figure. All the residua samples appear to contain sulfur bound in sulfidic and thiophenic forms, the amount of sulfidic sulfur increasing from sample 1 to sample 3. The asphaltene samples prepared from residua 2 and 3 also appear to contain both forms. Assuming that the composition of the sulfur... 

This work has demonstrated that organically bound sulfur forms can be distinguished and in some manner quantified directly in model compound mixtures, and in petroleum and coal. The use of third derivatives of the XANES spectra was the critical factor in allowing this analysis. The tentative quantitative identifications of sulfur forms appear to be consistent with the chemical behavior of the petroleum and coal samples. XANES and XPS analyses of the same samples show the same trends in relative levels of sulfide and thiophenic forms, but with significant numerical differences. This reflects the fact that use of both XPS and XANES methods for quantitative determinations of sulfur forms are in an early development stage. Work is currently in progress to resolve issues of thickness effects for XANES spectra and to define the possible interferences from pyritic sulfur in both approaches. In addition these techniques are being extended to other nonvolatile and solid hydrocarbon materials. 

C) The derivatives of In g and B are obtained analytically. This method is arduous since requires to determine first, second and third derivatives of all the structural parameters. It permits the use of a single nuclear conformation. The corresponding levels are called SET IV and the derivatives are shown in Table 3. Appendix I contains the formulas of the derivatives of the molecular Cartesian coordinates. 

The derivatives of these equations require one to obtain the first and second derivatives of the G matrix elements. This, in turn, requires to obtain the first, second and third derivatives of the d, R, a and P (equations 5) internal coordinates with respect to Y, and the first, second and third derivatives of the Cartesian coordinates with respect to the internal coordinates. 

From the last column of the table, we see that the ratio of the parallel-spin to the total correlation energy is remarkably independent of the size of the basis set. Contrary to expectation, the parallel-spin correlation contribution appears to be about as difficult to account for within a finite basis-set approach as the antiparallel-spin correlation. Our investigation does not provide a careful study of the basis-set saturation behavior in MP2 calculations, such as is given in Refs. [74,72,75,33]. However, our results show that, with small- and moderate-sized basis sets which are sufficiently flexible for most purposes and computationally tractable in calculations on larger systems, there is no evidence that the parallel-spin correlation contribution converges more rapidly than the antiparallel-spin contribution. A plausible explanation for this effect is that, for small interelectronic separations, the wavefunction becomes a function of the separation, which is difficult to represent in a finite basis-set approach for either spin channel. The cusp condition of Eq. (19) is a noticeable manifestation of this dependence, but does not imply that the antiparallel-spin channel is more difficult to describe with a moderate-sized basis set than the parallel channel. In fact, in the parallel correlation hole, there is a higher-order cusp condition, relating the second and third derivatives with respect to u [76]. 

By far, the most common procedure for the determination of heavy-atom positions is the difference Patterson method it is often used in combination with the difference Fourier technique to locate sites in second and third derivatives. 

The third derivative term (which is also the first time derivative of acceleration) also has a name the jerk. 

First-derivative spectrophotometry was used to identify chlorpromazine in the presence of other phenothiazines, while second derivative spectrophotometry enabled the direct determination of chlorpromazine in spiked blood samples [89], In addition, third derivative spectroscopy was used to determine chlorpromazine and its sulfoxide decomposition product in pharmaceutical dosage forms [90]. 

This can be thought about either in terms of hybrid orbitals , e.g., pd and sp hybrids as shown above, or alternatively in terms of a Taylor series expansion of a function (d functions are the first derivatives of p functions, p functions are the first derivatives of s functions). While the first way of thinking is quite familiar to chemists (Pauling hybrids), the second offers the advantage of knowing what steps might be taken next to effect further improvement, i.e., adding second, third,.. . derivatives. 

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