The Dirac equation for ${\mathrm{H}}_2^{+}$ is solved numerically using an iterative method proposed by Kutzelnigg [Z. Phys. D 11, 15 (1989)]. The four-component wave function is expanded in a newly introduced kinetically balanced exponential basis set. The ground-state relativistic energy is obtained with an accuracy of ${10}^{-20}$, which represents an improvement by several orders of magnitude, and is shown to be in good agreement with results obtained from perturbation theory. Highly accurate relativistic wave functions are obtained, which is a first step towards nonperturbative calculations of the one-loop self-energy correction in hydrogen molecular ions.

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• Atomic & molecular structure
• Relativistic & quantum electrodynamic effects in atoms, molecules,& ions