What happened during the election of 1800 to lead to passage of the 12th amendment?

The 1800 election exposed a defect in the original formula in that if each member of the Electoral College followed party tickets, there could be a tie between the two candidates from the most popular ticket. Both parties planned to prevent this by having one of their electors abstain from voting for the vice presidential candidate to ensure a clear result. Jefferson managed to secure a majority of pledged electors, but the margin in 1800 was so slim that there was little room for error if the Democratic-Republicans were to avoid repeating the Federalists' miscues of 1796. Given the technical limitations of 18th-century communications, Democratic-Republicans electors in all states were left to assume that an elector in another state was the one responsible for casting the one abstention necessary to ensure the election of unofficial vice presidential nominee Aaron Burr to that office. All Democratic-Republicans electors in each state were so reluctant to be seen as the one responsible for causing outgoing President Adams to be elected as vice president that every Democratic-Republican elector cast a vote for both Jefferson and Burr, resulting in a tie.

Consequently, a contingent presidential election was held in the House of Representatives. Federalist-controlled state delegations cast their votes for Burr in an effort to prevent Jefferson from becoming president. Neither Burr nor Jefferson were able to win on the first 35 ballots. With help from Alexander Hamilton, the gridlock was finally broken on the 36th ballot and Jefferson was elected President on February 17, 1801. This prolonged contingent election, combined with the increasing Democratic-Republican majorities in both the House and the Senate, led to a consequential change in the nation's frame of government, the requirement of separate votes for president and vice president in the Electoral College and end in the 12th Amendment.