What experiment did Rappaccini perform on his daughter?

Rappaccini is a scientist who toys with nature, experimenting with poisonous flowers and plants. He transfers the poison to his daughter Beatrice, and she herself becomes poisonous, deadly to anyone who would come near her. She is both beautiful and dangerous, like the gorgeous flowers in the garden. The poison is such a part of who she is, woven in the fibers of her being, that without the poison in her system, she would die. And this is what in fact happens. When Giovanni, a man who had fallen in love with her from a distance, gives her an antidote to rid her body of the poison, it kills her, because she IS poison.

How does Dr. Rappaccini conduct his scientific experiment in "Rappaccini's Daughter"?

The purpose of empirical investigation is to seek information and gain knowledge from actual experience and is based on observation instead of conjecture or belief. Dr. Rappaccini evidently believes that, just as his daughter's physical human nature has been affected by the poisons with which she is surrounded in the deadly garden, another human being who would come into constant contact with her would be likewise and equally affected. In order to test this theory empirically, the doctor seems to curate situations in which the handsome, young tenant of the building, Giovanni Guasconti, comes into direct contact with Rappaccini's beautiful daughter, Beatrice. In this way, by repeatedly exposing Giovanni to his poisonous daughter, the doctor will be able to see, empirically, if and how Giovanni begins to change as a result of his exposure. Rappaccini will be able to observe the changes within Giovanni's nature—the young man will be less negatively affected by Beatrice's poison breath, and so forth, and he will also begin to have a similarly deadly effect on creatures like insects or flowers that do not share his poisonous nature—and so this is how he uses empirical investigation in order to conduct his experiment.