This paper provides a numerical analysis of a procedure for determining the effects of ionization on the nonlinear susceptibility coefficients of the hydrogen atom. To solve the relevant system of Schrödinger-type equations, we have developed a multidomain pseudospectral code with high accuracy symmetric finite differences to update cell boundary points. Using a conservative time stepping, one is then able to resolve the oscillatory solutions to the underlying equations, compute the ionization probability, and accurately determine the polarization. To gain insight into the physical mechanisms involved, we have calculated the full susceptibility and one which depends solely on the electronic bound-bound transitions. Our analysis reveals that saturation of the susceptibility occurs without including bound-continuum transitions. We have also found that a linear extrapolation of the total susceptibility versus the ionization probability to obtain the instantaneous susceptibility is quantitatively unreliable.
"Numerical analysis of the ab initio computation of the effects of ionization on the nonlinear susceptibility coefficients of the hydrogen atom." Commun. Math. Sci. 4 (1) 53 - 80, March 2006.