Splitting molecule The photon reacts with the binding electrons orbiting the two atoms.
The photons have the 'correct' wavelength for Bond Dissociation Energy (BDE).
'Splitting' the molecule involves applying the photon wavelength to separate the electron from the molecule.
With the photons being applied between the binding electrons in between the two oxygen atoms, does 'splitting' occur when there is one photon reacting with one binding electron, even when there are two binding electrons?
 A: This isn't really how it works. A photon doesn't interact with a single electron, it interacts with the entire molecule.
Suppose you take the example of ozone photolysis to $O_2$ and an oxygen atom. We can do a calculation for ozone and come up with a series of molecular orbitals, then put two electrons in each orbital. So far so good. But if you remove an electron, or even just excite it to a higher energy orbital, then all the molecular orbitals change and you have to recalculate them all. You can't do anything to one electron without affecting all the others and changing the properties of the molecule as a whole.
In the case of ozone it can absorb a photon and the whole ozone molecule rearranges into a higher energy state. From this higher energy state it can relax back into the ground state and re-emit a photon, or it can split into $O_2$ and an oxygen atom. Like most things in quantum mechanics this is a probabilistic process. We can calculate the probabilities of relaxing and spliiting, but it's impossible to predict what any individual excited ozone molecule will do.
A: When a molecule absorbs a photon it reaches to an excited state and there are various mechanisms in which the molecule can relax. Dissociation of the molecule is just one of the possibilities.
It is not necessary to ionize (to separate the electron from) the molecule for dissociation to occur. What is necessary is to excite a bonding electron, that is, an electron in the molecule which is involved in bonding. This electron can still be bound to the molecule after absorption of the photon and of course it can leave the molecule. This is to be determined by the binding energy of the electron and the energy of the exciting photon.
One side information, except certain cases (dense laser light) in the process of absorption of light by mater only one photon and one electron is involved. One electron absorbing two photons or one photon exciting two electrons is not common.
After this prelude the answer to your question is that: one photon can excite one bonding electron and as a result a molecule can dissociate. Yes, even there are two bonding electrons this happens. Because as soon as one electron is excited the molecule has a hole in a bonding level and therefore in an unstable state.
A: I agree with what John Rennie said,
"A photon doesn't interact with a single electron, it interacts with the entire molecule."
The 'probabilistic process' is a better way of stating 'Give it a shot, and see what happens.'
The probability between relaxing and splitting, or whether the photon and the molecule reacts at all, 
sounds good to me.
Please correct me if I am wrong.
Let me partially explain my experiment.
The photons have the wavelength for 'Bond Dissociation Energy' (BDE).
The BDE should be energy where the electron is not connected to the molecule, and drifts away.
If the energy and the 'noisy' energy of the atom does not interact with the photon, then nothing will happen.
It might take a lot of trials before it will react.  But, after it separates the oxygen-oxygen atoms those 
wavelength dependent photons might react with the molecule.
Lets work with a molecule that looks like this, RPO-OH
The 'R' is nitrogen, hydrogen, carbon, etc., all taking about -300nm.
Removing the -OH is the goal.
This is a singular bond between the oxygen and oxygen.
The other bonds won't take -700nm for bonding purposes, they have to go lower, like about ~300nm.  For example, the phospholipid molecule
that could be modeled by RPO-OH, has the O-O around 700nm, where all of those other binds are above
~300nm.  
When the molecules absorbs one of the photons, at the right wavelength, and then there is molecular splitting.
The splitting attracts a lower ~300nm binding. 
The experiment is finished, even though the -700nm photon source keeps running.
I'm going to have to think about what linuxich wrote down.  It sounds good !!!
