The Bohr model for the hydrogen atom marks the historic paradigm shift in the development of the quantum theory of matter . Bohr's original attempt to extend his model to molecules  has long been regarded as a complete failure. More recently, however, his molecular model has been revisited , specifically in connection with Dudley Herschbach's dimensional scaling transformation of the Schrödinger equation.
In this Demonstration, we propose a modified version of Bohr's model for the lowest singlet and triplet states of the hydrogen molecule. The two protons, shown in blue on the left graphic, are separated by an internuclear distance
. The two electrons, shown in red, traverse circular orbits of radii
perpendicular to the internuclear axis and centered at distances
from their respective nuclei. For the
singlet state, the electron orbits are located between the nuclei and their electrons orbit in a contra-rotating sense. For the
repulsive triplet state, the orbits are outside
the internuclear axis and move in parallel.
The energy of the hydrogen molecule, referred to the energy of two separated hydrogen atoms (
), can be written in atomic units as:
Select the parameters
"by hand," using the sliders, so as to produce the best visible fit to the energy functions
obtained by accurate quantum-mechanical computations. (We have added an additional effective nuclear charge parameter
for greater flexibility.) We have not considered a more systematic procedure, in which the parameters can be determined variationally for each value of
. The resulting energy curves, along with the corresponding exact results, are plotted in the left graphic, for either the singlet or triplet states. Snapshots 2 and 3 show near-optimized parameters for the two states.