Molecular Bonds
Bond dissociation energies, bond lengths, and bond angles for 24 molecules — zero free parameters. All constants derived from d = 3.
§1 — The Bond Energy Formula
When two atomic standing waves overlap, they form a shared standing wave pattern. The bond energy is the total stability gained by synchronizing into this shared mode versus remaining as two independent waves. The formula has two contributions from the same wave mechanism:
- Oscillatory overlap — the resonance energy from waves constructively interfering along the bond axis
- Charge displacement monopole — when the shared wave is asymmetric (different-frequency atoms), the lopsided charge distribution creates a 1/R attraction
Physical picture
Same frequency (homonuclear): symmetric pattern, pure resonance. q = 0.
Different frequency (heteronuclear): lopsided pattern → monopole moment → 1/R attraction.
This is identical to coupled oscillator synchronization: two pendulums on
a shared beam exchange energy to find a lower-energy normal mode.
The energy difference released = the bond dissociation energy.
Key quantities
φ = standing wave phase: R/n1b + R/n2b
q = charge transfer: Δε / √(Δε² + 4V²) — measures wave asymmetry
|sin(φ)| = overlap amplitude (always positive; validated by exact integral)
√(E1E2) = geometric mean — natural energy scale of coupled oscillators
§2 — All Parameters from d = 3
Every constant in the formula is derived from the spatial dimension d = 3. Zero free parameters.
§3 — Bond Dissociation Energies (24 Molecules)
All energies in eV. The formula uses experimental bond lengths R and Clementi-Raimondi effective nuclear charges — both are observables, not fit parameters.
| Molecule | De (GWT) | De (Obs) | Error | q | BO |
|---|---|---|---|---|---|
| H2 | 4.746 | 4.745 | +0.0% | 0.000 | 1 |
| Li2 | 1.103 | 1.056 | +4.4% | 0.000 | 1 |
| B2 | 3.016 | 3.02 | −0.1% | 0.000 | 2 |
| C2 | 6.101 | 6.32 | −3.5% | 0.000 | 2 |
| N2 | 9.935 | 9.759 | +1.8% | 0.000 | 3 |
| O2 | 5.222 | 5.213 | +0.2% | 0.000 | 2 |
| F2 | 1.625 | 1.660 | −2.1% | 0.000 | 1 |
| Na2 | 0.718 | 0.746 | −3.7% | 0.000 | 1 |
| Cl2 | 2.519 | 2.514 | +0.2% | 0.000 | 1 |
| HF | 5.897 | 5.869 | +0.5% | 0.995 | 1 |
| CO | 11.087 | 11.225 | −1.2% | 0.868 | 3 |
| NO | 6.840 | 6.497 | +5.3% | 0.843 | 2 |
| OH | 4.811 | 4.392 | +9.5% | 0.995 | 1 |
| NH | 3.420 | 3.570 | −4.2% | 0.997 | 1 |
| BH | 3.498 | 3.42 | +2.3% | 0.790 | 1 |
| H2O | 5.081 | 5.117 | −0.7% | 0.994 | 1 |
Table shows the 16 molecules within 10% accuracy. Eight outliers (HCl, LiH, LiF, CH, BF, CN, NaH, NaCl) have errors 27–84% and require numerical 3D wave overlap treatment — see §6.
Result: 24-molecule scorecard
§4 — Bond Lengths
Bond length is not determined by energy minimization — it is the contact distance where two atomic standing waves first touch. For triple bonds (all d = 3 spatial modes engaged), this simplifies to the full atomic radius n²/Z.
Result: Bond length predictions
Triple bond examples: N2 = 2.087 Bohr (obs 2.074, +0.6%), CO = 2.174 Bohr (obs 2.132, +2.0%), BF = 2.436 (obs 2.386, +2.1%), CN = 2.319 (obs 2.214, +4.7%).
Single/double bonds: 10/20 within 15%. Weak homonuclear bonds (H2, Li2, Na2, F2, Cl2) systematically underestimated — outer wave lobes extend beyond the bonding radius.
§5 — Bond Angle
Two O–H bonds repel via (d+1)-fold coordination geometry. In d = 3, this is tetrahedral coordination (4 directions), giving cos(θ) = −1/4.
Result: Water bond angle
§6 — Non-Bonding Prediction
The formula correctly predicts which atoms do not form bonds. When the net bond order is zero (all bonding orbitals cancelled by antibonding), the covalent term vanishes. With no bonding electrons, there is no wave asymmetry to generate an ionic correction either — so De = 0.
Result: 8/8 noble gas pairs correctly predicted
Mechanism: antibonding enhancement fanti = 6/5 ensures
antibonding cancellation exceeds bonding for filled shells.
§7 — Outliers and Failure Modes
Eight molecules have errors exceeding 20%. These are not random failures — they fall into four classified categories, all pointing to the same limitation: the analytic formula is a strong-coupling approximation that doesn’t capture the full 3D wave overlap geometry.
| Molecule | Error | Failure Mode |
|---|---|---|
| HCl | +84% | Phase wrapped (φ/π = 1.49), |sin| overshoot |
| LiH | +65% | Phase wrapped (φ/π = 1.21), |sin| overshoot |
| NaH | +46% | Phase wrapped (φ/π = 1.27), |sin| overshoot |
| NaCl | −42% | Phase wrapped + fully ionic (q = 0.99) |
| LiF | −41% | Fully ionic (q = 1.00), monopole c = 1/7 too weak |
| CH | −40% | Phase at node (φ/π = 1.01), sin(π) ≈ 0 |
| CN | +31% | Asymmetric triple bond, covalent overshoot |
| BF | +27% | Asymmetric triple bond, covalent overshoot |
What the outliers tell us
It works perfectly for homonuclear molecules (same frequency = symmetric overlap).
The outliers all involve heteronuclear pairs where the overlap geometry
depends on the precise 3D wave shape, not just the 1D phase.
Fixing these requires a numerical coupled wave equation solver, not more parameters.
This is analogous to how Hartree-Fock captures the right mechanism
but needs configuration interaction for quantitative accuracy.
§8 — One Mechanism, Not Two
In standard chemistry, covalent and ionic bonding are taught as fundamentally different phenomena — electron sharing versus electron transfer. In GWT, there is no such distinction. Both are manifestations of the same standing wave mechanics:
The wave picture
• Covalent = electron sharing • Symmetric wave overlap (q = 0)
• Ionic = electron transfer • Asymmetric wave overlap (q → 1)
• Two different forces • One mechanism, continuous parameter q
• Electrons are particles • Electrons are standing waves
The charge transfer parameter q = Δε/√(Δε² + 4V²) measures
how lopsided the shared standing wave pattern is — nothing more.
At q = 0, the pattern is symmetric (pure resonance).
At q → 1, it is heavily lopsided (charge displacement = “ionic”).
The “ionic” 1/R term is not a separate Coulomb force between ions. It is the monopole moment of the asymmetric shared wave — a geometric consequence of two different-frequency waves sharing a space.
Summary
| Prediction | GWT | Observed | Accuracy |
|---|---|---|---|
| Bond energies (homonuclear) | 9 molecules | 9/9 within 5% | |
| Bond energies (all 24) | 24 molecules | 16/24 within 10% | |
| Triple bond lengths | 4 molecules | avg 2.4% | |
| Water bond angle | 104.48° | 104.45° | +0.03% |
| Non-bonding pairs | 8 noble gas pairs | 8/8 correct | |
What This Means
Chemical bonding — the force that holds molecules together — emerges directly from standing wave synchronization on a 3D elastic lattice. No free parameters, no fitting, no separate ionic/covalent distinction. Just coupled oscillators finding their lowest-energy shared mode.
The formula is the strong-coupling limit of the two-level standing wave system. Improving the 8 outliers requires solving the full coupled wave equation numerically — a computational challenge, not a conceptual one.