Do you know the model number of your heat pump? Do you know if it has one stage or two stages? How was the W2 wire connected on the old thermostat?

The 8580 appears to be a single stage thermostat. If your heat pump has two stages then this thermostat is not going to be able to control the second stage. You may need to buy another thermostat.

The old thermostat had seven wires, the Black wire was connected to the E terminal. The w2 was connected to the W2 on the old thermostat. It is an old Goodman Heat pump that is installed at my mother in laws apartment. She is blind and 80 yrs old and is having trouble adjusting the thermostat. That is why I installed it. I was told that the W2 could be on the terminal with the E for the emergency heat. I cannot get to the pump itself as it is not accessible to me. The thermostat installation manual says that the W2 or the E wire go to the E/Aux on the new thermostat. Since it is a small apartment on the 3rd floor and Texarkana, AR it never really gets extremely cold and I doubt that Emergency heat would be necessary. I was told that I could place both wires on the E/AUX and this would just make it warmer faster if needed.

The wiring you described will work. The problem is if the low stage of the heat pump can't keep up then the heat strips and the second stage will turn on together. That wastes energy but perhaps in this location it does not happen very often if the thermostat setting is kept at a constant temperature.

It would be better if the funds were available to spend a little more money on a 2-stage thermostat.

Sprung for a two stage Honeywell Lyric 5 today. Thanks for your help.
Jim