6 research outputs found
Economical, green, and safe route towards substituted lactones by anodic generation of oxycarbonyl radicals
A new electrochemical methodology has been developed for the generation of oxycarbonyl radicals under mild and green conditions from readily available hemioxalate salts. Monoâ and multiâfunctionalised Îłâbutyrolactones were synthesised through exoâcyclisation of these oxycarbonyl radicals with an alkene, followed by the sp3âsp3 capture of the newly formed carbonâcentred radical. The synthesis of functionalised valerolactone derivatives was also achieved, demonstrating the versatility of the newly developed methodology. This represents a viable synthetic route towards pharmaceutically important fragments and further demonstrates the practicality of electrosynthesis as a green and economical method to activate small organic molecules
Improved semiclassical density matrix: taming caustics
We present a simple method to deal with caustics in the semiclassical
approximation to the thermal density matrix of a particle moving on the line.
For simplicity, only its diagonal elements are considered. The only ingredient
we require is the knowledge of the extrema of the Euclidean action. The
procedure makes use of complex trajectories, and is applied to the quartic
double-well potential.Comment: 20 pages, 7 figures. Revised version, accepted for publication in
Phys. Rev.
Semiclassical time evolution of the density matrix and tunneling
The time dependent density matrix of a system with potential barrier is
studied using path integrals. The characterization of the initial state, which
is assumed to be restricted to one side of the barrier, and the time evolution
of the density matrix lead to a three-fold path integral which is evaluated in
the semiclassical limit. The semiclassical trajectories are found to move in
the complex coordinate plane and barrier penetration only arises due to
fluctuations. Both the form of the semiclassical paths and the relevant
fluctuations change significantly as a function of temperature. The
semiclassical analysis leads to a detailed picture of barrier penetration in
the real time domain and the changeover from thermal activation to quantum
tunneling. Deep tunneling is associated with quasi-zero modes in the
fluctuation spectrum about the semiclassical orbits in the long time limit. The
connection between this real time description of tunneling and the standard
imaginary time instanton approach is established. Specific results are given
for a double well potential and an Eckart barrier.Comment: 27 pages, 8 figures, to be published in Phys. Rev.