what causes the engine to run rough with ac on at idle runs good without ac on

These 2.4 four cyl. engines are not very powerful and you will notice a idle the AC stops working and they labor at idle on the automatics. They have designed the AC to turn off as not to over tax the engine. They all do it to a degree.If the Idle was adjusted higher you would have hold your foot on brake harder and the transmission would overheat unless you have a manual transmission then it is possible to do this.