Capital punishment is arbitrary. Mistakes are made. We occasionally hear of people released from prison, even death row, because of new evidence coming to light. We can't undo it . Even if errors are extremely rare, I believe any errors are sufficient reason to stop the practice.
People can change, at least sometimes.
We should give them the chance. I have played that scenario in my mind, and I hope that after a period of what I know would be intense anger, I would remain true to my principles. I don't believe in an afterlife, but that doesn't change my thoughts on capital punishment. In an ideal world, execution is the ideal punishment for certain crimes. I think we have the right to create rules for our society, rules that are established through democratic processes, and we have the neccessity of enforcing those rules.