Molecular Geometry & Bonding
what the shape decides
Concept

Orbit (group theory) — where it appears

A set of atoms carried onto one another by a group's operations, which is what a symmetry-resolved count counts. Two atoms in one orbit are indistinguishable by any measurement, however different they look on a drawing.

Named by 7 essays across 2 fields — each of them below, with the objects they name alongside it.

¹¹BF₃: what each mode is made of. ¹¹BF₃. Each row is one distinct frequency and each column one internal coordinate; the bar is the share of the motion in that coordinate. A mode whose largest share reaches nine tenths is a motion of one bond or one angle and is named for it. 1 of 4 here are.

Group frequencies, and where they stop

A carbonyl band sits near 1,700 wavenumbers in every ketone anybody looks at, and that regularity is real. Computed for a set of small molecules, four of twenty-one distinct frequencies belong to a single internal coordinate — and the four are exactly the coordinates symmetry leaves alone.

spectra · Normal mode phosphorus pentafluoride — D3h. 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.

Site symmetry, and what it constrains

A molecule's group is not the only group in the problem. Each atom sits at a position with a symmetry of its own, and what that local group permits decides how many kinds of environment a structure really has.

symmetry · Representation phosphorus pentafluoride: 3 environments. The atoms of phosphorus pentafluoride sorted into orbits under its own symmetry group — two atoms are in the same orbit when an operation of the group carries one onto the other, which makes them indistinguishable by any measurement. There are 2 of F, 1 of P, and a spectrum that resolves environments counts those rather than atoms.

A spectrum counts environments, not atoms

Phosphorus pentafluoride has five fluorines in two inequivalent sets, so its magnetic resonance spectrum should show two signals. It shows one — and the reason is not a symmetry the molecule has but a motion faster than the measurement.

spectra · Spectrum Where the explanation gives a negative number of vibrations. The usual formula against the answer, for the 15 molecules with internal coordinates here. It is right for 7 of them and wrong for 8, and for sulfur hexafluoride, benzene and both ferrocenes it predicts a negative number of totally symmetric vibrations — which is the clearest possible sign that the quantity being subtracted is not the one that should be.

A formula that predicts minus eleven vibrations

A molecule's count of totally symmetric vibrations is often explained as one per orbit of internal coordinates, less one per redundancy, with methane as the worked example. Computed for fifteen molecules it is right for seven and wrong for eight — and for sulfur hexafluoride, benzene and both ferrocenes it returns a negative number. Two independent corrections turn it into an identity.

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 Water's two hydrogens are closer than platinum's chlorine. For each molecule, the longest pair that is a bond and the shortest pair that is not, on a logarithmic length axis. A cutoff on the length has to sit to the right of every filled mark and to the left of every open one, and it cannot: the longest bond in the collection is 2.3200 ångström and the shortest non-bond is 1.5144. The two populations overlap by a factor of 1.53, so the rule in use is not a rule with a badly chosen number in it — it is a rule with no number that works.

No length separates them

A bond list here is one distance cutoff with a clause about hydrogen, added when peroxide came back with five bonds instead of three. The clause repaired one molecule. Across the twenty-three molecules drawn here, the longest bond is 2.32 ångström and the shortest pair that is not a bond is 1.51 — so no cutoff can work at all, and five molecules currently come back with no bonds.

symmetry · Point group The orbit identity holds on all nineteen, and the formula it replaced on eight. For each molecule in the census under the radius rule: its number of totally symmetric vibrations, the corrected count — symmetric orbits less symmetric redundancies — and the usual formula, orbits less redundancies. The corrected count lands on the molecule's own count every time. The usual formula is right for 8 of 19, and for ferrocene under the new bond list it predicts minus sixty-six.

The census a bond rule was hiding

Every internal coordinate, redundancy and totally symmetric count here is built on a bond list, and the bond list came from a length cutoff that gave five molecules no bonds. Rebuilt on the radius rule, the census reaches nineteen molecules instead of fifteen, the orbit identity holds on every newcomer, ferrocene's coordinates finally span all its vibrations — and a different gap appears: no bond rule can give a square-planar centre its two out-of-plane vibrations.

symmetry · Point group

Named alongside it

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

Internal coordinateModel limitPoint groupStabiliser (group theory)Symmetry operationCoordination numberDegeneracyGroup orderIrreducible representationsVibrational modesCharacter tableConvention

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