I just read your post about "procession". I must say that it was well put. It also leads me to ask you a question.... What is the difference between Divine Essence and the relations ad intra? I know there must just be some angle I am not getting about the question, but I just cant wrap my mind around it. Help if you can. Thanks, chrisWell, the "divine essence" is basically the nature of the three persons. The divine essence, or the nature, of the Father is divine. The essence of the Son is divine. The essence of the Holy Spirit is divine. One reason we can say that God is still one, even though He is three persons, is because he is one divine essence. All three persons share equally in the divine essence.
"Ad intra" however, refers not to the essence of the Persons, but of their inner-relatedness. An illustration may be helpful: This phrase is also used in reference to Church documents. For example, an "ad intra" document of the Church would be Sacrosanctum Concilium, because it governs her liturgy, her inner life of worship. It is concerned with her inner workings, not with her relations with things outside herself. Documents that pertain to her relationship with things outside of herself are called "ad extra" documents. An example of one of these would be Nostra Aetate, because it deals with the relationship between the Church and non-Christian religions.
Similarly, "ad intra" describes the inner life of the Trinity. For example, we know that the First Person relates to the Second Person as "Father" to "Son," and vice versa. They both relate to the Third Person as "Holy Spirit." Furthermore, the Father "begets" the Son, and the Spirit "procedes" or is "spirated" from the Father and the Son. All of this describes the Trinity "ad intra".
When we speak of the Trinity "ad extra" we speak of how God relates to man (not to Himself). Another name for this "ad extra" point of view is "economic Trinitarianism." So, for example, we too, by our incorporation into the Son by grace, relate to the First Person as "Father." We relate to the Second Person as "Savior," "Redeemer," "Brother," etc. We relate to the Third Person as "Sanctifier," "Forgiver," "Gift-Giver," etc. All of this describes the Trinity "ad extra".
So, to review, "divine essence" refers to the nature of God, in which all three Persons share equally. "Ad intra" refers to the inner life of the Trinity, or how God relates to Himself.
I hope that helps.