Application of the dual-kinetic-balance sets in the relativistic many-body problem of atomic structure

TL;DR

This paper extends the dual-kinetic-balance (DKB) basis set method to non-local potentials in atomic structure calculations, demonstrating improved accuracy over traditional methods for properties sensitive to small distances.

Contribution

The authors adapt and compare the DKB method with the Notre Dame approach using B-spline basis sets for the Dirac equation in atomic systems, showing enhanced performance for certain properties.

Findings

DKB outperforms Notre Dame method for properties near the nucleus.

Both methods have similar accuracy for properties far from the nucleus.

A basis set optimization strategy reduces computational effort while maintaining accuracy.

Abstract

The dual-kinetic-balance (DKB) finite basis set method for solving the Dirac equation for hydrogen-like ions [V. M. Shabaev et al., Phys. Rev. Lett. 93, 130405 (2004)] is extended to problems with a non-local spherically-symmetric Dirac-Hartree-Fock potential. We implement the DKB method using B-spline basis sets and compare its performance with the widely-employed approach of Notre Dame (ND) group [W.R. Johnson and J. Sapirstein, Phys. Rev. Lett. 57, 1126 (1986)]. We compare the performance of the ND and DKB methods by computing various properties of Cs atom: energies, hyperfine integrals, the parity-non-conserving amplitude of the $6s_{1/2}-7s_{1/2}$ transition, and the second-order many-body correction to the removal energy of the valence electrons. We find that for a comparable size of the basis set the accuracy of both methods is similar for matrix elements accumulated far from the nuclear region. However, for atomic properties determined by small distances, the DKB method outperforms the ND approach. In addition, we present a strategy for optimizing the size of the basis sets by choosing progressively smaller number of basis functions for increasingly higher partial waves. This strategy exploits suppression of contributions of high partial waves to typical many-body correlation corrections.