Do Atheists hope there is no God? I’ll give defining those words a try, but I cannot confirm that you will be satisfied with my definitions.
Atheist - Someone who does not think that God exists
God - That which atheists do not think exist
Exist - That which atheists do not think God exhibits
But seriously, I will try to give it a go.
God - A transcendent deity who created the universe and involves itself in the universe (still a very ambiguous term, sorry)
Exist - something that
could have an a effect on everything else that exists
Atheist - Someone who does not believe that God exists
There’s still many many problems with these definitions, but I am intrigued to see what you do with them.