Orbital overlap

Orbital overlap is a concept in the molecular orbital description of chemical bonding. In the simplest terms, a covalent bond can only form between two atoms if there is significant overlap between atomic orbitals on the atoms.

Orbital overlap can be expressed in qualitative terms through empirical rules, or quantitatively through overlap integrals S_ab (where a and b represent atomic orbitals).

Overlap integral

The overlap integral S_ab is the name given to an integral of the form

<math>S_{\rm ab} = \int \varphi_{\rm a}^\ast \varphi_{\rm b} {\rm d}\tau</math>

where φ_a and φ_b are the wavefunctions corresponding to the atomic orbitals a and b and dτ indicates that the integral is to be taken over all space. In Dirac notation, S_ab = <φ_a|φ_b>, usually written as simply <a|b>.

Basic results

1. If a = b, i.e. the two orbitals are the same (and on the same atom), S_ab = 1. This is a way of saying that each orbital overlaps perfectly with itself.
2. For any set of hydrogenic orbitals on the same atom, S_ab = 0 if a ≠ b. This might seem counterintuitive at first sight, as surely a 2s orbital must "overlap" with a 1s orbital.
3. Where a and b are orbitals on different atoms, S_ab lies between zero and one.