By treating the vibrational transition in the HCl spectrum from its ground to first excited state as a quantum harmonic oscillator, the bond force constant can be calculated. This transition frequency is related to the molecular parameters by:

The desired transition frequency does not show up directly in the observed spectrum, because there is no j=0, v=0 to j=0, v=1 transition; the rotational quantum number must change by one unit. It can be approximated by the midpoint between the j=1,v=0->j=0,v=1 transition and the j=0,v=0->j=1,v=1 transition. This assumes that the difference between the j=0 and j=1 levels is the same for the ground and first excited state, which amounts to assuming that the first excited vibrational state does not stretch the bond. This "rigid-rotor" model can't be exactly correct, so it introduces some error.

For the HCl molecule, the needed reduced mass is

Note that this is almost just the mass of the hydrogen. The chlorine is so massive that it moves very little while the hydrogen bounces back and forth like a ball on a rubber band!

Substituting the midpoint frequency into the expression containing the bond force constant gives:

A bond length for the HCl molecule can be calculated from the HCl spectrum by assuming that it is a rigid rotor and solving the Schrodinger equation for that rotor. For a free diatomic molecule the Hamiltonian can be anticipated from the classical rotational kinetic energy

and the energy eigenvalues can be anticipated from the nature of angular momentum.

Assuming that the bond length is the same for the ground and first excited states, the difference between the j=1,v=0->j=0,v=1 transition and the j=0,v=0->j=1,v=1 transition frequencies can be used to estimate the bond length. The separation between the two illustrated vibration-rotation transitions is assumed to be twice the rotational energy change from j=0 to j=1.