Editing Openai/69227557-0850-8003-9c19-8f26825e337d (section)

==== 1. Overlap is not an on/off switch — it becomes significant when it affects energy. ====

The overlap integral SSS matters because it appears directly in the energy expression for a molecular orbital formed from atomic orbitals.

For a simple diatomic (LCAO approximation), the energy of the bonding orbital is:

E+=HAA+HAB1+SE_+ = \frac{H_{AA} + H_{AB}}{1+S}E+=1+SHAA+HAB
and the antibonding orbital:

E−=HAA−HAB1−SE_- = \frac{H_{AA} - H_{AB}}{1-S}E−=1−SHAA−HAB
Here:
* HAAH_{AA}HAA is the energy of an isolated atomic orbital
* HAB=⟨ψA∣H^∣ψB⟩H_{AB} = \langle \psi_A | \hat{H} | \psi_B \rangleHAB=⟨ψA∣H^∣ψB⟩ is the resonance/exchange integral
* S=⟨ψA∣ψB⟩S = \langle \psi_A | \psi_B \rangleS=⟨ψA∣ψB⟩ normalizes the wavefunction because the orbitals are no longer orthogonal

A bond forms only if:

E+<EseparateE_+ < E_{\text{separate}}E+<Eseparate
So overlap is significant when it materially lowers the energy eigenvalue.