Step 1: Measure the emission spectrum for a diatomic molecule, then fit a function of vibrational state number G(v) to the measured vibrational energies and a function B(v) to its measured rotational energies (which will look something like the equations above). This step is complicated, but quick for spectroscopists!

Step 2: Pick a value for v. As we learned earlier, molecules have distinct vibrational levels (i.e., they can only have v=0,1,2,3,...). But here we can pick any positive number for v (a quirk of us using integrals where nature would use discrete sums). This determines the upper limit for the integrals and the value of G(v) in the denominator of each integral. 

Step 4: Solve for the turning points. Two equations in two unknowns can always be solved exactly; all we need to do is some basic algebra and we have our two turning points for that vibrational level (A1 and A2 on the figure)! Repeating steps 2-4 for a wide variety of vibrational levels lets us "fill in" the potential curve for a molecule point-by-point.