Q:

What determines the efficiency of electron production in photosynthetic bacteria?

Is there a specific gene involved, perhaps? Would one be able to genetically engineer a bacterium to oxidize water and generate electrons quicker? I am speaking about this biological problem in terms of an application to solar cells. 
Edit: I am not inquiring to the existence of a gene responsible for photosynthesis, I am inquiring as to why some exoelectrogenic bacteria can produce electrons in electron transport more efficiently than others. Is there a reason for this that can be traced to a certain gene that makes certain exoelectrogens superior to others?

A:

You may or may not consider this an actual answer to your entire question, but it's interesting nonetheless.
A physicist friend of mine did some work recently modelling the quantum dynamics of photosynthetic complexes, and their coupling constants for passing energy throughout the photosystem. His results showed that, (and I believe this is in agreement with the literature), that photosynthetic systems are actually about as efficient as it is possible to be (not only this, but they are the most efficient energetic systems known to man I believe - over something like ~90% efficient).
If you begin altering positions of chromophores and so on, the photosystems exhibit remarkable robustness and you get a more or less logistic decay in efficiency (i.e. removing one or 2 chromophores results in marginal reductions in efficiency, more still in considerably more severe debilitation, until ultimately it's efficiency is markedly lessened, but further manipulation has a plateau in efficacy reduction).
Much of the work that's been done on the quantum biology of photosystems has concerned the FMO Complex (Fenna-Matthews-Olson) which is a simplified model system from photosynthetic algae - you might find more detailed answers to your queries by reading up in this area.
As for discrepancies seen in photosynthetic effectiveness bought about genetically, there are theories that suggest that the organisms are actually protecting themselves. E.g. to avoid over-production of ROS species. I can't offer you much in the way of a well evidenced case for this though, it's just a conversation I had with a post-doc at work recently, who completed his PhD on photosynthetic cyanobacteria.
EDIT Here's one of the papers about optimality in the FMO:
http://aip.scitation.org/doi/pdf/10.1063/1.4930110

A:

@Joe Healey's answer is great, but I would like to expand on the subject, having done a thorough literature review on the subject before.
Before we do that, I would like to clear up some misunderstandings that you seem to have about the electron transfer. 
Just like @stords said, electrons aren't generated during photosynthesis. Where it comes from is from the oxidation of water mediated by the reaction center. The reaction center is part of the photosystem,  which contains all the light-dependent parts of the reactions. The oxidation is performed by the oxygen-evolving complex. This process of oxidation isn't very well understood. However, most photosynthetic bacteria contain similar complexes. Note that green sulfur bacteria doesn't use H2O for oxidation, but instead H2S.
Reaction center
Now, in the light-dependent reactions, we have what's known as a light-harvesting complex, which contains many pigments that absorb light and get excited. The pigments then transfer the energy to the reaction center, which also gets the electrons from the oxidation. So when the energy from the antenna reaches the reaction center, the energy excites the electron to a higher energy level. This higher energy electron is transferred to the electron transport chain, where it reduces various electron acceptors, and while doing so, using the energy released to pump electrons to make a proton gradient. This proton gradient is used to make ATP. Even in bacteria, the process is very similar. The exact details can be read here
Excitation efficiency
Now, answering your question, what really determines the efficiency of the electron transfer is actually the energy transfer occurring in the light-harvesting complex (LHC). Now, different organisms have different mechanisms, but all scientists agree that there is some quantum mechanics behind this. In the following discussion, I will be referring to the dynamics of the FMO complex, the antenna complex found the green sulfur bacteria, and commonly used to study the quantum mechanical interactions of energy transfer.  
Now when we have this transfer of energy from one pigment molecule (chlorophyll in plants, bacteriochlorophyll a in green sullfur bacteria), we call it an exciton. This exciton is a quantum mechanical state. Now, in quantum mechanics, there is a concept called superposition. Basically, superposition means that a single quantum mechanical state is composed of multiple states. In the case of the light-harvesting complex, when we say that the exciton is in a superposition, what this means is that the energy can travel in all possible pathways from a pigment to reaction center, and when the best pathway is found, the interactions in the complex causes the state to collapse into the best state. This is how the transfer of energy occurs in less than 1 ns!
This dynamics is very complicated, and very surprising, since this quantum mechanical dynamics is occurring in an open system. Typically, it is very hard to observe quantum mechanical dynamics because of something known as decoherence. This is when a mixed (superposition) state collapses into one state. However, interestingly, the complex uses this decoherence to collapse into the best path for the exciton to reach the reaction center.
This field is very interesting. If you have any questions, please leave a comment. I am a little rusty on this subject, but I am very interested in this subject. If you want to see more examples of quantum mechanics in biology, look up "quantum biology". Quantum biology is a new field studying the intersection of quantum mechanics and biology.
References:

https://en.wikipedia.org/wiki/Light-dependent_reactions
http://rsif.royalsocietypublishing.org/content/11/92/20130901
https://en.wikipedia.org/wiki/Fenna-Matthews-Olson_complex

