How to apologize for declining an offer from a potential PhD advisor? I sent many emails to prospective PhD advisors looking for a position in their research groups. Recently, I got two offers from professors in different universities and I choose one of them as his research interests fit  more with my experience. Now, I have to apologize to the second one that I won't to be able to work with him. How to do that without leaving a bad impression ?
The best thing you can do is be honest and explain the situation. You thank the second professor for the offer, explain you accepted another offer and wish the second professor well with their future research.
This is a common situation and the second professor should be understanding.