In hydrogen molecule, two electrons are shared between two nuclei, and we optimize the positions of the nuclei and electrons, as well as the size of the electrons, simultaneously.

h2.cfg: @params calc = minimize @nuclei 0 0 0 1 1.4 0 0 1 @electrons 0 0 0 1 2.0 1.4 0 0 -1 2.0 h2.cfg.restart: ... @nuclei -0.037024 0.000000 0.000000 1.000000 1.437024 0.000000 0.000000 1.000000 @electrons 0.699998 0.000000 0.000000 1 1.771522 0.700002 0.000000 0.000000 -1 1.771522 h2.eff: ... [total_energy] -0.955935

After the optimization, the electrons become bond-centered, and the wavefunction is closed-shell - that is, the up and down spin electrons have the same spatial wavefunction. The bond length is 1.47 bohr, near the exact value of 1.40 bohr. We can compute the eFF bond energy by taking the difference between the hydrogen molecule energy and double the hydrogen atom energy:

-(-0.955935 - 2 * (-0.424413)) = 0.107109 hartree = 67 kcal/molusing the conversion factor 627.51 kcal/mol = 1 hartree. In comparison, the bond energy computed using unrestricted Hartree-Fock (6-311g** basis) is 83 kcal/mol, and the exact bond energy is 109 kcal/mol - both the Hartree-Fock and these eFF calculations are missing a term that corrects for electron correlation.

Imagine breaking the bond by stretching it to longer distances. At some point, the bond breaks homolytically, so that each electron goes to a separate proton, and the electrons become nucleus-centered instead of bond-centered. To study this effect, we fix the nuclei to be separated a distance of 3 bohr, then repeat the optimization. Attaching the suffix # after a coordinate fixes it at that value during minimization or dynamics.

h2_fixed.cfg: @params calc = minimize @nuclei 0# 0 0 1 3# 0 0 1 @electrons 0 0 0 1 2.0 3 0 0 -1 2.0 h2_fixed.cfg.restart: ... @nuclei 0.000000 0.000000 0.000000 1.000000 3.000000 0.000000 0.000000 1.000000 @electrons 0.021826 0.000000 0.000000 1 1.904498 2.978174 0.000000 0.000000 -1 1.904498

We see that the electrons are now nucleus-centered upon optimization. In the Hartree-Fock model, this transition between open and closed-shell wavefunctions occurs at 2.3 bohr; in eFF, we observe the transition at 2.1 bohr instead.

By performing a series of calculations with nuclei constrained at different distances, we obtain a potential energy surface for the interaction of two hydrogen atoms in the ground state, which may be compared against ground state surfaces obtained using other methods. Overall, the eFF curve has a proper inner wall, bonding region, long range tail, and a correct transition between closed and open shell wavefunctions.