How do I go on about inviting my ex out?

My ex girlfriend lied to me and I got angry at her for it and told me not to speak to me, She dates this guy who hates me because he was jealous I dated her first (yeah..) anyways he spreads rumors about me saying I raped/beat her and than later starts telling her that I'm calling her a slut etc.. and eventually I message her and I asked if shes will to speak to me because I feel about about the argument (at this point I had no idea he was saying this ****) they break up and eventually I find out he said that stuff and I message her about it because she still wasnt talking too me and eventually we speak and hangout for the whole day and after weeks of awaiting for acceptance on facebook she finally adds me and I wanted to hangout with her, It's summer so shes off from school and I was wondering how I would go about inviting her over or too a movie or something?