Bellas Lullaby or something else?

I have on song on my computer called Bella's Lullaby, but i'm not sure if it's the "real" song. It's because it's not the song that Edward Cullen plays for her in the movie Twilight. So i wanna know if eneyone know the EXACT song that he plays in the movie; what it's called and who plays it. Thank you (: