It really depends on what type of corals you would like to keep. You mention "hard" corals, but are you wanting LPS or SPS corals? I have a single BoostLED PAR30 with 60 degree optics over a 15XH (13"x13"x20" deep) and the corals I have in it are doing great under only 10W of LED light, but they are relatively undemanding corals with regards to light - xenia, zooanthids, toadstool leathers, mushrooms, open brain and trumpet coral. The bulb only cost $65 plus shipping and I am quite pleased with it, but I am also not trying to grow any light-demanding corals like most SPS species.
You could aquascape the 14g correctly (center rock stack...) to allow you to try a PAR38 LED bulb with either 40 or 60 degree optics, and you could probably keep certain corals that demand more light, especially closer to the bulb. RapidLED makes a nice PAR38 for around $100, but the tightest lenses for it are only 60 degrees, so you would have to mount the bulb closer to the tank to gain enough par, and the light spread would be good over a 12"x 12" area.
Read up on the various PAR38 LED bulbs and the tanks they are being used on - you will find lots of info. Ecoxotic Panorama LED modules could also be an option, but they are $100 each, and you would probably need 2-3. You could also spend more for an AI Sol nano or other really nice fixture, but don't let the sticker shock get to you - these LED fixtures have a long lifespan, and you will recoup the cost by not buying new bulbs and in saving energy!