Accurate solution of the Dirac equation on Lagrange meshes

Area of Science:

• Computational Physics
• Quantum Chemistry
• Theoretical Atomic Physics

Background:

• The Lagrange-mesh method is an approximate variational technique.
• It utilizes Gauss quadrature, forming grid-based equations.
• Associated Laguerre polynomials form the basis for Lagrange functions.

Purpose of the Study:

• To apply the Lagrange-mesh method to the Dirac equation.
• To investigate its accuracy for potentials with 1/r singularities.
• To determine the efficiency in obtaining precise atomic properties.

Main Methods:

• Employing a basis of Lagrange functions with associated Laguerre polynomials.
• Applying Gauss quadrature approximation for grid-based calculations.
• Solving the Dirac equation for various atomic potentials.

Main Results:

• Numerically exact energies and wave functions for hydrogenic atoms with few mesh points (n+1).
• Accurate calculation of mean values of radial coordinate powers (-2 to 3) with n+2 mesh points.
• 15-digit agreement with benchmark energies for the Yukawa potential using ≤50 mesh points.

Conclusions:

• The Lagrange-mesh method provides a highly accurate and efficient approach for solving the Dirac equation.
• It is effective for atomic systems, including those with singular potentials.
• The method offers a robust tool for high-precision quantum mechanical calculations.