High order cross derivative computation for the differential cross section of double ionization of helium by electron impact

Overview & Synopsis

The double ionization of an atom or a molecule is strongly dependent of the quality of the description of the initial state of the target. Recently, absolute measurements have been reported for the double ionization of helium by 5.6 keV electron-impact. Since the incident (and scattered) electron is very fast, one may apply the usual first Born approximation. Calculations with the first Born approximation lead to an overall magnitude that is about 50% larger than experiment when a simple (one term) Hylleraas wavefunction describes the initial state. Two numerical approaches are available to tackle an accurate (18 terms) Hylleraas wavefunction: a 6-dimensional numerical quadrature (expensive in computer time), or a 2-dimensional quadrature applied to high order cross derivatives (up to the order 9). Automatic differentiation tecnhiques allow for high order derivative computations. Nevertheless existing differentiation tools do not deal with codes written in complex arithmetics and explicit cross derivative computations. This paper first describes the high order differentiation (based on recursive rules) and the extraction of cross derivatives. An operator oveloading library is constructed for the differentiation work. Numerical results, obtained at a lower cost than the sextuple integral, show the pertinence of our approach.