Development of the kinetically and atomically balanced generalized pseudospectral method

The kinetically and atomically balanced conditions have been extensively used in various basis-set expansion methods to remove the nonphysical spurious states appearing in the numerical calculation of the Dirac equation; however, they are generally not applicable for methods in discrete variable representation. In this paper we show that these conditions can be conveniently introduced into the generalized pseudospectral (GPS) method. Four types of balanced condition, the mono kinetic balance (MKB), dual kinetic balance, mono atomic balance (MAB), and dual atomic balance, are incorporated into the GPS method to eliminate the spurious states. Numerical calculations for a variety of bound states of H-like ions in point-charge models are compared with the analytical solutions to demonstrate the accuracy and efficiency of the developed methods. The application to highly charged ions with extended nuclear models is performed to show the flexibility of the balanced GPS methods in practical atomic structure calculations. It is concluded that the MAB-GPS and MKB-GPS methods, which are both free of any spurious states, show better performance and simpler implementation than the others in solving the Dirac equation with the potential in point-charge and extended nuclear models, respectively. The balanced GPS methods developed in this paper provide a useful tool for accurately solving the one-electron Dirac equation and efficiently constructing the multielectron relativistic wave functions.