Molecular Geometry & Bonding
what the shape decides
Series

Point group — the series

21 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. ammonia — C3v. The molecule with the point group found from its coordinates rather than looked up: every candidate operation was applied and kept when it permuted the atoms among themselves.

    Point groups from coordinates

    A molecule's symmetry is not a label to be looked up. It is decidable from the atom positions by searching for the operations that permute them, and the search either finds an operation or it does not.

    part 1 · symmetry
  2. What the group settles. For each molecule, the point group found from its coordinates and the two properties that follow from the group alone. Neither column required knowing anything about the bonds.

    Symmetry forbids a dipole

    Whether a molecule can have a dipole moment follows from its point group alone. The usual argument — adding up bond vectors — gets the right answer for easy cases by a route that does not generalise.

    part 2 · symmetry
  3. bromochlorofluoromethane — C1. The molecule with the point group found from its coordinates rather than looked up: every candidate operation was applied and kept when it permuted the atoms among themselves.

    Chirality is a symmetry statement

    A molecule is chiral when its group contains no improper operation at all. The four-different-groups rule is a useful special case that misses molecules with no stereocentre and wrongly condemns some that have several.

    part 3 · symmetry
  4. 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.

    part 4 · symmetry
  5. 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.

    part 5 · symmetry
  6. 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.

    part 6 · symmetry
  7. The group of benzene, against how much error is forgiven. Four distortions of benzene, none larger than six hundredths of an ångström, and the point group the search reports for each at nine tolerances. The exact structure is D6h at every one of them, so the staircases below belong to the distortions rather than to the search. The cell marked in warning colour is a step at which the order of the group named at the tighter tolerance does not divide the order of the one named at the looser: C6h of order 12, so the sequence is not a chain of subgroups.

    The tolerance is a decision

    A measured structure is never exactly symmetric, so assigning it a point group means deciding how much error to forgive. Sweep that decision from a thousandth of an ångström to a third of one and benzene, bent by a hundredth, is assigned five different groups — and at one step the group named is not even a supergroup of the one named before it.

    part 7 · symmetry
  8. Three counts for each molecule, and the barrier between them. Four molecules, with the number of operations available to each: the point group, which is what a rigid rotation or reflection can do; the feasible group, which adds whatever a crossable barrier makes available; and the whole permutation-inversion group, which is every rearrangement of identical nuclei whether a molecule can reach it or not. The middle column is the one that describes a real spectrum, and it equals the first only when nothing moves.

    The group of a molecule that will not hold still

    A point group is a group of rotations of space, and it presupposes a structure for them to act on. Ethane has no one structure — its methyl groups turn billions of times a second — and the group that describes its spectrum has thirty-six elements where the point group has twelve, out of two thousand eight hundred and eighty conceivable.

    part 8 · symmetry
  9. 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.

    part 9 · symmetry
  10. 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.

    part 10 · symmetry
  11. 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.

    part 11 · symmetry
  12. A species label is ambiguous for every one of 17. For each molecule, the share of its vibrations belonging to a symmetry species that appears more than once — the distortions a species label cannot price, because the label picks a space rather than a mode. Every molecule in the census has at least one such species, the share averages 71.4 per cent, and for 9 of 17 the repetition is not forced by the group's capacity — those molecules have fewer vibrations than their group could hold without repeating, and repeat anyway.

    A label that prices nothing

    Pricing a distortion by its symmetry species works when the species appears once. It appears more than once for every one of seventeen molecules — 71.4 per cent of their vibrations on average belong to a species that is not unique — and for nine of the seventeen nothing forces it: their groups have room to spare and they repeat anyway.

    part 12 · symmetry
  13. Where the explanation gives a negative number of vibrations. The usual formula against the answer, for the 15 molecules with internal coordinates here. It is right for 7 of them and wrong for 8, and for sulfur hexafluoride, benzene and both ferrocenes it predicts a negative number of totally symmetric vibrations — which is the clearest possible sign that the quantity being subtracted is not the one that should be.

    A formula that predicts minus eleven vibrations

    A molecule's count of totally symmetric vibrations is often explained as one per orbit of internal coordinates, less one per redundancy, with methane as the worked example. Computed for fifteen molecules it is right for seven and wrong for eight — and for sulfur hexafluoride, benzene and both ferrocenes it returns a negative number. Two independent corrections turn it into an identity.

    part 13 · symmetry
  14. Which torsions a molecule's own operations turn backwards. Every torsion orbit of the molecules here that have torsions: how many operations carry the torsion onto itself, how many of those are proper, and which operations reverse it. 7 of 9 orbits are reversed, and in every one the operation doing it is improper — the plane of a planar molecule, a mirror bisecting the torsion's bond, or a centre of inversion at that bond. Proper operations fix torsions too, benzene's twofold axes and hydrogen peroxide's among them, and never reverse one. Staggered ferrocene gives the same rows as eclipsed.

    Five coordinates for six vibrations

    A torsion is reversed by every improper operation that carries it onto itself and by no proper one, so adding torsions tests the orbit rule on a second kind of signed coordinate — and the rule holds on every molecule. It also moves hydrogen peroxide's count of totally symmetric vibrations from three to four, which a property of a molecule cannot do. Its five coordinates never spanned its six vibrations, and three of fifteen coordinate sets had been counting vibrations they did not describe.

    part 14 · symmetry
  15. Water's two hydrogens are closer than platinum's chlorine. For each molecule, the longest pair that is a bond and the shortest pair that is not, on a logarithmic length axis. A cutoff on the length has to sit to the right of every filled mark and to the left of every open one, and it cannot: the longest bond in the collection is 2.3200 ångström and the shortest non-bond is 1.5144. The two populations overlap by a factor of 1.53, so the rule in use is not a rule with a badly chosen number in it — it is a rule with no number that works.

    No length separates them

    A bond list here is one distance cutoff with a clause about hydrogen, added when peroxide came back with five bonds instead of three. The clause repaired one molecule. Across the twenty-three molecules drawn here, the longest bond is 2.32 ångström and the shortest pair that is not a bond is 1.51 — so no cutoff can work at all, and five molecules currently come back with no bonds.

    part 15 · symmetry
  16. The orbit identity holds on all nineteen, and the formula it replaced on eight. For each molecule in the census under the radius rule: its number of totally symmetric vibrations, the corrected count — symmetric orbits less symmetric redundancies — and the usual formula, orbits less redundancies. The corrected count lands on the molecule's own count every time. The usual formula is right for 8 of 19, and for ferrocene under the new bond list it predicts minus sixty-six.

    The census a bond rule was hiding

    Every internal coordinate, redundancy and totally symmetric count here is built on a bond list, and the bond list came from a length cutoff that gave five molecules no bonds. Rebuilt on the radius rule, the census reaches nineteen molecules instead of fifteen, the orbit identity holds on every newcomer, ferrocene's coordinates finally span all its vibrations — and a different gap appears: no bond rule can give a square-planar centre its two out-of-plane vibrations.

    part 16 · symmetry
  17. Two molecules whose coordinates reach seven of nine. What fraction of each molecule's vibrations its stretches and in-plane bends span, with bonds taken out to 2.5 ångström so that every molecule in the census raises coordinates at all. Most reach every one. Three are short because they need a torsion, which an earlier essay established. Two are short for a different reason and are the subject here: a square-planar centre has two out-of-plane vibrations that no angle in its plane changes to first order.

    The two coordinates a square plane was missing

    A square-planar centre has nine vibrations and its stretches and in-plane bends span seven of them: no angle in the plane changes to first order when the centre leaves it, or when the ligands pucker. The construction that closes the gap is two coordinates, one per pair of trans ligands, each the signed distance of the centre from its own pair's axis — and two coordinates raise the rank by exactly two. Why nothing measured this for three essays is the more useful half: both molecules with a square-planar centre have bonds longer than the census's default cutoff, so they were dropped before anything looked at them.

    part 17 · symmetry
  18. Half the distance from a coin to a decision. Every distortion whose symmetry species appears more than once, sorted by how concentrated the projection is across the occurrences of that species. The dashed line at a half is what a label that said nothing would give, since every one of these species appears exactly twice. The mean is 0.761 — so the label carries the answer about half the distance from a coin to a decision. Six of the twenty-four are above nine tenths and settled; ten are below six tenths and are not.

    Half the distance from a coin

    Every essay of this argument that priced a distortion by its symmetry species carried the same caveat: where a species appears more than once, the label cannot say which vibration the distortion goes into. With every coordinate set now complete or incomplete for a stated reason, the caveat can be measured against an actual projection on the five molecules that have force fields. The label carries the answer from 0.500 to 0.761 — about half the distance from a coin to a decision — settling six of twenty-four distortions and leaving ten essentially undecided, the worst of them ammonia's bond angles at 0.516.

    part 18 · symmetry
  19. One sentence, three displacements, three answers. For each symmetry-distinct internal coordinate of the five molecules, how concentrated the projection is across the occurrences of the species the label names — under the B row the published measurement used, under the displacement that changes the coordinate at the least kinetic cost, and under the one that changes this coordinate and no other. Ammonia's bond angle goes from 0.516, a coin, to 0.883 and 0.926. Boron trifluoride's angles are the worst case under one clean displacement and its bonds under the other, both at 0.573.

    Three ways to stretch a bond

    A symmetry label was measured at half the distance from a coin to a decision about which vibration a distortion goes into. The distortion was a B-matrix row, which moves a central atom as far as a ligand — and in the metric where the vibrations are orthogonal, three quarters of water's bond-angle distortion turns out to be the molecule moving rather than deforming. Two displacements that realise the same sentence and carry no rigid motion at all double what the label decides, from eleven coordinates of twenty-nine to twenty-two, and move the worst case from ammonia to boron trifluoride. Between them sits an exact identity that found a rounding error in a stored molecule.

    part 19 · symmetry
  20. Ammonia's E verdicts across the fields its spectrum allows. The concentration of each distortion on one occurrence of ammonia's E species — the share of its E projection on the favoured vibration — for every field in the walk, with the stretch–bend constant held and the rest refitted to both isotopologues. The band is where the fit cannot tell the fields apart and the dotted line is the published fit. Across the band the angle under the B row reads 0.502 to 0.579, the angle at least cost 0.840 to 0.923, the bond at least cost 0.890 to 0.957.

    The worst case was the loosest constant

    A symmetry label's verdict on a distortion is read off mode shapes, and mode shapes are set by a fitted force field. Walked along the direction its own frequencies see least, ammonia's field moves the verdict on its bond angle from 0.840 to 0.923 without the fit getting measurably worse — and boron trifluoride, the census's worst case at 0.573, has a whole curve of fields that reproduce its spectrum exactly, along which the same verdict runs from a coin to 0.862.

    part 20 · symmetry
  21. Ammonia's two Coriolis constants along the family of fields the spectrum cannot tell apart. ζ₃ and ζ₄ of NH₃ on every member of ammonia's family of force fields, each refitted to the frequencies of NH₃ and ND₃ with its stretch–bend constant held, against that constant. The shaded band is where the fit is no worse than its own noise, -1.000 to -0.634. Across it ζ₄ runs from -0.807 to -0.641 and ζ₃ from 0.418 to 0.584, while their sum stays at -0.222 on every member.

    Two Coriolis constants are one number

    Ammonia's frequencies leave its force field free along a band, and across that band a symmetry label's verdict on the bond angle changes. The measurement proposed to settle it was the Coriolis constant of each E vibration, which reads mode shapes directly — and it would settle it, sixteen times more sharply than the frequencies do. But the two constants are not two measurements. Their sum is fixed by the geometry and the masses, ζ₃ + ζ₄ = I∥/2I⊥ − 1 (the moments of inertia about the axis and across it), on every field there is, so the spectrum's Coriolis constants carry exactly one number about the field between them.

    part 21 · symmetry

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