Enantiosensitive Structure Determination by Photoelectron Scattering on Single Molecules

X-ray as well as electron diffraction are powerful tools for structure determination of molecules. Electron diffraction methods yield \r{A}ngstrom-resolution even when applied to large systems or systems involving weak scatterers such as hydrogen atoms. For cases in which molecular crystals cannot be obtained or the interaction-free molecular structure is to be addressed, corresponding electron scattering approaches on gas-phase molecules exist. Such studies on randomly oriented molecules, however, can only provide information on interatomic distances, which is challenging to analyse in case of overlapping distance parameters and they do not reveal the handedness of chiral systems8. Here, we present a novel scheme to obtain information on the structure, handedness and even detailed geometrical features of single molecules in the gas phase. Using a loop-like analysis scheme employing input from ab initio computations on the photoionization process, we are able to deduce the three dimensional molecular structure with sensitivity to the position individual atoms, as e.g. protons. To achieve this, we measure the molecular frame diffraction pattern of core-shell photoelectrons in combination with only two ionic fragments from a molecular Coulomb explosion. Our approach is expected to be suitable for larger molecules, as well, since typical size limitations regarding the structure determination by pure Coulomb explosion imaging are overcome by measuring in addition the photoelectron in coincidence with the ions. As the photoelectron interference pattern captures the molecular structure at the instant of ionization, we anticipate our approach to allow for tracking changes in the molecular structure on a femtosecond time scale by applying a pump-probe scheme in the future

Edge Detection by Adaptive Splitting II. The Three-Dimensional Case

In Llanas and Lantarón, J. Sci. Comput. 46, 485–518 (2011) we proposed an algorithm (EDAS-d) to approximate the jump discontinuity set of functions defined on subsets of ℝ d . This procedure is based on adaptive splitting of the domain of the function guided by the value of an average integral. The above study was limited to the 1D and 2D versions of the algorithm. In this paper we address the three-dimensional problem. We prove an integral inequality (in the case d=3) which constitutes the basis of EDAS-3. We have performed detailed computational experiments demonstrating effective edge detection in 3D function models with different interface topologies. EDAS-1 and EDAS-2 appealing properties are extensible to the 3D cas