You are correct.
I would suggest that sociopaths and psychopaths are not intimidated or dissuaded by the threat of capital punishment.
I would submit that such persons are EMBOLDENED by the knowledge that they don't face capital punishment. "Sociopaths are interested only in their personal needs and desires, without concern for the effects of their behavior on others." "Psychopath - (technical) someone with a personality disorder characterized by extreme callousness, who is liable to behave antisocially or violently in getting their own way, without any feelings of remorse."
I would also submit that swift execution of such persons puts a real crimp in the personal needs and desires of a sociopath, and that death kind of prevents a psychopath from getting his or her own way. Anybody, anybody? Mr. Bundy.
Ah yes, Mr. Bundy.
Do you recall what he did when Ted escaped from prison? He did not even attempt to flee the country. He immediately went out and killed again, although he knew(being an attorney) he faced the death penalty, before authorities tracked him down. The possibility did not deter or embolden him. Ted Bundy just continued his murderous behavior.
IMHO sociopaths and psychopaths are so deranged (Charles Manson comes to mind) that the threat of execution for criminal behavior doesn't affect their conduct. And yes, Manson surely deserved the death penalty.
Your descriptions of "sociopaths", and, "psychopaths", though true, fails to fully illustrate their criminal behavior. I would suggest that any time they are able to manipulate and control anyone, they are "EMBOLDENED".
Obviously, executing a killer puts a "crimp" in their personal needs, being that they are dead, if I understand you correctly. But as I had said in my previous post; just and capital punishment does not deter those who are truely evil.