Concept

Bond energy — where it appears

The energy needed to pull two bonded atoms apart, letting everything relax. It differs from a band gap or an orbital energy by the relaxation and by every repulsion term a one-electron picture omits.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

The gap against size: uniform against δ = 0.15. The HOMO–LUMO gap of a half-filled chain plotted against the number of sites, on log axes, for a uniform chain and for one whose bonds alternate. The uniform sequence falls without limit; the alternating one settles at four times the alternation.

A band gap is not a bond energy

Silicon's gap is 1.1 electronvolts and its Si–Si bond is 2.3. Both are quoted in the same units, both describe the same material, and neither is convertible into the other — one is the cost of promoting an electron and the other is the cost of taking two atoms apart.

wrong · Bands in a solid

⟨x²⟩ is the average number of neighbours. For each structure, the mean coordination counted off the edge list beside the mean of x² measured off the eigenvalues. They are equal by an identity about graphs, not by a limit — and the binding each bond supplies, in the last column, obeys no such rule and falls as neighbours are added.

The bond that weakens as neighbours multiply

Wrapped structures with two, four and six neighbours per site give a mean of x² of exactly 2, 4 and 6. Their binding per site goes 1.272, 1.611, 1.979 — and their binding per bond falls from 0.636 to 0.330, which is why a metal atom with twelve neighbours has weak bonds and a great many of them.

solids · Cohesion $Five sixths of the bonds, nine tenths of the binding. What a site in the outer layer of an open block keeps, measured three ways: the fraction of its bonds, the fraction the second-moment rule predicts of its binding, and the fraction the calculation gives. The last two agree and the first does not.$

A surface is not a count of broken bonds

Cut a crystal and every atom in the new face has lost one of its six neighbours. The standard estimate follows immediately: a surface costs one sixth of the cohesive energy per atom exposed. Computed, it costs a little over half that — the atom keeps 91.2 per cent of its binding while keeping only 83.3 per cent of its bonds, because the bonds that survive get stronger when their competitors are removed.

wrong · Cohesion

The covalent arrangement the inversion finds is Pauling's mean — except for hydrogen fluoride. For eight bonds, the covalent arrangement's binding found by solving the two-state problem exactly against the measured dipole and bond energy, against Pauling's covalent mean — the geometric mean of the two homonuclear bond energies — which the inversion never sees. The line is equality. HCl 0.97, HBr 0.98, HI 1.03, ClF 1.09 of the mean; HF 1.68. Open circles are the alkali halides, where the ionic weight is near 0.8 and the result is mostly the assumed repulsion.

Two measurements leave nothing to fit

A bond pictured as a mixture of its ionic and covalent arrangements has two unknowns — the coupling between them and the covalent arrangement's own energy — and a dipole fixes only one. Add the bond energy and the two-state problem inverts exactly, with no parameter left. The covalent arrangement it returns can then be held against something it never saw: Pauling's mean of the two homonuclear bond energies. For hydrogen chloride, bromide and iodide it lands on that mean to within three and a half per cent, and chlorine monofluoride within nine. Hydrogen fluoride lands 68 per cent above it. And the couplings rise and fall with length rather than falling — hydrogen fluoride's, on the shortest bond, is the smallest.

wrong · Electronegativity

Named alongside it

The objects these essays reach for when they reach for this one.

Bands in a solid Cohesion Model limit Tight-binding models Adjacency matrix Approximation Band filling Band width Closed form Coordination Coordination number Dipole moment

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