You guys are making it far to difficult. Use this method, its very easy and I got it done first try. Thank Sasholon for the link.
http://www.youtube.com/watch?v=gVfPfhBAVqA
Yes, easiest way.
If you don't want to watch the video, here's the rundown.
Take some gas bombs and mines with you to the vault. If necessary, buy some demolition charges from the vending machine.
After you get through all the speakers and holograms and finally make it to the vault, you'll want to open the vault and sneak in partially- make sure not to go all the way in because the door will shut. Access both terminals and shut down security. Also use melee to destroy the generator under one of the terminals. Then go and plant your mines in front of the steps that Elijah will come down. You can save and do a dry run if you wish to see where he walks.
Then after you plant your stuff, set your difficulty to V. Easy so elijah will be weaker, go into the vault, and steal everything. Then go to the far end terminal, read only the first message, and exit. Talk to elijah, and goad him into coming down to see you. You don't have to pass checks, you can just say you'll cooperate with him and he comes down.
As soon as you exit dialog, go over to the forcefield he has now activated and crouch walk against it. He'll deactivate the shields for a split second so he can pass through. You will then be flung through the other side of the field and he'll walk into your trap and die.
Now, you just walk over to the elevator and voila! You made it!