Molecular Geometry & Bonding
what the shape decides
Concept

Sign change — where it appears

The reversal of a wavefunction's sign across a node, which is what distinguishes a bonding combination from an antibonding one. A picture of the density has thrown it away, which is why every orbital here is coloured by sign.

Named by 4 essays across 3 fields — each of them below, with the objects they name alongside it.

The 2pz orbital. The 2pz orbital at the contour enclosing 90 per cent of its density — a level solved for by integration rather than chosen. The two colours are the two signs of the wavefunction, which is what distinguishes a bonding interaction from an antibonding one. Contours drawn: 2pz at 90% of its density, |ψ| = 9.48e-3.

What an orbital is

Not a region the electron occupies, not a path it follows, and for any atom but hydrogen not an exact anything. An orbital is a one-electron wavefunction, and almost every difficulty in this subject comes from forgetting that.

orbitals · Orbital The radial function of 3s. The radial part of the wavefunction, which changes sign at each node, and the radial distribution, which is the probability of finding the electron in a shell at that radius. The second vanishes at the nucleus and the first does not.

Nodes

An orbital with quantum numbers n and l has exactly n−l−1 radial nodes and l angular ones. That is a count, it is exact, and it is the fastest way to catch a drawing that is wrong.

orbitals · Orbital Levels of a ring closed with a half turn in it. The orbital energies of a ring with one resonance integral reversed in sign, which is what half a turn in the ribbon of p orbitals does to it. The levels come in degenerate pairs from the bottom up rather than singly, so the count that closes a shell is 4n rather than 4n + 2. Degenerate levels share a line. The electrons are placed by aufbau with Hund's rule, so a half-filled degenerate shell shows as two unpaired spins.

The ring with a twist in it

Reverse the sign of one resonance integral in a ring and its levels stop being one nodeless orbital above degenerate pairs and become degenerate pairs all the way up. The count that closes a shell changes from 4n+2 to 4n, and every aromatic ring size and antiaromatic ring size exchange places.

beyond · Aromaticity The bonding follows the overlap over, and turns where it turns. The stabilisation of three pairs of orbitals against how far apart they are, each scaled to its own largest, with the coupling taken from the computed overlap. Two orbitals with no radial node are stabilised less at every separation further out; the two with one turn over — and the maximum of the bonding is at exactly the separation of maximum overlap, to five decimal places, because a coupling proportional to the overlap makes the stabilisation a strictly increasing function of it.

The same overlap, a different bond

A 1s and a 2s change their overlap by two thirds between two and seven bohr, and change what they are bonded by by 0.154 per cent. The overlap turns over and the bonding turns over with it, at exactly the same separation to five decimal places — the arithmetic refused the expectation that the secular denominators would move it — and what separates one pair from another is not where the maximum is but how little of it there is.

wrong · Overlap

Named alongside it

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

AntibondingContour levelNodeQuantum numbersRadial nodeWavefunctionAngular nodeAntiaromaticityAromaticityBondingClosed formConjugation

All concepts