Short answers, pulled from the story.
Øyvind Burrau derived the first mathematically complete quantum description of a simple chemical bond in 1927. This work focused on one electron in the hydrogen molecular ion H2+ and proved that electrons behave as waves when they move between atomic nuclei.
Covalent bonds involve two electrons drawn into the space between two atomic nuclei while ionic bonding operates by transferring electrons completely instead of sharing them. Richard Abegg proposed his rule stating the difference between maximum and minimum valencies is often eight which helped scientists understand how one atom might transfer an electron to another.
Robert Boyle developed the concept of a chemical element as substance different from a compound working in the late 17th century. Antoine Lavoisier later redefined an element as a substance which scientists could not decompose into simpler substances by experimentation near the end of the 18th century.
Gilbert N. Lewis developed the concept of electron-pair bonds allowing two atoms to share one to six electrons forming single double or triple bonds in 1916. Walther Kossel put forward a similar model assuming complete transfers of electrons between atoms creating a model of ionic bonding also in 1916.
The hydrogen-hydrogen bond measures 74 picometers with energy of 436 kilojoules per mole while carbon-carbon bonds vary from 154 picometers in single bonds to 120 picometers in triple bonds carrying 839 kilojoules per mole.