Molecules have a much more complicated energy level structure than atoms. In this figure, we see a portion of the energy structure for diatomic iodine (I2), with the vertical axis representing the energy stored in the molecule and the horizontal axis representing the distance between its two nuclei. Each curve represents an allowed electronic energy state and each horizontal line represents an allowed vibrational mode (i.e., a frequency at which the molecule is allowed to vibrate).

Molecules in higher vibrational states have a higher maximum speed as they oscillate, meaning they can stretch (or compress) further from the equilibrium internuclear distance (the point where the electronic energy curve is at its lowest). The point at which a molecule can't stretch any further is known as a turning point - shown on the figure where the vibrational mode lines intersect their respective electronic energy curve.

There's a lot going on in this graph, so let's visualize a simple example: an iodine molecule in its ground state - that is, in the n'' electronic state and the 0 vibrational level. That molecule bounces back and forth between turning points (indeed, there is still some vibration even at the lowest possible energy!) - compressing until the distance between its atoms reaches the inner turning point (the leftmost intersection of the n'' curve and the v=0 line), expanding until it reaches the outer turning point (the rightmost intersection of the n'' curve and the v=0 line), then repeating the process. The total energy of the molecule never deviates, only changing forms - completely in the form of potential energy at the turning points, and completely in the form of kinetic energy at the equilibrium internuclear distance. Thus the total energy is equal to the height of the turning points.

Now let's talk about transitions (illustrated by the arrows in the figure). Iodine in the ground state strongly absorbs 532nm light (a shade of green), making a simultaneous transition to the n' electronic state and the 9 vibrational state. Like atoms, molecules in excited electronic states spontaneously decay to the lowest electronic state, emitting light in the process. But molecules do not have to transition to the lowest vibrational state - when decaying, they may transition to any vibrational state lower than their initial state. Critically, decaying to a higher vibrational state will release less energy than decaying to a lower vibrational state! Thus, many fluorescence wavelengths are observed for a single electronic transition - ranging from the same shade of green used to excite the molecule (the leftmost fluorescence arrow in the figure) to dull shades of red (the rightmost fluorescence arrow).

The spacing between adjacent fluorescence wavelengths for a molecule allows us to determine the allowed vibrational frequencies for that molecule - which can be used in a couple helpful calculations to determine important molecular properties and reproduce the molecular potential curves depicted in the figure. This second result takes some difficult calculations, but is of utmost importance in spectroscopic theory.

As it turns out, if we know the potential curves for a molecule, we know everything there is to know.