Bring two atoms close and their electron clouds overlap into new shared orbitals — a BONDING one that sits between the nuclei (lower energy) and an ANTIBONDING one (higher). Put the two electrons in the low one and the pair is more stable together than apart: that surplus stability IS the chemical bond. Slide the atoms together and watch the energy drop.
When two atomic orbitals of energy E overlap, they combine into two molecular orbitals: a BONDING orbital at E−β (electron density piled between the nuclei, screening their repulsion) and an ANTIBONDING orbital at E+β (a node between them). Two electrons fill the bonding orbital, so the molecule’s energy is 2(E−β), LOWER than the 2E of two separate atoms — the drop 2β is the bond energy; sharing electrons is energetically downhill. A fail-loud self-check throws unless bonding < atomic < antibonding. ◆ real quantum chemistry, node-verified.
A two-level LCAO / Hückel picture (the exact bonding/antibonding split); real bonds add overlap, ionic character and the internuclear repulsion that sets the equilibrium distance — the share-to-lower-energy mechanism is the exact core.