The problems that you face with your question is that two people with even identical machines might experience significantly different results.
If one person downloads a lot off the Internet, doesn't trim Windows proccesses, doesn't run system cleanup and defrag regularily, has a bunch of background appplications running and is pushing a big high resolution monitor their perfomance will be significantly less than someone doing the opposite on the same machine. So, to offer a minimum system then means that you also have to define your individual habits.
The second problem is your requirement for self shadows/smooth shadows which is a big frame killer. While most reasonably modern "good" computers of the past few years should run the game without issue with all other settings maxed, this single requirement pushes you from "good" computer to "leading edge" and the budget that goes along with that distinction.
In general I'd say a computer (just the box) priced in the 900-1200 range will handle AH without a problem regardless of your habits but when you take that next step to smooth shadows then the price rises exponentially as you have to move to crossfire/SLI with a bigger PSU, better cooling and all that goes with that.