Higher-order derivative spectra

Normalization of Higher-Order Derivative Spectra . Fresenius Z, Anal, Chem, 327, (1987) 83-84. 

The potential of UV-Vis derivative spectroscopy has not been fully exploited. Much of the spectroscopic analysis of aromatic compounds concerns the precise location of shoulders and maxima of hidden bands, a task ideally suited for derivative spectroscopy. The higher order derivative spectra can provide a fingerprint analysis that can distinguish compounds with the same chromophore, such as ketones (266). Isomeric compounds, such as o- and /7-xylene, which cannot be differentiated by their zero-order spectra, can be quantitatively analyzed in higher order spectra (375). The technique of differential second-derivative UV (SEDUV)... 

Using absorbance data collected as a function of time, the distribution function H(k,t) was calculated, and plotted versus In(time) to yield a curve providing a maximum for each first-order process in the reaction. The position of each maximum yields an estimate of the rate constant (t = 2/k), and the area under the maximum provides an estimate of the amount of metal dissociating by that process. Due to concerns of using higher-order derivatives to calculate rate constants for distributions of pathways in multiple first-order mechanisms as described in the literature (16-20), the kinetic spectrum method was used only to obtain initial estimates for the appropriate rate equations. The actual rate parameters reported herein were obtained from a simplex non-linear regression (21) of the original experimental data. When dissociation occurred in both fast (ti/2 > 30 s) and slow (t 1/2 < 30 s) time domains, each data set was treated independently. 

Derivative spectroscopy provides a means for presenting spectral data in a potentially more useful form than the zero th order, normal data. The technique has been used for many years in many branches of analytical spectroscopy. Derivative spectra are usually obtained by differentiating the recorded signal with respect to wavelength as thf spectrum is scanned. Whereas early applications mainly relied on hard-wired units for electronic differentiation, modem derivative spectroscopy is normally accomplished computationally using mathematical functions. First-, second-, and higher-order derivatives can easily be generated. 

In both cases, accurate results are only obtained if the most favorable measuring points are chosen, and no unknown substances or undefinable background interference is present [18]. One way to resolve a mixed spectrum more accurately is to operate in the derivative domain each standard spectrum, along with the mixture, is transformed to the first-, second-, or higher-order derivative. Linear combinations can then be computed [19]. For further information see [20-24]. 

The full information lies exclusively in the fundamental spectrum. Therefore, the derivative technique cannot give more information, but derivatives, especially higher-order derivatives, make this information more accessible, more visible, and easier to evaluate. 

In some cases the values of D and E given in a paper were derived from a spectrum fitting analysis including higher order terms. We shall indicate this by D. The original papers should be consulted for the higher order terms. 

The first derivative of an absorption spectrum, represents the gradient at all points of the spectrum and can be used to locate hidden peaks, since dA/dX = 0 at peak maxima (Fig. 4). However, second and higher even-order derivatives are potentially more useful in analysis. 

To date, most applications of TDDFT fall in the regime of linear response. The linear response limit of time-dependent density functional theory will be discussed in Sect. 5.1. After that, in Sect. 5.2, we shall describe the density-functional calculation of higher orders of the density response. For practical applications, approximations of the time-dependent xc potential are needed. In Sect. 6 we shall describe in detail the construction of such approximate functionals. Some exact constraints, which serve as guidelines in the construction, will also be derived in this section. Finally, in Sects. 7 and 8, we will discuss applications of TDDFT within and beyond the perturbative regime. Apart from linear response calculations of the photoabsorbtion spectrum (Sect. 7.1) which, by now, is a mature and widely applied subject, we also describe some very recent developments such as the density functional calculation of excitation energies (Sect. 7.2), van der Waals forces (Sect. 7.3) and atoms in superintense laser pulses (Sect. 8). 

The derivative spectrophotometry methods provide higher selectivity and higher sensitivity than do the methods based on normal (zero-order) absorption spectra. The increase in selectivity (with reduction or elimination of the effect of the spectrum of one substance on the spectrum of another one) results from reducing the band-width in the derivative spectra. An appropriate order of derivative spectrum may give complete separation of the spectra owing to the corresponding components of the system). 

Another approach that can be taken is the use of derivative spectra, that is, plotting the first, second, or even higher derivatives of the absorbance spectra. Derivative spectra can enhance the differences among spectra, resolve overlapping bands, and reduce the effects of interference from other absorbing compounds. The number of bands increases with higher orders of the derivative. The increased complexity of the derivative spectrum may aid in compound identification. For example, the absorbance spectrum of testosterone shows a single broad peak centered around 330 nm the second-derivative spectrum has six distinct peaks (Owen). 

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