If the president has not committed a crime, he cannot be impeached. If the president has committed a crime, he can be impeached. The constitutional phrase “high crimes or misdemeanors” is deliberately open-ended. It is a kind of punt, meant to give broad discretion to the House of Representatives. Impeachment is a political act, unbounded by law. Even if the president has committed a clearly impeachable act, the House of Representatives has discretion to refuse to impeach him.