Why can't we harness the energy from neutron + protium = deuteron reaction? The answer to this physics se question, established that 2.2Mev (Mega electron volts) of energy is emitted as a gamma ray photon when the hydrogen atom nucleus oka protium absorbs a slow moving free neutron causing the formation of a deuterium nucleus.  Deuterium is heavy hydrogen with both a proton and neutron in the nucleus and weighs twice as much as regular hydrogen. 
Deuterium is a more stable nucleus than protium, meaning it is more energetically favored over protium and neutron seperate. This is due to the strong nuclear force binding nucleons together.  The mass of the deuteron is less than the sum of the proton and neutron.  The deficit of mass in the reaction is released as energy; in this case in emission of the gamma ray photon of 2.2MeV.  This is what happens in nuclear fusion where lighter element nuclei merge together into a heavier element nuclei and releasing a great deal of energy.  
Now this reaction cited is a very special case because we don't need millions of degrees temperature to cause the fusion since there is no Coulomb repulsion. A slow neutron strikes a regular protium hydrogen nucleus (just a proton) and we have 2.2MeV of energy.  This happens at room temperature with no energy consuming magnetic fields used to contain high temperature plasma as is done with current fusion reactors.  Granted we are not getting 17.59MeV that is released when tritium and deuterium are fused, but we are not requiring 25MeV of input power to cause and maintain said reaction, resulting in net loss of power.
In the neutron-protium fusion we have a net 2.2MeV energy output with little or no energy input.  My main question is why this energy is not being harnessed for power generation?  2 somewhat challenging but not insurmountable reasons come to mind.
For one, we need free neutrons.  These can be obtained from a fission nuclear reactor, cyclotrons, alpha radiating radioactive isotopes with beryllium targets, and others.
If we want to avoid the fission reactor as the neutron source, we can use various other methods.  This physics se question probes an exothermic beryllium target process.  I had in mind a cyclotron to mildly accelerate alpha particles (helium nuclei) to the appropriate energy level to cause neutron emission when striking a beryllium target.  The beryllium target would be encased in liquid or super-critical hydrogen to maximize neutron capture.
My estimation is that the energy output from the n + p = d reaction will supply enough power for the neutron generating cyclotron and then some for useful output power.  My reasoning for this is the alpha particles don't need that much energy for this purpose. The process in and of itself is exothermic so we are getting free neutrons plus energy.  We are also not concerned with the enormous energy required to confine a plasma.
The other hurdle is harnessing the energy from the emitted gamma ray. This SE question explores the gamma ray energy conversion topic.  Then we have to figure out a way to use some of the output energy that was harnessed from the gamma ray to generate more neutrons to "keep the fire going".  
While this process is indeed challenging, I think a working system can be developed for a fraction of what is being spent on ITER, the most expensive science project in history to investigate surplus energy from fusion.
 A: Most stable nuclei have a "neutron separation energy" of eight or ten MeV --- that's the minimum cost to produce a neutron, before you factor in any inefficiencies.
You can't get ahead spending 10 MeV to free a neutron, having the neutron deposit its 1-2 MeV kinetic energy in some sort of thermalizing moderator so that it can actually capture on hydrogen, and then capturing the 2 MeV photon.
There are some exceptions.  For instance, if you put refined uranium in a big pile, neutrons come out of the pile for free.  But each fission event releases 200 MeV and only a couple of neutrons that you could use for your purpose; collecting the 4-6 MeV from putting those neutrons on hydrogen is basically rounding error.
There's also the $\rm {}^9Be(\alpha,n\gamma)$ reaction which you mention.  However that reaction is not very efficient.  In a high-quality AmBe or PuBe source, the number of alphas that undergo $(\alpha,n\gamma)$ on beryllium is about sixty per million.  And the americium or plutonium for those sources comes from ... reactor neutrons on uranium.  You don't win there.
Let's suppose that you accelerate some helium nuclei onto a beryllium target to set off this $(\alpha,n\gamma)$ reaction.  You magically overcome all the technical problems somehow every 3 MeV alpha particle turns into a 10 MeV beryllium disintegration, all of which energy you capture usefully.  That's a terrible  operating configruation for a power supply, where 30% of the energy (in the wildly optimistic case) has to be renivested in the operation of the power plant.  Compare to a campfire (also a pretty inefficient power source) where you find the fuel on the ground, light it with  a tiny match, and it keeps you warm for hours.
A: Your estimation that the reaction creates enough power to power an neutron producting cyclotron is incorrect. It costs far more energy than $2.2~\rm MeV$ per produced neutron, and so even if you could perfectly efficiently extract the produced energy you still wouldn't be producing any net energy. The main problem is that most alpha particles scatter without actually producing a neutron, and so all the energy spent accelerating those alphas is wasted.
A: Actually, we do on a rather large scale... in light water reactors. This is the only abundant supply of free (not bound to some nucleus) neutrons we have today. The cooling water absorbs some of the neutrons escaping the chain reaction and then absorbs the heat from the gamma-ray dissipation. Profit! (Probably way less than 1% of the total reactor heat output).
A: Well it will be impossible to be able to create some energy with future technology. 
The one major problem is magnetic confinement, it is incredibly difficult or even impossible to trap neutrons to magnetic field. There tiny magnetic moment is not big enough for them to be used in fusion reactors. They also sometimes change the chemical makeup of the container of whatever is holding the fusion materials. 
It is also hard to get a hold of neutrons and even the neutrons that is available will undergo beta decay where the neutron turns into a proton and then it will be impossible(you could do it but for more energy than you get back) to make the fusion reaction happen.
