Molecular Geometry & Bonding
what the shape decides
Concept

Selection rules — where it appears

The statement that a transition vanishes unless a product of representations contains the totally symmetric one. It is exact where it forbids, so a forbidden band is absent at every temperature rather than merely weak.

Named by 27 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 The density of states of a chain of 2000. The 2000 levels of a linear chain, binned into 34 intervals across the band, with the closed-form density drawn through them. The density piles up at both edges because that is where the level spacing turns over, and nothing periodic was assumed to get it.

A density of states is not a spectrum

A molecule's spectrum is a list of positions and a solid's is a shape, and the shape is not the density of states. Between the two sits everything the count leaves out — which transitions are allowed, how strongly, and from where to where.

solids · Bands in a solid Where the d–d band falls, and where the eye is. Nine ligands' measured splittings as wavelengths, on a logarithmic scale, with the visible range shaded. Three fall inside it; the halides sit in the near infrared and carbon monoxide in the ultraviolet. The splitting decides where the band is and does not decide what is seen.

Where a d–d band falls

A splitting is an energy and an energy is a wavelength, so the ligand field fixes where a complex absorbs. Only three of nine common ligands put that band inside the visible range at all — and the most intensely coloured transition-metal compound in the cupboard has no d electrons to excite.

applied · Ligand field 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 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 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 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 carbonyl sulfide: a linear. The principal axes of carbonyl sulfide drawn at its centre of mass, with the moment of inertia about each. Which kind of top this makes it is decided by comparing three numbers, and the point group forces the same answer independently: an axis of order infinity means a linear molecule, with one moment at zero forbids an asymmetric top.

The rotational spectrum is a moment of inertia

Every line in a microwave spectrum sits at a multiple of one number, and that number is a conversion constant divided by a sum of mass times distance squared. No bonding argument appears anywhere in it.

spectra · Rotation 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 Dipole selection rules in Oh. 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.

Why a d–d band is weak

In a centrosymmetric complex the transition between the two halves of a split d shell is forbidden — exactly, by parity, with no small quantity anywhere. What makes it visible at all is that the molecule is never quite centrosymmetric, and the computation says which vibrations do the work.

spectra · Representation 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 Every set of point groups here that shares a table. Found by comparing character matrices rather than named: five sets among the tabulated groups, at orders 2, 4, 6, 8, 20. The smallest is the most startling — a mirror plane, a centre of inversion and a twofold axis all have the two-row table with entries 1, 1 and 1, −1, and one of the three describes a chiral molecule. The largest is the eclipsed and staggered conformers of ferrocene, which are one molecule at two temperatures.

One table, three groups

A mirror plane, a centre of inversion and a twofold axis have the same character table — two rows, entries 1, 1 and 1, −1 — and one of the three describes a chiral molecule. A character table is a property of an abstract group; a point group is that plus an action on space, and the action is what a molecule has.

symmetry · Representation ammonia's rotational levels, sorted by K. The rigid rotational levels of ammonia up to J = 4, each drawn at its computed energy and grouped by J. Within a group the levels are pushed apart by the second rotational constant, so it is plainly there in the level pattern.

The constant a spectrum cannot see

A symmetric top has two rotational constants and its microwave spectrum reports one of them. Not badly, not with difficulty: ammonia's A of 6.3406 wavenumbers appears in none of its lines at any J and any K, because the term it belongs to cancels exactly out of every transition. The molecule turns about that axis, the energy is real, and the measurement is blind to it.

spectra · Rotation 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 A photoelectron band is a filter, and the group chooses the filter. The Huang–Rhys factor of every vibration of five molecules under a change of geometry that lengthens every bond alike — which is what removing an electron from a non-degenerate orbital does. On a logarithmic scale spanning sixteen decades, eight modes carry the whole of it and the rest sit on the floor at arithmetic noise. Which ones is decided by the point group: only a totally symmetric vibration can appear, whatever the size of the change.

A band is a filter on the modes

A photoelectron band's vibrational structure reports the frequencies of a few of the ion's vibrations and is silent about the rest, and which few is decided by the point group before any geometry is known. Under a change of shape that lengthens every bond alike, methane's totally symmetric stretch gets a Huang–Rhys factor of 0.905 and its other eight modes get between 10⁻²⁸ and 10⁻³⁵.

spectra · Photoelectron An effect that cannot distort a molecule, deciding how far it distorts. The energy along one distortion coordinate, three times. With only the first-order term the minimum is at 0.6; with only the second-order term there is no minimum away from zero at all, because the gap of 1.5 is above the critical 1 for a closed shell on its own. With both, the molecule distorts to 1.06 — well past the first-order answer — and gains 0.09 more than the two separate stabilisations add up to.

Two distortions in one coordinate

A second-order Jahn–Teller effect that cannot distort a molecule by itself — its gap is half again above the critical value — nearly doubles the distortion when a first-order effect is already acting. The molecule goes to 1.075 instead of 0.600 and gains twice the energy, and it does it while the gap the second-order term divides by is opening rather than closing.

shape · Peierls distortion Three quarters, exactly, for every mode that is not totally symmetric. The depolarisation ratio of every Raman-active mode of five molecules, computed from a bond-polarisability model. 18 of the 26 sit on three quarters to better than a millionth, and they are exactly the modes that are not totally symmetric: the mean polarisability derivative is a trace, a trace is invariant, and an invariant has no derivative along any other species. The polarised ones below the line are the totally symmetric modes, and where they sit is a property of the model rather than of the group.

The one intensity symmetry does fix

Symmetry says which bands exist and declines to say how strong they are. It makes exactly one exception, and it is a ratio: every Raman band that is not totally symmetric is depolarised by exactly three quarters, whatever the molecule and whatever the model — and methane's symmetric stretch is polarised by exactly zero, because its group is cubic.

symmetry · Spectrum 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 Two events, not one — and both below the n = 2 shell's. The field at which each coupled pair's shift stops being quadratic and starts being linear, for the n = 3 shell and for the n = 2 shell. The n = 3 shell has two, a factor of 3.66 apart, and both are far below n = 2's single one — so the linear effect returns is two events in a shell with a d, and it happens at a field thirty times weaker than in a shell without one.

Two events where there was one

A broken symmetry returns at a field where the coupling matches the gap it has to overcome, and in a shell with two levels that field is a single number. The next shell up has three, two gaps and two dipoles — so there are two crossovers, a factor of 3.66 apart, both of them below the single one of the shell below, and the exponent takes more than a decade of field to travel between them.

symmetry · Representation 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 Tilting the field raises the count, and then lowers it again. How many distinct fields the shell has an event at, against the angle between the field and the z axis. Along either axis there are 3; at a general tilt every one of the 6 coupled pairs has its own field and there are 6. In between the count comes back down at four angles where two events coincide, and at forty-five degrees two separate coincidences happen at once.

Four angles the shell chooses

A field along one axis gives a shell of nine functions three fields with an event, and tilting the field should separate the coincident ones and raise the count towards the number of coupled pairs. It does — from three to six. But not monotonically: at four angles two events collide again, and every one of those angles is the arctangent of a ratio of the shell's own angular integrals.

symmetry · Representation Six distortions, and the one the ratio cannot see at all. How far a depolarised band of boron trifluoride comes off three quarters when the bonds are stretched in six different combinations, all at the same size. Five move it. The one that stretches every bond by the same amount moves it by nothing whatever — not a small amount, exactly none — because that distortion keeps every symmetry operation the molecule had, and the band sits at three quarters by its species.

The distortion the ratio cannot see

A depolarisation ratio can be made into a structural probe: break a molecule's symmetry and a band fixed at three quarters comes off it by an amount that depends on the distortion. The question is whether two distortions the ratio can see separately might cancel into one it cannot. They do — and the cancellation is exact, at any size, because the blind direction is the one that keeps the symmetry.

spectra · Spectrum The sum is flat and the maximum is not. Both readings round the circle, each divided by its own average so the two can be drawn together. The maximum varies by 98 per cent; the sum over all four depolarised bands varies by 0.038. A sum cannot be changed by reordering, so if each band's departure is an isotropic quadratic form the sum must be constant — and to four parts in ten thousand it is.

The sum was flat all along

The depolarisation reading varies by a factor of two round a circle in the non-symmetric plane, and the anisotropy may belong to taking a maximum over four bands rather than to the physics. Summing the four instead gives a reading constant to four parts in ten thousand, against a maximum that varies by ninety-eight per cent.

spectra · Spectrum The zero at π/R follows parity, not bonding. The bonding and antibonding combinations of three atomic functions on nitrogen, each along the bond and each normalised to its own largest value, against momentum in units of π/R. For 2s the bonding combination is zero at π/R and the antibonding one is not. For 2p along the bond it is the other way round: the σ bond carries a sine, is zero at the origin and near its largest at π/R, and the antibonding combination carries the cosine. For 2p across the bond the π bond carries the cosine again. The factor is a cosine exactly when the orbital's inversion parity matches the atomic function's.

The zero is a parity, not a bond

A hydrogen-like σ bond has a momentum profile along its axis that vanishes at π/R, and it is natural to read that zero as a bond's signature. Built from 2p functions pointing along the axis, the σ bond carries a sine instead and sits at 83 per cent of its peak there. Which factor an orbital carries is decided by whether its inversion parity matches its atom's, and bonding has nothing to do with it.

orbitals · Orbital

Named alongside it

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

Irreducible representationsDegeneracyPoint groupVibrational modesModel limitCharacter tableRaman activitySymmetry-forbidden transitionsInfrared activityThe rule of mutual exclusionNormal modeSymmetry breaking

All concepts