For the longest time I did not understand how it would be possible that a molecule should not show fluorescence. You take a photon to push it up to a higher level. So how should it be able to move down in a different way, between two different quantized levels. The main errors in this thought are the static view point and the Born-Oppenheimer approximation.
The first thing to consider is that gaps between electronic states (in the Born-Oppenheimer approximation) are not the same at every geometry. At the ground state minimum there is typically a gap of several eV to the first excited state and correspondingly excitation is in the UV. If you distort the geometry, the ground state energy increases (since we were at the minimum). Since the minimum of the excited state does in general not coincide with the minimum of the ground state, it can relax to a lower excited state energy. Ground state and excited state move together and the gap decreases. Fluorescence will be at a lower energy than absorption. This is just the semi-classical explanation of the Stokes shift.
You can go on trying to decrease the gap. And it turns out that for many molecules of interest there are actually geometries where the ground and first excited states are degenerate. These are called conical intersections. And there is not just one such geometry but in the general case the crossing seam forms a hyperline in the space of all nuclear geometries (i.e. an N-2 dimensional subspace, where N is the number of nuclear degrees of freedom). Conical intersections are usually structures strongly distorted from the ground state geometry (hence the ground state energy increases). And they are adapted to the excited state (to keep its energy low).
Nucleic acid bases are of interest in this context. As I read it in a nice popular science article: When a photon from the sun is absorbed by a molecule, this essentially means that this molecule is heated to the temperature of the surface of the sun. Conjugated polar molecules like nucleic acid bases definitely absorb UV light. So why are our cell nuclei not scorching away when we are out tanning a little bit? Well the answer is that they are very efficient in giving away the excess energy and thereby returning to the ground state.
Actually people are arguing that photostability was one main points of selection pressure in the early biosphere. A support of this is that the nucleic acid bases have very short decay times compared to analogues with different substitution patterns. Analogues were destroyed by photochemistry while the bases that are now actually in use could protect themselves with photophysics (i.e. non-radiative decay).
Cytosine is an interesting example . In this case there are three excited states of interest and all of them cross with the ground state. The question is which one is actually the decay channel and there is not really consensus. Well I just want to show the geometries.
The ground state minimum is almost planar with same pyramidalization on the amino group.
This is one of the conical intersections, usally called "twist CI". The excited state is a biradical state localised mostly on the two C atoms in front (coming from the ππ* state). Through this twisting the biradical state is stabilized because the electrons move out of each other's way. I think you could also say that you have pyramidalized radicals just as usual in organic chemistry.
This on is the "sofa CI" which is related to the nNπ* state (the n orbital localized on the N that is not protonated). Ring puckering and stretching of the amino group stabilizes this state.
The last minimum of the crossing seam between first excited and ground state is this one. It comes from the nOπ* state.
Well these conical intersections exist. But the probability of actually reaching one is zero. As I said crossing seams are hyperlines. Imagine a line in the room where you are just sitting - any given random point will at least have a little bit of distance  from the line. The way out is that with close lying electronic states the Born-Oppenheimer approximation breaks down and a transfer between electronic states is possible through coupling with the nuclei.
 The method was MCSCF in Columbus but I cannot really give computational details here. This is more thought as a "community outreach" - just showing some nice pictures.
And you can for example check out this ref if you want to know more about cytosine.
 an ε > 0, if you like calculus
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