Accurate solution of the Dirac equation on Lagrange meshes.
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 or fewer mesh points.
Physique Quantique, C. P. 165/82, and Physique Nucléaire Théorique et Physique Mathématique, C. P. 229, Université Libre de Bruxelles (ULB), B-1050 Brussels, Belgium.
Daniel
D
Baye
Chimie Quantique et Photophysique, C. P. 160/09, Université Libre de Bruxelles (ULB), B-1050 Brussels, Belgium.
Livio
L
Filippin
Chimie Quantique et Photophysique, C. P. 160/09, Université Libre de Bruxelles (ULB), B-1050 Brussels, Belgium.
Michel
M
Godefroid