Molecular Geometry & Bonding
what the shape decides
Concept

Symmetry-forbidden transitions — where it appears

Vanishing because a product of representations lacks the totally symmetric one. Such a quantity is exactly zero rather than small, and computing it returns arithmetic noise, which is a different claim from being negligible.

Named by 13 essays across 7 fields — each of them below, with the objects they name alongside it.

1s with 2px at 2.8 bohr. The two orbitals in the plane containing both nuclei, with the regions where their product is positive and negative shown faintly. The overlap integral is the signed volume of that product, and where symmetry makes the two regions mirror images it comes out exactly zero. Contours drawn: 1s at 50% of its density, |ψ| = 1.48e-1; 2px at 50% of its density, |ψ| = 3.16e-2.

Exactly zero

Where symmetry forbids an interaction the overlap is not small. It is zero — and computing it and finding arithmetic noise is a different kind of statement from computing it and finding a small number.

bonding · Overlap What the group settles. For each molecule, the point group found from its coordinates and the two properties that follow from the group alone. Neither column required knowing anything about the bonds.

Symmetry forbids a dipole

Whether a molecule can have a dipole moment follows from its point group alone. The usual argument — adding up bond vectors — gets the right answer for easy cases by a route that does not generalise.

symmetry · Point group Dipole selection rules in Td. For every pair of symmetry species, whether an electric dipole transition between them is allowed and along which polarisation. An entry is allowed exactly when the triple product of representations contains the totally symmetric one.

Selection rules are one theorem

An integral over all space vanishes unless the integrand is totally symmetric. Every selection rule in spectroscopy is that sentence with a different integrand — and the rule of mutual exclusion falls out rather than being remembered.

symmetry · Representation 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 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 The total energy against the distortion, at several gaps. The elastic cost plus the second-order lowering, for five gaps between the ground state and the excited state it mixes with. The critical gap is 1.00: above it the symmetric structure is the minimum, below it the minimum has moved off zero, and nothing about the molecule is degenerate in either case.

A distortion needs two states

A degenerate electronic state cannot survive — that is the Jahn–Teller theorem, and it can be computed. A closed shell can fail to survive too, and the condition is a number: the symmetric structure holds only while the nearest excited state of the right symmetry lies above 2λ²/k, and one of ten symmetry species in an octahedron is the right one.

applied · Peierls distortion Four, however many ligands there are. For each molecule, the number of ligand σ combinations that find a partner among the central atom's four s and p orbitals, against the number of ligands and lone pairs it has. The matched count rises along the diagonal and then stops at four, because there are four orbitals; everything above the ceiling is a pair with nowhere on the central atom to go, and that is what the word hypervalent names.

Four is all that s and p can match

Reduce the ligand σ set of ten molecules in each one's own point group and ask how many of its components transform as one of the central atom's four valence orbitals. The answer is never more than four — not by arrangement, in every geometry from linear to octahedral — and what is left over is n + L − 4, with exactly twice that many electrons in excess of an octet.

beyond · Hypervalency A ratio that measures a distortion, and squares it first. How far ammonia's depolarised bands come off three quarters against how far one of its bonds has been stretched. Undistorted the departure is 3.1e-8, which is the rounding in the stored coordinates rather than a physical effect; at 0.1 Å it is 0.04239, and the slope is 1.999 — the departure goes as the square of the distortion, so a ratio measured to three decimals fixes a length to one and a half.

A ratio that squares what it measures

A depolarised Raman band sits at exactly three quarters because symmetry says its mean polarisability derivative is zero. Distort the molecule and it comes off — by 4.5 × 10⁻⁴ for a hundredth of an ångström and 0.042 for a tenth, going as the square of the distortion, which makes a ratio measured to three decimals a length known to one and a half.

spectra · Spectrum 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 The gap that makes sixteen special does not move. The gap above the sixteen-electron closure of a square plane and above the eighteen-electron closure of an octahedron, against the π strength. The octahedron's is 3eσ − 4eπ and moves at every value; the square plane's is exactly 2eσ until the π strength reaches a quarter of the σ one, because the orbital that sets it is d(z²) and a square-planar ligand set has nothing of that symmetry to offer. Past the threshold the two are the same number, which is not a coincidence: beyond it the square plane's gap is set by d(xy) and the expression is the octahedron's.

The orbital a ligand cannot reach

The sixteen-electron count of a square plane is a statement about an energy rather than about symmetry matching, so it was the count that ought to be sensitive to a π channel where the eighteen-electron one is not. It is not sensitive either — and for a sharper reason. The orbital that sets its gap is d(z²), and a square-planar ligand set contains nothing of that symmetry, so the gap is exactly 2eσ until the π strength reaches a quarter of the σ one.

applied · Electron count Every crossover a shell has. The field at which each coupled pair's linear behaviour returns — the gap between the two levels divided by twice the dipole joining them — on a logarithmic axis. The n = 3 shell has 3 distinct fields rather than four, because its m = ±1 half is a two-level ladder with one coupled pair. All three are below the n = 2 shell's single one.

Three events, and a ratio of two dipoles

The m = 0 half of a shell has two crossovers because it has three levels and two coupled pairs. The other half has two levels and one, so the whole shell has three distinct fields rather than four — and two of the three share a gap exactly, which makes the ratio between them a ratio of two dipoles, 2/√3.

symmetry · Representation A pair with no overlap, and a third orbital swept past it. The three levels of a trio in which the two outer orbitals have exactly no overlap with each other, as the third orbital's energy is swept. The middle line is at -13.6 at every point — the antisymmetric combination of the two, which has no partner of its own symmetry and cannot mix with anything. The other two move, so the pair is split by an orbital it has no direct contact through.

A bond order between atoms that do not interact

A diatomic held where its overlap changes sign has no interaction between its two orbitals at all — which is what the sign change of its overlap means. Put a third orbital beside it and the pair is still split, one line sits exactly at the free-atom energy at every third-orbital energy, and the bond order between the two runs to −0.9999. Three measures of the same bond disagree completely.

wrong · Overlap

Named alongside it

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

Selection rulesIrreducible representationsModel limitPoint groupDegeneracyTransition momentVibrational modesDipole momentInfrared activityCharacter tableThe rule of mutual exclusionNon-bonding orbitals

All concepts