yes they can to the nation, but it comes back to the old quandary that Romney faced - I can run to win the white house or I can run to win the GOP primary - but those are two different campaigns.
Nixon said something to the affect that you run to the right to win the primary, you runt to the left to win the general, and you government from the middle. That seemed to work for him but I don't know if that's valid today.