If you have to use capital punishment, and if you use lethal injection, I can't imagine why it's so hard to anesthetize the person before poisoning him. Surgeons routinely cut people open in ways that would be unbearably painful without proper anesthesia.
It should really not be that hard to make lethal injection basically painless.
So I am skeptical when I hear that the lethal injection drugs or procedures currently used can cause large amounts of pain. The story is that there are three drugs (anesthetic, paralytic, poison) but that the anesthetic can wear off before the person dies, leaving him paralyzed and in pain.
This would mean that somebody is really sadistic or incompetent.
Two simple solutions that may work: (1) don't use paralytic at all. What's the point, to make the killing look more serene for the witnesses? The person is going to be strapped to a gurney anyway, and if the anesthetic starts wearing off, you'll know right away and up the dose. (2) figure out the right dose of anesthetic, then double or triple it or whatever.
In any case, I am opposed to capital punishment on moral grounds.
Whether we believe this form of punishment is legitimate or not (a question outside the domain of science) ...
Sorry to go off your topic a bit, but I think that determining the "legitimacy" of capital punishment is not entirely outside the realm of science. Science can at least INFORM your decision on this question.
The question does need to be framed carefully first, and science may not be able to help with that.
But if "legitimate punishment" means "good crime deterrent" or "minimal cruelty" or "to the benefit of the greater good" then science can probably at least help you get to a good answer.
If "legitimate" means "aligned with religious doctrine X or philosophy X" then science may be of less help, but may still be able to provide data as inputs if needed.
Or of course, if we refuse to define "legitimate" then it will be difficult to get anywhere with science or not!