...One might actually argue that with such an automated ship, the only thing they would ever need is engineers. The automation takes care of commanding, piloting, researching, communicating and diplomacy. And the engineers take care of the automation!
Piloting you don't need for sure. You can fly that ship by voice command.
The need for a science department is pretty limited considering you could just transmit all the sensor and scan data to wherever you wanted.
You still need Command to make decisions. I'm sure you could program some sort of "command algorythm" or such, but I doubt you would make too many crewmembers comfortable with that.
Tactical again can, and probably should, be handled by the computer.
So Command personnel, Engineers, and Medical staff. Oh, and security staff to do... whatever it is they do.