Molecular Geometry & Bonding
what the shape decides
Concept

Transition moment — where it appears

The integral of an operator between two states, whose square sets a transition's intensity. It vanishes exactly where symmetry forbids it, and its size where allowed is a derivative rather than a property.

Named by 9 essays across 4 fields — each of them below, with the objects they name alongside it.

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 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 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 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 Three lines, then a hundred. The exact removal spectrum of a 6-site Hubbard ring at half filling: every final state of the ion, at the energy it costs to reach and with the intensity the matrix element gives it. With no repulsion there are 3 lines and they are the occupied orbital energies. At U = 8 there are 100, on a molecule with 6 orbitals — so the spectrum cannot be read as a list of orbital energies, because there are more bands in it than there are orbitals to name.

More bands than there are orbitals

A photoelectron spectrum is read as a list of orbital energies, one band per occupied orbital. Computed exactly for a six-orbital ring, it has three bands with no repulsion and a hundred with eight — and by then fifty-three per cent of the intensity is in lines that no orbital corresponds to. The total intensity is three at every repulsion, exactly, because that is a sum rule and not a fit.

spectra · Photoelectron 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 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

Named alongside it

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

Selection rulesModel limitSymmetry-forbidden transitionsDipole momentNormal modeVibrational modesInfrared activityIrreducible representationsPartial chargePoint groupAbsorptionEigenvalue

All concepts