The teaching on the just use of capital punishment follows the same circumstances that permit killing in self-defense – force, even lethal force, to protect innocent human life from an unjust aggressor is licit.
False, false, false. The Catechism treats of the death penalty under the sub heading of Legitimate Defense as distinguished from murder. 2263 -2267
http://www.vatican.va/archive/ccc_css/archive/catechism/p3s2c2a5.htm
Secondly it is Aquinas’ work in the 13th century that clearly defined state execution as lawful defense as far as the welfare of society is served.
ST II II Q64 Murder (Article 2)
Now every part is directed to the whole, as imperfect to perfect, wherefore every part is naturally for the sake of the whole. For this reason we observe that if the health of the whole body demands the excision of a member, through its being decayed or infectious to the other members, it will be both praiseworthy and advantageous to have it cut away. Now every individual person is compared to the whole community, as part to whole. Therefore if a man be dangerous and infectious to the community, on account of some sin, it is praiseworthy and advantageous that he be killed in order to safeguard the common good, since “a little leaven corrupteth the whole lump” (1 Corinthians 5:6).
However if society is harmed by the use of execution it is forbidden.
Our Lord commanded them to forbear from uprooting the cockle in order to spare the wheat, i.e. the good. This occurs when the wicked cannot be slain without the good being killed with them, either because the wicked lie hidden among the good, or because they have many followers, so that they cannot be killed without danger to the good, as Augustine says (Contra Parmen. iii, 2). Wherefore our Lord teaches that we should rather allow the wicked to live, and that vengeance is to be delayed until the last judgment, rather than that the good be put to death together with the wicked. When, however, the good incur no danger, but rather are protected and saved by the slaying of the wicked, then the latter may be lawfully put to death.
And yet again, the death penalty is referred to the common good as a defense. (Article 3)
it is lawful to kill an evildoer in so far as it is directed to the welfare of the whole community, so that it belongs to him alone who has charge of the community’s welfare. Thus it belongs to a physician to cut off a decayed limb, when he has been entrusted with the care of the health of the whole body. Now the care of the common good is entrusted to persons of rank having public authority: wherefore they alone, and not private individuals, can lawfully put evildoers to death.