A Stieltjes imaging method for molecular shake-off continua. Application to the core photoelectron spectrum of water

A Stieltjes imaging method for molecular shake-off continua is devised and numerically applied to the core photoelectron spectrum of water. Assuming the independent particle approximation, the full core photoelectron spectrum corresponding to an energy range of 0-100 eV, including main peak, shake-up peaks, and shake-off continua, is constructed in one ansatz. The shake-off cross sections are obtained in both length and velocity gauges and the effect of discrete and continuum interaction between channels with different spin coupling is investigated. The variation of the dipole transition moment vs excitation energy for the primary photoelectron is explicitly calculated, which gives the corrections to the sudden approximation also in the high-energy limit of x-ray excitation. By comparison, the plane wave approximation for the primary photoelectron is found appropriate only for relative cross sections in the shake spectrum, while for the absolute photoionization cross section it significantly deviates from results obtained from the variationally optimized photoelectron functions. The form of the variationally optimized continuum waves close to the oxygen nucleus is compared with the corresponding plane waves and Coulomb waves of different effective charges.