Molecular Geometry & Bonding
what the shape decides
Concept

Round-trip checks — where it appears

Recovering an input from an output as a test of a calculation. Assigning a point group to a structure whose group is known, or recovering bond lengths from moments of inertia, is the standard form of it here.

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

ammonia — C3v. The molecule with the point group found from its coordinates rather than looked up: every candidate operation was applied and kept when it permuted the atoms among themselves.

Point groups from coordinates

A molecule's symmetry is not a label to be looked up. It is decidable from the atom positions by searching for the operations that permute them, and the search either finds an operation or it does not.

symmetry · Point group The group of benzene, against how much error is forgiven. Four distortions of benzene, none larger than six hundredths of an ångström, and the point group the search reports for each at nine tolerances. The exact structure is D6h at every one of them, so the staircases below belong to the distortions rather than to the search. The cell marked in warning colour is a step at which the order of the group named at the tighter tolerance does not divide the order of the one named at the looser: C6h of order 12, so the sequence is not a chain of subgroups.

The tolerance is a decision

A measured structure is never exactly symmetric, so assigning it a point group means deciding how much error to forgive. Sweep that decision from a thousandth of an ångström to a third of one and benzene, bent by a hundredth, is assigned five different groups — and at one step the group named is not even a supergroup of the one named before it.

symmetry · Point group How far benzene is from its own group, against how far it has been pushed. The measure — the distance to the nearest structure with the ideal group — against the size of the distortion, for two ways of distorting the same molecule. Each curve is smooth and quadratic; the markers on it are where the point-group search changes its verdict, which happens at one step and says nothing about the steps either side of it.

How much symmetry is left

A point group is a verdict and every real structure fails it. Two distortions of benzene that are two per cent apart in how far they sit from D6h need tolerances 1.88 times apart before the search will call either of them D6h — and one tolerance, applied to six molecules, admits amounts of asymmetry differing by a factor of fifty-six.

symmetry · Point group Three parameters, two numbers, and a curve of answers. seven structures of formaldehyde, every one of which reproduces the measured rotational constants A and B exactly. The C=O length runs from 1 to 1.3 ångström, the C–H length from 1.56 down to 0.95, and the HCH angle from 74.58 to 164.47 degrees. The third constant is not a third number: for a planar molecule it is fixed by the other two, and it comes out at 1.14 for every member.

Three numbers is not a structure

Formaldehyde's rotational spectrum gives three constants, of which a planar molecule's are only two independent numbers, and its structure has three parameters. Seven structures are computed here that reproduce A and B to the last digit the solver carries: the C=O length runs from 1.000 to 1.300 ångström, the C–H length from 1.557 down to 0.952, and the HCH angle from 74.6 degrees to 164.5.

spectra · Rotation Every sign, lost. Formaldehyde in its own principal axes. Open circles are the atoms where they are; filled ones are where Kraitchman's equations put them, from the change in the three moments when each atom in turn is made heavier. The two agree to 7.6e-8 ångström — the equations are an identity for a rigid structure — but they return the square of each coordinate, so the two hydrogens at b = ±0.9348 both come back at +0.9348 and land on the same point.

The coordinate an isotope reports

Kraitchman's equations return an atom's position from the change in the moments when that atom alone is made heavier, and for a rigid structure they are an identity — formaldehyde's four atoms come back to a part in ten million. What they return is the square of each coordinate, so both hydrogens at b = ±0.9348 come back at +0.9348; every out-of-plane coordinate comes back imaginary at a moment error of one part in a hundred thousand; and the famous error cancellation, measured at a factor of thirteen, still leaves the answer two and a half times worse than a direct fit.

spectra · Rotation Four bonds, or one a₁ and three t₂. The same four occupied orbitals written in two bases. On the left each orbital sits on one bond; on the right one is shared over all four hydrogens and three follow the Cartesian directions. The transformation between them is orthogonal, so the density is unchanged.

The answer a search is most likely to give

If a cage has ten equally good localised descriptions, why does the literature agree about its picture? The hoped-for answer was that one basin is very large. Counting four hundred and eighty starting points says it is not: two independent searches agree one time in ten, and the description they most often return is not the best one.

beyond · Multicentre

Named alongside it

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

ConventionLeast-squaresModel limitGroup orderPoint groupStructureSymmetry operationEquivalent atomsImproper rotationIsotope substitutionIsotopologueMoment of inertia

All concepts