Not including student who contributed very little as coauthor of paper I am a researcher in condensed-matter physics, and I recently started a research project with another professor, for which I hired a student for a six-month undergraduate internship. The student was under my supervision. During the internship she showed good motivation, but did not make any actual contribution to the progress of the project.

Finally, she started a Ph.D. with someone else. I personally took over the project, taking care of both technical and conceptual parts, which requires a lot of time and work. 

I kindly invited her multiple times to double check some of my work, but she had not done anything on the project since the end of the internship.

In the past few months she started having an attitude that I find disturbing. She replies to emails once in a while, proposing new conceptual points to explore, as if I were the student and she was the supervisor. On top of that, she ignores the invitations to double check my calculations, which would require actual work on her side. 

Given that she did not make any actual contribution to the project, I am inclined to not include her as an author of any eventual publication. I think that it would be definitely fair to include her in the acknowledgments, but unfair to include her as an author. 

What do you think about this? How would you inform her of this decision?
This doesn't seem like an especially unusual situation to me. You had a student for a fixed-term internship, during which time she pursued an angle that did not result in a publication. The student then moved on to other things, and so did you. You kept her informed of your work that is related to what she did as a professional courtesy, but she isn't interested in participating after the internship ends (nor does she have an obligation to). This kind of thing happens often, and nobody behaved badly here...

Do you have any indication that she expects to be a co-author on the eventual publication? (Corresponding with you about the work out of interest doesn't mean she expects to be an author.) She may not have any expectation of authorship credit.

If at some point she asks about authorship, I think you can just gently point out that her work did not lead to publication, and that you started working on the current approach (that did lead to publication) after she left. 

If you do get the feeling that she expects authorship but she hasn't explicitly said so, you could say something to her like, "it has been interesting corresponding with you on this, X and I will be sure to thank you in the acknowledgement section of the eventual publication that he and I write." (If indeed you decide that you do want to do so.)