Molecular Geometry & Bonding
what the shape decides
Concept

Infrared activity — where it appears

Whether a vibration can absorb infrared light, decided by whether the dipole changes along it. A molecule with no dipole at all can have strong bands, and a mode that preserves the molecule's symmetry has none.

Named by 9 essays across 3 fields — each of them below, with the objects they name alongside it.

6 molecules, counted. How many vibrations each molecule has, how many symmetry species they fall into, and how many frequencies are infrared active, Raman active, both, or neither. Every column after the first counts frequencies rather than modes, because a degenerate pair is one line in a spectrum. The molecules with a centre of inversion are the ones with nothing in the both column — mutual exclusion as a computed count. Nothing here uses a force constant.

How many frequencies, not how many modes

Benzene has thirty vibrations. It has twenty distinct frequencies, and of those, eleven can be seen — four in the infrared and seven in the Raman, with no band in common. The other nine are invisible to both experiments, exactly.

spectra · Spectrum 4 molecules, counted. How many vibrations each molecule has, how many symmetry species they fall into, and how many frequencies are infrared active, Raman active, both, or neither. Every column after the first counts frequencies rather than modes, because a degenerate pair is one line in a spectrum. The molecules with a centre of inversion are the ones with nothing in the both column — mutual exclusion as a computed count. Nothing here uses a force constant.

Two structures, two spectra

A linear XY₂ gives two infrared bands, one Raman band and no band in common. A bent XY₂ gives three of each and three in common. Counting settles the shape, without a force constant, an assignment or a single measured frequency.

spectra · Spectrum benzene: 30 vibrations. The vibrational representation, obtained by subtracting the translations and rotations from the full set of Cartesian displacements, with the infrared and Raman activity of each species read off the same character table.

What an absence proves

A band that symmetry forbids is not weak. Its intensity is zero, exactly, by a theorem — while a band that is merely too faint to see is absent for reasons no theorem covers. The two look identical in a spectrum and support completely different conclusions.

spectra · Spectrum carbon dioxide: D∞h worked in D2h. The vibrations of carbon dioxide, computed in D2h because D∞h has infinitely many operations and the reduction formula divides by the order of the group. Each constructed operation is checked against the molecule before use. The count must come out at 3N−5 rather than 3N−6 — 4 for 3 atoms — because rotation about the molecular axis moves no atom and is not a motion of the molecule at all.

An infinite group, worked in a finite one

A linear molecule has infinitely many symmetry operations, and every formula in character theory divides by the number of them. The standard device is to work in a finite subgroup — and it is worth computing what that trade costs rather than putting it in a footnote.

symmetry · Point group methane, substituted 5 ways. CH₄, CH₃D, CH₂D₂, CHD₃, CD₄ — the same molecule with the same force constants, differing only in which nuclei are heavy. Every column is a consequence of which of the parent's operations survive the mass pattern: the surviving set is a subgroup, it is named from its own conjugacy classes, and the vibrations are reduced in it. The allowed and computed columns come from group theory and from the mass-weighted Hessian respectively, and they agree in every row.

A spectrum that changes when only a mass does

Methane has four distinct vibrational frequencies and two infrared bands. Replace two of its hydrogens with deuterium and it has nine and eight — with every force constant identical, every nucleus where it was, and the potential energy surface unchanged.

spectra · Spectrum Mutual exclusion across every group here. The 20 groups with tables here, each with its highest rotation order, whether it has a centre of inversion, which of its representations carry a coordinate, which carry a product of coordinates, and whether any carries both. Every group with a centre excludes, which is a theorem. 1 group without a centre excludes as well — D5h — so the rule does not run backwards, and the counterexample needs a fivefold axis.

Mutual exclusion does not prove a centre

A centrosymmetric molecule shows no band in both its infrared and its Raman spectrum. The rule is a theorem and its converse is read off as though it were part of it — but ferrocene in the gas phase has no centre of inversion and no coincidence either, and the reason is that a fivefold axis separates the coordinates from their products where a threefold or fourfold axis cannot.

spectra · Spectrum The dipole moment, and how many bands there are. For each of five molecules: the dipole moment of the point-charge model, the number of modes whose dipole derivative does not vanish, and the largest derivative. The molecules with no dipole at all have the most active bands, which is the whole of the argument.

A dipole is not what an infrared spectrum sees

Carbon dioxide has no dipole moment at all and three of its four modes are infrared active. Methane has none and six of nine; boron trifluoride none and five of six. Water, which has the largest dipole of the five, has three modes and three bands — and its dipole predicted neither number.

shape · Dipole 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 Four tables, one answer, and a reason it could not be otherwise. The intensity–motion correlation — infrared intensity against how far the atoms move — computed with charges from four published electronegativity tables. Every molecule gives the same number on all four, to machine precision, because each is made of two elements: its charges are one number times a fixed pattern, a change of table changes only that number, and a rank correlation does not notice a rescaling. The dipole moments beside them do notice, which is the check that the tables are genuinely different.

The table that could not have mattered

Charges taken from one of four electronegativity tables invite a worry, because the tables disagree with each other. They do disagree — hydrogen cyanide's dipole runs over a factor of eleven between them — and for the molecules in question the worry could not have applied, because a molecule of two elements has charges that are one number times a fixed pattern and a rank correlation does not notice a rescaling.

shape · Dipole

Named alongside it

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

Vibrational modesSelection rulesIrreducible representationsRaman activityCharacter tablePoint groupDegeneracyDipole momentThe rule of mutual exclusionNormal modeSymmetry-forbidden transitionsModel limit

All concepts