Molecular Geometry & Bonding
what the shape decides
Concept

Distortion — where it appears

A displacement of atoms away from a symmetric arrangement. It is adopted when it lowers the electronic energy faster than it raises the elastic energy, which for a degenerate state is guaranteed and for a closed shell is a comparison.

Named by 11 essays across 5 fields — each of them below, with the objects they name alongside it.

A chain of 100 is more stable alternating. The electronic energy gained by alternating the bonds of a half-filled chain, the elastic cost of doing it, and their sum, all per site and all against the alternation. The sum has its best value away from zero, so the evenly spaced chain is not the stable structure.

A chain cannot stay even

Diagonalise a half-filled chain, alternate its bonds slightly, and diagonalise again. The electrons gain more than the springs lose, and they do so for every spring constant whatever — because the gain is steeper than a parabola near the origin and a logarithm beats any constant.

solids · Peierls distortion d⁹: what a tetragonal distortion is worth. The electronic energy of d⁹, the elastic cost of the distortion, and their sum, against the fractional elongation of the axial bonds. The best distortion is at 0.14 and it is worth 0.54 in units of eσ; d⁶ in the same field gains 0, which is nothing.

Copper is never quite octahedral

A d⁹ ion in an octahedral field has three electrons in a doubly degenerate pair, which cannot be shared evenly. The energy it gains by distorting is linear in the distortion and the elastic cost is quadratic, so no stiffness holds the symmetric structure — and the four configurations with an even occupation gain exactly nothing.

applied · Peierls distortion A 6-ring at 111°: the twist-boat. A closed ring of 6 equal bonds meeting at 111°, drawn from the coordinates the closure conditions produce. Its torsions are 16.9°, -63.6°, 44.2° and repeat; its puckering amplitude is 0.508 bond lengths at a phase of 344°, with q₃ exactly zero, so it lies on the equator. Turning the ring changes none of those numbers, which is what makes them a description of the shape rather than of the view.

The ring that cannot hold still

Cyclohexane's chair is rigid and its boat is not, and that is a statement about the rank of a matrix rather than about strain. Hold every bond length and every bond angle fixed and count what is left: the chair has nothing, and the boat sits on a continuous loop of shapes with the same bonds and the same angles.

shape · Strain benzene: what alternation costs and gains. The π energy of benzene as its bonds are alternated by β(1 ± δ), the elastic cost of alternating them at a stated stiffness, and the sum. The π curve falls quadratically in δ, which is measured here and is the whole difference between a molecule that distorts and one that does not. The stiffness is a stated parameter, so where the minimum falls is not a claim; which power of δ each curve goes as is.

The hexagon is the frame's doing

Benzene's delocalisation energy is quoted as 2β and read as the reason its bonds are equal. Let the bonds alternate and the π energy goes down, not up — at every ring size, for 4n+2 as much as for 4n. The π electrons are not what keeps the hexagon regular; the σ frame is, and the margin between them is uncomfortably thin.

beyond · Delocalisation The twist the ring forces on the bond. For each ring size, the largest torsion about the double bond that the ring will close on — 180° being a flat trans arrangement and 90° being a π bond broken outright. The six-ring will not close at any torsion tested. The eight-ring reaches 139.3°, against 136° measured in trans-cyclooctene by diffraction: a model with bond lengths and bond angles in it and no energy anywhere agrees with the crystal to a few degrees.

The double bond a ring cannot hold

A trans double bond needs a ring of nine carbons to sit flat, and the eight-ring will hold one twisted by 40.7 degrees — against 136 degrees measured in trans-cyclooctene. The same eight-ring threshold, applied by counting ring sizes, sorts nine bridgehead alkenes correctly with no bridgehead anywhere in the argument.

shape · Strain 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 How far benzene is from its own group, against how far it has been pushed. The measure — the distance to the nearest structure with the ideal group — against the size of the distortion, for two ways of distorting the same molecule. Each curve is smooth and quadratic; the markers on it are where the point-group search changes its verdict, which happens at one step and says nothing about the steps either side of it.

How much symmetry is left

A point group is a verdict and every real structure fails it. Two distortions of benzene that are two per cent apart in how far they sit from D6h need tolerances 1.88 times apart before the search will call either of them D6h — and one tolerance, applied to six molecules, admits amounts of asymmetry differing by a factor of fifty-six.

symmetry · Point group One distortion, resolved into the species of the group it left. An arbitrary displacement of benzene, projected onto each symmetry species of D6h. The weights add to 1, which is the check that the projectors resolve the whole of it: the largest is E2g at 35.0 per cent, and the totally symmetric part — 3.4 per cent here — is the part that changes every distance and no symmetry at all.

How far, and along which coordinate

A continuous symmetry measure returns one number: how far a structure is from a shape. Projecting the same displacement onto the twelve symmetry species of benzene's group turns it into a list that sums to 1.000000000000, says which coordinates the structure left along, and shows that the totally symmetric part — 3.4 per cent of this one — moves every atom by 0.0200 ångström and lowers the symmetry measure by nothing at all.

symmetry · Point group 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 The shallowest slope, and the steepest. For three molecules, the softest and stiffest vibrations, the species each belongs to, and what a unit distortion along each costs. The cost is not taken from the frequency: it is computed by resolving the distortion onto the normal coordinates in the mass-weighted metric and adding up ω²q². That it comes back as the frequency is the identity the whole comparison rests on — the species decomposition uses only the coordinates and the group, and the cost uses only the masses and the force constants, and the two have to agree on this case before they can be asked to disagree on any other.

The coordinate it was already soft along

A distorted structure can be resolved into the symmetry species of its ideal group, and that resolution cannot say which coordinate a molecule fell down by itself and which one something outside pushed it along. The force field answers that, and the two halves agree on the case where they must — a unit distortion along a normal mode costs exactly that mode's frequency, to a part in a million, computed from masses and force constants by one side and from coordinates and characters by the other.

symmetry · Point group The whole difference lives at the two ends. The energy difference between the two dimerisations of an open chain, held at the same distortion, multiplied by the number of sites. It settles on a constant — 1.09 in units of the hopping — so the difference per site falls as one over the length, with a fitted exponent of -1. An end is a bond that is not there, and it is worth the same amount whatever it is attached to.

The distortion the ends decide

A chain of an even number of sites has an odd number of bonds, so its two dimerisations are different molecules rather than one molecule translated. Held at the same distortion they differ by 1.08715 in units of the hopping, whatever the length — a fixed amount of energy living at the two ends, with the per-site difference falling as one over the length at a fitted exponent of −0.99986. And below a hundred and twenty-eight sites the second dimerisation does not exist at all.

applied · Peierls distortion

Named alongside it

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

DegeneracyJahn–Teller distortionModel limitIrreducible representationsPeierls distortionPoint groupSymmetry breakingBond alternationConventionDescent in symmetryElastic energyHOMO–LUMO gap

All concepts