Oh you want more math?
On the high end 162mm/hr on my 6x12m roof would be dumped into the head
Discussion
For practical purposes, Hazen–Williams equation can be used for this instance. I found head loss (which is the term that total mechanical energy that is lost due to friction; measured by length) as 2,5 meters. With the initial height of 2,5 m you mentioned; 5 meters tall of a head is needed. You can check my math if you want.
Sorry for the late response. 
Late response? I have been playing with this for 3 months lol
You kinda confirmed what i thought my head needs to be larger/ taller than my water collection. I might need to use a a jet pump which i can i was just hoping to not worry about more complicated machines