Molecular Geometry & Bonding
what the shape decides
Concept

Dissociation — where it appears

The separation of a molecule into fragments, taking its shared electrons with them. It is the case a single determinant describes worst, because it forces the electrons to remain paired at every distance.

Named by 3 essays across 3 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 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 Two errors, opposite signs, four orders of magnitude apart. A finite basis makes H₂⁺'s binding too large by letting each atom borrow the other's functions, and too small by describing the molecule incompletely. Both are computed here against the exact binding of 0.102634 hartree. The second is thousands of times the first at every basis size, and it is the first that counterpoise removes — so the corrected number is further from the true one than the uncorrected at every row of this table.

The basis the other atom lent

Two atoms in a molecule are described in each other's functions and the separated atoms are not, so the molecule is treated better than the pieces and the binding comes out too large. That is the basis set superposition error, it is removed by a standard correction, and for H₂⁺ in four Gaussians a centre it is six tenths of a microhartree against an incompleteness error of twelve millihartree — a factor of eighteen thousand the other way.

orbitals · Basis

Named alongside it

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

Model limitApproximationBands in a solidBasisBasis setBond energyBond lengthClosed formConvergenceDouble occupancyEigenvalueElectron affinity

All concepts