I don’t agree with the notion that the death penalty is a deterrent. Do we honestly believe that these offenders would think “I may be killed for this, so I’m not gonna do it” like a sane person would? Any more than they must think “I may be locked in a cage for the rest of my life for this, I’m not gonna do it”?
I think if they’re not sane enough to think through the latter as any normal person would, it’s fair to assume they’re certainly not sane enough to think through the former, which basically defeats the “deterrent” angle re the death penalty.
I don’t think these offenders are in a state of mind where they can think this through and come to the rational conclusion, no matter what the potential consequence (be it life imprisonment or death). Or if they actually can, the threat of punishment is redundant anyway: as they actually truly believe that they’ll never be caught.
The death penalty may not deter them, but it would certainly stop them from doing it again.