Objection 1. It seems that Baptism can be conferred in the name of Christ. For just as there is "one Faith," so is there "one Baptism" (Ephesians 4:5). But it is related (Acts 8:12) that "in the name of Jesus Christ they were baptized, both men and women." Therefore now also can Baptism be conferred in the name of Christ.
Objection 2. Further, Ambrose says (De Spir. Sanct. i): "If you mention Christ, you designate both the Father by Whom He was anointed, and the Son Himself, Who was anointed, and the Holy Ghost with Whom He was anointed." But Baptism can be conferred in the name of the Trinity: therefore also in the name of Christ.
Objection 3. Further, Pope Nicholas I, answering questions put to him by the Bulgars, said: "Those who have been baptized in the name of the Trinity, or only in the name of Christ, as we read in the Acts of the Apostles (it is all the same, as Blessed Ambrose saith), must not be rebaptized." But they would be baptized again if they had not been validly baptized with that form. Therefore Baptism can be celebrated in the name of Christ by using this form: "I baptize thee in the name of Christ."
On the contrary, Pope Pelagius II wrote to the Bishop Gaudentius: "If any people living in your Worship's neighborhood, avow that they have been baptized in the name of the Lord only, without any hesitation baptize them again in the name of the Blessed Trinity, when they come in quest of the Catholic Faith." Didymus, too, says (De Spir. Sanct.): "If indeed there be such a one with a mind so foreign to faith as to baptize while omitting one of the aforesaid names," viz. of the three Persons, "he baptizes invalidly."
I answer that, As stated above (Question 64, Article 3), the sacraments derive their efficacy from Christ's institution. Consequently, if any of those things be omitted which Christ instituted in regard to a sacrament, it is invalid; save by special dispensation of Him Who did not bind His power to the sacraments. Now Christ commanded the sacrament of Baptism to be given with the invocation of the Trinity. And consequently whatever is lacking to the full invocation of the Trinity, destroys the integrity of Baptism.
Nor does it matter that in the name of one Person another is implied, as the name of the Son is implied in that of the Father, or that he who mentions the name of only one Person may believe aright in the Three; because just as a sacrament requires sensible matter, so does it require a sensible form. Hence, for the validity of the sacrament it is not enough to imply or to believe in the Trinity, unless the Trinity be expressed in sensible words. For this reason at Christ's Baptism, wherein was the source of the sanctification of our Baptism, the Trinity was present in sensible signs: viz. the Father in the voice, the Son in the human nature, the Holy Ghost in the dove.
Reply to Objection 1. It was by a special revelation from Christ that in the primitive Church the apostles baptized in the name of Christ; in order that the name of Christ, which was hateful to Jews and Gentiles, might become an object of veneration, in that the Holy Ghost was given in Baptism at the invocation of that Name.
Reply to Objection 2. Ambrose here gives this reason why exception could, without inconsistency, be allowed in the primitive Church; namely, because the whole Trinity is implied in the name of Christ, and therefore the form prescribed by Christ in the Gospel was observed in its integrity, at least implicitly.
Reply to Objection 3. Pope Nicolas confirms his words by quoting the two authorities given in the preceding objections: wherefore the answer to this is clear from the two solutions given above.
Source: Summa Theologica (New Advent)