I am working in a research project (not PhD and after Master's degree). I have completed a work which my adviser thinks is publishable. But he has such a heavy schedule that he hardly has time to respond. Can I take the whole charge of writing the manuscript and communicating the paper with his permission. (I have not yet asked him.)
There's no problem at all with a student being the principal author (in the sense of “being in charge of the writing”) of an academic paper. In fact, I think it's very good that you consider doing so. Here are, however, some thoughts on the matter:
Discuss this with your advisor before writing anything. Once both of you have established that there is sufficient matter for publication, and that you want to publish it, you should make plans for doing so. It includes determining (right now!) the authors list, the order of authors on the list, and plan who will do what.
The corresponding author (or corresponding authors), often indicated by an asterisk in the authors list and their contact details in a footnote, do not necessarily need to be the person(s) who actually did most of the writing. The contact author is the one who is the most central to the project, and who will be able to deal with correspondence about it in the near and not-so-near future. For that reason, some groups (including mine) do not consider students a good choice for corresponding author: in two years, they may be gone to a new job and not interested any more with the paper's research.
A few more comments for first time authors:
If you've never written a paper before, don't underestimate the task. Writing a paper from A to Z is a complex project, even for experienced authors. It requires using many different skills, and is truly an exercise in both scientific reasoning and communication, with very specific constraints (style, formatting, length limits, …). Assign a lot of time to writing, and don't hesitate to get help if you're stuck!
If in doubt while writing, put yourself in the reader's shoes. And specifically, a non-specialist reader's shoes. That should be the guiding principle.