Molecular Geometry & Bonding
what the shape decides
Concept

Valence bond — where it appears

A description built from localised structures — covalent and ionic — rather than from delocalised orbitals. It dissociates correctly where the molecular orbital picture does not, and the weights of its structures depend on a convention.

Named by 6 essays across 3 fields — each of them below, with the objects they name alongside it.

Splitting goes with overlap. For each pair, the atomic levels on the outside and the combinations they form in the middle, with the splitting drawn in proportion to the computed overlap. A pair that symmetry forbids does not split at all, because its overlap is exactly zero.

Molecular orbital and valence bond

Two frameworks, taught as rivals, describing the same molecules. One starts from delocalised orbitals and localises; the other starts from localised bonds and delocalises. Pushed far enough they meet.

bonding · Models How often two electrons are in the same place. Double occupancy per site in the exact ground state of a chain of 2, against the repulsion. It starts at the independent-electron quarter and falls to nearly nothing. The second curve is what the state actually pays in repulsion, which peaks and then falls because the electrons have stopped meeting.

Where molecular orbital theory dissociates

The molecular orbital description of a two-electron bond puts both electrons on the same atom half the time — at every bond length, including infinite. The exact answer falls from a half to 0.0039 as the atoms separate, and the point where the two standard models are equally wrong is exactly U = 4t.

bonding · Models The valence-bond and molecular-orbital directions in one plane. The valence-bond function and the molecular-orbital function as two directions, at the angle their overlap requires — 45.0°, since they overlap by 0.7071. The exact ground state lies in the plane they span at every repulsion, to twelve decimal places, and swings from one to the other as the repulsion grows without ever arriving. Neither picture is a special case of the other and the answer is not either of them.

Two pictures, one plane

Molecular orbital theory and valence bond theory are taught as rival descriptions of a two-electron bond. In a model small enough to solve exactly they are two vectors in a two-dimensional space, the exact answer lies in the plane they span at every repulsion, and it is neither of them at any repulsion but two.

bonding · Models The ionic weight against how much ionic structure is in the wavefunction. The percentage each convention calls ionic, at a fixed structure overlap, as the amount of ionic structure in the wavefunction is raised from none to the molecular orbital value. They meet at both ends of the sweep and disagree everywhere between. Every curve is a weight and every set sums to one.

A weight that depends on how it is weighed

The ionic character of a two-electron bond is quoted as a percentage. For one wavefunction at hydrogen's bond length, three conventions in the literature give 18.73, 34.74 and 5.88 per cent — a factor of six — and on a wavefunction with no ionic structure in it at all, one of them still reports a quarter.

bonding · Models An antibonding occupation fills the zero and moves the minimum. The profile along the bond near π/R, per electron, on a logarithmic scale, for five antibonding occupations of the same two orbitals. With nothing in the antibonding orbital the profile is exactly zero at π/R = 1.573. Two hundredths of an electron leave a minimum at 1.608, which reads the separation as 1.954 bohr instead of 1.997. At 0.104 the minimum becomes a flat shoulder, and the Heitler–London bond, at 0.236, has none.

The zero belongs to one determinant

A bonding orbital's momentum profile along the bond is exactly zero at π/R, and that zero reads a bond length with nothing fitted. It is a property of putting every electron into that one orbital. Any antibonding occupation fills it in linearly and drags the minimum outward, a tenth of an electron erases it, and the valence-bond wavefunction built from the same two functions never has one at any separation.

orbitals · Orbital 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.

Molecular orbitalBasisConfiguration interactionElectron correlationModel limitOverlap integralDouble occupancyExact diagonalisationHubbard modelMany-electron wavefunctionsOn-site repulsionPartial charge

All concepts