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The exchange energy, i.e. the splitting $\Delta E$ between gerade andungerade states in the hydrogen molecule has proven very difficult in numericalcalculation at large internuclear distances $R$, while known results are sparseand highly inaccurate. On the other hand, there are conflicting analyticalresults in the literature concerning its asymptotics. In this work we develop aflexible and efficient numerical approach using explicitly correlatedexponential functions and demonstrate highly accurate exchange energies forinternuclear distances as large as 57.5 au. This approach may find furtherapplications in calculations of inter-atomic interactions. In particular, ourresults support the asymptotics form $\Delta E \sim R^{5/2}e^{-2R}$, but withthe leading coefficient being $2\,\sigma$ away from the analytically derivedvalue.

Long-range asymptotics of exchange energy in the hydrogen molecule