  Aiming at efficient numerical treatment of tunneling ionization of atoms and
molecules by mid-infrared (IR) lasers, exterior time scaling (ETS) theory is
formulated as a generalization of the time-scaled coordinate approach. The key
idea of ETS is the division of the spatial volume into a small region around
the nucleus and its outside; the radial coordinates are time scaled only in the
outer region. The continuum components of photoelectron wave packets are
prevented from reaching the edge of the spatial simulation volume, enabling the
long-time evolution of wave packets with a relatively small number of basis
functions without concerns of electron reflections. On the other hand, the
bound-state components are free from shrinking toward the origin because of
non-time scaling in the inner region. Hence, the equations of motion in ETS are
less stiff than the ones in the original time-scaled coordinate approach in
which the shrinking bound states make the equations of motion seriously stiff.
For numerical implementation of ETS, the working equations are derived in terms
of finite-element discrete-variable-representation functions. Furthermore, the
stiffness-free Lanczos time propagator is introduced to remove any persistent
stiffness in the treatment of mid-IR lasers due to the involvement of hundreds
of angular-momentum states. The test calculations for atomic hydrogen
interacting with linearly polarized mid-IR pulses demonstrate the accuracy and
numerical efficiency of the new scheme, and exhibit its special capability if
there is no recollision with the parent ion. Hence, ETS will show its true
potential for the detailed analysis of photoelectron wave packet dynamics in
circularly or near-circularly polarized mid-IR fields.
