I don’t know with the switches and rail you are using. However, I can tell you that the minimum distance for HO scale between tracks is 2 inches. That’s an NMRA standard. Maybe you can convert that for your situation? I hope that helps.

at least 3.5″. probably 4″ is better to account for your eventual curves…and depending on the equipment you are going to run, you may have to allow more for overhang of the rolling stock, passenger cars and larger steam locomotives as they traverse the curves and maybe pass eachother as trains meet at or on a curve.