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

=== Assistant: Condensed question: ===
Condensed question:
What do Ψbonded\Psi_{\text{bonded}}Ψbonded and Ψseparate\Psi_{\text{separate}}Ψseparate mean physically?

==== They are two different quantum mechanical wavefunctions that describe the same set of nuclei and electrons under two different physical configurations: ====
* Ψseparate\Psi_{\text{separate}}Ψseparate: the wavefunction describing the system when the atoms are far apart and not interacting (just isolated atomic orbitals).
* Ψbonded\Psi_{\text{bonded}}Ψbonded: the wavefunction describing the system when the atoms are close enough that the electrons are shared/delocalized and a chemical bond exists (a molecular orbital).

These are not arbitrary labels—they are different eigenfunctions of the full Hamiltonian of the electron–nuclear system.

==== ### ====

H^=Te+Tn+Vee+Vnn+Ven\hat{H} = T_e + T_n + V_{ee} + V_{nn} + V_{en}H^=Te+Tn+Vee+Vnn+Ven
When the atoms are so far apart that they do not interact, the system factorizes:

Ψseparate=ψA(1)ψB(2)\Psi_{\text{separate}} = \psi_A(1)\psi_B(2)Ψseparate=ψA(1)ψB(2)
(or symmetrized/antisymmetrized depending on spin).

This describes:
* electron 1 around nucleus A
* electron 2 around nucleus B
* '''no molecular bonding'''
* just two separate atoms.

===== Ψbonded=12(1+S)[ψA+ψB]\Psi_{\text{bonded}} = \frac{1}{\sqrt{2(1+S)}}\left[\psi_A + \psi_B\right]Ψbonded=2(1+S)1[ψA+ψB] =====
where:
* ψA,ψB\psi_A, \psi_BψA,ψB are atomic orbitals
* S=⟨ψA∣ψB⟩S = \langle \psi_A | \psi_B \rangleS=⟨ψA∣ψB⟩ is the overlap integral

This wavefunction has:
* electron probability density between the nuclei
* constructive interference of the atomic wavefunctions
* a lower total energy eigenvalue

This is the bonding orbital.

==== You evaluate the expectation value of the Hamiltonian in each state: ====

Eseparate=⟨Ψseparate∣H^∣Ψseparate⟩E_{\text{separate}} = \langle \Psi_{\text{separate}}|\hat{H}|\Psi_{\text{separate}}\rangleEseparate=⟨Ψseparate∣H^∣Ψseparate⟩
Ebonded=⟨Ψbonded∣H^∣Ψbonded⟩E_{\text{bonded}} = \langle \Psi_{\text{bonded}}|\hat{H}|\Psi_{\text{bonded}}\rangleEbonded=⟨Ψbonded∣H^∣Ψbonded⟩
If:

Ebonded<EseparateE_{\text{bonded}} < E_{\text{separate}}Ebonded<Eseparate
then the molecule is more stable than separate atoms, and the difference:

ΔE=Eseparate−Ebonded\Delta E = E_{\text{separate}} - E_{\text{bonded}}ΔE=Eseparate−Ebonded
must be released as energy (heat, light, kinetic energy of atoms).

That difference is the bond energy.

==== <syntaxhighlight>Energy ====
│
│        Ψ_separate   ← higher energy (atoms apart)
│
│
│_________________     ← energy released
│
│   Ψ_bonded       ← lower energy (bond formed)
│
└─────────────────────── internuclear distance → 

</syntaxhighlight>

==== - Ψseparate\Psi_{\text{separate}}Ψseparate is the quantum state describing noninteracting atoms, ====
* Ψbonded\Psi_{\text{bonded}}Ψbonded is the quantum state describing the same atoms after their electronic wavefunctions reorganize into a lower-energy molecular orbital,

and comparing their energies tells us whether a chemical bond forms and how much energy is released.

If you want, the next logical step is:
“Show the explicit energy lowering term by expanding the LCAO normalization and overlap integrals (Coulomb integral JJJ, exchange integral KKK) for H₂.”