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

=== Assistant: Restated question: ===
Restated question:
What determines when the overlap integral S=⟨ψA∣ψB⟩S = \langle \psi_A | \psi_B \rangleS=⟨ψA∣ψB⟩ is “significant,” and how do we quantify that threshold physically—not heuristically?

==== 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.

==== 2. So what is the quantitative threshold? ====

A practical physical threshold emerges:

∣HAB∣HAA≳S\frac{|H_{AB}|}{H_{AA}} \gtrsim SHAA∣HAB∣≳S
Meaning:

: The overlap is significant when it is large enough that the change in energy due to constructive interference of the wavefunctions exceeds the normalization penalty.

Typical values:
* S<0.05S < 0.05S<0.05: negligible overlap → no bonding interaction
* 0.05≤S≤0.200.05 \le S \le 0.200.05≤S≤0.20: moderate overlap → weak bonding (van der Waals, H-bonding, weak covalent)
* S≈0.20–0.35S \approx 0.20\text{–}0.35S≈0.20–0.35: strong covalent bonding regime
* S>0.40S > 0.40S>0.40: rare; orbitals so close Pauli repulsion dominates

So “significant” does not mean geometric closeness only — it means:

ΔE=Ebonded−Eseparate<0\Delta E = E_{\text{bonded}} - E_{\text{separate}} < 0ΔE=Ebonded−Eseparate<0
and that only happens when the overlap term meaningfully perturbs the Hamiltonian eigenvalues.

==== 3. A deeper physical statement ====

The overlap integral matters because it controls the strength of the exchange interaction, which is a quantum effect with no classical analog. The exchange energy term is approximately:

Eexchange∝HAB1+SE_{\text{exchange}} \propto \frac{H_{AB}}{1 + S}Eexchange∝1+SHAB
and this is what lowers the energy enough to stabilize a covalent bond.

So the real physical meaning of “significant overlap” is:

: There is enough wavefunction overlap that the exchange term lowers the total energy more than repulsive Coulomb terms raise it.

==== 4. The point where attraction exceeds repulsion ====

If we treat the total interaction energy as:

E(r)=Ecoulomb(r)+Eexchange(r)+Ekinetic(r)+EPauli(r)E(r) = E_{\text{coulomb}}(r) + E_{\text{exchange}}(r) + E_{\text{kinetic}}(r) + E_{\text{Pauli}}(r)E(r)=Ecoulomb(r)+Eexchange(r)+Ekinetic(r)+EPauli(r)
then the bonding distance is found by:

dEdr=0\frac{dE}{dr} = 0drdE=0
This minimum corresponds to precisely the point where overlap has become large enough that:

∣ΔEexchange∣>ΔEPauli+ΔErepulsion|\Delta E_{\text{exchange}}| > \Delta E_{\text{Pauli}} + \Delta E_{\text{repulsion}}∣ΔEexchange∣>ΔEPauli+ΔErepulsion
That is the exact physical threshold.

==== 5. One-sentence definition ====

: Overlap becomes “significant” when the electron wavefunctions are close enough that the exchange integral 
: 𝐻
: 𝐴
: 𝐵
: H
: AB
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:  lowers the molecular orbital energy more than the normalization penalty from 
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: S and the Coulomb repulsion terms raise it, resulting in a net decrease in the system’s total energy.

==== If you want to go further: ====

Pick one:

A. Compute S(r)S(r)S(r) explicitly for two hydrogen 1s orbitals and plot it vs. distance.
B. Derive the approximate bonding/antibonding splitting magnitude using HABH_{AB}HAB and SSS.
C. Connect this to observable reaction rates and activation energy.

Which direction?