Symmetry operation — where it appears
Named by 49 essays across 7 fields — each of them below, with the objects they name alongside it.
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.
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.
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.
Group frequencies, and where they stop
A carbonyl band sits near 1,700 wavenumbers in every ketone anybody looks at, and that regularity is real. Computed for a set of small molecules, four of twenty-one distinct frequencies belong to a single internal coordinate — and the four are exactly the coordinates symmetry leaves alone.
Character tables and reduction
A molecule's point group is a list of matrices, not a label. Generating them, sorting them into classes, and dividing by the group order turns any set of orbitals into a statement about how many energies there can be.
A spectrum counts environments, not atoms
Phosphorus pentafluoride has five fluorines in two inequivalent sets, so its magnetic resonance spectrum should show two signals. It shows one — and the reason is not a symmetry the molecule has but a motion faster than the measurement.
Degeneracy is a group theorem
How many orbitals can share an energy is decided before any energy is computed. The dimensions of a group's irreducible representations are the only degeneracies it permits, and a molecule with no representation larger than one cannot have a degenerate level at all.
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.
The rotational spectrum is a moment of inertia
Every line in a microwave spectrum sits at a multiple of one number, and that number is a conversion constant divided by a sum of mass times distance squared. No bonding argument appears anywhere in it.
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.
One coordinate, three point groups
Hydrogen peroxide has four atoms and one soft internal coordinate. Turning it from nought to a hundred and eighty degrees takes the molecule through C2v, C2 and C2h — so it is chiral at every angle but two, and forbidden a dipole at exactly one of them.
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.
Why a character table stops where it stops
A character table is square, and both of its dimensions are forced. The number of representations equals the number of classes, the squares of their dimensions sum to the order of the group, and between them there is no room for another row.
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.
The projector is unique, the basis is not
A reduction says how many times each representation appears. It cannot say which combinations of orbitals they are — that takes a projection operator, and for a degenerate representation the operator returns a different pair of orbitals depending on which function it is handed first. Both pairs are correct, they span the same space, and every energy computed from either is identical.
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.
One table, three groups
A mirror plane, a centre of inversion and a twofold axis have the same character table — two rows, entries 1, 1 and 1, −1 — and one of the three describes a chiral molecule. A character table is a property of an abstract group; a point group is that plus an action on space, and the action is what a molecule has.
When a mode becomes a bond stretch
Water's two stretching modes are each exactly half in one O–H bond and half in the other, which is why neither of them belongs to a bond. Change one hydrogen to deuterium and the same force field at the same geometry gives two modes that are 99.5 and 99.7 per cent in a single bond each. Nothing about the bonding changed; a mass did.
Zero dipole is not no interaction
Benzene's dipole moment is exactly zero at every origin, and its quadrupole moment is large enough to decide a crystal structure. Two benzenes face to face repel by nine kilojoules a mole; edge to face they attract, and the sign changes on the way between.
The degeneracy no group predicts
How many orbitals can share an energy is decided by a group before any energy is computed, and the rotation group of a central potential permits one, three, five and seven. Hydrogen's second shell has four orbitals at one energy and its third has nine. The extra degeneracy is not an accident of the arithmetic — it is the signature of a symmetry that has not been named, and it survives only for a potential that goes exactly as one over r.
A full band is not an insulator
Two electrons per atom, two orbitals per atom, and the lower set exactly full: the count says insulator, and magnesium is a metal. The count is not wrong about the count. What it assumes is that the two sets of levels occupy separate ranges of energy, and whether they do is a comparison of four numbers that has nothing to do with how many electrons there are.
The symmetry that is not a rotation
Hydrogen's n = 2 shell holds four states at one energy and its rotation group accounts for at most three. The operator that accounts for the fourth is built here out of computed integrals: three matrices whose commutators close into the rotations, whose product with the angular momentum vanishes, and whose Casimir comes out at exactly n² − 1.
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.
Two rules that share no arithmetic
Hückel's rule is a statement about a gap. Put a magnetic flux through the same rings instead and ask which of them push the field out, and the answer is the same set — eighty cases, no exceptions — although the second calculation counts nothing and has no shells in it. And the rings the rule excludes turn out to have no magnetic susceptibility at all: their energy has a corner at zero field.
A band is a filter on the modes
A photoelectron band's vibrational structure reports the frequencies of a few of the ion's vibrations and is silent about the rest, and which few is decided by the point group before any geometry is known. Under a change of shape that lengthens every bond alike, methane's totally symmetric stretch gets a Huang–Rhys factor of 0.905 and its other eight modes get between 10⁻²⁸ and 10⁻³⁵.
More coordinates than motions
Methane has ten internal coordinates and nine ways to vibrate, boron trifluoride seven and six, formaldehyde seven and six. The excess is not bookkeeping: it is a combination of coordinates that describes no displacement of any atom, and adding twenty-five units of force constant along it moves every frequency by five parts in a hundred million.
How many descriptions a cage has
A localisation is a maximisation, and running it once reports the maximum it reached rather than the maximum there is. Run to exhaustion on a twelve-vertex borane it finds ten answers and then three batches of twelve starts in a row that find nothing new — and none of the ten is another one seen from a different side, because a symmetry of the cage cannot change a multiset of participation numbers and all ten multisets differ.
The one intensity symmetry does fix
Symmetry says which bands exist and declines to say how strong they are. It makes exactly one exception, and it is a ratio: every Raman band that is not totally symmetric is depolarised by exactly three quarters, whatever the molecule and whatever the model — and methane's symmetric stretch is polarised by exactly zero, because its group is cubic.
The current does not divide
A fused ring system's response was computed from the areas of its rings, and the obvious next question was whether the current divides between them the way it divides between two resistors. Giving each ring its own flux and taking the second derivatives says no: the response is a matrix, its off-diagonal entries are nearly half its diagonal ones, and anthracene's middle ring carries 1.18 times what its outer rings do.
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.
The angle that does not have to be searched for
A carbonyl's two bent components have no symmetry making them equivalent, so the mixing that best localises them looks like something to search for and their s characters look like two different numbers. Neither happens. The localisation functional is a quadratic with no linear term, so its maximum is at forty-five degrees or at the ends — bent bonds or canonical ones, decided by one inequality, with nothing in between.
The square that wastes an orbital
The orphan count that prices hypervalency was treated as a property of a molecule's composition — ligands plus lone pairs minus four. Run on twenty-six arrangements of the same ten molecules it is right for twenty-three and wrong for three, and all three are flat. A planar arrangement gives a main-group centre three usable orbitals rather than four, so the count is a property of the shape.
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.
The gap that only a tetrahedron closes
The sixteen-electron gap is 2eσ exactly, and a square-planar π set cannot touch it because d(z²) has no partner there. Fold the ligands out of the plane and the gap survives almost intact for fifteen degrees, closes to nothing only at the tetrahedron — and loses its exactness at the very first degree.
Expensive is not the same as unadopted
A counting formula right for twenty-three arrangements and wrong for three invites a reading: the failures are the arrangements nothing adopts, so the formula is reliable because chemistry stays away from where it breaks. Put the repulsion energy on the same axis and the reading fails — the most expensive arrangement in the census is one the formula gets right.
The ligand the rule was waiting for
A sixteen-electron complex is called reactive because it can add a ligand, and the gap that makes it sixteen points straight at where the ligand arrives. Bringing one in closes the gap exactly linearly — and leaves its exactness completely untouched, which is the opposite of what bending the same complex does.
An end effect with two signs
Neither of two separations accounts for the scatter in a fused ring system's response, and the natural guess is the end: pairs with more molecule outboard should behave differently from pairs at an edge. They do. In a straight chain an end pair responds a third less than an interior one, in a zigzag a quarter more, and in a chain of two straight arms meeting at one angular ring the anomaly is in the middle.
One integer, and everything it changes
Eight molecules with the same rings, the same carbons and the same graph distance between every pair, differing in which fusion's direction is turned — one angular ring when the turned fusion is at an end, two adjacent ones anywhere else. The scatter within a separation class runs from three to four and a half times, against a straight chain's one and a half — and the two ends of one molecule disagree by up to a factor of four.
The distortion that opens the gap
Two distortions close the sixteen-electron gap — one by bending all four ligands, one by adding a fifth. Folding two of the four makes it larger, by five per cent, before it makes it smaller. And it costs the exactness at the first degree, while the gap is still growing, so the size of a gap and whether it is exact are not one measurement.
A count that changes at one point
Where does the orphan count step along a distortion, relative to where the energy's minimum sits? A rule that depends on an exact symmetry may have no answer for a real molecule. On the path from a tetrahedron to a square plane the count is the same at every angle up to 89.99° and changes only at 90° exactly — which is the energy's maximum, not its minimum, and a single geometry out of a continuum.
The group nobody wrote a table for
The orphan count is predicted to be constant along the Bailar twist, because the twist keeps D3 the whole way. The premise is exactly right — at every angle strictly between the prism and the octahedron the symmetry finder assembles six operations that hold to four parts in 10¹⁶. The conclusion cannot be tested here, because a count is a reduction, and the calculation's nineteen character tables did not include D3.
A symmetry holds or it does not
One level of a three-orbital trio sits at the free-atom energy exactly, at every third-orbital energy, because the antisymmetric combination of the pair has nothing of its own symmetry to mix with. Detuning one of the two by a twentieth of an electron volt moves it by half of that — first order, immediately, with no protected regime at all.
The count the table was hiding
The interior of the Bailar twist looks uncountable — exactly D3 at every angle, and D3 is a point group whose character table is rarely written out. Writing it takes nine lines and settles the prediction made for the path: the orphan count is two at every geometry on the path that can be answered, in three different groups, out of three different decompositions.
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.
A level no symmetry was protecting
A three-orbital trio keeps one level at the free-atom energy exactly, and the reason given was that its two outer orbitals are equivalent. Make them inequivalent by changing one overlap rather than one energy and the level does not move at all — not to first order, not to any order, at any energy of the third orbital. It was never the symmetry. It is allyl's non-bonding orbital, held by a count.
The cage is on both sides
Two localisation criteria classify six cage-and-filling pairs differently, and a symmetry explanation for the six is the natural first guess. The icosahedron's graph has a hundred and twenty automorphisms, the most in the family, and supplies three of the six disagreements and nine of the agreements. What the six do have in common is sharper than a symmetry: in every one, one criterion's spread is not small but zero.
The gap found on purpose
A twist between two coordination arrangements passes through a point group with no character table among the twenty in use, and that was found by accident. Running every path between the arrangements — eighteen of them, two hundred and seventy geometries — finds exactly one more gap, at a group of order ten. It also finds something the search was not looking for: a group without a table had been reported as an infinite group.
The spin the count does not hold
Three orbitals in a row keep one level at the free-atom energy however the overlap of one end is changed, and at three electrons that level holds the radical's unpaired electron. Its energy does not move. Its spin does: from half on each end to 0.236 and 0.764 as one overlap goes from 0.25 to 0.45, exactly the squared ratio of the two overlaps, at every energy of the middle orbital. A coupling between the ends moves the level and cannot move the spin. The energy and the spin are answering to different things.
Folding the ring does not give the orbital back
Three arrangements break the orphan count n + L − 4, all flat, and the reason given was flatness: the p orbital perpendicular to the ring has no ligand combination of its species. Fold the ring into an umbrella at any angle and that orbital becomes totally symmetric — and the count stays wrong by exactly one, for rings of four to eight. The ring offers one symmetric combination to a centre with two symmetric orbitals. Writing C₅ᵥ to see it also counts the pentagonal pyramid at last, and the formula holds there.
Named alongside it
The objects these essays reach for when they reach for this one.
Point groupDegeneracyModel limitIrreducible representationsConventionCharacter tableGroup orderReduction formulaApproximationClosed formElectron countChirality