Even though there are three persons in the Sacred Trinity, there is complete unity within God because of His simplicity of His essence, substance or nature.
St. Athanasius’s Creed
The content of the Athanasian Creed stresses the affirmation of the Trinity in which all members of the Godhead are considered uncreated and co-eternal and of the same substance. In the affirmation of the Trinity the dual nature of Christ is given central importance. As the Athanasian Creed in one sense reaffirms the doctrines of the Trinity set forth in the fourth century at Nicea, in like manner the strong affirmations of the fifth-century council at Chalcedon in 451 are also recapitulated therein.
Whether God is the same as His essence or nature?
God is the same as His essence or nature. To understand this, it must be noted that in things composed of matter and form, the nature or essence must differ from the "suppositum," because the essence or nature connotes only what is included in the definition of the species; as, humanity connotes all that is included in the definition of man, for it is by this that man is man, and it is this that humanity signifies, that, namely, whereby man is man. Now individual matter, with all the individualizing accidents, is not included in the definition of the species. For this particular flesh, these bones, this blackness or whiteness, etc., are not included in the definition of a man. Therefore this flesh, these bones, and the accidental qualities distinguishing this particular matter, are not included in humanity; and yet they are included in the thing which is man. Hence the thing which is a man has something more in it than has humanity. Consequently humanity and a man are not wholly identical; but humanity is taken to mean the formal part of a man, because the principles whereby a thing is defined are regarded as the formal constituent in regard to the individualizing matter. On the other hand, in things not composed of matter and form, in which individualization is not due to individual matter—that is to say, to "this" matter—the very forms being individualized of themselves—it is necessary the forms themselves should be subsisting "supposita." Therefore "suppositum" and nature in them are identified. Since God then is not composed of matter and form, He must be His own Godhead, His own Life, and whatever else is thus predicated of Him.
Whether essence and existence are the same in God?
A thing can be in a genus in two ways; either absolutely and properly, as a species contained under a genus; or as being reducible to it, as principles and privations. For example, a point and unity are reduced to the genus of quantity, as its principles; while blindness and all other privations are reduced to the genus of habit. But in neither way is God in a genus. That He cannot be a species of any genus may be shown in three ways.
First, because a species is constituted of genus and difference. Now that from which the difference constituting the species is derived, is always related to that from which the genus is derived, as actuality is related to potentiality. For animal is derived from sensitive nature, by concretion as it were, for that is animal, which has a sensitive nature. Rational being, on the other hand, is derived from intellectual nature, because that is rational, which has an intellectual nature, and intelligence is compared to sense, as actuality is to potentiality. The same argument holds good in other things. Hence since in God actuality is not added to potentiality, it is impossible that He should be in any genus as a species.
Secondly, since the existence of God is His essence, if God were in any genus, He would be the genus "being", because, since genus is predicated as an essential it refers to the essence of a thing. But the Philosopher has shown (Metaph. iii) that being cannot be a genus, for every genus has differences distinct from its generic essence. Now no difference can exist distinct from being; for non-being cannot be a difference. It follows then that God is not in a genus.
Thirdly, because all in one genus agree in the quiddity or essence of the genus which is predicated of them as an essential, but they differ in their existence. For the existence of man and of horse is not the same; as also of this man and that man: thus in every member of a genus, existence and quiddity—i.e. essence—must differ. But in God they do not differ, as shown in the preceding article. Therefore it is plain that God is not in a genus as if He were a species. From this it is also plain that He has no genus nor difference, nor can there be any definition of Him; nor, save through His effects, a demonstration of Him: for a definition is from genus and difference; and the mean of a demonstration is a definition. That God is not in a genus, as reducible to it as its principle, is clear from this, that a principle reducible to any genus does not extend beyond that genus; as, a point is the principle of continuous quantity alone; and unity, of discontinuous quantity. But God is the principle of all being. Therefore He is not contained in any genus as its principle.
Whether God is altogether simple?
The absolute simplicity of God may be shown in many ways.
First, from the previous articles of this question. For there is neither composition of quantitative parts in God, since He is not a body; nor composition of matter and form; nor does His nature differ from His "suppositum"; nor His essence from His existence; neither is there in Him composition of genus and difference, nor of subject and accident. Therefore, it is clear that God is nowise composite, but is altogether simple.
Secondly, because every composite is posterior to its component parts, and is dependent on them; but God is the first being, as shown above (I:2:3).
Thirdly, because every composite has a cause, for things in themselves different cannot unite unless something causes them to unite. But God is uncaused, as shown above (I:2:3), since He is the first efficient cause.
Fourthly, because in every composite there must be potentiality and actuality; but this does not apply to God; for either one of the parts actuates another, or at least all the parts are potential to the whole.
Fifthly, because nothing composite can be predicated of any single one of its parts. And this is evident in a whole made up of dissimilar parts; for no part of a man is a man, nor any of the parts of the foot, a foot. But in wholes made up of similar parts, although something which is predicated of the whole may be predicated of a part (as a part of the air is air, and a part of water, water), nevertheless certain things are predicable of the whole which cannot be predicated of any of the parts; for instance, if the whole volume of water is two cubits, no part of it can be two cubits. Thus in every composite there is something which is not it itself. But, even if this could be said of whatever has a form, viz. that it has something which is not it itself, as in a white object there is something which does not belong to the essence of white; nevertheless in the form itself, there is nothing besides itself. And so, since God is absolute form, or rather absolute being, He can be in no way composite. Hilary implies this argument, when he says (De Trin. vii): "God, Who is strength, is not made up of things that are weak; nor is He Who is light, composed of things that are dim."
The Sacred Trinity is simply part of the Divine Essence of God.