Basis-set methods for the Dirac equation

Abstract

The pathologies associated with finite basis-set approximations to the Dirac Hamiltonian H Dirac are avoided by applying the variational principle to the bounded operator 1/ (H Dirac − W ) where W is a real number that is not in the spectrum of H Dirac . Methods of calculating upper and lower bounds to eigenvalues, and bounds to the wave-function error as measured by the L 2 norm, are described. Convergence is proven. The rate of convergence is analyzed. Boundary conditions are discussed. Benchmark energies and expectation values for the Yukawa potential, and for the Coulomb plus Yukawa potential, are tabulated. The convergence behavior of the energy-weighted dipole sum rules, which have traditionally been used to assess the quality of basis sets, and the convergence behavior of the solutions to the inhomogeneous problem, are analyzed analytically and explored numerically. It is shown that a basis set that exhibits rapid convergence when used to evaluate energy-weighted dipole sum rules can nevertheless exhibit slow convergence when used to solve the inhomogeneous problem and calculate a polarizability. A numerically stable method for constructing projection operators, and projections of the Hamiltonian, onto positive and negative energy states is given. PACS Nos.: 31.15Pf, 31.30Jv, 31.15-p Résumé : Nous évitons les pathologies associées aux solutions approximatives de l'équation de Dirac qui utilisent une base finie, en appliquant le principe variationnel à l'opérateur 1/ (H Dirac − W ), où W est un nombre réel n'apparaissant pas dans le spectre de H Dirac . Nous décrivons ici comment calculer des limites inférieures et supérieures des valeurs propres et comment évaluer les limites d'erreur sur la fonction d'onde dans la norme L 2 . Nous démontrons la convergence et étudions le taux de convergence et les conditions limites. Nous présentons sous forme de table les énergies test et leurs valeurs moyennes pour le potentiel de Yukawa seul et avec le potentiel de Coulomb. Nous analysons analytiquement et sondons numériquement la convergence des règles de somme du dipôle avec poids en énergie, qui sont traditionnellement utilisées pour vérifier la qualité d'une base, ainsi que la convergence des solutions du problème inhomogène. Nous montrons qu'une base qui fait converger rapidement la règle de somme du dipôle peut ne faire converger que lentement le problème inhomogène. Nous proposons une méthode numériquement fiable pour construire les opérations de projection et les projections du Hamiltonien sur les états d'énergie positive et négative. [Traduit par la Rédaction]

Numerical solution of the Dirac equation by a mapped Fourier grid method

“…While the relative error for the sum rules given in equations (18), (19) and (21) only approaches about 10 −7 , this represents the achievable accuracy for those computations, since it is at their level of numerical noise. This is evident from the random pattern of points for s 2000.…”

“…Matrix representations of the Dirac equation are known to be plagued by the appearance of spurious roots for κ 1 [18,19]. This is related to the symmetric treatment of the large and small components, the dependence of the structure of the matrix representation on |κ|, while the true spectrum of bound states in the Dirac equation is different for the case of κ > 0 and κ < 0.…”

“…The radial integral is approximated by a finite sum terminating at r max which depends on N and s. Thus, the inner product in Hilbert space is replaced in the matrix representation by finite sums with features stemming from the finiteness of the representation. In order to test for orthogonality the relative angles (18) and (19), the dashed line for equation (20), the circle for equation (17) and the crosses for the energy eigenvalue (with the rest mass subtracted).…”

Accurate solution of the Dirac equation on Lagrange meshes

“…Nevertheless, thanks to condition (19), these functions are orthonormal at the Gaussquadrature approximation denoted with the subscript G,…”

“…If non regularized Lagrange functions were used, the optimal choice would be α ′ = 2γ like the one adopted in Refs. [1,19] for the B-spline expansions.…”