Solution of the two-center Dirac equation with 20 digits precision using the finite-element technique

We present a precise fully relativistic numerical solution of the two-center Coulomb problem. The special case of unit nuclear charges is relevant for the accurate description of the ${\mathrm{H}}_2^{+}$ molecular ion and its isotopologues, systems that are an active experimental topic. The computation utilizes the 2-spinor minmax approach and the finite-element method. The computed total energies have estimated fractional uncertainties of a few times ${10}^{-20}$ for unit charges and a bond length of 2 atomic units. The fractional uncertainty of the purely relativistic contribution is $1\times {10}^{-17}$. The result is relevant for future precision experiments, whereas at present the uncertainties arising from the quantum electrodynamic treatment of the rovibrational transition frequencies. are dominant.