Where entry 01 left off
Entry 01 followed the idea that the electrons of a molecule are standing-wave modes. Of everything checked there, one sentence came out strongest:
A chemical bond is a phase relationship between modes.
That sentence sounds like something this framework would say. It also closely matches how many chemistry textbooks introduce bonding. This entry looks at why — and at where the sentence stops being useful.
One caveat up front. This is the molecular-orbital way of describing a bond. The older valence-bond picture (Heitler and London, 1927) describes the same bond as two electrons pairing their spins. Both are approximations to the same underlying equations, and both work. “Phase” belongs to the first description, not to the bond itself.
1 · Same two waves, two outcomes
Take the simplest molecule there is: two protons sharing one electron (the hydrogen molecular ion, H₂⁺). Near each proton, the electron’s wave looks like the wave of a hydrogen atom. The molecule’s wave is built, to a good approximation, by adding those two atomic waves together.
There are two ways to add them:
- In step. The two waves reinforce each other in the region between the protons. Extra electron density gathers there. This is a bonding orbital: the molecule holds together.
- Out of step. The two waves cancel exactly at the midpoint, leaving a node. Density is pushed out of the region between the protons. This is an antibonding orbital: the molecule is pushed apart.
Nothing about the waves themselves changes between the two cases — same shape, same size, same atoms. Only the relative phase differs. The figure below lets you turn that one dial.
Density at the midpoint vs. no interference: +26%
In step (0°): the two waves reinforce between the nuclei. Extra electron density gathers between the nuclei — the textbook signature of a bonding orbital.
Simple textbook model (H₂⁺, two hydrogen 1s orbitals, nuclei 2 bohr apart), drawn along the bond axis. It shows one thing: with the same two waves, only the relative phase decides whether density gathers between the nuclei or is pushed away.
In this simple model, the density at the midpoint is about 26% higher than it would be if the two waves did not interfere at all, and exactly zero when they are out of step. Real molecules are more complicated — the simple picture uses unchanged atomic waves, and with more than one electron it also leaves out the way electrons avoid each other — but the contrast between in step and out of step survives in accurate calculations.
Why the in-step combination holds the molecule together is subtler than “density between the nuclei pulls them in”. The phase picture tells you which orbital bonds; it does not by itself tell you why.
The figure uses the textbook LCAO model: ψ± = (ψA ± ψB)/√(2(1 ± S)), with hydrogen 1s orbitals ψA, ψB centred R = 2 bohr apart (close to the experimental bond length of H₂⁺; the minimal LCAO model itself puts its energy minimum near 2.5 bohr) and overlap S = e^(−R)(1 + R + R²/3) ≈ 0.586. For intermediate phases it draws |ψA + e^(iφ)ψB|²/(2(1 + S cos φ)); such states are superpositions of ψ+ and ψ− with different energies and are not stationary. The midpoint ratio to the interference-free density (ψA² + ψB²)/2 is (1 + cos φ)/(1 + S cos φ): 1.26 at φ = 0, 0 at φ = π. On the mechanism: analyses since Ruedenberg (1962) point to a lowering of the electron’s kinetic energy as the key ingredient of the bond, not simply electrostatic attraction to the extra density.
2 · Counting in-step and out-of-step waves
Once a molecule has more than one electron, the modes fill from the bottom up, two per mode (entry 01, section 2). Bonding modes lie lower in energy, so they fill first; antibonding modes fill after them. Chemists summarise the balance in one number, the bond order: the number of electrons in bonding modes minus the number in antibonding modes, divided by two.
That simple count explains a surprising fact:
- Hydrogen (H₂) has two electrons, both in the bonding mode. Bond order one: a stable molecule.
- Helium (He₂) has four electrons. Two go into the bonding mode, and the other two have nowhere to go but the antibonding mode. The out-of-step pair undoes — and slightly more than undoes — the in-step pair; bond order zero. Two ground-state helium atoms do not form an ordinary chemical bond.
Apart from an extraordinarily weak pairing held together by the van der Waals attraction, first detected in cold helium beams (Luo and colleagues, 1993; Schöllkopf and Toennies, 1994), ground-state He₂ indeed does not exist as a chemically bonded molecule. The count changes when the electron count changes: the ion He₂⁺, with one antibonding electron fewer, does hold together. The phase picture gets all of this right with nothing more than counting.
3 · Light can move an electron into an antibonding mode
Bonding and antibonding modes differ in their relative phase — and in energy. So anything that moves an electron from one to the other changes the bond. Light can do exactly that. When a photon with the right energy is absorbed, an electron can be lifted into a state with antibonding character. The bond weakens, and often the molecule falls apart.
This happens on a planetary scale. High in the atmosphere, ozone absorbs ultraviolet light; the absorption lifts the molecule into states with antibonding character, and the molecule breaks apart. Together with ordinary oxygen higher up, this is a large part of why so little of the sun’s harmful UV-B and UV-C reaches the ground.
4 · What “phase” means here — a careful note
The word phase has to be used with care, because it means slightly different things in different places:
- Here, phase is a sign. For the stationary orbitals in this example, the two atomic waves are added with either a plus or a minus. “In step” and “out of step” mean exactly that, not a clock angle running in time. The figure’s slider also shows angles in between; those mixed states are not stationary orbitals and are there only to show the transition between the two signs.
- Only the relative phase counts. Multiplying the whole molecular wave by −1 changes nothing measurable (entry 01, section 6a). What matters is the sign of one atom’s contribution relative to the other.
- It is not entanglement. Elsewhere on this site (Bridge 4, entanglement as phase lock), phase appears in discussions of entanglement between separated particles — a use that page itself marks as limited by Bell’s theorem. The bonding phase is a property of one electron’s wave spread over two atoms; it says nothing about signals or correlations between distant objects.
Where this leaves the idea
| Claim | Status |
|---|---|
| A bond is a phase relationship between atomic waves | Textbook chemistry — in the molecular-orbital picture |
| In step → bonding, out of step → antibonding | Textbook; survives in accurate calculations |
| Bond order explains why two ground-state helium atoms do not bond | Textbook, and matches experiment |
| Light can lift an electron into an antibonding state and break a bond | Textbook (photodissociation) |
| This shows the framework is right | No. It shows that resonance vocabulary fits chemistry well. |
Honest status. Chemistry already is a theory of interfering waves, so it is no surprise that a framework built on waves can describe it. The test of a framework is not whether it can redescribe known results, but whether it predicts something the standard theory does not. This entry finds nothing of that kind. What it does give is a clean, checkable example of the language this framework uses — a place where “phase relationship” has a precise meaning that anyone can verify.
Sources. W. Heitler & F. London, Z. Phys. 44, 455 (1927) (valence-bond treatment of H₂) · J. E. Lennard-Jones, Trans. Faraday Soc. 25, 668 (1929) (molecular-orbital treatment) · K. Ruedenberg, Rev. Mod. Phys. 34, 326 (1962) · F. Luo et al., J. Chem. Phys. 98, 3564 (1993) · W. Schöllkopf & J. P. Toennies, Science 266, 1345 (1994). Bond order, the LCAO model and the He₂ argument are standard in introductory physical-chemistry texts.
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