I spent a lot of time thinking about this question and a few related questions yesterday, and spent some of that time reading some fine books. Here's what I scratched out at my favourite cafe yesterday.
If we want to get what happens at the Cross right, we do well to remind ourselves of some Trinitarian building blocks. So, we must affirm the unity of God, while carefully preserving the distinction of the Persons. No reckoning of the Atonement will suffice which rips apart the identity of God.
So, God propitiates himself in the Son's voluntary obedient sacrifice to the Father who both sends and offers him, made in the Spirit. It is the united act of the Triune God, yet it is in some sense an inner-Trinitarian act, to which we are parties by the assumed humanity of Jesus the God-Man. The Atonment only has effect in that Jesus is the God-Man, one persons, two natures, distinct yet not divided. His human nature is our sinful humanity, which he assumes for us, heals and sanctifies for us, and offers up to the Father for us.
At the cross, the impossible occurs - 'God dies'. For God is Life, and Life in himself, and the Living God. Yet Jesus truly dies. He dies 'in his humanity', which must be in no way separated from his divinity. That is why we do best to affirm that the person of the Son dies in (or with respect to) his human nature, and not to say that the 'human nature dies'.
Since Jesus is fully God, and the fullness of God, in some way beyond my understanding and comprehension, the Living God tastes, sees, experiences, participates, indeeds suffers death in the assumed humanity of the Son.