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

==== For two atoms, the relevant Hamiltonian includes: ====

H^=Te+Tn+Vnn+Vee+Ven\hat{H} = T_e + T_n + V_{nn} + V_{ee} + V_{en}H^=Te+Tn+Vnn+Vee+Ven
Where:
* Te,TnT_e, T_nTe,Tn: electron and nuclear kinetic energy terms
* VnnV_{nn}Vnn: nucleus–nucleus repulsion
* VeeV_{ee}Vee: electron–electron repulsion
* VenV_{en}Ven: electron–nucleus attraction

When atoms are infinitely separated, the terms are mostly localized.
As they approach, the electron wavefunctions overlap and the expectation value of the Hamiltonian changes.

A bond forms only if:

⟨Ψbonded∣H^∣Ψbonded⟩<⟨Ψseparate∣H^∣Ψseparate⟩\langle \Psi_{\text{bonded}} | \hat{H} | \Psi_{\text{bonded}} \rangle < \langle \Psi_{\text{separate}} | \hat{H} | \Psi_{\text{separate}} \rangle⟨Ψbonded∣H^∣Ψbonded⟩<⟨Ψseparate∣H^∣Ψseparate⟩
i.e., the bonded state is a lower energy eigenstate of the Hamiltonian.

The difference in energy must be emitted as photons or transferred to kinetic motion → that is the “released energy.”