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To be clear, these circumstances are the rare exception; don't take this as license to throw the two-week rule out the window. But if you find yourself in this situation, what's the best way to handle it? Two things are key:
1. First, you must be apologetic. The fact is that leaving in less than two weeks does violate professional convention. Right or wrong, it's seen as unprofessional. So if you're matter-of-fact about it, you'll come across as if you don't care. This will make matters worse. Instead, you want to be explicitly apologetic. Sounding genuinely sorry, even mortified, will make most people want to cut you some slack.
2. Offer to work extra hours in whatever remaining time you have. Also, be available for questions for a short period after you're gone. The whole reason employers expect two weeks notice is so that they have time to transition the work to someone new. If you go out of your way to help with that, it can negate the impact of the shorter notice period.
If you don't do the above, you'll burn bridges, jeopardize future references, and potentially even impact your professional reputation. But if you do, you'll likely be able to leave the situation on good terms.
Alison Green is the author of Managing to Change the World: The Nonprofit Leader's Guide to Getting Results. She is chief of staff for the Marijuana Policy Project, a nonprofit lobbying organization, where she oversees day-to-day management of the staff as well as hiring, firing, and staff development. Her writings have been published in the Washington Post, the New York Times, Maxim, and dozens of other newspapers. She blogs at Ask a Manager.