Kronig-Kramer

In other words, we have expressed the interaction between the adsorbate and the metal in terms of A(e) and /1(e), which essentially represent the overlap between the states of the metal and the adsorbate multiplied by a hopping matrix element A(e) is the Kronig-Kramer transform of A(e). Let us consider a few simple cases in which the results can be easily interpreted. 

From the Kronig-Kramer relation it immediately follows that A(e) = 0, as the function to be integrated is odd, and hence the resulting projected density of states becomes... 

ORD and CD are connected through the Kronig-Kramers integral transform. Thus, for the -th transition it follows that ... 

The Kronig-Kramers relation is of fundamental importance for optics and for physics in general13). Here, these equations do not seem practical because of the integration of the wavelength from 0 to oo. However, these are very useful for calculating the molar ellipticity magnitude from the observed ORD curve 14). 

From these relations we see that the width and shift of the power spectrum and consequently the spectroscopic lines are related through the Kronig-Kramers dispersion relations. Exactly the same arguments apply to the Laplace transform of the time-correlation function, H(/co). The real and imaginary parts, C H(co) and C"//(/(0), are related by Kramers-Kronig dispersion relation. 

Rigorously, ORD and CD spectra are related through the Kronig-Kramers theorem, a well-known general relationship between refraction and absorption, i.e. nL - nR is determined by eL - % for A from zero to infinity [128], (The analogous relationship between refraction and reflection applies to cholesteric liquid crystals.) Hence, in order to maximize ORD in the transparent region, Cotton effects, associated with exciton coupling (both intramolecular and intermolecular), have... 

The electronic absorption characteristics of chromophores within potential gela-tors can provide an important experimental monitor of the microscopic environment in which they reside. This is especially true when the information includes optical rotatory dispersion (ORD) and circular dichroism (CD) data for potential gelalors that arc chiral. Dichroism relates to the absorptivity difference between the two components of circularly polarized light, w-hich constitutes the incident plane of linearly polarized light as described by the Kronig-Kramers transform. The intensity of UV/vis absorption depends on corresponding quantum transition. The wavelengths at which nonzero circular dichroism may be observable in the CD spectrum can be discerned from the shape of the absorption bands. The... 

Linear viscoelasticity theory predicts that one component of a complex viscoelastic function can be obtained from the other one by means of the Kronig-Kramers relations (10-12). For example, the substitution of G t) — Ge given by Eq. (6.8b) into Eq. (6.3) leads to the relationship... 

The beauty of the linear viscoelastic analysis lies in the fact that once a viscoelastic function is known, the rest of the functions can be determined. For example, if one measures the comphance function J t), the values of the components of the complex compliance function can in principle be determined from J(t) by using Fourier transforms [Eqs. (6.30)]. On the other hand, the components of the complex relaxation moduh can be obtained from those of / (co) by using Eq. (6.50). Even more, the real components of both the complex relaxation modulus and the complex compliance function can be determined from the respective imaginary components, and vice versa, by using the Kronig-Kramers relations. Moreover, the inverse of the Fourier transform of G (m) and/or G"(co) [/ (co) and/or /"(co)] allows the determination of the shear relaxation modulus (shear creep compliance). Finally, the convolution integrals of Eq. (5.57) allow the determination of J t) and G t) by an efficient method of numerical calculation outlined by Hopkins and Hamming (13). 

Anomalous ORD and CD both originate from light absorption by a chiral species and as such contain the same information. A mathematical equation, the Kronig-Kramers transform, relates one to the other over the wavelength range of the absorption, namely, [ (/I)] = -2/71 - X )dX. When the... 

B. A. Boukamp and J. R. Macdonald, "Alternatives to Kronig-Kramers Transformation and Testing, and Estimation of Distribtuions," Solid State Ionics, 74 (1994) 85-101. 

B. A. Boukamp, "A Linear Kronig-Kramers Transform Test for Immittance Data Validation," Journal of The Electrochemical Society, 142 (1995) 1885-1894. 

The Cotton effects can be either positive or negative. ORD and CD are related phenomena and data obtained by each method can be transformed to the other by means of the Kronig-Kramers relations [e.g., Ref. (18)]... 

One such link between semiempirical theory and experiment that appeared about that time was the development of calculational methods for optical rotatory dispersion. Moffitt s theoretical work with Kronig—Kramers transforms coupled with Djerassi s experimental data on steroids gave rise to rules for the prediction of the sign of optical rotation. Computer calculations with semiempirical methods played a role. i Wavefunctions of at least an approximate sort were needed for the dipole and dipole velocity matrix elements of the theory. 

The local absorption bands for the ot-helix were taken from the resolved data of Quadrifoglio and Urry, and the partial refractive indices were calculated using the Kronig-Kramers transforms. 

As a consequence of a CD the normal plain curve of the wavelength dependence of a, the "optical rotatory dispersion" (ORD) becomes anomalous and the plain curve is superposed by an S-shaped curve. Both, CD and anomalous ORD, are called "Cotton effect". Quantitatively CD and ORD are linked through the Kronig-Kramers relationship. No stringent coupling of the extinction coefficient (Emax) with the CD effect exists, because contains... 

Sucrose (DeTar, 1969), tris(ethylenediamine)cobalt triiodide hydrate (DeTar, 1969), (+ )-10-camphorsulfonic acid (DeTar, 1969 Cassim and Yang, 1969), and d-(-I-)-camphor have been used as preliminary standards for the calibration of CD instruments. In essence, the methods consist of calibrating the circular dichrometer against a calibrated spectropolarimeter using the Kronig-Kramers transform (Moscowitz, 1960 Krueger and Pschigoda, 1971) to compare ORD and CD curves. 

The use of (+)-10-camphorsulfonic acid as a calibration standard has been pioneered by instrument manufacturers. The quality of this commercially available standard varies (DeTar, 1969 Cassim and Yang, 1969), and recrystallization and proper storage are necessary. The calibration of circular dichrometers with (-l-)-l 0-camphorsulfonic acid using the Kronig-Kramers transform to compare ORD and CD curves was reexamined (Cassim and Yang, 1970), and the results were compared with the existing literature data. It was concluded that calibration of a CD instrument with this standard or any other standard requires that its purity be known. 

Impedance is by definition a complex quantity and is only real when 0=0 and thus Z(m) = Z(a>), that is, for purely resistive behavior. In this case the impedance is completely frequency-independent. When Z is found to be a variable function of frequency, the Kronig-Kramers (Hilbert integral transform) relations (Macdonald and Brachman [1956]), which holistically connect real and imaginary parts with each other, ensure that Z" (and 9) cannot be zero over all frequencies but must vary with frequency as well. Thus it is only when Z(linear resistance, that Z(m) is purely real. 

Boukamp BA. A linear Kronig-Kramers transform test for immittance data validation. J Electrochem Soc 1995 142(6) 1885-1894. 

It Is interesting to note that the QRD curve resembles the first derivative of the CD curve. ORD and CD are therefore coupled phenomena which in principle are mathematically commutable. It is possible to calculate CD spectrum of a given compound from its ORD spectrum by applying the mathematical relationship known as Kronig-Kramer transform. Interpretation... 

The set of expressions to which we refer are known as the Kronig-Kramers relationships, and we shall discuss these now . 

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