This is my interpretation of this gorgeous, heartbreaking tune. The original guitar
are a bit weird if you wanna play it and sing a long to the song so I made a tab
for strumming/picking.
I actually think the 4th chord sounds better as an C#m instead of E during the verses 
but I like the sound of the open strings there and they are relative major/minor anyway,
doesn't really make a big difference. Hope you like it. It sounds good to me but please 
comment if you have anything to add!

Artist: The National
Album: High Violet (2010)

A (x02220@1)    C#m (x13321@4)           F#m (244222@1)     ESorrow (022100@1)found me when I was young

A (x02220@1)    C#m (x13321@4)           F#m (244222@1)    ESorrow (022100@1)waited, sorrow won

A (x02220@1)         C#m (x13321@4)         F#m (244222@1)    ESorrow (022100@1)they put me on the pill

A (x02220@1)        C#m (x13321@4)            F#mIt's (244222@1)in my honey it's in my milk

E (022100@1)                  A (x02220@1)        C#mDon't (x13321@4)leave my hyper heart alone

F#mOn (244222@1)the water

E (022100@1)         A (x02220@1)         C#m (x13321@4)      F#mCover (244222@1)me in rag and bones, sympathy

E (022100@1)                     A (x02220@1)      C#m (x13321@4)   F#mCause (244222@1)I don't wanna get over you

E (022100@1)               A (x02220@1)      C#m (x13321@4)   F#mI (244222@1)don't wanna get over you

E (for one extra measure)

A (x02220@1)        C#m (x13321@4)       F#m (244222@1)     ESorrows (022100@1)my body on the waste

A (x02220@1)         C#m (x13321@4)          F#m (244222@1)     ESorrows (022100@1)the girl inside my cake

A (x02220@1)         C#m (x13321@4)       F#m (244222@1)     EI (022100@1)live in a city sorrow build

A (x02220@1)        C#m (x13321@4)             F#mIt's (244222@1)in my honey, it's in my milk

E (022100@1)                  A (x02220@1)        C#mDon't (x13321@4)leave my hyper heart alone

F#mOn (244222@1)the water

E (022100@1)         A (x02220@1)         C#m (x13321@4)      F#mCover (244222@1)me in rag and bones, sympathy

E (022100@1)                     A (x02220@1)      C#m (x13321@4)   F#mCause (244222@1)I don't wanna get over you

E (022100@1)                A (x02220@1)      C#m (x13321@4)   F#mI (244222@1)don't wanna get over you

E (022100@1)                  A (x02220@1)        C#mDon't (x13321@4)leave my hyper heart alone

F#mOn (244222@1)the water

E (022100@1)         A (x02220@1)         C#m (x13321@4)      F#mCover (244222@1)me in rag and bones, sympathy

E (022100@1)                     A (x02220@1)      C#m (x13321@4)   F#mCause (244222@1)I don't wanna get over you

E (022100@1)               A (x02220@1)      C#m (x13321@4)   F#mI (244222@1)don't wanna get over you

Same chords during the "choir" parts, stay on E for an extra measure and then end on A.