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

=== Assistant: Condensed version of your question: ===
Condensed version of your question:
What physically attracts two atoms into forming a bond, why do they not bond at long distance, and what exact event/threshold triggers the reaction?

==== At the most fundamental level, the attraction comes from the Coulomb force within the electron–nucleus system: ====

F=14πϵ0q1q2r2F = \frac{1}{4\pi\epsilon_0}\frac{q_1 q_2}{r^2}F=4πϵ01r2q1q2
Specifically:
* The electrons of one atom are attracted to the nucleus of the other.
* If the electronic wavefunctions overlap constructively, the system can reorganize into a lower-energy quantum state (a molecular orbital).

This attraction competes with:
* electron–electron repulsion
* nucleus–nucleus repulsion

A bond forms only if the net energy decrease (from electron–nucleus attraction + quantum delocalization) outweighs the repulsive terms.

==== Because at long distances: ====
* The electron wavefunctions do not overlap significantly.
* The Coulomb attraction between nuclei and the other atom’s electrons falls off as 1/r21/r^21/r2.
* The system’s quantum mechanical state remains essentially the direct product of isolated atomic orbitals.

Without overlap, there's no possibility of forming a lower-energy molecular eigenstate—so there is no bond, just weak long-range interactions (van der Waals, dispersion).

Mathematically:

S=⟨ψA∣ψB⟩≈0⇒no bonding orbital availableS = \langle \psi_A | \psi_B \rangle \approx 0 \quad \Rightarrow \quad \text{no bonding orbital available}S=⟨ψA∣ψB⟩≈0⇒no bonding orbital available

==== Bonding becomes energetically favorable when electron wavefunction overlap becomes non-zero. ====

This occurs when interatomic separation approaches approximately one Bohr radius scale (10⁻¹⁰ m)—in practice, when atoms are close enough that the overlap integral SSS is significant.

At this distance:
* The bonding molecular orbital becomes lower in energy than the separated-atom state.
* The system now has a quantum mechanically available state that is lower in total energy.

This distance is where the attractive part of the potential energy curve dominates.

==== Two things must happen simultaneously: ====

===== Atoms must be close enough that: =====

ΔE=Ebonded−Eseparate<0\Delta E = E_{\text{bonded}} - E_{\text{separate}} < 0ΔE=Ebonded−Eseparate<0
That is: a lower-energy quantum state becomes available.

===== To transition into the lower-energy state, the system must shed energy: =====
* via photon emission
* via collisions with nearby particles
* via redistribution into vibrational/rotational kinetic energy

If the excess energy cannot be removed, the atoms approach → accelerate → and then rebound (like charged elastic balls).

So the key trigger is:

: 

Without dissipation, bonding cannot complete.

This is why:
* cold atoms bond more easily than hot ones
* chemistry slows at low pressure
* catalysts don't force bonding—they provide energy pathways and intermediate states

==== If the atoms get too close, then: ====
* nucleus–nucleus repulsion rises steeply (Coulomb)
* Pauli exclusion forces the electron wavefunctions into antibonding configurations

So the total energy curve looks like:

<syntaxhighlight>Energy
│         ↑ Pauli + nuclear repulsion
│        /
│       /      ← too close → repulsive
│      /
│     /      minimum (bond length)
│    /
│   /  ← attractive region where bond forms
│  /
│ /
│/ _______________________________________ distance →
   far apart (no overlap)

</syntaxhighlight>

==== Two atoms form a bond when they come close enough for their electron wavefunctions to overlap, allowing a lower-energy molecular orbital to exist, and when the system can dissipate the excess energy; before that critical distance there is no significant attraction, and past the equilibrium distance repulsive forces dominate. ====

If you'd like, next step could be:
* A full Lennard-Jones/Morse potential derivation
* A simulation with actual overlap integrals
* How chemistry differs for ionic, covalent, and metallic cases when viewed at this fundamental level

Which direction do you want to go?