Degeneracy — where it appears
Named by 114 essays across 9 fields — each of them below, with the objects they name alongside it.
Normal modes are not bond stretches
Water has two stretching frequencies and two O–H bonds, and it is almost irresistible to pair them off. Computed, each mode is exactly half in one bond and half in the other, and neither frequency belongs to a bond at all.
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.
Delocalisation
Benzene does not alternate between two structures. It has one structure, and the two Kekulé forms are basis functions in a description of it — which is a different and much less exciting claim than the one usually made.
Two models, one ratio
A tetrahedron splits a d shell by four ninths of what an octahedron does. Two models that share nothing but the ligand directions — an integral over a point-charge potential and a rotated diagonal matrix — both produce that number to eight decimal places, and neither was told it.
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.
How many frequencies, not how many modes
Benzene has thirty vibrations. It has twenty distinct frequencies, and of those, eleven can be seen — four in the infrared and seven in the Raman, with no band in common. The other nine are invisible to both experiments, exactly.
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.
Hückel theory and what it gets right
A conjugated system's orbital energies are the eigenvalues of a matrix of ones and zeroes. Nothing about carbon enters, no geometry enters, and the results that survive are exactly the ones that depend on neither.
The band limit
Every level of a ring of n atoms lies between −2β and +2β, however large n gets. The levels do not spread out as the molecule grows; they crowd into a fixed interval — and that crowding, computed, is a band with its density of states diverging at both edges.
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.
Aromaticity as a computed shell closure
Hückel's 4n+2 rule is not a rule. It is what comes out when every ring size is filled and asked whether its highest occupied shell came out full — and the calculation that answers does not know the phrase.
Complex harmonics against real ones
The p orbitals every chemist draws are not eigenfunctions of anything. They are real combinations of the complex solutions, chosen because they point along axes — and the choice is invisible until a magnetic field makes it matter.
What an absence proves
A band that symmetry forbids is not weak. Its intensity is zero, exactly, by a theorem — while a band that is merely too faint to see is absent for reasons no theorem covers. The two look identical in a spectrum and support completely different conclusions.
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.
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.
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.
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.
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.
A moment counts electrons, not orbitals
A magnetic moment is one of the few chemical measurements that returns an integer. Feed the count of unpaired electrons into √(n(n+2)) and nine first-row ions come back within a hundredth for five of them — and the five that miss all miss the same way, which is what a missing term looks like.
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.
The ring with a twist in it
Reverse the sign of one resonance integral in a ring and its levels stop being one nodeless orbital above degenerate pairs and become degenerate pairs all the way up. The count that closes a shell changes from 4n+2 to 4n, and every aromatic ring size and antiaromatic ring size exchange places.
What a photoelectron spectrum measures
The bands of a photoelectron spectrum are routinely read off as orbital energies. They are ionisation energies, which is a different quantity — and the identification rests on two errors of about an electronvolt each that happen to have opposite signs.
A cage needs one pair more than it has corners
Every closed borane holds n+1 skeletal electron pairs for n vertices, and the extra one is a theorem about connected graphs rather than an observation about boron. A cage's radial orbitals have exactly one nodeless combination, always, whatever its shape.
A filled shell has no shape
Sum the angular densities of a complete p shell and the answer is 3/4π in every direction, to sixteen decimal places. A filled d shell gives 5/4π. The lobes are in the decomposition and not in the density, and nothing that measures a closed-shell atom can see them.
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.
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.
The smallest many-electron calculation
A Hückel energy comes from a model with one electron in it. Add a single term — a cost for two electrons on the same site — and the problem stops being a matrix of size n and becomes a matrix over configurations. Four sites give thirty-six of them, which is small enough to solve exactly, and the answers correct two things the one-electron model got wrong.
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.
An orbital carries no angular momentum
The d orbitals every chemist draws carry exactly no orbital angular momentum, and the proof is one line about a matrix being antisymmetric. A set of three of them carries a whole unit, which is why the spin-only formula works for most ions and fails for cobalt by nearly a Bohr magneton.
Fewer bands than electrons
Methane has eight valence electrons and two photoelectron bands. Ammonia has eight and three; water has eight and four. The count is not of electrons, not of bonds and not of orbitals — it is of the symmetry species the occupied orbitals fall into, and it falls as the symmetry rises.
The ring that cannot hold still
Cyclohexane's chair is rigid and its boat is not, and that is a statement about the rank of a matrix rather than about strain. Hold every bond length and every bond angle fixed and count what is left: the chair has nothing, and the boat sits on a continuous loop of shapes with the same bonds and the same angles.
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 same ring, three charges
Three carbons in a ring are aromatic with two π electrons, a doublet with three, and a triplet with a delocalisation energy of exactly zero with four. Nothing about the molecule changed but the count, and adding electrons to a π system can make its π binding energy fall.
VSEPR does not reach a transition metal
Four ligands minimising their repulsion give a tetrahedron, whatever the metal. Half the four-coordinate complexes of the platinum group are square planar, which has larger repulsion, and the term that overrules it is largest at d⁸ and exactly zero at d⁰ and d¹⁰ — which is where the repulsion rule works again.
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.
The g-value is the orbital coming back
A ligand field quenches the orbital angular momentum of a d electron, and spin-orbit coupling gives some of it back — upward for a shell more than half full and downward for one less than half full. Which way a resonance line moves counts the electrons, and the size of the move comes from two matrix elements and one optical splitting.
The ring's levels are its group's characters
The Hückel energies of a cyclic system are twice the real part of the characters of its own rotation group, so the level pattern — one level, then pairs, then one more if the ring is even — is a group theorem rather than a calculation. Hückel's 4n+2 rule is a statement about the representations of a cyclic group and about nothing else.
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.
A moment between two integers
A magnetic moment is celebrated as one of the few chemical measurements that returns an integer: count the unpaired electrons, feed the count into a formula, and nine first-row ions come out right. That works when one state lies far below the others. Sit a complex at its own crossover and the same measurement returns 0.30 at 80 K and 3.61 at 400 K — a quantity that counts nothing and is a temperature in disguise.
A third kind of correlation
The two-site model gave an exact identity — the power of the correlation energy in the repulsion equals the occupation of the bonding natural orbital — and asked whether anything like it survives with more orbitals. It does not, and the way it fails is better than the identity was: a ring of four gives a power of one where every closed-shell system gives two, at repulsions fifty times weaker than the hopping, because its reference was never a single state.
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.
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.
How much of a band is a bond stretch
Boron trifluoride's 1454 cm⁻¹ band is 36.7 per cent B–F stretch, or 49.9, or 97.9, depending on which of four standard ways of sharing a mode out among internal coordinates is used. All four are defensible, all four sum to one, and the spread between them is sixty-one percentage points on one band of one molecule.
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.
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.
A vacancy is not an impurity
An impurity is a site whose energy has been changed, and everything about the level it produces depends on by how much. A vacancy is a site that is not there, and the levels it leaves sit at exactly zero for a reason that cannot be tuned, weakened or moved — the count of them is a difference between two numbers of atoms, and two vacancies do not split however far apart they are put.
The top that reports all three
A symmetric top hides one of its two rotational constants in every line of its spectrum. Break the symmetry and the hiding stops: for water, twenty-three of the twenty-five levels up to J = 4 move when A is changed, and the two that do not are the ground state and the one at B + C. There is no formula for any of them.
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.
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.
The metal a thermometer cannot find
A metal is a system with excitations of arbitrarily small energy, which is a claim about a sequence of finite systems rather than about any one of them. Put a temperature on it and the claim needs a second limit, and the two do not commute: a uniform ring of forty-two at kT = 0.002 carries exactly as much as an alternating one, and at kT = 0.1 a ring of three hundred and twenty-two carries only four times as much.
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 distortion the ends decide
A chain of an even number of sites has an odd number of bonds, so its two dimerisations are different molecules rather than one molecule translated. Held at the same distortion they differ by 1.08715 in units of the hopping, whatever the length — a fixed amount of energy living at the two ends, with the per-site difference falling as one over the length at a fitted exponent of −0.99986. And below a hundred and twenty-eight sites the second dimerisation does not exist at all.
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.
How nearly a broken symmetry survives
The hydrogen shell's extra symmetry is what makes its Stark effect linear, and a real atom does not have it. Screening splits the shell, and the field needed to overcome the splitting and restore the linear behaviour is a curve — from forty thousand volts a centimetre at a quantum defect of 0.0004 to thirty million at a defect of 0.21.
An anomaly that is not the first of a series
Naphthalene's shared bond behaves differently when the geometry is allowed to answer back, and the obvious reading is that two rings feeding one bond is what does it. Run it up the acenes and the effect falls at every step and changes sign: +0.02036 for two rings, +0.00678 for three, −0.00029 for four, −0.00603 for five. Two rings feeding a bond is not the beginning of anything.
Ten directions no frequency can see
A redundant force field has one obvious flat direction. There are ten: methane has fifty-five independent force constants and a ten-dimensional subspace of them that no spectrum can touch. The literature's repair is to project — and the metric a vibrational analysis is naturally written in cannot define the projection at all, because a redundancy is a null vector of it.
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.
The correction that was invented
A standard error model puts the zero-point error in a rotational constant at a few tenths of a per cent, shared between the three moments by invented weights, with the parent and its deuterated form differing by five. Computed from a force field it is 1.88 per cent, one of the three shares is negative, and the mismatch is 35 — so the cancellation the substitution method rests on is worth a factor of 2.5 and not thirteen.
Two models that disagree about the shape
The identity says how much charge a hypervalent molecule's ligands must share and nothing about how. Working out which arrangement shares it most evenly puts the σ model and the repulsion model on one axis for the first time: the even sharer is the pentagonal plane, which is the arrangement the repulsion likes least — and along the interchange chemistry actually uses, one model sees 0.15 per cent of a change and the other sees 54.
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 boundary belongs to the gap
A satellite stops being tellable from a fundamental somewhere, and it can be located on one ring at one filling. Four systems put it at repulsions of 2, 4 and 8 — ordering exactly with each one's own one-electron gap and not with its band width — and the fourth, whose gap is zero, has no boundary at all: its satellites are indistinguishable at every repulsion including none.
The count that cannot be broken by strength
Back-donation puts electrons into orbitals that are not the metal's, and the eighteen-electron rule counts the metal's nine. Turning the π channel up as far as it will go never breaks it: the counted orbital's metal share falls from 100 per cent to 54.67 and approaches a half from above without reaching it. What does flip it is not strength but order.
A particle in a box the alloy made
A random alloy's gap is set by the longest run of like sites. What sits at the edge of that gap turns out to be the simplest state in quantum mechanics: a particle in a box of L sites, occupying 2(L+1)/3 of them, to within three per cent and with nothing fitted — except for the two states in forty that found a second run to share.
The forty-five that are fixed
Methane's fifty-five-dimensional space of force constants has ten directions no frequency can see, and the useful thing to report is the forty-five that are fixed. It is a table: every stretching constant is fixed on its own, no bending constant is, and projecting a fitted field onto the determined subspace takes four fits that span 0.88 mdyn per ångström down to four that span 0.0027.
Where the count stops being an effort
A cage's localised descriptions were counted by a search that stopped when it stopped finding new ones, which makes the count a property of the stopping rule. Run to four thousand starts the count is fourteen and the last new description appears at start 124 — after which three thousand eight hundred and seventy-six starts add nothing. The four rarest are found six times in a thousand, which is what says nothing rarer is hiding.
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.
One number was one direction
The departure of a depolarisation ratio from three quarters goes as the square of a distortion, and the exponent was first measured along one bond stretch. Computed along every non-symmetric coordinate the exponent is always two and the coefficient is not: it spans a factor of thirty inside methane. And the ratio is not preferentially sensitive to the coordinates a molecule is soft along — in two molecules of three the stiffest coordinate is the most sensitive.
Two events where there was one
A broken symmetry returns at a field where the coupling matches the gap it has to overcome, and in a shell with two levels that field is a single number. The next shell up has three, two gaps and two dipoles — so there are two crossovers, a factor of 3.66 apart, both of them below the single one of the shell below, and the exponent takes more than a decade of field to travel between them.
A contrast with a closed form
Below half filling the satellite test flattens instead of failing, and the value it flattens at could be above or below the factor of two the test needs. It is — on all eight systems, by between 1.25 and 3.7 times. And on a ring the limit is (1 + 2cos(π/n))², to six figures, on every ring tried.
A satellite that never loses its place
The repulsion at which a satellite stops being tellable from a fundamental orders exactly with the one-electron gap across four systems. Changing the gap by the filling instead is the sharper test, and the ordering does not survive it: a six-site chain has a larger gap at half filling and a smaller boundary. Below half filling there is no boundary at all, at any repulsion up to sixty-four times the hopping.
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.
Three shapes from one search
A cage's fourteen localised descriptions fell into two groups with a factor of five between them and nothing in the gap, which made counting the big basins look like a stopping rule. Two more cases from the same family give a smooth tail over sixty-three descriptions and a single basin holding ninety-eight per cent. One search, one family, three shapes.
The axis that goes the other way
One dimensionless coefficient turned a zero-point correction into an expression a spectroscopist could evaluate, and the three principal axes were averaged over to get it. Split by axis it gets worse, not better — the coefficients span 2.3 where the molecule-averaged ones span 1.36 — and water's smallest moment does not grow at all. It shrinks.
Three events, and a ratio of two dipoles
The m = 0 half of a shell has two crossovers because it has three levels and two coupled pairs. The other half has two levels and one, so the whole shell has three distinct fields rather than four — and two of the three share a gap exactly, which makes the ratio between them a ratio of two dipoles, 2/√3.
Fifty descriptions of one molecule
A search whose largest basin takes ninety-eight per cent of its starts will report one description however long it is run, and nobody runs four thousand starts when the first fifty agree. Whether the rare ones are worse descriptions or merely rarer is one number per description, already computed and never looked at. On two of three cages they are not worse — they are the same answer, to parts per million.
The half of the square a ring of four cannot show
There is a warning that says in advance whether a transferred correction will hold: the mean field's own symmetry breaking, which collapses at a definite site-energy modulation and takes the transfer with it. The other axis of the same square has a threshold too — on the other side — and the ring of four it was all measured on is the one system with no threshold to find.
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.
Two moments about two different lines
Asked axis by axis, the substitution method's near-cancellation gives numbers as large as 163 per cent. The arithmetic is the smaller half of the answer. A principal axis is an eigenvector of a tensor built from the masses, so one deuterium turns water's frame by 21.12° — and the two moments being compared are not moments about the same line.
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.
A bond order between atoms that do not interact
A diatomic held where its overlap changes sign has no interaction between its two orbitals at all — which is what the sign change of its overlap means. Put a third orbital beside it and the pair is still split, one line sits exactly at the free-atom energy at every third-orbital energy, and the bond order between the two runs to −0.9999. Three measures of the same bond disagree completely.
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.
Seven points that looked like a switch
A constant belongs to a square net and not to a triangular one, which points at the coupling graph. Seven wrapped nets say which property of it: the third moment, and neither the dimension nor the coordination. They also say it is a switch — and made continuous, it is a crossover that every one of the seven sits twenty-five times past.
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.
A ratio of exactly one is a tie
Does the intensity contrast fall below two at half filling? It does — it falls to exactly one. But one is the floor of a ratio between two ranked quantities, and it is reached here because the cut between fundamental and satellite lands between two lines of identical weight. The guard installed to catch that case tests the wrong degeneracy, and the guard installed to license the extrapolation cannot tell an exact answer from a divergent one.
Four angles the shell chooses
A field along one axis gives a shell of nine functions three fields with an event, and tilting the field should separate the coincident ones and raise the count towards the number of coupled pairs. It does — from three to six. But not monotonically: at four angles two events collide again, and every one of those angles is the arctangent of a ratio of the shell's own angular integrals.
Counting was right except where it mattered
A cage whose localisation gives dozens of descriptions that are all the same answer raises a worry: if degeneracy is common across the family, counting descriptions is the wrong measure of ambiguity. Across forty-eight cage-and-filling pairs it is the right measure on eleven of the thirteen that have anything to count — and it fails on the one leaned on hardest.
None of the six was a crossing
Counting events in a tilted field finds the count rising from three to six, dropping again at four angles that are exact arctangents of the shell's own integrals. Every one of those statements is true of the two-state estimates. Diagonalising the five-level problem exactly finds one avoided crossing at every tilt, in a field that moves by a third across ninety degrees, and no feature whatever at any of the four angles.
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 sign a frustrated ring changes
There is a sharper question than whether a low-temperature feature buys back a fourth parameter: does it change which combination is free? It does, and by a sign. Every odd ring of spins leaves free the monomer fraction times a negative power of the temperature-independent term, and every even ring a positive power, with no case in between.
The composition that is hard is not the full one
A landscape of arrangements measured at one composition on each of two nets raises the question of where the hardest one sits — and the natural guess is the half-filled one, where there is most to arrange. It is the easiest. On thirty-six sites at a contrast of one, half filling has one local optimum and nine billion arrangements, and a quarter filling has six optima and a hundredth as many.
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 number the tie got right
On a ring of six the contrast at half filling comes out exactly one, and the one is an artefact — the rank cut had landed between two lines of identical weight, so the ratio was a quantity divided by itself. The chain of six has no such pair anywhere, at any repulsion, at any filling. Its contrast at half filling converges to one anyway.
The variation was the basis
Solved in the five functions a field in one plane couples, the tilted Stark problem has one avoided crossing at every tilt and a crossing field that moves by a third across ninety degrees. Solved in the whole nine-function shell the field does not move at all — the same number at every direction, to eleven decimal places — and the thirty-four per cent was the truncation.
One number was a direction too
The depolarisation probe is exactly blind to the totally symmetric direction, which suggests it reads the distortion's component outside that species. What is left is a plane, and over a circle in it the reading varies by a factor of 1.98 — smallest for one bond against another, largest for two bonds against one, repeating every sixty degrees.
Consistently wrong is not a limit
Does the two-state picture of a tilted Stark shell become right as the quantum defect closes the l degeneracy? Swept over two decades it does not move: one avoided crossing against eight estimates at every defect. The reason is that every dimensionless quantity settles — the crossing sits at 0.04000 of the zero-field gap and the nearest estimate at 0.9067 of the crossing, and neither is heading anywhere.
It was the count, not the frustration
The free combination's exponent comes out negative on every odd ring and positive on every even ring and on an open chain, and the sign was put down to frustration. The control was a chain of eight. A chain of five is not frustrated in any sense — a chain is bipartite and every bond can be satisfied — and its exponent is −0.792.
The direction the gap cannot see
A two-ligand fold opens the sixteen-electron gap and a four-ligand bend closes it, so a distortion mixing the two must pass through a direction the gap does not move along. It does, and the direction is the antisymmetric fold — one pair of trans ligands up, the other down. Its blindness is exact, because exchanging the two pairs is a symmetry of the arrangement.
The sum was flat all along
The depolarisation reading varies by a factor of two round a circle in the non-symmetric plane, and the anisotropy may belong to taking a maximum over four bands rather than to the physics. Summing the four instead gives a reading constant to four parts in ten thousand, against a maximum that varies by ninety-eight per cent.
The frustrated cluster with an even count
A parity account of the exponent's sign replaced a frustration account and left the two still confounded: every case tested had frustration and odd parity aligned. A tetrahedron of four spins is frustrated and even. Its exponent is +0.2084, and so are those of three more clusters the two accounts disagree about — but the mechanism proposed with the parity rule is refuted along with the account it replaced.
The crossing nothing couples
A tilted Stark shell's one avoided crossing settled at 0.04000 of the zero-field gap, and the natural guess was a ratio of angular integrals. With the quantum defect taken out the limit is 0.0399865, not four hundredths — and the two levels at that minimum belong to different symmetries about the field, which no element of the field connects. It was never an avoided crossing. The one minimum between levels that do interact sits at half the field, behind a level of the other kind.
A blindness that is inherited
There is a direction the sixteen-electron gap does not move along, and it is tempting to call it the one a complex is softest along without paying for it. That second half is a claim about an energy the gap model has no term for. Put the ligand repulsion on the same family and the answer splits: where a symmetry fixes the blind direction the repulsion is blind to it too, exactly, and slightly downhill — and where no symmetry fixes it, the two are nearly perpendicular.
Two levels cannot make a minimum
The one minimum between coupled levels in a Stark shell sits at 0.0196 of the s–p gap, and the nearest two-state estimate at 0.0192 — two per cent away, which reads as the estimates having been aimed at the right feature all along. They were not. Two coupled levels only ever separate, so no estimate can be where its own pair is closest. The minimum belongs to a third level, exists only while the d level sits within a quarter of the s–p gap, and meets the estimate by crossing it.
The quarter, generalised
A shell's one coupled minimum exists because its d level sits within a quarter of the s–p gap, and the quarter was found by bisecting a numerical search on one shell. It is exactly 2(n² − 4)/(5(n² − 1)) for every shell — zero at the second, a quarter at the third, two fifths in the limit — while the offset it is compared against is one fifth whatever the shell.
A second criterion left a gap too
The localisation spreads are bimodal — an empty factor of two hundred around the degeneracy threshold — and the gap might belong to the criterion rather than to the cages. Boys localisation leaves a gap of seven decades on the same forty-eight pairs, so it belongs to the cages. But the two criteria disagree about six of them, in both directions, and the family's most ambiguous cage under one is exactly degenerate under the other.
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 limit of one is a parity
A half-filled chain of six has no degenerate removal lines, and its intensity contrast still goes to one — after dipping near U = 32 and rising again to U = 128. Followed line by line, every removal line is smooth and monotone; the dip is an exact tie between two lines at U = 28.916, and the rise ends where two satellites change places. At large repulsion the lines pair up at ±E with equal weights, so the contrast goes to one exactly when the cut falls inside a pair. On a chain of four it does not, and the limit is 1.3125.
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.
Six disagreements and three calculations
Two localisation criteria classified six of the family's forty-eight cage-and-filling pairs differently, and three of the six were consecutive fillings of one icosahedron with spreads identical to three figures. They are not three coincidences and not one degeneracy being filled: they are one calculation, because the survey's forty-eight rows are thirty-three distinct questions.
The basis a diagonaliser happened to return
Six of the cage family's thirty-three inputs have their occupied set cut through the middle of a degenerate shell, and the members of a degenerate shell are interchangeable. Re-orienting the shell changes nothing about the cage, the filling or the criterion — and it moves the best localisation functional by up to twenty-one per cent, moves the count of descriptions from ten to fourteen, and flips the degeneracy label on three of the six.
Named alongside it
The objects these essays reach for when they reach for this one.
Model limitIrreducible representationsConventionSymmetry operationApproximationClosed formEigenvaluePoint groupSymmetry breakingVibrational modesd orbitalsHückel theory