Numerical solution of the time-independent Dirac equation for diatomic molecules: B-splines without spurious states

TL;DR

This paper compares two B-spline based numerical methods for solving the relativistic Dirac equation in diatomic molecules, demonstrating accurate results without spurious states in the spectrum.

Contribution

It introduces and validates two B-spline based numerical approaches for the Dirac equation, avoiding spurious states in diatomic molecule calculations.

Findings

Both methods yield accurate relativistic spectra.

No spurious states are observed in the discretization.

Methods are applicable to H2+ and Th2+ molecules.

Abstract

Two numerical methods are used to evaluate the relativistic spectrum of the two-centre Coulomb problem (for the $H_{2}^{+}$ and $Th_{2}^{179+}$ diatomic molecules) in the fixed nuclei approximation by solving the single particle time-independent Dirac equation. The first one is based on a min-max principle and uses a two-spinor formulation as a starting point. The second one is the Rayleigh-Ritz variational method combined with kinematically balanced basis functions. Both methods use a B-spline basis function expansion. We show that accurate results can be obtained with both methods and that no spurious states appear in the discretization process.