With this purchase, Qatar might be swapping soft power for military might. The gas-rich emirate gambled on the region-wide success of the Muslim Brotherhood in the years after the "Arab Spring" protests. But its strategy toppled with the military coup that removed Mohammed Morsi in Egypt. Qatar's neighbors also became increasingly suspicious of its support for Islamist movements throughout the Middle East, leading to one of the biggest diplomatic crises in the history of the Gulf monarchies.
Qatar's been unable to insulate itself from regional chaos through its diplomatic outreach, which has had substantial blowback. But Patriot missiles will do just fine: They're perhaps the most advanced projectile of their type, and have the ability to intercept incoming missiles and destroy enemy tanks and planes.
The Qatari monarchy has yet another hard power asset insuring its survival: Qatar is home to Al Udeid Air Base, one of the most important U.S. military installations in the Middle East.