Molecular Geometry & Bonding
what the shape decides
Concept

Descent in symmetry — where it appears

What happens to a level when a molecule loses symmetry operations: each representation of the parent group splits into representations of the subgroup. The correlation between the two is fixed by the characters and is where a distortion's spectroscopic consequences come from.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

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 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 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 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

Named alongside it

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

DistortionJahn–Teller distortionPoint groupIrreducible representationsNormal modeReduction formulaSubgroupSymmetry breakingZero modeBond lengthChiralityClosure

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