On the construction of minimum information bivariate copula families

Copulas have become very popular as modelling tools in probability applications. Given a finite number of expectation constraints for functions defined on the unit square, the minimum information copula is that copula which has minimum information (Kullback–Leibler divergence) from the uniform copula. This can be considered the most “independent” copula satisfying the constraints. We demonstrate the existence and uniqueness of such copulas, rigorously establish the relation with discrete approximations, and prove an unexpected relationship between constraint expectation values and the copula density formula.     

Peri-Decoration of a Tetraazaperylene with Urea Units: Chiral Octaazaperopyrenedioxides (OAPPDOs) and Their Optical and Chiroptical Properties

Octaazaperopyrenedioxides (OAPPDOs) are a new class of fluorescent polycyclic aromatic hydrocarbons based on a tetraazaperylene core that is formally condensed with N-substituted urea units in the two opposite peri positions. Here, we report the synthesis of series of substituted OAPPDO derivatives with different N-substitution patterns (H, alkyl, benzyl) in the peri positions, including bay-chlorinated OAPPDOs. Starting from the latter, a series of bay-arylated OAPPDOs was synthesized by Suzuki cross coupling, which resulted in the formation of helically chiral OAPPDO derivatives. The electrochemical and photophysical properties were investigated by UV/Vis and fluorescence spectroscopy as well as cyclic voltammetry. The P and M enantiomers of a phenylated OAPPDO were separated by semipreparative HPLC and further analyzed by CD spectroscopy. The frontier orbital energies, the mechanism of the isomerization, the electronic excitation and the CD spectrum (TD-DFT) were computed and compared to the experimental data. The reversible 1e⁻ oxidation of the OAPPDOs generates the corresponding radical cations, one of which was characterized by EPR spectroscopy. The reversible oxidation process was also systematically investigated by spectro-electrochemistry.