A method for desulfurizing natural gas includes contacting the natural gas with an adsorbent which preferentially adsorbs at least one of hydrogen sulfide, COS, sulfur odorants, or combinations thereof, at a selected temperature and pressure, thereby producing desulfurized natural gas and an at least one of hydrogen sulfide/COS/sulfur odorant/combinations thereof-rich adsorbed component. The adsorbent includes a copper species adapted to form π-complexation bonds and direct metal-sulfur bonds with the at least one of hydrogen sulfide, COS, sulfur odorants, or combinations thereof, and wherein the preferential adsorption occurs by π-complexation and direct metal-sulfur bonding.