Molecular Geometry & Bonding
what the shape decides
Concept

Perturbation theory — where it appears

A way of computing how a small change to a system moves its levels, by expanding in the size of the change rather than solving the changed problem. The first-order shift is the change averaged over the unperturbed state, and it is exact only in the limit where the change is small.

Named by 20 essays across 8 fields — each of them below, with the objects they name alongside it.

Seven nets, and the one column that sorts them. Every wrapped net here, with its dimension, coordination, third moment and band shape, beside the exponent the coupled-band measurement returns for it. The 4 nets whose third moment vanishes all give an exponent within 0.03 of −2 and a quotient constant to a fiftieth of a per cent; the 3 that do not all give one near −1 and no constant at all. Dimension does not sort them and neither does coordination — each takes values in both groups.

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.

solids · Bands in a solid The two overlaps, squared, at the measured bond lengths. For each chromium(III) donor: the σ overlap squared, a metal 3d(z²) against the donor's p(z), and the π overlap squared, a 3d(xz) against its p(x). Everything is computed — the radial functions from Slater's rules, the separation from the measured bond length, the integral by quadrature. Chloride's π overlap is 3.5 times fluoride's, which is the opposite of what the overlap argument required of it.

The overlap the model is not proportional to

Without a computed π overlap, the natural argument reasons about one instead: the denominator over-predicts the halide trend, so the overlap must shrink down the group to cancel part of it. Computed, it grows — 3.5 times from fluoride to chloride. And the fitted parameter changes sign across the series, which no ratio of squared overlaps can do.

applied · Ligand field Nine combinations, and the column that sorts them is not the bands'. Two bands and a coupling, varied separately. The composite graph's third moment splits into triangles that lie inside a band and triangles that use two coupling bonds, and only the second sorts the table: every row with no gap-crossing triangle gives an exponent near −2 and a nearly constant quotient, whatever its bands are made of. Triangular bands carrying an intra-band moment of 7.296 behave exactly like square ones when the coupling is a matching.

The triangles that were never in the bands

A switch in how a gap scales is usually attributed to a band's third moment, and the attribution cannot be tested while the coupling runs along one of the bands. Separated, the bands turn out to decide nothing. Two triangular bands coupled along a matching — which cannot close a triangle across the gap — behave exactly like square ones.

solids · Bands in a solid One crossing, and the six fields that are not one. Open marks: every two-state crossover field, at each tilt. Filled line: the field at which the exact spectrum's one avoided crossing actually sits. The estimates scatter over a factor of four to twenty; the real crossing moves by a factor of 1.34 across the whole ninety degrees, and passes the four coincidence angles — the dashed verticals — without any feature at all.

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.

symmetry · Representation What happens to the level that was exact. The three levels of the trio as one of the two outer orbitals is raised. At zero detuning the middle one sits at -13.6 exactly — it is the antisymmetric combination, and nothing of its symmetry exists for it to mix with. The moment the two are made inequivalent that statement is gone: the level leaves linearly, and the other two barely move by comparison.

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.

wrong · Overlap Two charges, and three ligands no charge reaches. The metal effective charge each ligand would need on its own for the model's ratio to equal the fitted one. The band is the range Slater's rules allow chromium. Two ligands have an answer, both far outside it and 2.30 apart from each other. Chloride needs more than the overlap rule can be trusted to compute. Ammonia's fitted parameter is exactly zero and cyanide's is negative, and a quotient of squares is neither.

A contraction that cannot reach three of them

The angular overlap model's own derivation gives a π/σ ratio that disagrees with the fitted parameters by up to sixfold, and the metal's contraction is the obvious candidate to account for it. The whole range Slater's rules allow moves the ratio by a factor of two. Two ligands need charges far outside it, one needs a charge past where the overlap rule can be trusted at all, and two are unreachable at any charge because a quotient of squared overlaps cannot be zero or negative.

applied · Ligand field One and eight, over two decades of defect. The number of avoided crossings the whole shell has, against the number of two-state crossover fields its coupled pairs supply, as the quantum defect is swept towards zero. The question is whether the estimated count falls to meet the exact one as the l degeneracy closes. It does not move: one against eight at every defect tried, from 0.02 down to 0.0002, with the estimates spanning a factor of 7.08 throughout.

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.

symmetry · Representation One spectrum along three directions, and three different sets of estimates. The two-state estimate gap ÷ 2d for every pair of the shell's functions the field couples, with the field along z, along x and at the tilt the defect sweep used. Along z there are three distinct estimates, along x three different ones and at the tilt eight, none equal to any of the axial three. The exact spectrum is identical along all three directions, and its two minima are drawn as vertical lines: the one between coupled levels at 0.0196 and the tangency of uncoupled levels at 0.0400. An estimate is a property of the axes the functions were written along, and a feature is not.

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.

symmetry · Representation A fixed offset against a rising threshold. The offset at which a shell's coupled minimum disappears, against the principal quantum number, with the offset every shell's own d level actually has. The threshold is 2(n² − 4)/(5(n² − 1)) — zero at the second shell, a quarter at the third, and rising to two fifths. The shell's own offset is one fifth whatever the shell, because it comes from the reciprocal of l plus a half and carries no n at all. So the comparison is a constant against a curve, and it changes answer exactly once.

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.

symmetry · Representation Two interactions, and they push the metal in opposite directions. Each ligand's filled π and empty π against a metal d level, on one energy scale with the vacuum at zero. The π lies below the metal and pushes it up, which is the only interaction the model's derivation has; the π lies above and pushes it down, which is the one it lacks. Both level positions are measured — an ionisation energy and an attachment energy — and the metal's is the single quantity nothing here measures, drawn at -8.0 electronvolts and swept elsewhere.

The channel that points at the metal

Two ligands in the spectrochemical series carry a fitted π parameter no quotient of squared overlaps can produce, because it is negative. Giving the derivation the second interaction it lacks makes both of them negative at every metal level — and the reason is not the energy denominators, which favour the donor channel in all three cases. It is where each orbital keeps its amplitude.

applied · Ligand field The measured cubic and quartic overshoot what Morse fell short of. Each molecule's αₑ as a fraction of the measured value, from its Morse curve and from a quartic potential built with the cubic and quartic coefficients the measured αₑ and ωₑxₑ imply. Every Morse curve is short, by 4 to 15 per cent. Every quartic is over, by 1 to 5 per cent: the repair moves αₑ past the measurement rather than onto it.

Two coefficients are not a potential

A Morse curve built from measured constants gets the vibration–rotation constant αₑ short by four to fifteen per cent, and the measured αₑ and anharmonicity imply the cubic and quartic a real potential should have. Built with exactly those two coefficients and solved, the potential overshoots instead. The reason is that the Morse curve's own series, cut after its quartic, moves αₑ by a sixth to a half of the error being repaired — so a quartic cannot tell whether the shortfall was the cubic.

spectra · Rotation The A–C overlap changes by a fifth and the exact level does not move. The three levels of the trio as the overlap between A and C is raised from 0.25 by up to 0.2, with B's overlap to C held and both site energies at -13.6 eV. The two outer orbitals stop being equivalent at the first step. The lowest level falls by 0.80 eV and the highest rises by 5.24, and the middle one stays at -13.6 eV — its largest departure over the whole sweep is 2.7×10⁻¹⁴ eV, which is rounding.

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.

bonding · Overlap The orderings in use move the splitting by 0.47 per cent between them. The change in NH₃'s ground inversion splitting under each ordering of the kinetic operator, relative to BenDaniel–Duke, with the bond-conserving mass throughout. The bars are exact solves and the ticks are first-order perturbation theory. The five span 0.469 per cent, from −0.407 to 0.060; the change from a constant mass to the bond-conserving one, in the same well and box, is 43.1 per cent, 92 times as large.

An ordering worth half a per cent

A mass that varies along a coordinate has no unique quantum kinetic energy, and the choice among the Hermitian orderings in use was the one thing left that could undo a forty-three per cent correction to ammonia's splitting. It cannot. The five orderings anybody uses span 0.47 per cent between them, a ninety-second of the correction, and the family only reaches the measurement at exponents three times larger than any of them.

shape · Inversion Three molecules the family cannot reach, and one it reaches twice. Along a family of potentials that all hold the measured ωₑ and ωₑxₑ, the vibration–rotation constant αₑ as a fraction of the measured one, against the parameter λ that bends the Morse curve's range; λ = 0 is the Morse curve. Hydrogen chloride and deuterium chloride rise to 0.957 and 0.957 and fall again; hydrogen fluoride to 0.909. None reaches one. Carbon monoxide crosses one twice.

Three constants, two potentials

A Morse curve falls short of the measured vibration–rotation constant, and a quartic built from the two measured coefficients overshoots it. A Morse curve whose range bends with the separation has one more shape parameter, and fitted to ωₑ, ωₑxₑ and αₑ together it should say whether 'the shortfall is the cubic' was ever a well-posed statement. For carbon monoxide it is: two members of the family fit all three constants, and both carry the leading-order cubic to five figures. For hydrogen chloride, deuterium chloride and hydrogen fluoride no member fits at all — and every member that gets close has stopped being a curve that dissociates.

spectra · Rotation Every level decides what three levels could not. For each alkali's valence shell, the polarisability of the s level and of the p level with no angular momentum about the field, each summed over every s, p and d level. Where the p bar is longer the s–p separation falls at first in a field. It is longer in sodium, potassium, rubidium and caesium — by 365, 506, 779, 1484 atomic units — and shorter only in lithium, whose valence shell has no d level of its own. The three-level threshold called potassium's, rubidium's and caesium's valence shells shells without a minimum.

The shell was the wrong unit

Whether an alkali's s and p levels approach each other in a weak field was decided from three levels of one shell. The criterion that holds every level is a difference of two polarisabilities, and polarisabilities are measured. Summed over every s, p and d level with integrals that reproduce five measured ground states to two per cent, the difference keeps sodium's verdict at thirty times the margin and reverses potassium's, rubidium's and caesium's at every shell — because the d level beside p belongs to the shell below.

symmetry · Representation Carbon monoxide's fourth constant picks the fit that turns over. Along the Morse curves with a linearly bent range that hold carbon monoxide's measured ωₑ and ωₑxₑ, the ωₑyₑ read from the lowest four levels, against the bend λ. The two members that also have the measured αₑ are marked: the one at λ = −0.221 dissociates and predicts 0.0182 cm⁻¹; the one at λ = −0.453 turns over at 2.10 bond lengths and predicts 0.0110. The measured value, read from the published level expansion the same way, is 0.0110 — the second, to 0.0 per cent.

The fourth constant chose a curve, not a depth

Read from its lowest four levels the way every potential here is read, carbon monoxide's measured ωₑyₑ is 0.01101 cm⁻¹ — and of the two bent-range Morse curves that fit its first three constants, the one that predicts 0.01100 is the one that never dissociates. Let the range bend and stay bounded and every molecule, the hydrides included, is fitted to all four constants by a curve that does dissociate. The depths those curves carry are 6 to 31 per cent too deep, further from the measurement than the plain Morse curve's on all four, and hydrogen chloride and deuterium chloride — one potential — are fitted by two different ones.

spectra · Rotation With its measured Ritz series potassium's p level stops being the more polarisable at the thirty-fourth shell. Potassium's α(np₀)/α(ns), every s, p and d level summed, against the principal quantum number, with three readings of the defects. The four-decimal asymptotic defects put the turn below one at n = 36.93; the measured δ₀ alone at 36.99; the measured Ritz series, whose corrections at those shells are a ten-thousandth to a thousandth, at 33.72. The ratio falls by about 0.0074 a shell, which is why corrections that small move it three shells.

Potassium's turn was a fine-structure question

Summed over every level, potassium's p level stops being more polarisable than its s level somewhere past the thirtieth shell, and a thousandth in any quantum defect moved the turn by shells. With the Ritz series measured on ultracold ³⁹K in 2019 the inputs are fixed to a millionth, and the turn lands at the thirty-fourth shell, not the thirty-seventh. But the two fine-structure components of the p series differ by three thousandths: with either one in place of their average the turn is at the twenty-third or the forty-fifth. A sign decided by the spinless sum is decided by an interval a hundred and fiftieth of the s–p gap.

symmetry · Representation The chain of four's strongest line is a hole's standing wave read against the singlet's ends. For the half-filled chain of four at infinite repulsion, each removal line's weight is a sum over the four sites of two factors. The first is the hole's standing wave squared on each site: the inner levels at ±0.62 put more of the hole on the end sites, the outer levels at ±1.62 more on the middle ones. The second belongs to the spin chain alone: removing an up electron at an end site leaves the stronger spin doublet with weight (2 + √3)/8 = 0.4665, at a middle site 1/4. Their sums are 0.4067 and 0.3098, which are 1/4 + √3/16 ± √15/80, and the limit of the contrast is their ratio, 1.312498.

The limit was two chains at once

At infinite repulsion every removal line of a half-filled chain is a hole's standing wave read against the spin chain the hole moves through, and every hole level carries exactly half an electron. On the chain of four that makes the contrast's limit (20 + 5√3 + √15)/(20 + 5√3 − √15) = 1.3124978 — not 21/16, which it misses in the sixth figure. On chains of eight and twelve, which no exact spectrum had reached, the limit is 1.0458 and 1.0195, and an energy that vanishes in the limit decides it.

wrong · Photoelectron The energy sees the frame, not the missing orbital. Every census arrangement of four, five and six ligands, in the order its σ-only bond energy puts it at weak coupling, most stable first. The bars are the fourth-order cost — the ligands' net direction and their excess over a tight frame, weighted as the series weights them — and they rise down each group without exception. The right-hand column is the Gram matrix's smallest eigenvalue, the measure of how nearly an orbital is unreached: it does not fall down each group, and the square with an eigenvalue of zero sits above the seesaw.

What a missing orbital costs

A flat ring of ligands leaves one central orbital unreached, and a continuous measure of how nearly an orbital is unreached — the smallest eigenvalue of a Gram matrix — goes to zero there. Whether that makes a weak bond is a question a σ-only bond energy can answer exactly. It answers no, twice. At second order the energy is the same for every arrangement of the same ligands. At every order it sees the ligand directions only through their sum and their second moment, so an octahedron and a suitably tilted prism have one energy at every coupling — and a square with an orbital missing is more stable than a seesaw that reaches all four, until the coupling is strong.

beyond · Hypervalency Five ligands: which shape the σ energy wants, by one number. The fourth-order cost of the trigonal bipyramid (flat at one sixth) and of the best square pyramid, against q — the ratio of the net-direction coefficient to the frame coefficient, which with equal overlaps is r(1 + r) for p at r times s's height. The two cross at q = 0.862. The bipyramid is a minimum only above q = 2/3 and the pyramid only below 0.955, so between them both are minima. The model's own heights give q = 3.75, and any main-group centre, with p above s, has q above 2.

Five ligands, and no ideal to aim at

A σ-only bond energy prefers, at the order where geometry first enters, an arrangement that points nowhere and treats every direction alike — a spherical 2-design. Four ligands have one and so does every count from six. Five does not, and the energy has to trade one virtue against the other. One number decides the trade, and across it the trigonal bipyramid and the square pyramid exchange the roles of minimum and transition state on Berry's pseudorotation path.

beyond · Hypervalency

Named alongside it

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

Model limitClosed formDegeneracyMatrix elementOverlap integralStark effectQuantum defectReference stateUnderdeterminationAnharmonicityApproximationAvoided crossing

All concepts