The majority of this sounds like keys with a lot of processing rather the 
guitar but playing the barre chords in Drop D sounds decent enough. 

Intro/Verse 
e|----------| 
B|----------|    I don't know how many times it actually plays but you get 
G|3---------|    the idea; it's Gm for the whole verse. 
D|5---------|  
A|5-(x120)--| 
D|5---------| 

If that's not exciting enough for you there's always the backing vocal melody 

e|----------------|                |-------------------| 
B|----------------|                |-------------------| 
G|----------------|  Or an octave  |-----7-10-8-7------| 
D|----------------|  higher        |---8----------10---| 
A|----5-8-6-5-3---|                |10-----------------| 
D|5-8-------------|                |-------------------| 

There are some extra harmonies on top of this through the second verse 

Chorus 
e|--------------------------------| 
B|--------------------------------|   
G|3-3-3-3-7-7-7-7-0-0-0-0-0-0-0-0-|   
D|5-5-5-5-8-8-8-8-1-1-1-1-1-1-1-1-|   
A|5-5-5-5-8-8-8-8-1-1-1-1-1-1-1-1-| 
D|5-5-5-5-8-8-8-8-1-1-1-1-1-1-1-1-| 

Gm - Bb - Eb in other words 


Bridge 

e|------------------|    
B|------------------|   This is a weird bit. Sounds like Russell's sliding back and forth 
G|------------------|   between these two notes while tremolo picking, and catching the chromatic 
D|5/8/5/8/5/8/5/8---|   notes in between. Knowing him it could well be a pitch shifting pedal though. 
A|------------------|   Play around with it a bit I'm sure you'll get something that's roughly correct. 
D|------------------| 

That bit also turns up quite low in the mix in the final bit 

e|------------------------------------------------10-11-12-13-14-15-15-15-15-| 
B|--------------------------------9-10-11-12-13-14---------------------------| 
G|-------------------8-9-10-11-12--------------------------------------------| 
D|5-6-7-8-9-10-11-12---------------------------------------------------------| 
A|---------------------------------------------------------------------------| 
D|---------------------------------------------------------------------------| 

More tremolo picking. The timing you can pick up from the song and you might find  
it easier to change on which string you play some notes. I'd say Russell's definitely  
using some pitch effects here because there's certainly a fifth above that last note 
but the fretboard on a telecaster isn't that long. Curse him and his fancy technology.  

e|-------------------------------------------------| 
B|---------------------------------------------6---| 
G|----3--------7---------------3-------7-----------| 
D|---------------------8---------------------------| 
A|-----------------6-----------------------6-------| 
D|5-------8----------------5-------8---------------| 

This last bit then carries on in to the final section and you finish on that Gm chord a few times for 
good measure.