Objection 1. It would seem that to believe is not to think with assent. Because the Latin word "cogitatio" [thought] implies a research, for "cogitare" [to think] seems to be equivalent to "coagitare," i.e. "to discuss together." Now Damascene says (De Fide Orth. iv) that faith is "an assent without research." Therefore thinking has no place in the act of faith.
Objection 2. Further, faith resides in the reason, as we shall show further on (4, 2). Now to think is an act of the cogitative power, which belongs to the sensitive faculty, as stated in I, 78, 4. Therefore thought has nothing to do with faith.
Objection 3. Further, to believe is an act of the intellect, since its object is truth. But assent seems to be an act not of the intellect, but of the will, even as consent is, as stated above (I-II, 15, 1, ad 3). Therefore to believe is not to think with assent.
On the contrary, This is how "to believe" is defined by Augustine (De Praedest. Sanct. ii).
I answer that, "To think" can be taken in three ways. First, in a general way for any kind of actual consideration of the intellect, as Augustine observes (De Trin. xiv, 7): "By understanding I mean now the faculty whereby we understand when thinking." Secondly, "to think" is more strictly taken for that consideration of the intellect, which is accompanied by some kind of inquiry, and which precedes the intellect's arrival at the stage of perfection that comes with the certitude of sight. On this sense Augustine says (De Trin. xv, 16) that "the Son of God is not called the Thought, but the Word of God. When our thought realizes what we know and takes form therefrom, it becomes our word. Hence the Word of God must be understood without any thinking on the part of God, for there is nothing there that can take form, or be unformed." In this way thought is, properly speaking, the movement of the mind while yet deliberating, and not yet perfected by the clear sight of truth. Since, however, such a movement of the mind may be one of deliberation either about universal notions, which belongs to the intellectual faculty, or about particular matters, which belongs to the sensitive part, hence it is that "to think" is taken secondly for an act of the deliberating intellect, and thirdly for an act of the cogitative power.
Accordingly, if "to think" be understood broadly according to the first sense, then "to think with assent," does not express completely what is meant by "to believe": since, in this way, a man thinks with assent even when he considers what he knows by science [Science is certain knowledge of a demonstrated conclusion through its demonstration.], or understands. If, on the other hand, "to think" be understood in the second way, then this expresses completely the nature of the act of believing. For among the acts belonging to the intellect, some have a firm assent without any such kind of thinking, as when a man considers the things that he knows by science, or understands, for this consideration is already formed. But some acts of the intellect have unformed thought devoid of a firm assent, whether they incline to neither side, as in one who "doubts"; or incline to one side rather than the other, but on account of some slight motive, as in one who "suspects"; or incline to one side yet with fear of the other, as in one who "opines." But this act "to believe," cleaves firmly to one side, in which respect belief has something in common with science and understanding; yet its knowledge does not attain the perfection of clear sight, wherein it agrees with doubt, suspicion and opinion. Hence it is proper to the believer to think with assent: so that the act of believing is distinguished from all the other acts of the intellect, which are about the true or the false.
Reply to Objection 1. Faith has not that research of natural reason which demonstrates what is believed, but a research into those things whereby a man is induced to believe, for instance that such things have been uttered by God and confirmed by miracles.
Reply to Objection 2. "To think" is not taken here for the act of the cogitative power, but for an act of the intellect, as explained above.
Reply to Objection 3. The intellect of the believer is determined to one object, not by the reason, but by the will, wherefore assent is taken here for an act of the intellect as determined to one object by the will.
Source: Summa Theologica (New Advent)