The simplest possible molecule is the hydrogen molecule ion, consisting of two protons and one electron. However, even this example cannot be solved analytically, as it is a three-body problem. However, it is possible to utilise the fact that the protons are four orders of magnitude heavier than the electron, so that the dynamics of the protons are frozen to a very good approximation from the point of view of the electrons. This allows a separation approach in which nuclear dynamics and electron dynamics are considered separately. This transformation into a restricted three-body problem is known as the Born-Oppenheimer approximation . The potential in which the electron is located can then be described, for example, by a soft-core Coulomb potential:

In contrast to the hard Coulomb potential, the additional parameter ε is present here, which cancels the divergences at the location of the protons (x = ±R/2, z = 0). The numerical solutions of the corresponding Schrödinger equation for the electron are shown below. The protons are the red dots in the x-z plane, with the wave function of the electrons plotted in the y direction. Positive values of the wave function are shaded blue, negative values red.

The Schrödinger equation obviously produces two solutions that differ drastically in their consequences for molecular bonding. The above solution, also known as the binding solution, has a non-vanishing residence probability for the electron between the protons. This means that the electron can act attractively on both protons and shield the repulsive Coulomb force between the protons. This creates an energy minimum, which makes molecular bonding possible.

In contrast, the solution shown below, also known as antibonding, always has a vanishing probability of residence between the protons. This means that the repulsive force between the protons cannot be shielded, i.e. there is no energy minimum and no molecular bond is formed.