Molecular Geometry & Bonding
what the shape decides
Concept

Observable — where it appears

A quantity a measurement can return, which is not what most of this subject's vocabulary names. Orbitals, charges, hybrids and structure weights are all features of a chosen description rather than things that can be measured.

Named by 5 essays across 4 fields — each of them below, with the objects they name alongside it.

One double bond, two descriptions. A carbon–carbon double bond drawn twice in the plane perpendicular to the molecule: as a σ orbital along the axis with a π orbital above and below it, and as two equivalent bent bonds tilted 50.8 degrees either side of the axis. The two descriptions are related by a rotation and have the same density everywhere.

Two bent bonds, or a σ and a π

A carbon–carbon double bond is drawn two ways and the two look like rival claims about what is there. They are one occupied space in two bases, related by a rotation of exactly forty-five degrees: the density is identical to the last bit a double holds, the bent pair are sp⁵ hybrids at 50.77° to the axis, and their charge sits 0.235 Å off the plane of the molecule where neither canonical orbital's does.

bonding · Hybrids The Compton profile, exact and fitted. The momentum density integrated over the two perpendicular directions, for the exact 1s and for three fitted bases. The exact curve is 8/3π(1 + q²)³ in closed form; the fitted ones are sums of Gaussians and are cusped differently at the origin, which is the position-space cusp showing up as a shape in momentum.

The measurement a basis was not fitted to

Six Gaussians reproduce hydrogen's energy to eleven parts in a hundred thousand and its Compton profile to three parts in a thousand — twenty-five times worse, on a quantity an X-ray scattering experiment measures directly. The gap between the two errors widens as the basis is improved, because the energy is the one property a variational fit is best at.

orbitals · Basis What boron trifluoride's bands are strong in, and what they move. Every infrared-active mode of boron trifluoride, with its band strength and the root-mean-square displacement of its atoms in the zero point, each scaled to its own largest. The two do not order the modes the same way — the rank correlation between them is 0.2 — and the strongest band belongs to the mode at 719 cm⁻¹, in which 89.93 per cent of the motion is the lightest atom's. Every mode moves the same weighted amount of mass, exactly, so that is not what separates them either.

The mode that moves least radiates most

Boron trifluoride's strongest infrared band is the one in which the fluorines barely move: ninety per cent of the motion belongs to the boron, which is a fifth of the molecule's mass. The mode that moves the most mass is nine and a half times weaker. Across five molecules the rank correlation between band strength and how far the atoms actually go runs from +1 to −0.66, and every normal mode carries exactly the same weighted motion by construction.

shape · Dipole Which numbers survive a change of frame, and which are coordinates. Every quantity this field quotes, against the four things that can be changed without changing the molecule: the zero of energy, the unit, the reference state a stabilisation is measured from, and what a "per" quantity is divided by. A filled mark is a quantity that moves. three of the 9 survive all four, and every one of them is a property of the eigenvectors rather than of the energies. The first two changes are exact symmetries of the model, so a quantity that moves under either is a coordinate and not a quantity at all.

Which numbers carry a frame

A Hückel calculation is written in two numbers nobody computes and quoted against reference states nobody measures, so every quantity it prints is a quantity in a frame. Nine of them, tested against four changes of frame: three survive all four, and the one that survives both exact symmetries and looks safest — a dimensionless ratio between two molecules — is the most fragile of all, because against one reference its denominator is exactly zero.

bonding · Delocalisation Three quarters, exactly, for every mode that is not totally symmetric. The depolarisation ratio of every Raman-active mode of five molecules, computed from a bond-polarisability model. 18 of the 26 sit on three quarters to better than a millionth, and they are exactly the modes that are not totally symmetric: the mean polarisability derivative is a trace, a trace is invariant, and an invariant has no derivative along any other species. The polarised ones below the line are the totally symmetric modes, and where they sit is a property of the model rather than of the group.

The one intensity symmetry does fix

Symmetry says which bands exist and declines to say how strong they are. It makes exactly one exception, and it is a ratio: every Raman band that is not totally symmetric is depolarised by exactly three quarters, whatever the molecule and whatever the model — and methane's symmetric stretch is polarised by exactly zero, because its group is cubic.

symmetry · Spectrum

Named alongside it

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

Model limitNormal modePartial chargeApproximationBasisBond orderCharacter tableClosed formConventionConvergenceDelocalisationDensity matrix

All concepts