Objection 1. It would seem that excommunication is unsuitably defined by some thus: "Excommunication is separation from the communion of the Church, as to fruit and general suffrages." For the suffrages of the Church avail for those for whom they are offered. But the Church prays for those who are outside the Church, as, for instance, for heretics and pagans. Therefore she prays also for the excommunicated, since they are outside the Church, and so the suffrages of the Church avail for them.
Objection 2. Further, no one loses the suffrages of the Church except by his own fault. Now excommunication is not a fault, but a punishment. Therefore excommunication does not deprive a man of the general suffrages of the Church.
Objection 3. Further, the fruit of the Church seems to be the same as the Church's suffrages, for it cannot mean the fruit of temporal goods, since excommunication does not deprive a man of these. Therefore there is no reason for mentioning both.
Objection 4. Further, there is a kind of excommunication called minor*, by which man is not deprived of the suffrages of the Church. [Minor excommunication is no longer recognized by Canon Law.] Therefore this definition is unsuitable.
I answer that, When a man enters the Church by Baptism, he is admitted to two things, viz. the body of the faithful and the participation of the sacraments: and this latter presupposes the former, since the faithful are united together in the participation of the sacraments. Consequently a person may be expelled from the Church in two ways. First, by being deprived merely of the participation of the sacraments, and this is the minor excommunication. Secondly, by being deprived of both, and this is the major excommunication, of which the above is the definition. Nor can there be a third, consisting in the privation of communion with the faithful, but not of the participation of the sacraments, for the reason already given, because, to wit, the faithful communicate together in the sacraments. Now communion with the faithful is twofold. One consists in spiritual things, such as their praying for one another, and meeting together for the reception of sacred things; while another consists in certain legitimate bodily actions. These different manners of communion are signified in the verse which declares that those who are excommunicate are deprived of-- "os, orare, vale, communio, mensa." "Os," i.e. we must not give them tokens of goodwill; "orare," i.e. we must not pray with them; "vale," we must not give them marks of respect; "communio," i.e. we must not communicate with them in the sacraments; "mensa," i.e. we must not take meals with them. Accordingly the above definition includes privation of the sacraments in the words "as to the fruit," and from partaking together with the faithful in spiritual things, in the words, "and the general prayers of the Church." Another definition is given which expresses the privation of both kinds of acts, and is as follows: "Excommunication is the privation of all lawful communion with the faithful."
Reply to Objection 1. Prayers are said for unbelievers, but they do not receive the fruit of those prayers unless they be converted to the faith. In like manner prayers may be offered up for those who are excommunicated, but not among the prayers that are said for the members of the Church. Yet they do not receive the fruit so long as they remain under the excommunication, but prayers are said for them that they may receive the spirit of repentance, so that they may be loosed from excommunication.
Reply to Objection 2. One man's prayers profit another in so far as they can reach to him. Now the action of one man may reach to another in two ways. First, by virtue of charity which unites all the faithful, making them one in God, according to Psalm 118:63: "I am a partaker with all them that fear Thee." Now excommunication does not interrupt this union, since no man can be justly excommunicated except for a mortal sin, whereby a man is already separated from charity, even without being excommunicated. An unjust excommunication cannot deprive a man of charity, since this is one of the greatest of all goods, of which a man cannot be deprived against his will. Secondly, through the intention of the one who prays, which intention is directed to the person he prays for, and this union is interrupted by excommunication, because by passing sentence of excommunication, the Church severs a man from the whole body of the faithful, for whom she prays. Hence those prayers of the Church which are offered up for the whole Church, do not profit those who are excommunicated. Nor can prayers be said for them among the members of the Church as speaking in the Church's name, although a private individual may say a prayer with the intention of offering it for their conversion.
Reply to Objection 3. The spiritual fruit of the Church is derived not only from her prayers, but also from the sacraments received and from the faithful dwelling together.
Reply to Objection 4. The minor excommunication does not fulfill all the conditions of excommunication but only a part of them, hence the definition of excommunication need not apply to it in every respect, but only in some.
Source: Summa Theologica (New Advent)