Ideal basis sets for the Dirac Coulomb problem: Eigenvalue bounds and convergence proofs

Abstract

Basis sets are developed for the Dirac Coulomb Hamiltonian for which the resulting numerical eigenvalues and eigenfunctions are proved mathematically to have all the following properties: to converge to the exact eigenfunctions and eigenvalues, with necessary and sufficient conditions for convergence being known; to have neither missing nor spurious states; to maintain the Coulomb symmetries between eigenvalues and eigenfunctions of the opposite sign of the Dirac quantum number κ; to have positive eigenvalues bounded from below by the corresponding exact eigenvalues; and to have negative eigenvalues bounded from above by −mc2. All these properties are maintained using functions that may be analytic or nonanalytic (e.g., Slater functions or splines); that match the noninteger power dependence of the exact eigenfunctions at the origin, or that do not; or that extend to +∞ as do the exact eigenfunctions, or that vanish outside a cavity of large radius R (convergence then occurring after a second limit, R→∞)....