Limitations of variational methods for hydrogenic systems

Theoretical methods that give good approximations of the behaviours of hydrogen-like impurity states in semiconductors in uniform external magnetic fields are by now well established. With variational methods in mind the sizes of the determinants that give sufficiently accurate energies greatly depend on the quality of the wave functions chosen as these affect the rapidity of the series convergence. The recently developed hybrid functions enable hydrogenic energies to be calculated up ton=7 using fairly small determinants. However, it has emerged that there is a restricted region of variational parameters which maintains the numerical symmetry of the hamiltonian matrix elements for each group of states. Although each such parameter regime appears to narrow as the magnetic quantum number increases, the accuracies of the calculated eigenvalues remain satisfactory.     

Determination of the absolute configuration of vibsanin F by asymmetric synthesis via π-allylpalladium complex

The absolute configuration of vibsanin F (1), one of the simplest 11-membered ring vibsane-type diterpene, was established to be 6R, 7S and 11S by asymmetric synthesis. A diastereomer (1a) with a 6S, 7S and 11S configuration was synthesized starting from myrcene by procedures featuring a Sharpless asymmetric epoxidation and a high diastereoselective eleven-membered ring formation between a π-allylpalladium complex and a β-ketoester nucleophile. Subsequent reduction of epoxide in 1a with LiAlH₄ gave a diol, which was identical with that derived from vibsanin F.     

Ground state of aD − ion in semiconductors

The mechanisms that feature in the formation ofD ^– ions in spherical band semiconductors are discussed. Analogies are drawn with H^– formation (a proton binding two electrons). The variational method is applied in calculating the binding energy of the extra electron in the absence of external magnetic field. The binding energies obtained with different trial functions are compared with that extrapolated from experimental measurements. From such a comparison it is clear that a simple trial function with only one variational parameter involving the screening of the positive charge of the nucleus gives good results.