What is the physical significance of the overlap integral?

Question

What is the physical significance of the overlap integral in the covalent bond formation (say, the $H_2$ molecule)?

as far as I can understand, even if the overlap integral is zero, the Heitler-London theory of $H_2$ atom, it still admits a lower energy state compared to the total energy of two isolated H-atoms.

What is the significance of the following statement from Wikipedia?

In chemical bonds, an orbital overlap is the concentration of orbitals on adjacent atoms in the same regions of space. Orbital overlap can lead to bond formation.

Is it necessary to have non-zero orbital overlap for bond formation? Or does it suffice to have a lower energy level (compared to the total energy of two isolated H-atoms) to say a bond is formed?

Answer 1

The eigenstates of a quantum mechanical system are all orthogonal to each other. This means that the overlap matrix between these states is diagonal. If you consider orbitals for a single atom and bring it in the vicinity of another atom you obtain a nonzero overlap between the states of the different atoms. This implies that the orbitals for the combined system are not the same as the orbitals for the two isolated atoms as the orthogonality condition is broken.

When you move the two atoms nearer and nearer to each other you can observe that the overlap between the states increases. The orbitals located on each of the atoms then combine and form bonding and antibonding states. The bonding state is lower in energy, the antibonding state higher.