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

We present a precise fully relativistic numerical solution of the two-center Coulomb problem. The special case unit nuclear charges is relevant for accurate description ${\rm H}_2^+$ molecular ion and its isotopologues, systems that are an active experimental topic. computation utilizes 2-spinor minmax approach finite-element method. computed total energies have estimated fractional uncertainties few times $10^{-20}$ bond length 2 atomic units. uncertainty purely contribution $1\times10^{-17}$. result future precision experiments, whereas at arising from quantum electrodynamic treatment rovibrational transition frequencies. dominant.