I knew that Thomas Jefferson was the second Vice President, but I had always assumed that Adams chose him as a running mate. (Oh, assuming and the trouble it will get you into.)
The truth is….
Presidents didn’t always run with an official running mate – The Constitution in 1796 required presidential electors to place the names of two individuals on their ballots; the candidate with the highest vote count, if a majority, became the president and the runner up the vice president.
This proved to be a bit of a predicament in 1796, when Federalist John Adams won the majority vote and Democratic -Republican Thomas Jefferson was the runner up. (This is the only instance that the President & VP were not from the same party.) Naturally , the two clashed over issues such as states’ rights and foreign policy.

Adams (Federalist) & Jefferson (Republican_Democrat)
I learned about this Constitution loop hole from Ken Burns’ documentary on Thomas Jefferson. I would recommend viewing this documentary to learn more or click here