Molecular Geometry & Bonding
what the shape decides
Concept

Improper rotation — where it appears

A rotation followed by a reflection in the plane perpendicular to its axis. A structure with any such operation is superimposable on its mirror image, which is what makes chirality a group-theoretic property.

Named by 6 essays across 2 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 bromochlorofluoromethane — C1. 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.

Chirality is a symmetry statement

A molecule is chiral when its group contains no improper operation at all. The four-different-groups rule is a useful special case that misses molecules with no stereocentre and wrongly condemns some that have several.

symmetry · Point group One dihedral, three point groups. Hydrogen peroxide built at 13 values of its dihedral angle, with the point group searched for from the coordinates at each. The group is C2v when the hydrogens eclipse, C2h when they are anti, and C2 at every angle strictly between. The rails below are what the group settles on its own: the molecule may be polar except at the anti arrangement, and it is chiral except at the two ends. No energy is computed anywhere, and the marked angle is the measured one rather than a minimum found here.

One coordinate, three point groups

Hydrogen peroxide has four atoms and one soft internal coordinate. Turning it from nought to a hundred and eighty degrees takes the molecule through C2v, C2 and C2h — so it is chiral at every angle but two, and forbidden a dipole at exactly one of them.

shape · Dipole How many groups a molecule can fall to. For each molecule, every subgroup of its point group, found by closing subsets of the operations recovered from its atom positions — beside the number the corresponding abstract group is known to have. The two agree in all 4 cases. The last column is how many of those subgroups this site holds a character table for, which is a minority in every row but the first.

Every group a molecule can fall to

A distortion takes a molecule's symmetry away, and what is left is not a free choice — it has to be a group. Closing subsets of methane's twenty-four operations returns thirty of them, which is exactly the number the symmetric group on four letters has.

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 Which torsions a molecule's own operations turn backwards. Every torsion orbit of the molecules here that have torsions: how many operations carry the torsion onto itself, how many of those are proper, and which operations reverse it. 7 of 9 orbits are reversed, and in every one the operation doing it is improper — the plane of a planar molecule, a mirror bisecting the torsion's bond, or a centre of inversion at that bond. Proper operations fix torsions too, benzene's twofold axes and hydrogen peroxide's among them, and never reverse one. Staggered ferrocene gives the same rows as eclipsed.

Five coordinates for six vibrations

A torsion is reversed by every improper operation that carries it onto itself and by no proper one, so adding torsions tests the orbit rule on a second kind of signed coordinate — and the rule holds on every molecule. It also moves hydrogen peroxide's count of totally symmetric vibrations from three to four, which a property of a molecule cannot do. Its five coordinates never spanned its six vibrations, and three of fifteen coordinate sets had been counting vibrations they did not describe.

symmetry · Point group

Named alongside it

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

Symmetry operationChiralityPoint groupGroup orderModel limitPolarityRound-trip checksSubgroupApproximationBond dipoleCharacter tableConformation

All concepts