Chemo-, regio-, and enantioselective Pd-catalyzed allylic alkylation of indolocarbazole pro-aglycons

[reaction: see text] Monosubstituted isomerically pure indolopyrrolocarbazole precursors have been prepared via palladium-catalyzed asymmetric allylic alkylation methodology, employing both achiral cyclopentenyl electrophiles and chiral glycal derivatives. Chemoselective allylation of (bis)indole lactam pro-aglycon 3 allows access to N-distally substituted indolopyrrolocarbazole derivatives; glyoxamide precursor 14 provides entry into N-proximally substituted derivatives.     

Mechanisms of tetrazole formation by addition of azide to nitriles

It is well-known that azide salts can engage nitriles at elevated temperatures to yield tetrazoles; however, there is continued debate as to the mechanism of the reaction. Density functional theory calculations with the hybrid functional B3LYP have been performed to study different mechanisms of tetrazole formation, including concerted cycloaddition and stepwise addition of neutral or anionic azide species. The calculations presented here suggest a previously unsuspected nitrile activation step en route to an imidoyl azide, which then cyclizes to give the tetrazole. The activation barriers are found to correlate strongly with the electron-withdrawing potential of the substituent on the nitrile.     

Enolstannanes as electrofugal groups in allylic alkylation

Enolstannanes serve as nucleophiles towards allylic acetates under the influence of palladium(O) catalyst.     

Ergodic semigroups of positivity preserving self-adjoint operators

We prove that an ergodic semigroup of positivity preserving self-adjoint operators is positivity improving. We also present a new proof (using Markov techniques) of the ergodicity of semigroups generated by spatially cutoff P(?)₂ Hamiltonians.     

Geometric methods in multiparticle quantum systems

Technically simple proofs are given of the HVZ theorem on the bottom of the essential spectrum of multiparticle systems and of Combes' result on completeness below the lowest three body threshold. The first proof is a variant of a proof of Enss and a decendent of Zhislin's original proof. Finally, we apply our methods to the bound state spectrum.On leave from Princeton University. Research support in part by U.S. National Science Foundation under grants MPS-75-11864 and MPS-75-20638     

Aspects of supersymmetric quantum mechanics

We review the properties of supersymmetric quantum mechanics for a class of models proposed by Witten. Using both Hamiltonian and path integral formulations, we give general conditions for which supersymmetry is broken (unbroken) by quantum fluctuations. The spectrum of states is discussed, and a virial theorem is derived for the energy. We also show that the euclidean path integral for supersymmetric quantum mechanics is equivalent to a classical stochastic process when the supersymmetry is unbroken (E₀ = 0). By solving a Fokker-Planck equation for the classical probability distribution, we find P_c(y) is identical to |Ψ₀(y)|² in the quantum theory.