Renormalization of the inverse square potential

The quantum-mechanical D-dimensional inverse square potential is analyzed using field-theoretic renormalization techniques. A solution is presented for both the bound-state and scattering sectors of the theory using cutoff and dimensional regularization. In the renormalized version of the theory, there is a strong-coupling regime where quantum-mechanical breaking of scale symmetry takes place through dimensional transmutation, with the creation of a single bound state and of an energy-dependent s-wave scattering matrix element.     

Sodium cation complexation in a macrocycle containing thymines as sidearm donor groups

Previous studies of macrocycles having alkyl sidearms terminated in adenine or thymine have been focused on intermolecular association. Electrospray mass spectrometric analyses suggested that sodium complexation could be a major alternative process in the presence of the appropriate cation. Sodium cation complexation by N,N-bis(3-(1-thyminyl)propyl)-4,13-diaza-18-crown-6 has been confirmed by X-ray crystallographic analysis. A sodium cation is bound in the center of the macroring. The four ether oxygens and two thyminyl carbonyl donors have the shortest contacts with the two, macroring nitrogen atoms being somewhat more remote. Neither the iodide counterion nor an acetone of solvation appears to interact with the sodium cation. Crystal Data: Triclinic, , a = 11.9371(12), b = 12.1588(13), c = 13.7885(14) Å, = 78.083(2)°, = 83.213(2)°, = 78.997(2)°, and D_calc = 1.392 g cm^–3, Z = 2.