The opamp can oscillate and drive a "charge pump" made with a couple of diodes and a couple of capacitors. Then the negative DC output will be about -3.6V if the load current is low.
Use an ICL7660 charge pump IC to create a negative DC supply voltage for the opamp. The output will be -5.0V if the load current is very small and will drop due to its 23 ohms source resistance. So a 10mA load will drop the output voltage to -4.77V.