Long-range asymptotic behaviour of the exchange energy in the hydrogen molecule

The exchange energy is of fundamental importance for the understanding of interatomic interactions. Its long-range behavior in H$_2$ was investigated in the early days of quantum mechanics\footnote{W. Heitler and F. London, Z. Physik \textbf{44}, 455 (1927)}. Those results were met with criticism\footnote{C. Herring, Rev. Mod. Phys. \textbf{34}, 631 (1962)} and asymptotically correct form was derived by Herring and Flicker\footnote{C. Herring and M. Flicker, Phys. Rev. \textbf{134}, A362 (1964)}. Despite further work by many authors\footnote{K. T. Tang, J. Peter Toennies, and C. L. Yiu, J. Chem. Phys. \textbf{99}, 377 (1993)}\textsuperscript{,}\footnote{B. L. Burrows, A.~Dalgarno, and M. Cohen, Phys. Rev. A \textbf{86}, 052525 (2012)}, long-range asymptotics of exchange energy in H$_2$ still raises controversy. \\ We present high-precision variational calculations of energy splitting between the lowest hydrogen states performed with our H2SOLV\footnote{K. Pachucki, M. Zientkiewicz, V.A. Yerokhin, Comput. Phys. Commun. \textbf{208}, 162 (2016)} package, which utilizes explicitly correlated exponential basis\footnote{K. Pachucki, Phys. Rev. A \textbf{88}, 022507 (2013)}. Due to correct asymptotic behavior of our basis, we claim that our numerical results resolve this controversy.