I am going to design a circuit which should be able to achieve the following function:
If the voltage applied to this circuit is negative, exactly the same negative voltage should be sent to the output; If the voltage applied to the circuit is positive, zero voltage should be delivered instead.
What you need is a negative half wave rectifier without the diode drop. Place the rectifying diode inside the opamp feedback loop to get around the diode drop problem. https://en.wikipedia.org/wiki/Precision_rectifier