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A Galerkin method is developed to solve the time-dependent Dirac equation inprolate spheroidal coordinates for an electron-molecular two-center system. Theinitial state is evaluated from a variational principle using a kinetic/atomicbalanced basis, which allows for an efficient and accurate determination of theDirac spectrum and eigenfunctions. B-spline basis functions are used to obtainhigh accuracy. This numerical method is used to compute the energy spectrum ofthe two-center problem and then the evolution of eigenstate wavefunctions in anexternal electromagnetic field.

Galerkin method for unsplit 3-D Dirac equation using atomically/kinetically balanced B-spline basis