Accurate solution of the Dirac equation on Lagrange meshes

Abstract

The Lagrange-mesh method is an approximate variational method taking the form of equations on a grid because of the use of a Gauss quadrature approximation. With a basis of Lagrange functions involving associated Laguerre polynomials related to the Gauss quadrature, the method is applied to the Dirac equation. The potential may possess a $1/r$ singularity. For hydrogenic atoms, numerically exact energies and wave functions are obtained with small numbers $n+1$ of mesh points, where $n$ is the principal quantum number. Numerically exact mean values of powers $-2$ to 3 of the radial coordinate $r$ can also be obtained with $n+2$ mesh points. For the Yukawa potential, a 15-digit agreement with benchmark energies of the literature is obtained with 50 mesh points or less.

References in corpus (8)

• CODATA Recommended Values of the Fundamental Physical Constants: 2010
• Dual kinetic balance approach to basis set expansions for the Dirac equation
• A B-spline Galerkin method for the Dirac equation
• Application of the dual-kinetic-balance sets in the relativistic many-body problem of atomic structure
• Computational investigation of static multipole polarizabilities and sum rules for ground-state hydrogen-like ions
• Lagrange mesh, relativistic flux tube, and rotating string
• Calculation of the two-photon decay rates of hydrogen-like ions by using B-polynomials
• Bound state equivalent potentials with the Lagrange mesh method

Cited by in corpus (5)

• Relativistic calculations of quasi-one-electron atoms and ions using Laguerre and Slater spinors
• Relativistic polarizabilities with the Lagrange-mesh method
• Relativistic two-photon decay rates with the Lagrange-mesh method
• Towards the Theory of the Yukawa Potential
• Relativistic semiempirical-core-potential calculations in Ca${}^{+}$, Sr${}^{+}$, and Ba${}^{+}$ ions on Lagrange meshes