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Additionally, the present formulation is very flexible and general, and it allows for use of an arbitrary correlation factor in the electronic structure calculations with minor or no changes.

The exchange energy, i.e. the splitting between gerade and ungerade states in the hydrogen molecule has proven very difficult in numerical calculation at large internuclear distances, while known results are sparse and highly inaccurate. On the other hand, there are conflicting analytical results in the literature concerning its asymptotics. In this work we develop a flexible and efficient numerical approach using explicitly correlated exponential functions and demonstrate highly accurate exchange energies for internuclear distances as large as 57.5 au. This approach may find further applications in calculations of inter-atomic interactions. In particular, our results support the asymptotics form ∆E(R) ∼ R 5/2 e −2R , but with the leading coefficient being 2 σ away from the analytically derived value.

TABLE S1. Calculated BO energies, expectation value of potential, ⟨∇1 • ∇2⟩, and dE/dR of the 1 1 Πg state in atomic units (hartree). Uncertainties originate purely from extrapolation to the complete basis set limit. United atom values at R = 0 are taken from Ref. [? ].TABLE S2. Calculated BO energies, expectation value of potential, ⟨∇1 • ∇2⟩, and dE/dR of the 2 1 Πg state in atomic units (hartree). Uncertainties originate purely from extrapolation to the complete basis set limit. United atom values at R = 0 are taken from Ref. [? ]. 2 1 Π g R E ⟨V ⟩ 10 6 • ⟨∇1 • ∇2⟩ dE/dR 0.0 −2.031279846178687(7) − 129.1751887(8) − TABLE S3. Calculated BO energies, expectation value of potential, ⟨∇1 • ∇2⟩, and dE/dR of the 3 1 Πg state in atomic units (hartree). Uncertainties originate purely from extrapolation to the complete basis set limit. United atom values at R = 0 are taken from Ref. [? ].5.1 −0.615570031381760(6) −1.265759260899(3) −65636.86120(2) −0.0067880780657(5) 5.2 −0.61623904916449(3) −1.266714068969(4) −66903.26601(2) −0.0065838405077(5) 5.3 −0.616883401355604(9) −1.267158926281(4) −67962.20188(2) −0.0063004006735(6) 5.4 −0.617496744203276(9) −1.267219536365(4) −68856.97913(2) −0.0059677866589(6) 5.5 −0.6180853196896611(4) −1.266994774604(4) −69621.493243(10) −0.0056080245862(6) 5.6 −0.61861725175936(8) −1.2665617579874(8) −70282.0815747(3) −0.0052370097265(2) 5.7 −0.61912201191664(8) −1.2659803415671(8) −70859.1559792(3) −0.0048660206551(2) 5.8 −0.61959001904326(8) −1.2652968411745(8) −71368.5422729(2) −0.0045028970841(2) 5.9 −0.62002234241467(9) −1.2645470073349(8) −71822.5397140(2) −0.00415293601788(10) 6.0 −0.6204304827255747(5) −1.263758359(3) −72230.89(8) −0.0038162323(5) 6.5 −0.62197421146385(3) −1.2598165566(10) −73799.2527(5) −0.0024412514(2) 7.0 −0.62294868122491(2) −1.2565668374(5) −74882.642(2) −0.00152421072(8) 7.5 −0.62355517660007(2) −1.2542213627(3) −75675.0084(5) −0.00094813461(5) 8.0 −0.623934198222452(9) −1.2526481546(2) −76263.5451(3) −0.00059746978(3) 8.5 −0.624175920351951(6) −1.251644951966(5) −76700.1251(3) −0.0003874248547(9) 9.0 −0.624335617890345(5) −1.251029658832(4) −77022.11251(2) −0.0002620470059(7) 9.5 −0.624446191820649(10) −1.250663874871(7) −77258.445184(9) −0.0001885780243(9) 10.0 −0.624526858556839(2) −1.250451608377(3) −77431.645411(5) −0.0001397891264(4) 11.0 −0.6246382508422657(3) −1.25026097210325(2) −77653.0312283(2) −0.000089497310795(3) 12.0 −0.62471380189591186(10) −1.25019536682094(9) −77776.64663864(4) −0.000063980252425(5) 13.0 −0.624769358751530723(10) −1.25016520828807(10) −77849.6363596(2) −0.000048191598845(6) 14.0 −0.624811753081569110(6) −1.25014420717773(3) −77895.65769984(6) −0.000037192929615(3) 15.0 −0.624844707349533346(4) −1.250126062678238(3) −77926.55327888(9) −0.0000291098652784(5) 16.0 −0.624870637584202693(2) −1.250109782346086(2) −77948.41483907(9) −0.0000230316986052(3) 17.0 −0.6248912526172034632(10) −1.250095383081320(2) −77964.53144884(7) −0.0000184045792303(3) 18.0 −0.624907802583200445(2) −1.250082877226923(8) −77976.7884072(6) −0.00001484...

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