What's the name of that song Edward plays on the piano in Twilight?

It's the one after Bella meets Edward's family that he plays on the piano. It's not "Bella's Lullaby"... I have "Bella's Lullaby" on my playlist & I heard the one in the movie & it's NOT the same one. So what's the name of the song in the movie & who does it? Thanks! :)