Based on your description of the symptoms, the most likely cause for the hesitation/ rough idle after sitting for several hours would be a fuel pressure loss, possibly through a leaking fuel injector.

The fues system on your car contains pressurized fuel; when the car is shut off, it must maintain pressure so that there is fuel immediately available to the injectors the next time the engine is started. If the fuel pressure is bleeding down overnight, especially if the fuel loss is into a cylinder through an injector that drips after shutdown, it will cause symptoms much like you describe.

To check for this condition, your mechanic will want to attach a fuel pressure gauge to the service port on the fuel rail and allow the vehicle to sit overnight to check for any pressure loss.

Since you are using the correct octane fuel for the vehicle, it is not a fuel volatility issue you are experiencing.

Although a vacuum leak may be present, it is not likely the cause of the symptoms you describe. The engines used in this model are prone to developing intake plenum gasket leaks; typically hte symptoms you will see with a vacuum leak of this type would be a rough idle (which does not go away), constant hesitation, fluctuating idle speeds, and a malfunction indicator lamp illuminated with a lean fuel mixture and/ or idle speed control fault stored.

The "emissions" light being on may also yield some important clues upon investigation. According to the service information I have, this is a malfunction indicator; when it is on, there should be service codes and a frozen snapshot of the vehicle's data stream taken at the moment of the initial failure stored in memory. A leaking gas cap is only one of literally hundreds of possible causes for this light to come on; your mechanic will want to connect a scan tool to the vehicle to retrieve the information that is stored associated with this light; this will give more pieces to the puzzle.

Unfortunately without having any test equipment, there really is not much you can do at home to diagnose or repair this type of problem; usually problems of this type are not to difficult to resolve with the right test equipment, provided you are willing to allow your mechanic to have the vehicle overnight a couple of times to diagnose the condition and verify any repairs.

I hope this is helpful; if so, please click my accept button! Thanks!