Kramer-Kronig transformation

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. 

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... 

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. 

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. 

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. 

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... 

A Kronig—Kramers transform of the two CD bands observed in cyclohexane accounts for the observed positive ORD spectrum. In contrast, a third large and negative ORD band centered at 155.5 nm is responsible for the negative ORD spectrum observed in HFIP. In the latter solution as well as in benzene, the ORD spectrum was found to fit the Drude one term equation with Xq=150 nm. 

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