Finite basis set approach to the two-center Dirac problem

Owing to the recent experimental advances in ion accelerator and storage ring techniques, more possibilities arise to study formation of quasi-molecules in (relatively) slow collisions of highly-charged, heavy ions. Extremely strong electromagnetic fields produced in these collisions are expected to cause a ''decay'' of unstable physical vacuum and a spontaneous creation of electron-positron pairs. Theoretical understanding of such an overcritical-field phenomenon requires, in general, solution of the two-center time-dependent Dirac equation. For low velocities of colliding ions this equation may still be treated adiabatically and, hence, can be traced back to the static (two-center) problem. In our work we developed an efficient method for dealing with this problem by utilizing finite basis sets constructed from B-splines. We argue that B-spline analysis can be performed most naturally in Cassini coordinates that are very efficient for the description of two-center Coulomb potential. To underline the advantages of the present approach, detailed calculations will be presented for quasi-molecular energy spectra obtained for slow symmetric (Z1=Z2) as well as asymmetric (Z1>Z2) ion-ion collisions.