Part B · How stuff sticks

5 · Bonds: one electrical story

Textbooks hand you three kinds of bond as if they were three different forces: covalent, ionic and metallic. They're one force, electrical attraction, with the electrons arranged in different ways.

In chapter 1, argon atoms stuck together with a well 0.01 eV deep. Two hydrogen atoms stick with a well of 4.5 eV, about 400 times deeper. Salt holds itself together more strongly still. What changes from one case to the next isn't the force. It's what the electrons do.

Sharing: one electron, two protons

Take the simplest bond there is: H2+, two protons and one electron. In isolation the electron sits in a fuzzy 1s cloud around a single proton. That cloud is an orbital, the electron's standing wave. Bring a second proton close and the electron's wave can spread over both.

Quantum mechanics lets us add waves. A good first guess for the shared electron is a sum of the two atomic orbitals, φA + φB, or a difference, φA − φB. This is called LCAO. Slide the protons together and compare the two.

Adding the waves in phase (φA + φB) or out of phase (φA − φB): which one gives a bond?

In phase. The two waves interfere constructively in the gap, so the electron is more likely to be found between the protons, where it is attracted to both at once. That lowers the energy, giving a well. Out of phase, they cancel in the middle (a node). The electron is pushed to the outsides, and the energy only goes up: no well, no bond.

Top: the electron's wave along the line through both protons (thin lines: each atom's own 1s orbital; shaded: probability density |ψ|²). Bottom: total energy of H2+ against distance for both combinations, measured from "H atom + lone proton far apart". Distances are in Bohr radii (1 a0 = 0.53 Å). The curves are the standard LCAO formulas, computed exactly.

The sum φA + φB is a crude guess. It puts the bond at 1.32 Å with a depth of 1.76 eV; the exact answer for H2+ is 1.06 Å and 2.79 eV. The crude guess still gets the main thing right: interfering in phase creates a well, and interfering out of phase doesn't.

That's a covalent bond: electron density piled up between two nuclei, attracting both. With two electrons (H2) the same in-phase orbital holds both, one spin up and one down, and the well deepens to 4.5 eV. A third electron would have to go into the out-of-phase orbital and would undo much of the bonding. That's why bonds come in pairs of electrons, and why two helium atoms (four electrons: two bonding, two antibonding) form no chemical bond. They cling together only through the faint dispersion force from chapter 1.

Physics bridge

A covalent bond is constructive interference. The in-phase superposition of two atomic orbitals has lower energy than either atomic orbital alone; the out-of-phase one has higher. Diagonalise the 2×2 Hamiltonian in the {φA, φB} basis and you get two eigenstates split by the coupling: the bonding and antibonding orbitals. Every "molecular orbital diagram" in chemistry is a bigger version of this 2×2 problem.

Many textbooks say the bond forms because the electron's kinetic energy drops when it spreads out. That holds for H2+ and H2 but fails for bonds between heavier atoms such as C–C and F–F, where the kinetic energy often goes up. Levine & Head-Gordon (2020) argue that constructive interference is the general origin.

Taking: the spectrum from sharing to stealing

When the two atoms are different, the sharing is unequal. Some nuclei pull on shared electrons harder than others. That pull is called electronegativity, χ. Fluorine is the strongest (3.98 on Pauling's scale) and sodium is weak (0.93). The bigger the difference Δχ between two atoms, the more lopsided the shared cloud.

A cartoon of the shared electron cloud between two atoms as the electronegativity difference grows. The left atom is the one that pulls harder. "Ionic character" uses Pauling's empirical formula 1 − e−Δχ²/4.

There's no sharp line between covalent and ionic: the cloud just slides over. The "% ionic character" number is a rough empirical rule of thumb, not something you can measure directly, and different electronegativity scales disagree on the numbers (though not on the trends).

At one end, equal sharing (H–H, Cl–Cl). In the middle, polar bonds such as O–H, where oxygen hogs the electrons and the hydrogen end is left slightly positive. Water's whole personality comes from this (chapter 6). At the far end, sodium and chlorine: the electron is essentially handed over, leaving Na+ and Cl− ions. That's an ionic bond, but it's the same electrical attraction as before, now between whole charges.

Why does salt exist? (Not because atoms "want" eight electrons)

You may have been taught that sodium "wants" to lose an electron to get a full shell, and chlorine "wants" to gain one. Let's check that against the energy bookkeeping, per mole, for atoms in the gas phase:

Energy ledger for making solid salt from separate gas-phase Na and Cl atoms. Values in kJ/mol, rounded: ionisation energy of Na (496), electron affinity of Cl (349), lattice energy of NaCl (787). Up = costs energy, down = releases it.

Pulling an electron off sodium costs 496 kJ/mol. Putting it on chlorine gives back only 349. So the swap alone is uphill: two isolated ions are less stable than two neutral atoms. Sodium doesn't "want" anything. In fact, 97% of students in one study rated Na+ as more stable than a sodium atom, which this ledger contradicts.

What pays for the swap is plain electrical attraction. Once the ions exist, each Na+ sits surrounded by six Cl− and vice versa, in a cubic crystal. That arrangement releases 787 kJ/mol, the lattice energy. The full-shell story is a useful rule for predicting which ions form. Coulomb's law is the reason anything forms at all.

Common trap

"Atoms want a full outer shell (the octet rule)." The octet rule is a good bookkeeping rule for light elements, but it isn't a force or a desire. Na+ on its own is higher in energy than Na. Ions form when the electrical attraction they then enjoy more than pays the cost. And many stable molecules break the octet rule outright.

The third kind: metals

In a metal, each atom gives up one or more outer electrons to the whole crystal. Those electrons spread over the entire lump in orbitals that are delocalised across billions of atoms. That's the chemist's version of the band structure you may know from solid-state physics. The positive ions sit in a "sea" of shared electrons that holds them together. Because the electrons aren't tied to any one pair of atoms, they move freely, and that's why metals conduct electricity, shine and bend instead of shattering.

So all three kinds are one story. Electrons arrange themselves to sit where positive nuclei attract them, while quantum mechanics (interference, the Pauli exclusion principle) decides which arrangements are allowed:

Tell it to a 6-year-old
Look
Pour a little table salt onto a dark plate. Look at the grains through a magnifying glass (or a phone camera zoomed all the way in).
Wonder
"What shape are the grains? Are they round like sand, or something else?"
Try
Find the tiny cubes, little boxes with flat sides. Then try two fridge magnets: feel them pull and snap together, then feel how hard you have to pull to get them apart.
True sentence
"Salt is made of two kinds of tiny pieces that pull on each other very strongly. They stack up in a neat box pattern, so the grains come out as little cubes. Pulling pieces apart always takes effort, just like the magnets."

The magnets stand in for the pull. Salt's pull is electric, not magnetic, but "pulling apart takes effort, snapping together gives off a bit of energy" is exactly right. Avoid "the atoms want to be together". Some table salt grains are chipped or clumped; the cubes show best with coarse or sea salt.

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