Bidding a very special hand
My concerns are:


That the opponents aren't in this auction yet despite a ton of distribution, and so suits are likely to be breaking badly for us; and
My hand is far too strong to make a descriptive splinter call and appoint partner captain of the hand. 


Therefore I wish to take advantage of the opponents silence to describe my hand, so partner can make an intelligent bid again or double decision on the next round. Therefore I bid a simple 3D.

This bid:


Is forcing one round (by virtue of being a new suit by responder); 
Promises strength to play 3H or 4D, or better, safely;
Promises a fifth heart (or slam interest, or both); and 
Promises either a fourth diamond or hidden club support. 


It is intended to give partner the greatest possible room to complete the description of his hand, which so far sounds like the only balanced hand at the table with 3-4-2-4.

It sounds like RHO is long in clubs and diamonds, and LHO is long in hearts and clubs, and no-one is holding better than honour-fourth in spades. At least two, if not three or four, spades are missing from this deck unless the opponents are red against white.

What is the vulnerability, and who is the dealer?

Update From OP above, both vulnerable and partner dealt.

Continuation. Partner will take 3D for a game try and bid one of:


3H to decline - you will continue with 3S game forcing, showing a control in spades, requesting partner to show a control. If partner shows the club ace you will continue with 5NT GSF as below;
4H to accept, no extra values - you will continue with 4S control showing; if partner then bids 5C you will bid 5NT GSF and continue to 7 if he shows 1 with or without a side king.
4C to show the club control - continue with 5NT GSF as above.
3S to show the spade control then bid 4D to reveal the missing club control. If partner then bids 4H to deny it, you will have to guess - and rue not responding 1D originally.


Note how much easier the auction is if the original response is 1D receiving a raise; hearts can be cue-bid without ambiguity to locate the other missing card.  Ease in correctly deciding between 5 or 6 r 7 is worth far more than the difference between a major or minor denomination.