Optimization of the imaginary time step evolution for the Dirac equation

Abstract

Taking the single neutron levels of 12C in the Fermi sea as examples, the optimization of the imaginary time step (ITS) evolution with the box size and mesh size for the Dirac equation is investigated. For the weakly bound states, in order to reproduce the exact single-particle energies and wave functions, a relatively large box size is required. As long as the exact results can be reproduced, the ITS evolution with a smaller box size converges faster, while for both the weakly and deeply bound states, the ITS evolutions are less sensitive to the mesh size. Moreover, one can find a parabola relationship between the mesh size and the corresponding critical time step, i.e., the largest time step to guarantee the convergence, which suggests that the ITS evolution with a larger mesh size allows larger critical time step, and thus can converge faster to the exact result. These conclusions are very helpful for optimizing the evolution procedure in the future self-consistent calculations.