I have heard a religious definition of death as separation from God. This makes the death of Christ--God himself--a matter of incomprehensible existential crisis. Adding to this the imputation of sin on the Son, and the wrath of the Father poured out on him, it makes me wonder if this event does not affect the relationship of the members of the trinity in some permanent way. Is there any evidence of this in scripture?
I have never thought about it that way. Here is my view:
The idea of separation is the loss of fellowship and peace due to sin. Christ, as a man lost this peace and fellowship from God when becoming sin for us, but as God he lost nothing and can never loose anything. As God, Christ could endure the whole weight of sin, which as a man he never could have. As God he could bring the human soul and body back up from the separation. I see no temporary or permanent change in the Son with the Father. This was human nature taking the punishment of human sin, but as God this could not destroy the God-Man. The result of this work was the exaltation of the human nature of Christ into his glorious kingdom. The Eternal Son was not exalted, per se, as God can’t be made higher than He eternally is.
I think the immutability of God is the controlling idea here.
Phillippians 2 says that because of Jesus' death, God the Father exalted Jesus. As 2:9 specifically says (and note the therefore):
This theme is picked up by Athanasius, in De Incarnacione. His soteriology says our redemption was a gift that the Father gave the Son for his willingness to go through this separation. It's probably my favorite theory of how salvation works. The idea is that being God, both God the Father and God the Son already had everything. But out of an abundance of love, God the Father gave God the Son the opportunity to redeem that which was lost.