Point group — where it appears
Named by 49 essays across 5 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.
The splitting is a symmetry statement
Put six ligands round a metal and the five d orbitals stop being degenerate. Which of them stay together, and how many sets there are, follows from the point group alone — before any account of what the ligands are made of, and before any number is computed.
Five sites are not alike
Every other common arrangement has one or two distinct angles. Five has three, because two of its positions are on an axis and three are round an equator — and a molecule built that way does something about it.
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.
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.
Two structures, two spectra
A linear XY₂ gives two infrared bands, one Raman band and no band in common. A bent XY₂ gives three of each and three in common. Counting settles the shape, without a force constant, an assignment or a single measured frequency.
Site symmetry, and what it constrains
A molecule's group is not the only group in the problem. Each atom sits at a position with a symmetry of its own, and what that local group permits decides how many kinds of environment a structure really has.
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.
Which angles are symmetry and which are the model
VSEPR says electron pairs repel and never says by what law. For four, five and six domains it makes no difference whatever — change the exponent by a factor of twelve and not one angle moves. For seven it decides the answer.
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.
Six bonds and four orbitals
The six fluorine σ functions of sulfur hexafluoride span a₁g ⊕ eg ⊕ t₁u. Sulfur's 3s and 3p supply a₁g and t₁u and nothing else, so four bonding orbitals hold twelve electrons across six bonds — a bond order of two thirds, computed from characters with no energy anywhere in it.
What a dipole cannot tell apart
A dipole moment is three numbers extracted from a whole charge distribution, and enormously many distributions give the same three. The moment above it is not even a property of the molecule unless the one below it vanishes — which is why a quadrupole is quoted with an origin and a dipole is not.
Why a d–d band is weak
In a centrosymmetric complex the transition between the two halves of a split d shell is forbidden — exactly, by parity, with no small quantity anywhere. What makes it visible at all is that the molecule is never quite centrosymmetric, and the computation says which vibrations do the work.
A spectrum that changes when only a mass does
Methane has four distinct vibrational frequencies and two infrared bands. Replace two of its hydrogens with deuterium and it has nine and eight — with every force constant identical, every nucleus where it was, and the potential energy surface unchanged.
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.
The moment that is the sum of the other two
Three numbers computed from the coordinates and the masses decide whether a molecule is flat. For water, sulfur dioxide, formaldehyde, benzene and every other planar structure here the largest moment of inertia is the sum of the other two exactly; for ammonia it misses by 0.76 and for methane by 3.18.
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.
Mutual exclusion does not prove a centre
A centrosymmetric molecule shows no band in both its infrared and its Raman spectrum. The rule is a theorem and its converse is read off as though it were part of it — but ferrocene in the gas phase has no centre of inversion and no coincidence either, and the reason is that a fivefold axis separates the coordinates from their products where a threefold or fourfold axis cannot.
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.
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 constant a spectrum cannot see
A symmetric top has two rotational constants and its microwave spectrum reports one of them. Not badly, not with difficulty: ammonia's A of 6.3406 wavenumbers appears in none of its lines at any J and any K, because the term it belongs to cancels exactly out of every transition. The molecule turns about that axis, the energy is real, and the measurement is blind to 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.
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.
Four is all that s and p can match
Reduce the ligand σ set of ten molecules in each one's own point group and ask how many of its components transform as one of the central atom's four valence orbitals. The answer is never more than four — not by arrangement, in every geometry from linear to octahedral — and what is left over is n + L − 4, with exactly twice that many electrons in excess of an octet.
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.
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.
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 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.
A ratio that squares what it measures
A depolarised Raman band sits at exactly three quarters because symmetry says its mean polarisability derivative is zero. Distort the molecule and it comes off — by 4.5 × 10⁻⁴ for a hundredth of an ångström and 0.042 for a tenth, going as the square of the distortion, which makes a ratio measured to three decimals a length known to one and a half.
The table that could not have mattered
Charges taken from one of four electronegativity tables invite a worry, because the tables disagree with each other. They do disagree — hydrogen cyanide's dipole runs over a factor of eleven between them — and for the molecules in question the worry could not have applied, because a molecule of two elements has charges that are one number times a fixed pattern and a rank correlation does not notice a rescaling.
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 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.
The orbital a ligand cannot reach
The sixteen-electron count of a square plane is a statement about an energy rather than about symmetry matching, so it was the count that ought to be sensitive to a π channel where the eighteen-electron one is not. It is not sensitive either — and for a sharper reason. The orbital that sets its gap is d(z²), and a square-planar ligand set contains nothing of that symmetry, so the gap is exactly 2eσ until the π strength reaches a quarter of the σ one.
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.
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.
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.
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.
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.
The leftover changes sides
A main-group centre brings four valence orbitals against six ligand combinations, so two are orphaned. A transition metal brings nine, so the arithmetic inverts and three metal orbitals are left instead — three at every geometry of the Bailar twist, out of decompositions that share no species. Run past the whole arrangement census, exactly one arrangement orphans anything at a metal, and it needs an f orbital to fix.
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.
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.
Irreducible representationsSymmetry operationModel limitDegeneracyCharacter tableReduction formulaGroup orderSelection rulesConventionHypervalencyNormal modeVibrational modes