Howard Shore - Far Over The Misty Mountains Cold chords

This is how I worked out the chords of the original movie scene.
If you're playing the chords in Full-step-down tuning, they are played like Am, G 
and Em. (These easy chords sound quite good in Standard tuning as well, if you 
don't need the original key)
Improvements are welcome!

[ Tab from: http://www.guitaretab.com/h/howard-shore/340307.html ]
Gm           F               Gm
Far over the misty mountains cold
            F                Gm
To dungeons deep and caverns old
         Dm               F
We must away ere break of day
            Gm      Dm     Gm
To find our long forgotten gold

                       F      Gm        
The pines were roaring on the height
               F              Dm  
The winds were moaning in the night
                             F
The fire was red, it flaming spread
               Gm      Dm          Gm
The trees like torches blazed with light