Molecular Geometry & Bonding
what the shape decides
Concept

Jahn–Teller distortion — where it appears

The theorem that a molecule in a degenerate electronic state distorts to remove the degeneracy. The energy gained is linear in the distortion where the elastic cost is quadratic, so no stiffness whatever holds the symmetric structure.

Named by 11 essays across 6 fields — each of them below, with the objects they name alongside it.

Hückel levels of cyclobutadiene. The orbital energies of the pi system, computed as the eigenvalues of the molecule's adjacency matrix. Degenerate levels share a line. The electrons are placed by aufbau with Hund's rule, so a half-filled degenerate shell shows as two unpaired spins.

Delocalisation is stabilising, and other things that are false in general

Spreading electrons over more centres is supposed to lower their energy. Cyclobutadiene is delocalised in exactly the same sense as benzene, gains exactly nothing by it, and the arithmetic says so before any experiment does.

wrong · Aromaticity 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 cyclobutadiene: what alternation costs and gains. The π energy of cyclobutadiene 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 linearly 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 vibration that lowers the symmetry

A molecule in a degenerate electronic state distorts until the degeneracy is gone. Which distortion it needs is a direct product; whether it wins is a race between a π energy falling linearly and a σ frame resisting quadratically, and both powers are measured here.

beyond · Aromaticity Oh descending to D4h. Every irreducible representation of Oh restricted to D4h and reduced there — A1g, A2g, Eg, T1g, T2g, A1u, A2u, Eu, T1u, T2u. A representation that arrives in more than one piece is a degeneracy the lower symmetry cannot hold, so a level carrying it must split when the molecule distorts.

Descent in symmetry

Lower a molecule's symmetry and its labels stop being available. Which of them survive, which split, and into what, is decided by restricting characters to the operations that are left — arithmetic, not a table to be looked up.

symmetry · Point group How many groups a molecule can fall to. For each molecule, every subgroup of its point group, found by closing subsets of the operations recovered from its atom positions — beside the number the corresponding abstract group is known to have. The two agree in all 4 cases. The last column is how many of those subgroups this site holds a character table for, which is a minority in every row but the first.

Every group a molecule can fall to

A distortion takes a molecule's symmetry away, and what is left is not a free choice — it has to be a group. Closing subsets of methane's twenty-four operations returns thirty of them, which is exactly the number the symmetric group on four letters has.

symmetry · Point group 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 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 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 carriers a distortion was hiding. A half-filled ring of 40 — one of the 4m rings that carry exactly one pair of carriers at every temperature — allowed to distort. Cold, it alternates by 0.1232, opens a gap of 0.4927 and carries 3.6e-15 carriers rather than one pair. The alternation is undone continuously at kT = 0.1358, and the carrier count comes back as it goes.

The carriers a distortion was hiding

A half-filled ring of 4m carries exactly one pair of thermal carriers at every temperature, which is true only of a ring held rigid. Allowed to move, it does not carry them: it alternates, opens a gap of 0.4927, and carries none at all until a temperature that undoes the distortion.

wrong · Metal

Named alongside it

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

DegeneracyDistortionIrreducible representationsPoint groupBond alternationDescent in symmetryElastic energyHOMO–LUMO gapHückel theoryModel limitPeierls distortionVibrational modes

All concepts