The 1801 presidential transition
On February 17, 1801, Thomas Jefferson is elected the third president of the United States. The election constitutes the first peaceful transfer of power from one political party to another in the United States.
It was said Adams so bitterly resented that Jefferson had bested him in the election that he refused to come to his aid, and that if the truth were known he preferred Burr. But Adams expressed no enmity toward Jefferson; and in private correspondence, he and Abigail both said they preferred---and expected---Jefferson to be the one chosen.
It was not that they had recovered their former affection for Jefferson, but they knew his ability, "all the splendid talents, the long experience," in Adams's words....
On January 1, 1801, the Adamses held the first New Year's Day reception at the President's House. Several days later, they invited Jefferson to dine, one of several events that belie claims made then and later that Adams and Jefferson refused to speak.
"Mr. Jefferson dines with us and in a card reply to the President's invitation, he begs him to be assured of his homage and his high consideration," Abigail wrote Thomas[Adams] in a letter dated January 3. 1801. Adams having made the overture, Jefferson had responded graciously (page 556).
Adams didn't stay for Jefferson's inauguration.
On inauguration day, Wednesday, March 4, John Adams made his exit from the President's House and the capital at four in the morning, traveling by public stage under clear skies lit by a quarter moon. He departed eight hours before Thomas Jefferson took the oath of office at the Capitol, and even more inconspicuously than he had arrived...
To his political rivals and enemies Adams's predawn departure was another ill-advised act of a petulant old man. But admirers, too, expressed disappointment.
A correspondent for the Massachusetts Spy observed in a letter from Washington that numbers of Adams's friends wished he had not departed so abruptly. "Sensible, moderate men of both parties would have been pleased had he tarried until after the installation of his successor. It certainly would have had a good effect."
...After so vicious a contest for the highest office, with party hatreds so near to igniting in violence, a peaceful transfer of power seemed little short of a miracle. If ever a system was proven to work under extremely adverse circumstances, it was at this inauguration of 1801, and it is regrettable that Adams was not present....
No president having ever been defeated for reelection until then, there was no tradition of a defeated president appearing at the installation of the winner. It's also quite possible that Adams was not invited to attend, or made to feel he was not welcome....(page 564).