Complex rotation in two-dimensional mesh calculations for quantum systems in uniform electric fields

A computational method for calculations of quasi-steady states of quantum systems is developed on the basis of a complex coordinate transformation and multi-dimensional mesh solution of Schrödinger equations. The two-dimensional implementation of the method is applied to the hydrogen atom, a hydrogen-like system with a screened Coulomb interaction potential and the hydrogen molecular ion H₂⁺ in external electric and parallel electric and magnetic fields. Detailed calculations are carried out for the ground state of the hydrogen atom in parallel electric and magnetic fields. All the systems are considered in cylindrical coordinates (ρ,z). The complex coordinate transformation is applied to the coordinate z, both in the form of a classical complex rotation (atom-like systems) and in the form of a smooth exterior complex scaling. The boundaries of applicability of these methods are studied. In comparison with the boundary-condition method, developed previously, the complex coordinate transformation approach allows one to obtain more precise and stable results for the strong electric field regimes with |ImE|≈|ReE|.