Is it permissible to eat a deer killed by a rifle, and what is the legitimate method of hunting with it?

If you shoot a gazelle with a rifle and mention the name of Allah, and it dies from the shot, then it is permissible (halal). However, if you reach it while it is still alive, then it must be slaughtered (tadhkiyah). Birds shot with a rifle that die are permissible if Allah's name was mentioned when the shot was fired, based on the Prophet's (peace be upon him) saying: "Whatever causes blood to flow and the name of Allah is mentioned upon it, then eat." But if they are still alive with stable life, then they must be slaughtered and Allah's name mentioned over them. The mention of Allah's name must be at the time of firing the shot, not when loading the bullet into the rifle.