h_atom.cfg: @params calc = minimize @nuclei 0 0 0 1 @electrons 0 0 0 1 1.0 h_atom.cfg.restart: .. @nuclei 0.000000 0.000000 0.000000 1.000000 @electrons 0.000000 0.000000 0.000000 1 1.879971 h_atom.eff: .. [total_energy] -0.424413
In hydrogen atom, the optimized electron radius is a compromise between opposing forces, the kinetic energy which attempts to make the electron larger, in accordance with Heisenberg's uncertainty principle, and the electrostatic interaction between the electron and the proton, which attempts to shrink the electron on top of the proton.
The final electron radius is 1.88 bohr. The energy of the atom, -0.424413 hartree is higher than the variational limit of -0.5 hartree because the single Gaussian wavefunction does not capture the proper electron-cusp at the nucleus, or the correct falling away of the electron density far from the nucleus. However, energy differences between bound and separated atoms are expected to be more accurate, as these errors partially cancel.